European and Chinese Cognitive Styles and their Impact on Teaching Mathematics (Studies in Computational Intelligence)

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Filippo Spagnolo and Benedetto Di Paola (Eds.) European and Chinese Cognitive Styles and Their Impact on Teaching Mathematics

Studies in Computational Intelligence, Volume 277 Editor-in-Chief Prof. Janusz Kacprzyk Systems Research Institute Polish Academy of Sciences ul. Newelska 6 01-447 Warsaw Poland E-mail: [email protected] Further volumes of this series can be found on our homepage: springer.com Vol. 258. Leonardo Franco, David A. Elizondo, and José M. Jerez (Eds.) Constructive Neural Networks, 2009 ISBN 978-3-642-04511-0 Vol. 259. Kasthurirangan Gopalakrishnan, Halil Ceylan, and Nii O. Attoh-Okine (Eds.) Intelligent and Soft Computing in Infrastructure Systems Engineering, 2009 ISBN 978-3-642-04585-1 Vol. 260. Edward Szczerbicki and Ngoc Thanh Nguyen (Eds.) Smart Information and Knowledge Management, 2009 ISBN 978-3-642-04583-7 Vol. 261. Nadia Nedjah, Leandro dos Santos Coelho, and Luiza de Macedo de Mourelle (Eds.) Multi-Objective Swarm Intelligent Systems, 2009 ISBN 978-3-642-05164-7 Vol. 262. Jacek Koronacki, Zbigniew W. Ras, Slawomir T. Wierzchon, and Janusz Kacprzyk (Eds.) Advances in Machine Learning I, 2009 ISBN 978-3-642-05176-0 Vol. 263. Jacek Koronacki, Zbigniew W. Ras, Slawomir T. Wierzchon, and Janusz Kacprzyk (Eds.) Advances in Machine Learning II, 2009 ISBN 978-3-642-05178-4 Vol. 264. Olivier Sigaud and Jan Peters (Eds.) From Motor Learning to Interaction Learning in Robots, 2009 ISBN 978-3-642-05180-7

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Vol. 277. Filippo Spagnolo and Benedetto Di Paola (Eds.) European and Chinese Cognitive Styles and Their Impact on Teaching Mathematics, 2010 ISBN 978-3-642-11679-7

Filippo Spagnolo and Benedetto Di Paola (Eds.)

European and Chinese Cognitive Styles and Their Impact on Teaching Mathematics

123

Filippo Spagnolo, PhD Professore Associato Matematiche Complementari Dipartimento di Matematica e Informatica Università di Palermo Via Archirafi, 34 90123 Palermo Italy

Benedetto Di Paola, PhD Dipartimento di Matematica e Informatica Università di Palermo Via Archirafi, 34 90123 Palermo Italy

ISBN 978-3-642-11679-7

e-ISBN 978-3-642-11680-3

DOI 10.1007/978-3-642-11680-3 Studies in Computational Intelligence

ISSN 1860-949X

Library of Congress Control Number: 2009943590 c 2010 Springer-Verlag Berlin Heidelberg

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typeset & Cover Design: Scientific Publishing Services Pvt. Ltd., Chennai, India. Printed on acid-free paper 987654321 springer.com

Preface

Recently, mathematics educators have shown enthusiasm about conducting comparative studies involving East Asian countries, primarily because students of East Asian, such as Chinese students, show outstanding performance in various international assessments. Culture, teachers’ knowledge, instruction, curriculum, and assessment have been identified as factors contributing to the achievement gap in various studies. However, the authors in this book argued that cognitive processes in the socio-cultural context play an important role in the differences in student mathematics learning. They analyzed the cognitive processes based on the comparison of Italian and Chinese cultural contexts both in historical and in epistemological aspects. The investigation of the nature of the difference in student learning between Italy and China is the focus of this book. For so many years, international comparative studies in mathematics education focused on the difference in external aspects in student learning, and less attention was on the differences in the linguistic language structure in a socio-cultural context. The authors of this book broke new ground by contributing to a better understanding of how Chinese students acquire algebraic reasoning from an epistemological and linguistic aspect. By thoroughly examining the representative text of ancient Chinese mathematics culture (The Jiuzhang Suanshu, Nin Chapters on the Mathematics Art) and investigating the linguistic structure of the Chinese written language, the authors identified differences in the hierarchies of the reasoning models generally used by Chinese students and Aristotelian-Euclidona mould by Italian students in five experimental situations. The deeper study of the Chinese cultural origin led to an analysis of historical aspect of mathematics and specifically the evolution of algebraic thoughts over the different cultural traditions. Notably, both qualitative and quantitative methods were well-employed in this study. The instrument was well-designed based on the purpose of this study and based on European and Chinese cultural and historical contexts. The authors used a great deal of effort on the data collection which used multiple sources, such as survey and semi-structured interviews, documents, observations, to enhance the validity of the design. The five sessions of the research experiments were sequential, building upon one another. Cohesive hierarchical implicative classification was used appropriately for quantitative analysis of experimental data. The analytical narrative and discussion in the presentation focused on the diversity of cognitive styles, solution strategies, and learning methodologies as opportunities in mathematics. The book provides an excellent model in which it identifies linguistic, historical, neuro-scientific, psychol-lingusutic, and the didactic aspects in study of acquisition of algebraic competences in mathematics for the

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Preface

students as it is shown in discussion and analysis of the Chinese particular system of graphemic fixation and specific characteristics of Chinese writing. The charts and tables of the participant demographic information were very helpful and informative, and the statistical illustration in writing used various figures and tables, which organizes the information of the study in a visually meaningful way and provides readers with a deep understanding of the study. The authors used rigorous and unique methods to analyze and draw conclusions from the data by structuring data analysis in different types and chapters. More commendable, treating cultural diversity as an opportunity in mathematics by deeply studying Chinese mathematics from the arithmetic to the algebraic content in the historical aspect-cultural puzzle, is excellent approach in culture-related studies. The book provides strong evidence that research on the cognitive processes, from arithmetic thought to algebraic thought, should take into consideration the socio-cultural context. It is an important contribution to the literature on linguistic structure in comparative studies related to Chinese student mathematics learning. This book not only makes a great contribution to research in mathematics education, but the findings of this study also addressed insightful approaches and thoughts of understanding the development of algebraic thinking in cultural contexts for classroom teachers. Using written Chinese language from different theoretical references provided wonderful approaches for understanding student algebra cognitive development in a different way and calls educators to pay special attention to an epistemological and linguistic view of algebraic development. The findings inform classroom teachers that the cultural context plays an important role in student learning mathematics. A typical analysis of the cognitive dimension involved in some in the historical and cultural contexts is a great resource for classroom teachers. I really enjoyed reading this book and learned a lot from its compelling analysis. I am fascinated by the topic and the method of analysis of data from historical and cultural perspectives. The authors’ efforts in investigating student cognitive process from arithmetic to algebraic are extraordinary, especially the connection to linguistic, historical, the neuro-scientific, psycho-linguistic, and the didactic aspects. This study is very valuable to the mathematics education field and a remarkable contribution to international related studies. As a Chinese scholar, I admire Dr. Benedetto Di Paola’s brilliance in investigating the relationship of Chinese linguistic structure and student algebra thinking. I greatly appreciate his hardwork and excellent contribution to the mathematics education field. Special thanks go to Dr. Filippo Spagnolo’s great effort and dedication to the mathematics education field and his wisdom of advice and support to Dr. Benedetto Di Paola’s research relating Chinese culture and language. Their team has been very successful and fruitful. Shuhua An, Associate Professor Director of Graduate Program in Mathematics Education California State University, Long Beach

Introduction

The increase of different culture pupils represents, for the last decade, the greatest novelty in the Italian school. This growth will probably be constant for the next future. After a first period of simple notation of the situation, the Italian scholastic system and, inside of it, the teachers begun to warn the necessity to intervene and, consequently, to consider the possibility to foresee actions that kept in consideration the change of the cultural context, but also the social one, of the classroom. In the large majority of the cases, these actions are shaped in the form of training in-service courses on general themes of the interculturality and the intercultural pedagogy for the teachers. Parallelly, absolutely inevitable, different initiatives of teacher training to the didactics of the Italian language as second language, were made. For many years, the presence of different culture pupils in the classes was considered as a “problem” regarding only the general pedagogy, for the teachers, and regarding the language, for the pupils. Namely, the people thought that, it was not necessary for other things except to furnish these tools that are essentially communicativeness. The reality of the classroom shown and still now show how this assumption was and is enormously wrong. It shown this to the most careful teachers and to the ones that, for different reasons, warned only recently the “problem”. Among these last ones we have to include almost the totality of the mathematics teachers! The assumption that the “problem” of the presence of different culture pupils in the classroom and the “problem” of the multicultural classes are, exclusively, a problem of communication is nearly found on the belief of the teachers that, from one side, a specific subject is a “corpus” rigidly structured in their contents and in their formality of transmission and that, from another side, the socio-cultural background of the student does not influence, in general, his learning process if not for what it achieves from his not complete or scarce knowledge of the vehicular language, or from the different level of schooling with which he entered the classroom. This assumption is very rooted among the mathematics teachers, from the great majority of the teachers who see the subject taught as a “universal language” and “culture free”, and so not determined and conditioned, also in its contents, from the culture of the different societies. The research works of D’Ambrosio, since the first years 1980, and the research works of Bishop, from the beginning of the second halves of the same years, clearly shown instead as such reading and vision of the mathematics are to be considered limitated because they essentially reduce the mathematics to a complex of

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Introduction

notions and knowledge that are transmitted in a scholastic ambit. Complex of notions and knowledge that, even if it is vast and remarkable from the academic point of view, it cannot be considered as comprehensive of all the mathematical knowledge, presented in various forms and used in different social and cultural contexts, also not scholastic. Correctly, D’Ambrosio and Bishop have shown instead, even though with different approaches, that we have to speak of mathematics or in a better way of mathematical activities, rather than of mathematics. If it is accepted then to consider the mathematics as a complex of activities (to count, to measure, to locate, to draw, to play, and to explain), we see the mathematics as a cultural product, gotten by the exercise of such activities. On the other hand, even if such activities can be sights how common to all the societies as categories, they often differ themselves in their formalities of carrying out. In effects, these differences are not only revisable when very distant society (macrocultures), are observed in their complex system but also when single societies are analyzed (micro-cultures). A reading in cultural key of the mathematics inevitably asks a more punctual analysis on the single societies has developed their own cultures. It is necessary to keep in mind of different factors, beginning from the principal ones in anthropological ambit, historical ambit, linguistic ambit, religious ambit, and technological ambit. These factors are integral part of a culture, that produces them and it is developed from them, in a continuous process. They are also integral part, therefore, of a correct and complete analysis of the mathematics, as cultural product. To these factors it is necessary to refer, therefore, also when we intend to examine how the mathematical knowledge is transmitted and receipted inside one determined society or social group. Without doubt and meritoriously, Filippo Spagnolo and Benedetto Di Paola worked in this direction in their work of analysis and comparison, as specifically regards the phase of teaching/learning of the mathematics of two different cultural realities as Italy and China, so differ among them. The deep difference of these two cultures is also warned evidently in clear way from the teacher, who often see these represented more and more and in meaningful way in his/her own classroom. Such difference often means, also for the teacher of mathematics, an element of difficulty in comparison to the use of the usual didactic methodologies. A teacher of mathematics, aware of this difficulty, will find in this research work elements of knowledge and sprouts of reflection that should allow them to better facing the novelty of the didactic intercultural situation. Reality in which they operate and to make more profitable, for the whole class, the process of teaching /learning of the mathematics subject.

Prof. Franco Favilli Prof. Associato “Matematiche Complementari” Dipartimento di Matematica Università di Pisa

Contents

Introduction ......................................................................................... 1 1

2

A General Framework and Theoretical References ………………………..3 1.1 Principal Objective of the Book ................................................... 3 1.2 The Experimental Epistemology of the Mathematics as Paradigma of Reference ...............................................................................................3 1.3 The Differences and the Analogies of the Carrying Cultural Coordinates.............................................................................................4 1.4 What the References with the Mathematics, with to Deduce and to Conjecture (and to Conceptualize) .........................................................5 1.5 Experimental Epistemology of Mathematics, Didactics of Mathematics, Communication of Mathematics, Semiotics: Theoretical Possible Interpretation ........................................................ 6 1.5.1 Conclusions ...............................................................................10 The Chinese Written Language as Tool for a Possible Historical and Epistemological Reflections on the Mathematics and the Impact of Teaching/Learning of Mathematics...…………………………………....13 2.1 The Written Chinese Language as Possible Tool to Acquire Algebraic Competences in Mathematics...............................................13 2.1.1 The Point of View of the Neurosciences ...................................14 2.1.2 Origin and Development of the Chinese Writing: Some Fundamental Step ......................................................................15 2.1.3 The Pinyin and the Chinese Language in the Chinese School ........................................................................................17 2.1.4 Observations Cognitive on the Chinese Language ....................19 2.1.5 Chinese Language and Mathematics .........................................22 2.1.6 Rules of Composition of a Chinese Character ...........................24 2.1.7 The Classification of the Chinese Characters ............................25 2.1.7.1 Constitutions of Ideograms and Algebraic Thought ....27 2.1.8 The Research of the Characters on the Dictionary.....................30 2.1.9 The Parametric Structure of the Written Chinese Language......30

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Contents

2.1.10 The Meta-rules of the Language and the Mathematics ..............32 2.2 Can the Natural Language Influence the Educational System of the Area Logical-Mathematics? Which Is the Impact in the Teaching of the Mathematics? ..................................................................................34 2.2.1 Contents and Objectives in the Historical Evolution of the Teaching of the Mathematics in the Chinese School: A Brief Panning ......................................................................................34 2.2.2 The Teaching of the Mathematics in the Today's Didactic Practice ......................................................................................39 2.3 A Resume Chart on the Relationship Natural Language Mathematics..........................................................................................44 3

The Meta-rules between Natural Language and History of Mathematics……………………………………………..............................47 3.1 The Algebra between the History and the Didactics, a Variety of Perspectives ................................... …………………………………...47 3.1.1 The Algebraic Tradition in Antiquity: An Overview.................52 3.1.2 The Elements of Euclid as a Canon of Reference for the Western Mathematical Tradition ...............................................57 3.1.3 An Initial Comparison between the “Elements” of Euclid and the “Jiuzhang Suanshu”.......................................................58 3.1.4 One Possible Comparison between Diofanto and Babylonians ...............................................................................59 3.1.5 The Algebraic Tradition among the Arabs and the Indians .......60 3.1.6 Significant Developments for the Symbolic Algebra ................69 3.2 A Look at the Ancient Chinese Mathematics .......................................73 3.2.1 The Historical-Cultural ..............................................................73 3.2.2 The Jiuzhang Suanshu ...............................................................75 3.2.3 The Rule of Fangcheng as Meta-rule for the Chinese Algebra ......................................................................................80 3.2.4 Argue, and Demonstrate in Conjecturing Jiuzhang Suanshu an Example through the Algebra ...............................................84 3.3 Conclusion ............................................................................................87 3.4 How to Summarize the Role of History? ..............................................89

4

Common Sense and Fuzzy Logic……………………………………….....91 4.1 Fuzzy Logic, Fuzzy Thinking and Linguistic Approach.......................91 4.2 Fuzzy Sets and Their Representations ..................................................92 4.3 The Representative Point of View of Kosko ........................................95 4.4 Some Epistemological Reflections on the Approaches to the Fuzzy Logic.....................................................................................................98 4.5 Some Experimental Observations on Common Sense and Fuzzy Logic.....................................................................................................99 4.5.1 Common Sense and Fuzzy Logic, M. Ajello and F. Spagnolo ....99 4.5.2 Fuzzy Logic and Complexity...................................................100 4.5.3 The Hypothesis ........................................................................101

Contents

XI

4.5.4 The Assignment a Computer Science Activity .......................102 4.5.5 A Priori Analysis .....................................................................103 4.5.6 Implicative Analysis Using the Chic Program.........................104 4.5.7 From the Pupil Records ...........................................................105 4.5.8 A Possible Interpretation Key..................................................106 4.5.9 Conclusions..............................................................................107 4.5.10 Open Problems.........................................................................108 Conclusions .........................................................................................................117 5

The Experimental Epistemology as a Tool to Observe and Preview Teaching/Learning Phenomena...…………………………………………………..119 5.1 The Experimental Context ..................................................................119 5.1.1 Choice of the Problematic Situations in Accordance with the Theoretical Framework and the Hypotheses of Research...................................................................................119 5.1.2 The First Experimental Investigation: "Sudoku Magic Box" .........................................................................................127 5.1.2.1 A Priori Analysis........................................................130 5.1.2.2 Quantitative and Qualitative Analyses of the Data ....131 5.1.2.3 Second Experimental Investigation: "The Problem of Fermat", "Varying Questionnaire and Parameter in Different Semiotic Context" ................136 5.1.2.4 A Priori Analysis: “Fermat Problem” ........................145 5.1.2.5 Quantitative and Quantitative Analysis: “Fermat Problem” ....................................................................146 5.1.2.6 A Priori Analysis: “Questionnaire Variable and Parameter in Different Semiotic Context” .................151 5.1.2.7 Quantitative and Qualitative Analysis: “Questionnaire Variable and Parameter in Different Semiotic Context” .....................................................................154 5.1.2.8 Third Experimental Investigation: "The Sequence"," The Grid of Numbers" ...............................................165 5.1.2.9 A Priori Analysis “The Sequence”.............................170 5.1.2.10 Quantitative and Qualitative Analysis of “The Sequence” ..................................................................171 5.1.2.11 A Priori Analysis of “The Grid of Numbers” ............174 5.1.2.12 Qualitative Analysis of “The Grid of Numbers”........174 5.2 From Natural Languages to Logical Linguistic Aspects: Cognitive Styles of European and Chinese Students, M. Ajello and F. Spagnolo .........................................................................................185 5.2.1 Some Reflections on “Arguing, Conjecturing and Demonstrating” in Chinese Culture with Relation to Occidental Culture ...................................................................186 5.2.2 The Algorithm as Fundamental Element of Arguing and Demonstrating?........................................................................186

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5.2.3 What Are the Stable Reasoning Patterns in Chinese Culture?....................................................................................187 5.2.4 How Were the Situations/Problems Chosen? ..........................189 5.2.4.1 The a Priori Analysis of the Situations. We Report the a Priori Analysis of the Hypothesized Behaviours ................................................................190 5.2.4.2 Presentation of the Experimental Work in the Italian Classes ............................................................191 5.2.4.3 The Quantitative Analysis: Implicative and Factorial .....................................................................191 5.2.4.4 Implicative and Similarities Analysis ........................192 5.2.4.5 General Considerations on Factorial Analysis, Implicative Analysis and Similarities ........................193 5.2.5 Interviews with Two Chinese: Qualitative Considerations......194 5.2.6 The Experience in the Chinese Classes....................................195 5.2.6.1 Factorial Analysis ......................................................196 5.2.6.2 Supplementary Variables ...........................................196 5.2.6.3 Considerations on the Analysis of the Data Relative to the Chinese Sample .................................197 5.2.7 General Conclusions and Future Perspectives .........................197 6

Strategy and Tactics in the Chinese and European Culture: Chess and Weich'I , G. D’Eredità and F. Spagnolo ……………..…………….201 6.1 Introduction ........................................................................................201 6.2 Strategy and Tactics.............................................................................203 6.3 Notes on the Conceptions of Strategy and Tactics in the Orient and the Occident .................................................................................205 6.4 Historical Games of Strategy in the Orient and Occident: Chess, Wei-ch’i and the Different Conceptions of Strategy and Tactics .......206 6.5 Connection to Didactics and Open Problems .....................................211 6.6 Conclusions ........................................................................................212 References ...........................................................................................................212 7

Rhythm and Natural Language in the Chinese and European Culture, D. Galante and F. Spagnolo…………………….………………219 7.1 Rhythm and Natural Language in the West ........................................219 7.2 Rhythm and Natural Language in China ............................................222 7.3 Conclusions ........................................................................................227 References ...........................................................................................................227 8

Conclusions………………………………………………………………...229 8.1 The Conclusions of the Experimental Work.......................................229 8.1.1 Topics Such as Research Agreements?....................................230 8.2 The Experimental Epistemology of Mathematics: Concluding Observations on the Experimental Investigations Discussed..............231 8.1.2 The First Experimental Investigation.......................................232

Contents

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8.3 The Second Experimental Investigation .............................................234 8.4 Conclusions and Open Problems Related to "Strategy and Tactics" ...............................................................................................238 8.4.1 Links with Education ...............................................................239 8.5 Conclusions and Open Problems Related to Music ............................240 8.5.1 The References to Mathematics, the Reasoning and the Conceptualisation ....................................................................240 References ..........................................................................................................243

Introduction

During the years 1984–1986 four young Chinese graduates of the Guanxi region came to Palermo with a project of international cooperation. Among them was Ou Ye Lin a Chinese mathematician. In that occasion I gave some lessons of mathematics to a Chinese emigrant to Palermo and with the help of the dott. Ou Ye Lin and have analyzed, for the first time, the relationships among history of the mathematics, natural language, and learning of the mathematics in a different cultural situation from that of western (Spagnolo, 1986). My first amazement has allowed then to deepen this matter and to analyze it in the following years as field of research in didactics of the mathematics. Meaningful in this sense the 13° meeting of the I.C.M.E.1 (International Congress on Mathematics Education) on the comparative study2 east–west in didactics of the mathematics has been. The majority of the participants was of Oriental origin understood some transplanted that were found in USA by some generations. Beginning from 2002 I have started, also with the help of my group of research,3 to study in a systematic way the comparation of schemes of reasoning in the resolution of mathematical problems drawn by the Chinese and western tradition (Spagnolo F., M. Ajello, Z. Xiaogui, 2005a, 2005b). The work have been conducted in Italy (Palermo) and in China (Nanchino). Since 2002 I initiated my collaboration with Benedetto Di Paola on the passage from the arithmetic thought to the algebraic thought. His thesis of PhD has been written on this matter. The work of B. Di Paola has given a notable acceleration to this project. Resulted as that tied all the parametric vision of the language written Chinese they are at work due his. We have followed together courses of Chinese and we have consult some linguists for power better bearing our ideas on the matter. 1

The ICME is the subcommission of the International Mathematical Union, IMU. The comparative studies of the ICME are organized by narrow groups of researchers who are tightly interested to the theme and that for 5 days they discuss critically the work presentati from the participants. Every job has to be approved by with referee at least two months before. The participants receive one month first all the work. The discussions during the meeting concerning examination, critical analysis, and comparison among different points of view. The title of the meeting was Mathematics Education in Different Cultural Traditions: To Comparative Study of East Asia and the West. 3 G.R.I.M. (research group on the teaching/learning of the Mathematics) http://dipmat. math.unipa.it/~grim. Particularly the people who have collaborated to the experimental work and of theoretical reflection on the matter are Prof. Maria Ajello and Dr. Benedetto Di Paola. 2

F. Spagnolo and B. Di Paola (Eds.): European and Chinese Cognitive Styles, SCI 277, pp. 1–2. springerlink.com © Springer-Verlag Berlin Heidelberg 2010

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Introduction

Assistants to the project were Giuliano D’eredità, PhD student and expert in chess and Mario Ferreri expert in Neuroscience and reference point for this matter and for the whole doctorate that I have the pleasure to actually coordinate4 to today. Without the contribution of these people the present work would not have been able to come up. Palermo March 2009

4

Filippo Spagnolo

The doctorate is "History and Didactics of the Mathematics, of the Physics, and of the Chemistry". The doctorate is pooled with five Italian universities and five Europeans: http://math.unipa.it/~grim/dott_HD_MphCh/dott_HD_index.htm The reflections on the curriculum are been published in the proceedings of the Discussion Group 12 of the I.C.M.E. (International Commission Mathematics Education) in Mexico 2008: http://math.unipa.it/~grim/dott_HD_MphCh/icme11_dg12_proceedings_final.pdf

Chapter 1

A General Framework and Theoretical References

1

Introduction 1.1 Principal Objective of the Book To reflect for a long time on the comparison of cognitive styles of very different cultures the possibility to consider to your culture of reference. Has this happened to us when we started to work on the matter. Things can mean to be Aristotelian today in the European culture? Actually to that point today an European is aware of its Aristotelian being? To try to interpret the phenomenon of teaching/learning of the mathematics in situations of multiculturalty has brought us to make these reflections of theoretical–experimental nature. In the progress of formation for teachers, in the international conferences of Mathematics Education one of the considerations that were done when classes were analyzed with Chinese and European students was almost always the same one: "the Chinese students are better in mathematics" or also "they know how to make better calculations in arithmetic and algebra of ours". Any justification of the specific behaviors, any possibility to anticipate behaviors in class on the decisive strategies of situation/problem. The objective of this book is to furnish, even though under certain aspects, some interpretative tools on the phenomenon of teaching/learning of this type. In relationship to the indications theoretical/experimental already courses ahead from our group of research1 makes him necessary to take back a Theoretical Framework of reference as The Experimental Epistemology of the Mathematics.

1.2 The Experimental Epistemology of the Mathematics as Paradigma of Reference Our searches lean him on a paradigm of research in didactic of the mathematics that keeps in mind the Theory of the Didactic Situations of Guy Brousseau (1998) 1

Discussed in the supplemetary rewiew n. 1 to the “Quaderni di Ricerca in Didattica (Sez. Matematica)” del 2009: http://math.unipa.it/~grim/quaderno19_suppl_1.htm


4

1 A General Framework and Theoretical References

and R.Gras–E.Suzuki–F.Guillet–F.Spagnolo (2008) as it regards the statistic analysis of the experimental data. For the qualitative analysis of the experimental data we will refer to the study of the phases argumentative of the students in relationship to the study of the fundamental nucleuses to deduce and to conjecture analyzed through indicative semantic defined previously, of couples interviews, of analysis of cases, and analysis of video of the situations in class. The two analyses, those quantitative and those qualitative, have always been correlated and integrated to deduce the phenomenon of teaching/learning. We can consider everything as a sort of Experimental Epistemology (Spagnolo, 2006; Spagnolo et al., 2009). Why the Experimental Epistemology? In the perspective of research of the trials, of the teaching/learning the expression Experimental Epistemology explains it in a good point of view. There are no theoretical paradigms of reference: the experimental phenomenon are so complex not to be able to afford him/it. The only reference to the experimental routine could not allow alone a theoretical elaboration brief times. The point of view that is put forward in evidence is that the modeling "theoretical–experimental" of the phenomenon of teaching/learning could be a solution to the problem. The expression "theoretical–experimental" must be seen how a dialectical relationship among the two components and it represents therefore the overcoming of this opposition (in the sense of the dialectical Hegelian). In this perspective the word "Epistemology" takes back what had been elaborate in the teaching and in the learning past and that could be useful to interpret the phenomenon of teaching/learning. Everything will broadly be argomentated in the text. The research in didactics of the mathematics introduces him the facts as a sort of experimental epistemology. It keeps in mind the references epistemology and historical of the discipline and of the references cognitive as well as of the references to today's neurosciences, to be able to try to interpret the processes of teaching/learning in class.

1.3 The Differences and the Analogies of the Carrying Cultural Coordinates An analysis of the coordinates cultural retentions today foundational for the western thought and that Chinese Oriental allows us to be able to define a first classification of the fundamental disciplines in the two cultures; classification that will be more well-documented then in the text in detailed relationship on the writing (chap. 2) and the Wei-ch'ì (chap. 6) to Rithm and Music (Chap. 7) seen how culture strongly connected to the philosophical circle, scientific, mathematical of the Chinese thought:

1.4 What the References with the Mathematics, with to Deduce and to Conjecture

5

Table 1.

WEST2 Arts of the Trivio (artes sermocinales): grammar, rhetoric and dialectics. Arts of the Crossroads (artes reales): arithmetic, geometry, astronomy and music.

CHINA3 1. Music as disciplines tied up to the hearing but to sophisticated trials that regulate the vibrations. It is the practice of the internal qì. It has to do with the reported embodiment to the musical rhythms. Today we know that this is also in relationship with the learning of the number and the natural language. 2. Wei-ch'i 3. writing 4. painting

Can this chart be interpreted as fundamental for understanding the to deduce and to also conjecture in mathematics? 1.4 W hat the References with the Mathe matics, w ith to Deduce and to Co njecture

1.4 What the References with the Mathematics, with to Deduce and to Conjecture (and to Conceptualize) In our opinion the cognitive Chinese styles related to the mathematical learning can be reread in relationship to a whole series of competences acquired in the four cultural circles underlined in the chart of the preceding paragraph. In the text it will refer particularly us: 1. to the algebra as carrying element for the construction of new characters of the type "ideograms" for association, type logical. This aspect has not been much studied but we think both of the elements that it brings us to individualize one "logic" different from that of Aristotelian matrix. Perhaps, the reference to the logical fuzzy, recalled by Kosko and by some experimental jobs of Nisbett, he can be an interpretative key. In the western culture he has been the geometry that has had relationships with the logic. At least actually at the end of the eight hundred. 2. to the order with which the characters are written. The order has some wellprecise rules that behave as an exercise on the non-banal arrangement, both on the alphabetical arrangement and on that of symbols. The arrangement that the Westerners generally favor with rules, colors, etc. Here he is performed on abstract symbols and with rules, that, then they succeed in having meant only at the end of the writing of the character. The programs to the computer force to the order of the lines (Welin 3.0). 3. to the study of the to deduce, to conjecture and to show from a historical– epistemological point of view (K. Chemla 2001, 2004). The mathematical 2

The western classification, according to Martian Capella, philosopher of the late Latin (IV–V century AD). 3 Interview to Mrs. Nan Xi Jin (Scott A. Borman, pp.19–20).

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text of the Nine Chapters analyzes you, equivalent of the Elements of Euclide for the Chinese people, inside which the algorithm is considered as a demonstrative tool of the Chinese cultural tradition. Particularly, the demonstrative procedures have meant in the resolution of classes of problems. The key concept that organizes the description of the Nine Chapters is that of "class" or "category" (her), it plays a primordial role in the commentaries. The procedures serve for understanding the categories. To set a problem (wen) related to a category and, with this subterfuge, to understand ten thousand situations, what calls "to know the road". It is, therefore, through a work on the procedure that the classes of situations are determined. The perfection is defined in terms of simplicity and generality. 4. this last consideration involves, presumably, another. To work for concepts and for conceptual maps should almost be a consequence of the fact to always seek a "fundamental algorithm", with relative procedures to resolve "classes of problems" more and more ample and for "analogy" (Vd. association in the construction of the characters), being able "to check" and "to foresee" complex situations. This is that, then, Nisbett sustains to be the approach holistic of the oriental cultures. 1.5 Ex perimental Episte molog y, Didactics of Mathe matics, Co mmunicatio n

1.5 Experimental Epistemology, Didactics of Mathematics, Communication of Mathematics, Semiotics: Possible Theoretical Interpretation 1.5 Ex perimental Episte molog y, Didactics of Mathe matics, Co mmunicatio n

To reference model for an Experimental Epistemology. The diagram consists of the Verbal Linguistic Plane - Not verbal language and strictness - imprecision. Perpendicular to this plane there is the one of time which allows king-seeing the semiotic phenomenon (metalogic) in to dynamic vision.

Fig. 1.

1.5 Experimental Epistemology, Didactics of Mathematics, Communication

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The a priori analysis of the Theory of didactic Situations Khan allow, according to the depth of the analysis, to "forecast" of the temporal evolution of the didactic phenomenon. Notes Fig. 1. Note 5: In the semantic of possible world of Kripte, The intention is the same thing that the Sinn was in Frege mind. The intention of a word is not defined like a function which can give us the extension. If we consider the phrases "the ant drives the car" and "dog flies" a conventional semantic can say that they are impossible objects, an intentional semantic establish that their intention in the real world establish that it is the same, so, nothing. But, their extension is not anything in other possible worlds where the two terms have the same extension. In the moment in which they need to recognize contradictions, the intentional semantic creates problem because it has a useless extension in all the possible worlds. The formalization of intentional logic drives to expression like functions of functions, each of them is functions of functions. The order of calculus establishes the comprehension which we can have. The intentional logic, and so, the intentional semantic inspiring to an ecological realism (Putnam), substitute to the possible worlds of the intentional logic (Montague) the “situations”. Note 6: the term inaccuracy can be integrated with vagueness, ambiguity, and indecidibility. The notion of “ambiguity” is not only bounded to the presence (with) of more than one meaning, but it can also be associated to the same message or also to the ambiguity is propriety of every language. Note 7: In the sense of Tarski and Gödel: “…in relation to particular formal system.” • In the 1st quadrant Verbal Languages – Rigour, we can indentify the three currents of thought relative to the systemisation of mathematical languages and that is logicism, formalism, and structuralism. However, at the moment in which this quadrant is seen in the Time dimension (third axis) we find ourselves having the historical–epistemological evolution of the mathematical languages according to the interpretations given by the various currents of thought. The analysis that is done in this context is that of “Knowing” established and codified in a specific historic period and at the same time the evolution of Knowing in history. • In the 2nd quadrant Verbal Languages – Imprecision, we find non formalized semantics and the pragmatics of verbal communication. Also in this case, the Time dimension allows us to historically analyze the evolution of semantics and of the tools relative to the pragmatics of verbal communication (analysis of the text, hermeneutics, etc.). • In the 3rd quadrant Imprecision – non Verbal Languages, we find, for example, the visual languages, body languages, and mental images. In this quadrant the Time dimension is difficult to frame in an activity of the historical– epistemological nature, even if in the history of mathematics mental images have had an important role in the setting up of the languages. This quadrant covers an importance as regards the learning subject and his history.

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• In the 4th quadrant non Verbal Language – Rigour, we find the pragmatics of non verbal communication: analyses of non verbal behaviours. In the history of mathematics a meaningful example is supplied by the “Italian School of Algebraic Geometry” in the last century in the instance in which setting up a grammar was tried relative to mental images for the solving of geometric problems. Inserted in this quadrant are psychology, sociology, the science of education overall for that which regards the study of behaviors and for that which is concerned with verbal language we refer to the 2nd quadrant. In short, these disciplines are between the 2nd and the 4th quadrants. • The 3rd and 4th quadrants in the Time dimension can recall the History of Mathematics seen in the dimension of Art. • In the reorganizational process of the mathematical languages are the 2nd, 3rd, and 4th quadrants which verge on the 1st (syntactic access). This process can be easily explained with the statement that all formal languages are constructed and set up, thanks to Natural Language. The model set up by Rotman and Peirce foresees, for the mathematical subject, a division in three parts: a Person, a Subject, and an Agent. “The Person is who does mathematics in an informal way; the Subject responds to the imperatives of proving and defining bringing to life a world with imagination, and assigning to a double; Agent, the task of carrying out various imagined actions. The Agent is an idealized version of itself which carries out the imperatives as a robot working only with signs without meaning. In the passage from Person to Subject the indicant signs are forgotten. In the passage from Subject to Agent sense and meaning are forgotten.”4 Can this triadic model of the Subject who does mathematics also be used for the Subject who learns mathematics? The passage from Semantic Fields to syntax has not been sufficiently described or, sometimes, it has been passed over. Perhaps it is the Theory of Situations that regains some controllable and reproducible models for a return to “sense”. In the mathematics community today there are many declared Agents and few persons. This reference diagram can be read in a Experimental Epistemology of Mathematics and taking into account the Time dimension of the following points of view: • From the point of view of “Knowing”: Historical–epistemological itinerary. The analysis of extensional semantics: Historical–cultural evolution. The analysis of intentional semantics: epistemology of the possible worlds. • The point of view of the Pupil: Evolutions of the conceptions (in a semiotic sense which takes account of the reference of plan of the 4 quadrants) as a function of a context (Didactic Situation). Intentional semantics intervene in the research of the referent (2nd quadrant) and in the research of the conditions of truth. • From the point of view of the Teacher: Evolution of its own epistemology and control of the one relative to the pupil–knowing relationship. • From the point of view of the Researcher: The evolution of the didactic situation. The epistemological obstacles are framed in this perspective. Intentional 4

G. Lolli, Capire la matematica, ed. Il Mulino, 1996, Bologna.


9

semantics intervene in the moment in which the expected behaviors are analyzed in an a-priori analysis of a didactic situation (Conditions of signification). It seems interesting to us to refer to a model regarding algebraic5 language which considers three dimensions of algebraic language: • Axis x: natural-written symbolic language; • Axis y: semantic Syntax; • Axis z: Relational-procedural. The three dimensions highlight the background in which the processes of algebraic thought are carried out. The following diagram highlights the existing relationships between the dimensions.

Fig. 2

Extracting for the preceding diagram only two axes, we obtain the following relational plane which allows us to better analyze several aspects of algebraic language according to the noted classification of mathematics historians: rhetorical Algebra, syncopated Algebra, and symbolic Algebra. These two diagrams are derivable from the diagram of Fig. 1 of this Appendix and in particular with regard to the first two quadrants. The analysis of Pragmatics of the communication of algebraic thought is then managed by a description of didactic situations on several appropriately chosen “cases”. The analysis is based on a use of the triangle of Frege Sense (Sinn) –Denotation (Bedeutung) – of an expression (Zeichen) applied to algebraic language. The semantics are intentional. The changes of concept are interpreted by means of transformations of the triangle of Frege. 5

Ferdinando Arzarello - Luciana Bazzini - Giampaolo Chiappini, The algebra as tool of thought (theoretical Analysis and didactic considerations), Notebook strategic n.6 Project C.N.R. Technologies and didactic Innovations, Pavia, 1993.

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Fig. 3.

1.5.1 Conclusions The presentation of this diagram poses a series of questions regarding the problem of the distinction between a structural logic typical of a use of semiotics as an interpretive tool of linguistic phenomena in a static vision (research of invariants) and a dialectical logic which takes account of the historical evolution of the semiotic systems. In this context, the use is of the second type in that one tries to analyze and interpret the didactic phenomena in the diachronic and synchronic evolution. In all this, the problem of adaptation to the environment, both neurophysiological and cultural has a considerable place. This interpretation of the mathematical languages seen in their semantic– syntactic–semantic evolution through the system of reference of semiotics by a greater possibility of interpretation of the “didactic phenomena” in all the possible dimensions from the formalized languages to the non verbal languages (behavior analysis) and all this takes into consideration the historical evolution of the mathematical languages. An evolution which has different planes of reading from the recovery of “meanings” to the syntactic reconstruction of the languages. Can we consider this approach Constructivist (Piaget, Bishop, Brousseau) or Logicist Pragmatic (Lolli)? We have already seen that the classical points of view of the philosophy of mathematics can be inserted into a more general vision. Today, the problem is not that of referring oneself, at least temporarily, to a single theoretical interpretation of mathematics and of the Communication of Mathematics. Perhaps it is necessary to be more obliging in the use of, according to the situations, different theoretical reference systems keeping the generalizations without, however, becoming too attached to them. In the 1900s semiotics had, and in part still has, generalization


11

as its aim. Can all the phenomena of the Mathematical and Communicative Languages be interpreted with Semiotics? The problem, for example, remains open as to whether genetic and neurophysiological phenomena are subjects for the semiologist. Umberto Eco’s answer is negative while informational theories of genetics and of neuro-physiology are subjects for the semiologist. At this point, the neuro-physiological phenomena should be confronted using paradigms of research in the experimental sciences. And, what will the relationship between these theories and the paradigms be?

Chapter 2

The Chinese Written Language as Tool for a Possible Historical and Epistemological Reflections on the Mathematics and the Impact of Teaching/Learning of Mathematics 2 The Chinese Written Langua ge as Tool

2.1 The Written Chinese Language as Possible Tool to Acquire Algebraic Competences in Mathematics 2.1 The Written C hinese La nguage as Po ssible Tool

Strong and undisputed vector of the Chinese cultural tradition has been (till now), as told in the preceding chapters, its peculiar system of fixation grafemica. The Chinese characters (hanzi, characters of the Hans) in different areas of the Asian continent have been used for fixing to level grafemico you systematize linguistic distant tipologicamente among them; examples are two languages of it insulating what Chinese and the Vietnamese one and two languages agglutinants what Korean and the Japanese. The same Chinese people have, as mentioned in the chapter, the different variations of language, from south north, from west east; the problems of mutual understanding among speaking Chinese varieties have always been the norm in China and, as in past, only the appeal to a system grafemico "superior" what that writing has allowed to overcome the possible communication gap. The written language represents therefore a unique element, has been in the history and still exists. It is therefore a strong cultural element1. Before to enter into the treatment of some of those that rap-introduce fundamental characteristic for the graphemic system of Chinese characteristics that, as it is logical, we will not treat from a linguistic point of view; on the contrary, through a critical reading of die "mathematical", it seems interesting to ask us what a language is and therefore: What to give to this from the point of cognitive view meaning? Is there perhaps a specific difference among the different types of writing? Can the different types of writing favor the birth of particular strategies cognitive? 1

The whole linguistic varieties labeled as "Chinese" belongs to the great family of the until-Tibetan languages, compete over that from Chinese, from the Tibetan, from the Burmese one and from some languages spoken by Chinese minorities (Mair, 1991, p. 4). As it regards the analysis of the Indo-European languages instead, you will not be done a punctual analysis as Chinese writing. We think that information can be recovered complementarily. Where he/she will introduce the occasion we will produce some comparative charts; but a systematic treatment would make the text incisions on bamboo fillets heavy, compared to the writing on paper (Karlgren, 1923).


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2 The Chinese Written Language as Tool

The language is generally defined as a tool of communication and of expression of the thought. However, it is also tool of construction and transmission of a culture. From the psycho-linguistic point of view it is defined as an object multifunctional, you find stratified (the cognitive plain, affective, social, contextual tightly) and multi-channel (it uses manifold channels as the gesture, the mimicry, etc. what they compete to the construction of the meaning of a message). We are not able to affirmatively answer the questions that are set by Derrick de Kerckhove.

2.1.1 The Point of View of the Neurosciences Although we are aware of the importance of the reference to the neurosciences for the theoretical picture of reference that we consider, there are limits in the text to the most meaningful results on the interpretation neuroscience––learning of the natural language today. Writings exist today for everything, "lateralized" to the right, to the left and vertical; each of these has one peculiarity of his to cognitive level and therefore of learning. Schematizing the complex treatment, we can certainly affirm that the orientation of the writing essentially depends on what it results more "impelling" to help the reader to decipher a text. It is because every writing suggests the forms of the characters to the eye of the reader as well as their sequence, within certain rules of "choice." In the reading, the visual field of pertinent reference is that of the optic chiasm. In the man the visual field of every eye results separated vertically in two equal parts was governed by the hemisphere that celebrates her opposite. The fundamental principles that distinguish the privileged operations of the two hemispheres can reassume, than being said, that the left hemisphere effects an analysis temporalized of the connections and the sequences of the gestures, of the objects and of the perceptions, while the right has one perception global, unitary and, for some verses, spatial. What happens to the Chinese writing is that in some cases (for the characters pictographic) a rapid individualization of the forms is more impelling than the survey of their sequence; in other cases (for the composed characters), it results in more discriminating analysis detailed of the forms in comparison to that of the sequence. Recent Neuroscience researches have shown through the functional magnetic resonance as "reading Chinese is characterized by extensive activity of the neural systems, with strong left lateralization of frontal (BAs 9 and 47) and temporal (BA 37) cortices and right lateralization of visual systems (BAs 17-19), parietal lobe (BA 3), and cerebellum. The location of peak activation in the left frontal regions coincided nearly completely both for vague - and precise-meaning characters as well as for two-character words, without dissociation in laterality patterns. In addition, left frontal activations were modulated by the ease of semantic retrieval. The present results constitute to challenge to the deeply ingrained belief that activations in reading single characters macaws right lateralized, whereas activations in reading two-character words macaws left lateralized", "[…] read logographic characters […] to distributed network of brain areas was activated" (Tan, Liu, Perfetti, Spinks, Fox, Gao, 2001).

2.1 The Written Chinese Language as Possible Tool

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The Neurosciences, through researches being conducted for several years on the organization of the linguistic system in the brain, seem to confirm that an inversion of tendency in comparison to the past on the analysis of the areas celebrate involves them in the recognition, in the reading and in the processazione of the Chinese characters. If in the past in fact it was thought to an exclusive use or almost of the hemisphere celebrate her right (Cheng and Tang, 1998; Tang, 2007), today research set with the use of the fMRI “suggest that, although brain activations during reading aloud of Chinese characters are bi-lateralized (Tan et al., 2001), silent reading of Chinese is dominated by the activity of the left hemisphere (Chen et al., 1998, 2002; Tan et al., 2001). Neuroimagine findings with Japanese Kanji, which is similar to Chinese characters in orthography, seem to provide a corroboration of the results from Chinese reading (Fujimaki et al., 1999; Uchida et al., 1999)”. In the case of the alphabetical writing it is almost sure that sequence is more literal than their form to constitute the critical factor of the reading. Other interesting consideration, also confirmed by different searches of neuroscience, concerns the discriminating factor of the context on the phase of writing. In the case of the alphabetics, it is almost certain that the literal sequence prevails on the context in which this is inserted, based on the fact that in the brain he finds the tendency to privilege the analysis, the subdivision and the reductions to the minimum commune denominator, as well as the classifications and the research of the native identities, the universal ones etc. (de Kerckhove, 2002, p.272). we have described these aspects in detail in the chapter 7.

2.1.2 Origin and Development of the Chinese Writing: Some Fundamental Step To date, the origin of the ideographical Chinese writing is not certainly easy. According to some studious ones, the first signs of writing were found by Needham him around the fifth century B.C., by others between the fourteenth and the fifteenth century B.C. Needham, arranges with this second vision and puts the beginning of the Chinese writing in the last period of the dynasty Shang. The structural characteristics that fondant the actual system grafemico derive, however, from the dynasty Han under which was fixed the canonical form of every single character2. We must be specified since immediately that, for the description of the Chinese writing, the term "ideogram" it is not correct entirely. It is composed from Ideo and gramma: the first one is "idea", the second, derived from the Greek and means note "writing". The ideograms therefore “I am” "writings that represent ideas". But the ideograms are only one of the “you are” categories of Chinese writings characters. We will see this aspect subsequently in detail.

2

Such trial was probably determined by the change of the support of writing: incisions give on bamboo fillets to the writing on paper (Karlgren, 1923).

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Every single character is a unity grafemica, whose power badge contemporarily resides in the being an entity iconic (cultured to visual they-fleece), a syllabic unity (phonological, cultured to acoustic level) and a semantic unit (bearer to cognitive level of a meaning). The to converge, therefore, of form, sound and meaning within a unitary element, it confers entirely to every single character an absent versatility in unity grafemiche proper of languages defined through alphabetical systems. From the epoch Han to today the system Chinese grafemico is being unchanged, apart from a series of interventions, in the history of the Chinese people, aimed to rationalize the use of the judged characters "deviant." Some of these recent interventions schematized by him are as follows: Table 1. Some of the interventions of setup of the Chinese writing

1935 1936/ 1948 1955 1977 1980

Proposal of the nationalistic government of simplification of twothousand four-hundred characters. Discussion on the possible adoption of a system of writing founded on Latin characters (Latinxua xin wenzi, “new latin writings”). Publication of the Committee for the Reform of the Writing (Wenzi gainge weiyuanhui) of the list of over thousand non-correct characters. Proposal (ever applied) of simplification of 853 further characters. Definitive list of the characters in use.

Other matter concerns then the different attempts of "transformation" of the characters ideographic through an alphabetical system. Before penetrating us in the structural description of the system Chinese grafemico it seems us in fact interesting, as completeness of information, to spend some word on the system of transliteration of the Chinese characters. Examples of important attempts are able to be summerised this way: Table 2. Some of the interventions of "alphabetical transformation" of the Chinese writing

1605

1626

1687

Publication of the Jesuit Matteo Ricci of a brochure (today lost) to facilitate the learning of the Chinese writing of the missionaries, transcribes a Chinese text in Latin alphabet. Work of the French Jesuit Nicolas Trigault on the publication of a dictionary based on the transcript in Latin alphabet of the pronunciation of the Chinese characters. Editing of the manuscript (Digressio de Sinarum Literis) inside which is possible to find exposures detailed of some types of Chinese characters, a brief history of the writing and a treatment of the different types of lines badges of the characters.

A further meaningful reference in this sense is then the manuscript Confucius Sinarum Philosophus, text devoted to the exposure of the doctrine of the great Chinese thinker through the translation of some footsteps drawn by the classical


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Confucians and trasversal to the diffusion of the Chinese culture to the European people3. The systems of transliteration of Chinese amount overall to about thirty4 and they generate more confusion. An example can be the transcript of the term shang ("to climb", "above") that is transcribed "shang" in English, "chang" in French and "schang" in German, "cah" in Russian. In 1918, with the objective note to represent her/it "official pronunciation", that of the national language (guoyu), the zhuyin was diffused zimu (what translated it literally means alphabet of phonetic transcript), till today in use. In 1926 a group of Chinese linguists elaborated the guoyu luomazi pinyin ("Romanization of the national language using the Roman alphabet"), and after numerous changes of transcript, in February of 1958, it was approved and spread the hanyu pinyin fang'an ("system of phonetic transcript of the Chinese language") commonly known as pinyin (what it is simply translated in "to put together the sounds").

2.1.3 The Pinyin and the Chinese Language in the Chinese School The pinyin was not anticipated, initially as it systematizes able to put in discussion the use of the ideograms, considered cultural patrimony Chinese identity, but rather as orchestrate for the diffusion of the model of unitary language (the putonghua). Today, as it brings Liu Xiqin (1991) the traditional writing remains the official written language and the Chinese characters, in how much also legal writing, are used in the superior teaching and in that university, as in the press and in the literary texts. The standard official language is seen by the Chinese people as necessary in some circles, and above all "as formal symbol of the linguistic unity of the country" (Abbiati, 1992). Currently the pinyin is parallel, used to the writing conceive-graphics, in the first two years of the primary school, in the books destined to his/her/their children and, in general, in the texts of language with the finality to specify the formalities of pronunciation of a rare character. Middy, as it regards the books of primary school of Chinese language, through a series of contemplated exercises, comes first introduced the alphabetical writing and the tonal linguistic system and only later the characters are introduced but always accompanied by the writing alphabets.

3

The text of the manuscript, with translation and comment, has been published by K. Lundbaek in 1988. 4 Currently the systems of origin not Chinese increased used I am the Wade-Giles spread in UK; the EFEO used by the French sinologis; the systems Lessing and RÜDENBERG near the Germans, Cyrillic systems near the Russian.

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Fig. 1. Examples of tonal variation in Chinese mandarin (Abbiati, 1992, pp. 82-86)

Fig. 2. Example of text of Chinese language for the primary school. The simplest characters are introduced trying to recover their reported historical meaning to some pictograms.

In the texts of mathematics, the linguistic registers of presentation are almost always three: the pinyin, the writing ideographic that refers to the writing simplifies today's and the mathematical linguistic register.


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Fig. 3. Text of mathematics for the primary school

Fig. 4. Text of mathematics for the primary school

2.1.4 Observations Cognitive on the Chinese Language On the cognitive implications (mainly the phase of coding and decoding in the phase of writing and reading) of the use of the pinyin and the ideographical writing, different interesting researches of Neuroscience exist. Chen and colleagues (2002) seem to confirm that “Chinese offers a unique tool for testing the effects of word form on language processing during reading. The processes of lettermediated grapheme-to-phoneme translation and phonemic assembly (assembled phonology) critical for reading and spelling in any alphabetic orthography are largely absent when reading nonalphabetic Chinese characters. In contrast, script-to-sound translation based on the script as a whole (addressed phonology) is absent when reading the Chinese alphabetic sound symbols known as pinyin, for which the script-to-sound translation is based exclusively on assembled phonology. […] Results demonstrate that reading Chinese characters and pinyin activate a common brain network including the inferior frontal, middle, and inferior

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temporal gyri, the inferior and superior parietal lobules, and the extrastriate areas. However, some regions show relatively greater activation for either pinyin or Chinese reading. Reading pinyin led to a greater activation in the inferior parietal cortex bilaterally, the precuneus, and the anterior middle temporal gyrus. In contrast, activation in the left fusiform gyrus, the bilateral cuneus, the posterior middle temporal, the right inferior frontal gyrus, and the bilateral superior frontal gyrus were greater for nonalphabetic Chinese reading. […] both alphabetic and nonalphabetic scripts activate a common brain network for reading. Overall, there are no differences in terms of hemispheric specialization between alphabetic and nonalphabetic scripts. However, differences in language surface form appear to determine relative activation in other regions. Some of these regions (e.g., the inferior parietal cortex for pinyin and fusiform gyrus for Chinese characters) are candidate regions for specialized processes associated with reading via predominantly assembled (pinyin) or addressed (Chinese character) procedures.” (Chen et alii, 2002). We must be remembered that traditionally the characters were vertically layouts, from the tall one downward, to start from the right superior angle of the support of writing. The horizontal disposition with left reading toward right, although you are found in the history (but with right readings toward left), it derives from the imitation of the European uses. Today this type of writing is generalized (Alleton, 1976). The organization of the sequences of characters to form words (normally bisillabic, in modern Chinese) or superior unity as sentences or texts, determines notable implications, dependent cognitively, from the attitude of psicolinguistics through which the Chinese perceive their language. In accordance with Goody, "such attitudes are deeply different from those masses in evidence from subjects that they originate from linguistic-cultural areas in which the linguistic systems are encoded through alphabets or sillabaries" In such systems, the sequences graphemes represent sequences of sounds which are connected, through the reference to data experimental, semantic values: the level of "semantic opacity" of the chain phonic and grafemica, for what it refers to the decoding of the meaning, it is, in such systems, normally tall. In the system Chinese grafemico, contrarily, the single characters directly represent unity of meaning distributed in relationship one-to-one with their relative phonological realizations" (Goody et alli, 2002, pp.217-218). The macros differences among realizations alphabetical grafemiche and conceive-graphics are described by Rumihan (1991) as models of contrast among forms "that they result you trace within the component phonologic of a language" and forms "you trace within the component semantic." Meaningful, in this sense they are the words of Yuen Ren Chao: "If you look for anything in a page written in English, it needs that you, note, puts you in the order of ideas to look for her/it. If you have to make the spread out thing in a page of Chinese characters, the thing that you look for, if it is in the page, he/she will jump you to the eye, in evident way" (Yuen Ren Chao, 1968, p.12). To compile and to read a text in Chinese characters rigorously foresees the coding of articulated information in logic-sequential form; on the plain cognitive, the single sequences, subtended to the organization of the text; they express result


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"iconically" in the sequence of the characters. The literal translation of this sequence results entirely however incomprehensible from the semantic point of view. To decode the semantic knot needs that the reader organizes in sequences of hierarchized subsets the text and, through an interpretation often not linear and that he/she makes use of values that go beyond the simple decoding ideographical, gives a coherent interpretation of it. For Ryjik, "[…] this written language is not a writing", its code of artificial expression, able to express "things" and "concepts" they make her/it something more of a simple system grafemic. If we for instance take the character that means "old" hair lǎo ( ), "The modern form is an extreme corruption of a seal containing changing (color): old" (Karlgren, Wenlin Institute 1997-2002). As Granet declares “Chinese was able to become a powerful language of civilization and a great literary language without having to worry about either phonetic wealth or graphic convenience, without even trying to create an abstract material of expressions or supplying itself with a syntactic armament. It managed to maintain for its words and sentences a completely concrete emblematic value. It knew how to reserve only for rhythm the care of organizing the expression of thought. As if, above all, it wanted to liberate the spirit from the fear that ideas can become sterile if expressed mechanically and economically, the Chinese language refused to offer these convenient instruments of specification and apparent coordination which abstract signs and grammatical artifices are. It kept itself obstinately rebellious against formal precisions for the love of the concrete, synthetic adequate expression. Chinese does not seem organized for noting concepts, analyzing ideas or conversationally expressing doctrines. In its completeness, it is constructed for communicating sentimental behaviors, for suggesting conduct, for convincing, for converting.” (Granet, 1988). For the one who does not know Chinese, without a deep study of the "hangs" to give to the single characters in an analytical-global vision, it almost seems to be able to get endless translations, endless interpretations, all deeply different ones. It is as to compose a puzzle without a fixed on the contrary varying scheme, up to that he does not succeed in getting a reasonable result, subjective. In reality, in accordance with Demiéville (1965) it is the rhythm that, with the aid of some grammatical particles, helps the reader to divide the sequence of characters in sentences and periods, "to recognize you the articulations of the thought, dissimulates behind the uniformity of the monosyllables and their polysyllabic mixtures"5”. In a Chinese text the unities graphemes separated by white spaces, each corresponds to a character; every character corresponds to a syllable, but the single syllables do not constitute autonomous syntactic elements having identical functional value. The analysis in subsets of text ("parts of the discourse") allows to individualize both groups disyllabic how monosyllabic. The reorganization of the text, according to least unity of meaning and the fusion of the single values communicated by the single characters, allows the semantic values of every subset to emerge from the text.

匕

5

老

Then the music is also profit for the interpretation of the text.

毛

22


Different researches (also in neuroscience) that are interested in the analysis of the visual space in the reading and in the decoding of a Chinese text have been conducted, and the trials are involved cognitive, for example Chen &Tang, 1998; Osaka, 1992, Pollatsek, 1981. Also, in these cases, the thought put the decoding of the text into effect which seems, not only analytical but also holistic and global.

2.1.5 Chinese Language and Mathematics We think that it is possible to conduct trials, in learning of the language, to define the destination-rules to simplify for gathering them, to homogenize, equalized to make to communicate them. Destination-rules are found by him in the history of the mathematics and particularly in the Jiuzhang Suanshu (for the resolution of the equations and the systems of equations); we will see, perhaps the composition of the single lines of each of the characters ideographic. All of this is not able to put in evidence a strong cultural value of the learning. The binomial mathematics-language is, in our opinion, a binomial, meaningful for the study that are proposed to relatively discuss to the Chinese culture, and we consider the experimental analysis. Another particularity of the Chinese writing is its explanatory ability, especially in the case of homophonic terms. The alphabetical writing is in fact ambiguous in its presentation, and not ideographical. Two explanatory examples can be: the terms àn and shí for how much it relooks at the word àn, which means ( ) "bank", "shore", "coast"; ( ) "to press", "in accord with"; ( )"dark"; ( ) "rectangular stand to sustain vases of wine"; ( ) "amine" etc. The term shí can be brought back ideographic instead to characters as ( ) "ten", ( ) "time", ( ) "stone", ( ) "to recognize", ( ) "harvest" etc. Words therefore, in both cases, with equal pronunciation but with deeply different meaning (very frequent cases in the Chinese language because of the a lot of monosyllabic terms) that you/they are distinguished by the writing ideographical (Wenlin Institute, 1996) that is often used, in partnership to a opportune gesture, as only mean of understanding among speaking "different dialects". Words of Matteo Ricci are meaningful in this sense: "We arrived in this I bring of China in August and we were few more than one month in sea... you Immediately dictate me to the language cina... As for the to speak is very equivocal that holds a lot of words that mean more than thousand things et to the times there is no other difference between the one and the other that to pronounce him with taller or lower voice in four different de tones; and so, when they speak, to the times among them to be able him to intend they write what you want to say; that in the letter are different the one from the other... it engages her Chinese it has so many letters how many the words or the things are, of way that you/they pass of seventy thousand, and all are very different and “cheat”... theirs to write more I toast it is “paint”; and so they write with brush as our painters" (Letter to Claudio Acquaviva, February 13 th 1583; trad. Fontana, 2005). This aspect, analyzed by a point of view scientific neuro, results in interesting relationship to the nature of the mirror neurons and to their use as bridge among the observer and the actor in the process

胺

暗

时

石

案

岸

识

按

拾

十


23

of transformation of the relative visual information to an object in the necessary motor actions to interact with it. Recent studies (Iacoboni, 2008; Rizzolati, Sinigaglia, 2006) have put in evidence the role determined of the mirror neurons that, we hypothesize, are meaningful for the learning of the Chinese language more than for that alphabetical. This aspect would be described in detail. Of other song the jobs on the neurons mirror they are enough recent to have resulted meaningful on the matter (Iacoboni, 2008). The tones of meaning that in other languages you/they are made with the bending of the word (the unusual one and the plural of the nouns, the different times of the verbs, the declination) they are made then in Chinese adding opportune parti) "luxuriant", cles, also them easily ideograms specific associabili: àn'ǎi ( / "prosperous"; ànbà ( ) "underwater dike"; ànbái ( ) "dirty white", etc… The use of opportune classificatory categorize then the partnership thought to a character and it communicates on it a thought type concrete functional-relational (Abbinati 1992; Xiqin, 1991; Mair, 1991). Kyril Ryjik, in an important wise man devoted to the nature of the ideograms, vindicates the autonomous character of the same and it refers to the structure of the language as "a written language constituted by mnemography" (Ryjik, 1980, p.14). This aspect is meaningful. We believe in fact that the strong value of memorization given to the mathematical practice, as discussed more before, can also derive from the strong tradition cultural linguistics. To experimental level, as we will discuss more before, this aspect not only results in meaningful students Chinese-palermitans that are originated from a preceding education in China but also in those immigrants and frequenting the Italian school from the first year. The words of a Chinese student frequenting the inferior secondary school in Palermo "G. Garibaldi" are meaningful to this intention, who had been inserting in the Italian scholastic context for the first year of school, met us out of the institute for a single interview (brought following in some meaningful phases).

暗坝

暗白

暗黯蔼

S. "When I study mathematics, I always try to memorize everything, I repeat, I repeat, I repeat, also with my sister, because then if I have well memorized, it comes me easy immediately to understand thing to do for solve a problem that gives me the Prof. Me in fact I would always like to lift the hand." T. "Can you make me an example?" S. "Your mathematics is logical… I have only to learn to understand her/it and to use my head. To resolve a problem of arithmetic, for example always: one, to examine the problem; two, to see what it wants the problem; three, to use a formula to find the solution; four, to resolve the problem. If I have memorized all the formulas, I immediately find the solution. It is easy. If I don't find the formula… that happened me so many times not to understand the correct formula… but because I had not seen well things again she asked the problem… and therefore what formula to use… If I don't know what formula to choose, then I didn’t understand the problem. If you succeed in memorizing the whole book, chapter for chapter, is a genius. Also because the teacher leaves the exercises similar to the they-bro." The words of the student seem to observe as the process of memorization has bilaterally been used not only in the definition and the memory of the various use-

24


ful steps to the solution of a situation problematic proposal but also and above all for the definition of a decisive mathematical formula, tightly connected to the context of a proposed problem and the whole class of problems hierarchized arithmeticians and to it connected. This aspect, even though partly, validate our hypothesis on the relationship mathematical language/schema of reasoning. Through his/her words, it would seem to perceive a key of memorization of the "various formulas", scheme of reasoning pragmatic, concrete, procedural, categorized in line with how much it seems to find himself/herself inside the writing ideographic. Same observations are been able to do, even though in less marked way, for all the Chinese students involved in the experiment. These, in fact, in quite a lot cases, left free to converse you/they have put in evidence, in the phase of reasoning on mathematical contents, a continuous appeal to the "repetition" mnemonics of the "rules" that they intervened in the various situations didactic proposals, memorization able to drive them in the research of the solution. The concrete aspect, algorithmic for the definition and the carrying out of a practice mathematics, aspect of strong value epistemological, for how much I also dictate in the preceding chapters, it comes out then, to our opinion, even though unconsciously, also from the iron rules of construction of a character ideographic, that a Chinese boy has to learn and to follow for the phase of writing (Chan &Siegel, 2001, Ho & Bryant, 1997, Wu, 2000). Rules do not leave space to possible individual choices. In the first chapter we have already mentioned this aspect; in this chapter we will deepen him/it through also a critical reading of those that you/they can present as elements key in the inside organization of a single character and therefore the coding of this.

2.1.6 Rules of Composition of a Chinese Character Any is the tool of writing, the direction of the layout of the single lines must rigorously have respected always following simple step of composition in succession: 1. it is begun downward to trace to the right the left character and from the tall one; 2. they trace the horizontal lines and there after vertical; 3. when the extremity of a line goes to end on another line, it is traced first the first line and after the second; 4. when the whole or a part of the character is contained in a closed space, this cannot be closed until so much that all the inside lines have not been traced. 5. if the character is composed from more elements, first they trace the lines of the first one and only after having finished this it passes to the second character. To trace the different characters among the mixtures, it is traced first that taller and, in the choice right-left had a preference to the left.


25

The chart under brought it underlines the order of succession of some Chinese characters (Abbinati, 1992, p.92)

Fig. 5. Order of succession of some Chinese characters (Abbiati, 1992, p.92)

The rules of composition must be performed in the respect of precise dimensions that are established previously both for the global realization of the character and for the single lines. The character in fact must be traced inside an ideal square (as in the case under brought) and the dimensions of the single lines have to be "absolute"; least changes determine substantial differences in sound and meaning. Examples in such sense can be: tǔ ( ) "earth" and shi ( ) "pupil", "lettered" or does he/she anchor jǐ ( ) "himself" and yǐ ( ) "stop", "arrest", "already."

己

土

已

士

走

Fig. 6. Character of “to go” zǒu ( )

2.1.7 The Classification of Chinese Characters The first classification of the writing Chinese characters was in 121 with the realization from Xu Shen of the first great dictionary, the Shuowen jiezi. The work has a strong historical importance in how much it is really inside the fifteen volumes

26


of the work that for the first time the classification of the ideograms introduced him according to the criterions of their formation. Classification is mostly followed by the linguists till today (Li, 1999, pp.7-13). The introduction of the radicals for the research of the ideograms is particularly interesting in our job. 541 characters "key" (bushou) then meeting places, with the time, to the actual 214. ), the pictograms or symbols The first one constituted by the xiàngxíng ( iconic representative natural or physical elements, represented in relationship to "man, pertheir profile or to the contour. Examples can be: son"; "sun"; "moon"; "tree"; "elefant"; "mountain"; "horse"; "bird"; "crow" Like “bird”, but missing the dot in the head; the eye is invisible because a crow's eye is black like the feathers; Karlgren, Wenlin Institute 1997etc. 2002); hands ; field ; mouth ; center; Other examples “with higher degrees of mutation: a dog; a dragon; a deer; a bear; etc.” (Karlgren, Wenlin Institute 1997-2002). Numerically the pictograms represent a limited part of the whole the Chinese characters. , (literally The second category represented by the ideograms6, the zhǐshì "indication-object") deals with symbols ideographic signed abstract objects (spatial relationships, numbers etc.). The principle of codification does not foresee some relationship among the symbol ideographic and its phonological realization. "one"; "two"; "three"; "four" ("Even Examples can be the numbers: number, which is easily divided into two halves. The old form graphically repreinto halves", Wieger Wenlin Institute 1997-2002); sents the division of "five" (“The two principles yīn and yáng, [with the internal radical ] begetting the five elements, between heaven and earth.”, Wieger Wenlin Institute 1997“sixt” (“that comes after four; marked with a dot. Note that in the 2002); other simple even numbers, the divisibility is also indicated: two; eight”, Wieger Wenlin Institute 1997-2002); “seven”; “eight”; “nine”; “ten”; “eleven” (what is represented as ten more one) … “twenty” (what as ten is represented for two)… … "forty-sixth" (written as four for ten more) etc.; "tall", "above", "to climb" "fund", "under", "to go down" etc. In accord with Karlgren "The shapes of these characters macaws symbolic diagrams rather than pictures of any physical object." (Karlgren, Wenlin Institute 1997-2002). ("move-notation" or "extension The third category that of the zhuǎnzhù shown up of meaning"). It deals with characters derived by an image indicante a concrete object that communicates an abstract but connected idea with the object in matter or suggested by it. Examples can be:

象形

鸟

日

月

門

鸟

木

田

象

人

山

口

中

犬

熊

马

鹿

龙

指事

四

五

十一

一

六

四十六下

上

二

七

四

三

四

二

八

二十

二九

八十

转注

北

- běi ( ): "north" that it represents two people supported back against back: the north in how much opposite to the south. 6

What as confirmed more times, they only represent a class of Chinese characters.


老考

27

- lǎo ( ): "old" - kǎo ( ): "to give an examination, to be examined, to study" the character introduces him as "union" of ( lǎo) e qiǎo that is phonetic. The writing of this character can be due to different motivations: “to give students examinations and inspect their work, since that is what the old man did. By further extension, kǎo means to study, to ponder” (Wenlin Institute 1997-2002). - wǎng ( ): "to capture" as image of a net. - néng ( ): "power, to be able" it derives from the image of a bear (?), symbol of power and strength

耂老

丂

考

网能

The last three categories those related to the composition of characters are the most meaningful for our studies on the algebraic thought. 2.1.7.1 Constitutions of Ideograms and Algebraic Thought As said previously the learning of these characters they can assume, according to an important role in the study of the algebraic competences, in how much through their learning, the memorization and the techniques of composition discussed before, the students discover in naive way entirely a first approximate concept of symbolic equation. Even though, therefore, you don't render explicit him to the boy that the realized linguistic writing represents an equation; he, in implicit way, can acquire some algebraic competences that are able, under certain aspects, to reveal himself of help in the process of memorization of the algebraic thought "formal." As it brings Needham, "Such equations constitute to semiconscious mental foundation for whoever is acquiring familiarity with the language." (Needham, 1981, pp. 35-36, vol. The). that Is The fourth category of Chinese characters constituted by the huìyì they are literally translated as "union of meaning". The characters classified in this way derive from the combination of two or more autonomous characters. The union of their single values determine a new unity of meaning. The Chinese describing this rule of composition speak of "logical composition" of approach of meaningful semantic units. But what type of logic does it subtend this practice? Can they be significant for the learning of the algebra? We have already mentioned in the precedents paragraphs and to such intention we have spoken of destination-rules of composition and acquisition, even though not aware of the concept of variable. What must certainly be confirmed, since it immediately is as to level cognitive all the relationships on which a Chinese student reasons for the coding and decoding of these characters are shown in an icon that ties her in a formal expression. To determine the representative sign, and to individualize its meaning means therefore to operate on its syntactic and semantic. Processes of metacognizione that, according to us, you/they can also influence schemes of reasoning of algebraic or pre-algebraic die directly manifested in the esplicitation of the phase

会意

28


of reasoning and autonomous verification (Di Paola-Spagnolo, 2008a, 2008b; Spagnolo, 1986; Spagnolo-Ajello, 2008; Ajello-Spagnolo-Xiaogui, 2005). As said, these activities of linguistic learning that, you/they can schematize him in activity of recognition of pattern, coding and decoding of the same, memorization and relative graphic tracing of the lines composed of the various characters according to algorithmic fixed schemes; the Chinese children is proposed to during the primary school, parallel to the study of a whole series of meaningful didactic situations for the learning of the mathematics (of the arithmetic, of the algebra, of the Geometry). This aspect is, in our opinion, as said, meaningful in terms of specific competences key that you/they can be acquired in mathematics as the arrangement of the greatness, the recognition of particular geometric pattern and the sense-motor implications on the activities of manual realization. The implications cognitive are, in our opinion, evident: a functional dynamic vision both type semantic that syntactic on the formal writing, an ability "manipulative" not indifferent, a tall recognition of regularity discriminated, a slow and gradual passage from a reasoning type local circumstantial to a generality of thought, through the use of the destination-rules for the research of the invariant ones. The characters classified in this fourth grade category, can be codificated from the students, before the memorization, through different actions practices on the same and really thanks to these; they develop a thought serial on the single line, in partnership parallel to one global for the reading of the character inserted in a written text. Text written that as seen it asks then for a further abstraction. The destination rules that make reference to the aspect of "functional relationships and part-whole relationships" proper, according to Nisbett (2001), of the Chinese thought, of his logic and of his to deduce and to conjecture, in our opinion, the key of time for a learning aware of the language and, as said previously also some Chinese algebraic thought. We think it can be meaningful to propose the words of Pierce and to read its parallel on the language Chinese conceive-graphics "algebraic expectation which consists of recognition of conventions and basic properties, and identification of structure and of key features; and ability to link representations" (Maureen Hoch and Tommy Dreyfus, 2005, p.145, Pierce & Stacey, 2001; Warren and Pierce, 2004). Examples of characters that countersign for form, structure and composition, this category, can be:

明男好林休算

日田木亻人廾

月力女木木

“moon”); - míng ( ) “shining” = ( “sun”+ - nán ( ) “man” = ( “field”+ “force”); - hǎo ( ) “prosperity, wellness”= ( “woman” + “child”); - lín ( ) “forest” = ( “tree” + “tree”) = ( *2) - xiū ( ) “rest” = ( [ “man”] + “tree” - suàn ( ) “abacus” = ( “two working hand” + “on a abacus” + “on a bamboo" ); ) “teacher”. Signifacnt as renard the justification of Wieger: - jiàoshòu ( “From xiào “filial piety” and ( pū) “beat”. xiào “filial piety”, made of ( lǎo) and (zǐ) “child” means to respect and obey ones elders; this is TAUGHT by ( pū) beating, according to the most enlightened

教授孝耂老

子攵攴

攵攴

木

子

目

竹

孝


教

29

教育

teaching methods.” (Wieger Wenlin Institute 1997-2002); linked to this ) “teaching caracche we have also: jiāo ( ) “teaching” e jiàoyù ( education”, ) “pirates” = ( “tief” + “see”) - hǎidào ( - wēijī ( ) “cryses” = ( “dangerous + “opportunity”).

海盗危机

盗危

海

机

假借

The fifth category, sensitive to the phonological dimension of the monosyllabic represented by the character ideographic, does understand the jiǎjiè (literally translated it means "form-sound"). The characters that appear in this category are always composed characters formed by the union of autonomous characters one of which it has the assignment to signal the way according to which the character must be pronounced. The characters that compose this class are therefore for the more homonyms or omonomus. Seems significative the example discussed by for the Wieger: “Examples: ( ) lái meant "barley"; it was borrowed word lái "come" (of course the sound has changed over time, it wasn't really exactly lái). ( ) wéi meant "monkey"; it was borrowed for the word wéi "be". In these two cases, the original words (lái "barley" and wéi "monkey") became obsoyòu meant "right"; it was borrowed for the word yòu lete. Another example: "again". In this case, the word yòu "right" still exists, and it is represented by the to ” (Wieger Wenlin Institute character , which was formed by adding 1996).

为為

来來

假借

又

右

口又

形声

The sixth and last category, in Chinese as xíngshēng is translated ("semantics-phonetics") It is a hybrid category inside which the central element that plays a role of classification is that-him of "radical", term done already mention more times and indeed meaningful for the treatment of the binomial mathematics language and therefore algebraic thought, object of our investigation. Before penetrate us in the specification of our mathematical point of view of the linguistic concept of "radical" it is opportune to define, also in this case, some explanatory examples. We will limit there to only give three of it (Goody, 2002) from the mo-chin whether to retrieve others is enough simple: the 80%–90% of the Chinese characters belongs to this class of reference. These characters are generally constituted by a radical, which the general semantic mediation and an element having function is submitted (often latently) phonological, that suggests note of it the pronunciation. So for instance the characters ma ( ) "to insult", ma ( ) "mother", ma ( ) "sign, numerical code", ma ( ) "interrogative particle" have visibly meant different but do they contain to the inside a same character ( ) that it points out the phonological content of it. Equally in the characters qing ( "to ask"; "feeling, love"; "clear, clear"; "dragonfly") the phonetic element is contained ( ). For how much it concerns then to the semantic dimension. You can be recognized inside each some characters quoted a least unity of meaning, sets to the left generally, that communicates the analysis semantic of the whole character. In the case of but ma ( ) "to insult" it is ( ) "mouth"; in the case of ( ) "mother" it is ( ) "woman"; in the case of ma ( ) "sign is ( ) "stone" and

吗

清

妈

骂

妈马请

蜻

女

骂

口码

码

情

石

青

30


吗

finally in the case of but ma ( ) "interrogative particle" the key of interpretation (radical) it is again ( ) "mouth." In the second series of characters examined above the situation that finds again him can be schematized this way:

口

请情清蜻

讠

qing ( "to ask"): radical ( ) "word." qing ( "feeling, love"): radical ( ) "heart." qing ( "clear, clear"): radical ( ) "sun." qing ( "dragonfly"): radical ( ) "bug."

忄日虫

2.1.8 The Research of the Characters on the Dictionary The radical assumes then a central role in the research of the various characters on the dictionary. Using this criterion of research (bushou), it is considered what of the elements that compose him/it are listed in the list of the keys. If the character is composed from an only element, the key and the character coincide. (an example can be : "mouth): all it takes is counting the number of the constituent lines to find the section of the dictionary, where the character is recorded. If the elements are more than one, he goes for attempts: it occurs what some elements can be possible keys and, calculated the number of the lines, it is looked for under the list of the recorded characters in the dictionary under the keys if the character is present or less. (Goody et alii, 2002)

口

Fig. 7. Table of the Chinese radicals

2.1.9 The Parametric Structure of the Written Chinese Language From how much we observed, the final writing of a character composed of this type is inserted in a structure type parametric (the radical is the parameter that communicates sound or meant). Can an example be really the character kǒu ( ). It communicates the meaning of "mouth" in different characters; it is easier to decode as: ( ) "to approve"

口

可


口喜

丂

31

(“From (kǒu) 'mouth' and (qiǎo) 'exclamation': approve: okay”, Wieger Wenlin Institute 1996); ( ) “word. Language”; ( ) “name”; ( ) “sound, melody”; ( ) “felicity”; to these more complex that to be decoded they are in need of a knowledge more deepened of the Chinese culture in the philosophical aspects, logical, religious etc. Examples are: ( ) “past” (“That which has passed through ten mouths, i.e. a tradition dating back ten generations”, Wieger Wenlin Institute 1996); ( ) “fortune” (“From (bǔ) cracks on a tortoise shell (used for fortune telling in ancient times), and (kǒu) 'mouth'”, Wieger Wenlin Institute 1996) ( ) “good things” (the character is linked to that is linked to and this one vehicle the semantic aspect of ) ; ( ) “to cry” (“A howling dog" Wieger Wenlin Institute 1996). An other example could be ( ) “uncle”, ( ) “scolar” e ( ) “falbe”. The common component ( rén) “man” vehicle the meaning. The radical ( ) “field” is even tied up to well 138 other characters, semantically connected. The process of abstraction, other remarkable aspect from take in consideration for the mathematical thought that we are discussing, is, in this delicate context, and it would deserve greater close examinations. As confirmed a long time ago also by Rygaloff in a released interview some, on the language and the Chinese culture, inside this last it would seem more disposition and accordingly less difficulty, to abstract the semantic value of the characters rather than that phonological. The "hangs" cognitivo of the characters inserted in the categories 5 and 6 is different therefore. "Chinese character writing permits, without making it to condition, the abstraction of sounds while not prohibiting the same process to take place for the meaning, being however understood that it would be better suited to the former than the latter process" The "key", the radical, according to a mathematical reading would underline, on one hand an unknown (a specific sign but "inde-finished" that it assumes sense in relationship to the linguistic context in which is inserted ("word to ask, heart feeling" etc.)7; on the other hand, however, it would be able to assume the meaning of generalized sign, able to communicate the character in which it is inserted8 and to allow the reader the identification of this last. Last possible interpretation is that it comes out of the analysis of the existing functional relationships inside a composed character, among the various parts that compose him/it. In this sense, for what specifically interests us, her "key" assumes the role of parameter. Arranging then with Chiarugi on the strong cognitive value of the parameter for the mathematical thought and particularly for that algebraic (Chiarugi et alii, 2003), the acquisition, her "manipulation" autonomous and the memorization of an engages some kind would represent, since the primary school, one first strong

言

十口

仙

7

古卜口善哭

占

善

言

口

亻人

田

响

吅

伯

言

犬

生

This aspect also finds him again in the characters not radicals. Can example be the composed character píngděng ( ) "equal" and the semantic analysis of the two characters that you compose it (Rick Harbaugh, http://www.indiana.edu/~ealc /). 8 This aspect comes out in the research of an ideogram in the Chinese dictionary.

平等

32


exposure to the algebraic thought. As previously confirmed if we reflect there, it seems that, to reach this formal writing, the student metabolizing the concept of "variability" as initial relationship among "quantity" and subsequently as "quantity" dynamics reported through one "formula" to others "quantity" also them dynamics, try to gather the principal aspect of the algebra, its being language, tool of thought, mathematical tool to strengthen the resolution of problems and to individualize and to compare relationships and structures. To reach this level of metacognizione owes however to pass different step that departs from the simple manipulation and they arrives, after quite a lot years to a level of completeness organized by a continuous balancing among a thought serial, local, sectarian, to one global, holistic able to operate categorization type cognitive and possible generalizations that however they have to strongly remind you strongly hook to a procedural and concrete thought. Taking back into consideration one of the examples discussed in the paragraph (the case of the term shi), the situation that introduces him to a student that is approached to the study of the language can reassume him regarding this sketch:

Fig. 8. Complex relationships among orthographic aspect, phonetic and Semantic in the Chinese linguistic system (Li Hai et alii, 2001)

As previously confirmed, we hold this aspect, inserted in an ampler picture of the learning of the algebra, interesting for the study of the phase of transition from the arithmetic thought, for tests and errors, and that algebraic report her in how much able to conduct in almost natural way to the trial of "generalization" as "capability of noticing something general in the particular" (Love, 1986; Mason, 1996).

2.1.10 The Meta-rules of the Language and the Mathematics Other consideration can be made in relation to the destination-rules that recognize him to the structure of composition and therefore to the reading of these characters. Destination rules that are brought back, in our opinion to those discussed by


33

the point of view historical-epistemological in the second chapter for the ancient tradition of resolution of the equations: "homogenized" and "equalized to make to communicate them". Destination-rules of that a base has commune in the aspect "functional relationships and part-whole relationships"; characteristics these that, according to Nisbett, as regard of the Chinese thought, of its logic, of his to deduce and to conjecture (Nisbett, 2001). Every composed character, as seen, through the various examples, show then to his/her inside a further difficulty: a plurality of "readings"9, both as isolated character and, as says, inside a text. This complicates of it notably the memorization. An interesting job on the definition of a possible genealogy of the language Chinese ideographic, genealogy besides facilitative in the process of acquisition and memorization of the characters has been discussed, by now some years ago by Rick Harbaugh. His research, through a hierarchization of the characters, seems to underline the parametric aspect of the radicals. “If you take Xu Shen's etymological dictionary from 2000 years ago, put all the connections into a computer, and generate trees showing the connections, what do you find? That every part of every character can be traced back to less than 200 root characters (wen). This is not the bushou system (literally "section-heading" but often mistranslated as "radical") which only connects one part of each character with 214 characters. Rather, this new zipu system shows how every part of every character is itself a character. As a method for organizing a dictionary, it generalizes the bushou system by allowing any character to be found if the reader knows any component of the character or knows any character which shares the same component. Students can quickly locate characters while also better remembering the relations between characters.” (Harbaugh, 1998) Besides, for how much previously seen treating the historical evolution of the writing ideographic and the Chinese characters, with the thick stylization to which have been submitted you/they have suffered a trial that the relationship has made more and more indirect among significant (the form, the image) and referent (the represented thing, or better the mental image of it). Consequence of this is that although the circle of meaning of the written message results more and more delimited and conventional, and therefore less ambiguous and easier understanding and communication, nevertheless, requires to the student that learns the linguistic register, one always greater intellectual abstraction to be inclusive. Examples can be10: The appeal to the hierarchization and to a parametric vision of the structure can facilitate, in our opinion, to accent the strong one connection significante-referente emphasizing the semantic or phonetic aspect of the various characters.

9

Overlapped characters, placed side by side, one inside the other, repeated more times etc. (Shek et alii, 2006). 10 Further examples can be viewed on the site web of "Continent China" of the Rai (Radio Italian television) edited by the Prof. Yuan Huaning of the university You Catholic of Milan: [email protected]

34


Fig. 9. Examples of composition of complex Chinese characters

The Chinese research in didactics, in the recent years is interested in the problem list, defining in this sense, new didactic practices for the learning of the language. A meaningful example can be that brought in Tse, Marton, Wah ki & Ka Yee Loh in 2006 that “illustrates principles of the perceptual approach of teaching that aims to integrate language learning and structural awareness development” (Tse, Marton, Wah ki & Ka Yee Loh, 2006, pp.15-17). 2.2 Can the Natura l La ng uage I nfluence the Educatio na l Sy stem

2.2 Can the Natural Language Influence the Educational System of the Area Logical-Mathematics? Which Is the Impact in the Teaching of the Mathematics? 2.2 Can the Natura l La ng uage I nfluence the Educatio na l Sy stem

The learning of the language is already favored in the students in the primary school; this is contemporary to the learning of some mathematical notions. And' our hypothesis to believe that the competences acquired for the learning of the language natural writing can favor a first acquisition of some competences key of area logical-mathematics. We have been able to experimentally verify that this inference has been such in the historical evolution of the Chinese educational system.

2.2.1 Contents and Objectives in the Historical Evolution of the Teaching of the Mathematics in the Chinese School: A Brief Panning The Chinese mathematical education can consider a clear reflection of the oriental culture in its globalist, its very variegated historical evolution tightly appears connected to the tumultuous context of the partner-political changes of the Country. In this chapter, we will try to delineate its fundamental footsteps, reporting us in particular way to what currently he shapes as the Chinese scholastic curriculum of Mathematics.

2.2 Can the Natural Language Influence the Educational System

35

Aware of the reported didactic problem list to the Italian curriculum and the temporal evolutions of this (through the Curriculum of the school of base of De Mauro, 2001; the Recommendations for the Primary school of the Moratti, 2003; the indications of the UMI, 2001/2003/2004; the Indications for the curriculum of the first cycle of Fioroni, 2008 and Rule of the obligation of education; DM 139/07 for the 2008 second cycle first two years), the study of the didactic practices of the mathematics and therefore of the phenomenon of teching/lerning of this in a different cultural context what that Chinese, can result meaningful for our job in special way for the understanding of some behaviors of students involved in the experimentation of the work of research and not inserted on the scholastic palermitan system and with years of frequency of the scholastic system of the country of origin. As said previously, in the years, Chinese civilization, is shown very skilled in the study and in the "assimilation" of the western culture, enclosing to its inside those that could be the elements key of possible discordant visions and different approaches in political field, economic, social and therefore educational. The scholastic curriculum today, also preserving his/her native cultural identity, picks up therefore to his/her inside quite a lot of the experiences and of the external cultural traditions to China, in a continuous balancing among different approaches East-west. In China, did the study of the Mathematics and its didactics, have his "first" de, 581–618), Táng ( , 618–907) e velopment during the dynasties Suí ( , 960–1279), with the progress of the Imperial College (Guózǐjiàn Sòng ( ) the most greater academic institution of the period (Zhang, 2005). In the years, were these studies plain pian then "placed side by side" and then replaced in thick way by those for the calligraphy, subject of teaching done appropriately hold a strong cultural identity of the country and for this object of close examinations of pedagogic nature. The teaching of the mathematics, in a meaning more western, taken back his/her progress thanks to the studies of the missionaries that respectively marked with M. Ricci (1552-1610) and A. Wyle (1815-1887), the first true introduction of the western mathematics in China (Zhang, 2005). The translations in Chinese11 (and their following diffusion in the schools of the country) of fundamental texts for the western tradition what him "Elements" of Euclide, him "Elements of Geometry and differential and Integral Calculation" of Loomis (1811-1889), him "Elements of Algebra" of De Morgan (1806-1872) etc., laying the bases for a meaningful integration of the western and oriental mathematics in an international vision of the disciplinary didactics. This aspect of integration has always been, in the years, an element held by the Chinese of fundamental importance for a "improvement" meaningful. In this first brief excursus on the formation of the mathematical education in China, remarkable they are then the 1862 dates, year in which was founded, in the

宋朝國子監

11

隋朝

唐朝

To work for instance of Xu Guangqi (1562-1633), Them Shanlan (1811-1882), Xia Luanxiang (1823-1864).

36


capital, the institute of Astronomy and Mathematics and the 1898 year of foundaJīngshī Dàxuétáng), today tion of the Imperial university in Peking University in Peking Běijīng Dàxué) (Zhang, 2005). The influences with the Japanese educational tradition transversally find again then him through the Chinese translation of some of the most important texts of Japanese Mathematics and the publication of some writings of pedagogy in famous magazines as the Journal of Education, The World of Education etc. Examples are of it: the translation of the "plain Geometry" of D. Kikuchi (1855-1917), of the "New Geometria" of Nagasawa and the publication of the "Arithmetic and its pedagogy" (1895) of F. Rikitaro (1861-1933). As it specifically regards the algebraic disciplinary context, mathematical context on which we are detaining there, the press of the books of text of Algebra, up to 1906, was relegated to the left to a traditional pagination with texts written from top down in column and from right. The use of the variable was strongly limited to the use of some Chinese characters that seems you were used as in relationship to the contexts of presentation of the problematic situations12. After the 1911 revolution, the schools of the Republic almost all founded a big number of courses of elementary mathematics shaped mainly on the European and American example, also through the direct introduction of the books of text English and American imported in the Chinese educational system. Among the characters I detach some historical period that approximately develops him from the 1922 meaningful publication ("1922 Education Regulation") to 1963 with the "Syllabus of Full-time School Mathematics", they for instance find again him, F. Zhongsun (1898-1962) the "first mathematician who introduced the research of western mathematics fundamentals and mathematical logic in China", the teacher Hu, headmaster of the department of Mathematics (19471956) and deputy president of the Normal university in Peking (1949-1957), nominator, together to other researchers, of quite a lot courses of formation for teaches him, planned according to logical of balancing among the different traditions cultural westerners and Oriental, W. Zaiyuan and H. Dunfu, one of the pioneers of the mathematical education in China in the definition of books of text and manuals of Arithmetic, plain Geometry, solid Geometry etc. able "to enclose" together different approaches East–west to the mathematical knowledge held essential to the formation of the students, etc. From 1930 to 1950 the Chinese translations of some American texts (accompanied by the original version not translated), as those on the plain Geometry of A. Schultze, F. Sevenoak, E. Schulyer and of Algebra of H.B. End, were adopted as texts of reference for then the Chinese secondary school, gives this not to underestimate in terms of integration of the saperis inside the scholastic formation, and not only.

北京大学

12

京师大学堂

The actual situation is well different. The registers semiotics of representation that find again him in almost all the books of text of Arithmetic and Algebra for all the scholastic orders, introduce an alphabetical writing to the inside (pinyin), an algebraic symbolic writing and a symbolic type ideographic. On this aspect and the reverted Implicative cognitive for the learning we will return subsequently in the next chapter.


37

Other meaningful element is the introduction in the Chinese scholastic system, around 1940, of the ideas of didactics of the mathematics of F. Kline (1849-1925) promoter of the Program of Erlangen. In relationship to this renewal curricular, almost exclusively goes always however to consideration a practice didactics type transmission, with the teacher to the center of the didactic action and the simple students "louder". Her philosophical implications, previously discussed also, that do head to a Confucian philosophy strongly felt by the Chinese people, they are to consider themselves connected to the didactic and methodological choice for the disciplinary teaching. This for years has remained (and perhaps has never been abandoned entirely even today) strongly anchored to the learning of hundreds of students. To fully understand the point of view of the learning would need in fact to analyze the role of the student in the culture Confucian and the philosophical-moral virtues of the strong bond between the student and the teacher. As confirmed before, also today, the research in didactics of the mathematics is still questioned, through an international debate, on these aspects of the Chinese curriculum. (Greene, 2000; Lan et alii, 1999, Nisbett et alii, 2001; Zhang Diaozhou, 2003). With the 1949 Chinese Popular Republic institution the scholastic system mainly turned him to the Soviet model that emphasized the logic and the deduction (ICME 6, Chinese Delegation, 2008). Contextually, in the following years, the disciplinary contents were reorganized in more organic way through a system of those that the fundamental elements were held for a more careful didactics to the disciplinary learning: "combination of theory with practice, guides of teacher with participant of students, strictness with flexibility and consolidation with development" (ICME 6, Chinese Delegation, 2008). The 1963, with the publication of a new syllabus for all the scholastic orders of the Chinese educational system, laying the bases for a general reflection on the formative needs of the school, in relation to the partner-cultural characteristics of the epoch. Her knowledge and the competences of base aimed to mainly develop the abilities of calculation of the students, their spatial imagination and the analytical logical thought to hold relationship to an intense activity of practice and exercise. The ten years of the cultural revolution, from 1966 to 1976, to brought the country on the edge of the civil war; the most greater part of the Chinese cultural institutions and the system were destroyed it returned back quickly. As reverted, the preparation of the students was decidedly scarce. The 1977 also saw the reintroduction of the examinations of admission to the university and graces to intense exchanges with different western Countries, in the 1980s, he returned to the diffusion of new ideas and new didactic methodologies among which the practice of the tests of access standardized for the universities and the theory of Gorge Polya on the Problem Solving, became popular among the Chinese teachers of Mathematics (ICME 6, Chinese Delegation, 2008).

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In 1986, with the publication of the "Compulsory Educational Act", the obligatory scholastic system was reorganized that was brought by six to nine13. The developments of the years 1990s underline the interest of the country for the educational research, considered disciplinary research since then, among the national priorities Chinese, through the review of the procedures of university selection, the computer science introduction to school and the definition of new didactic addresses and books of text. Among 2001 and 2003, with the publication of the "standard Mathematics Curriculum of Full-time Compusory Education" and the "Mathematical standard Curriculum High Senior of School" have been introduced, after a first phase of test that has involved around 470.000 students of 38 different districts, different new addresses (AA.VV., 2001) and new materials for the teachers. The fundamental elements of the document enact an obligatory education divided in three phases, for each of which knowledge are established, ability, formality of mathematical thought, typology of problematic situations, attitudes of learning etc. Among the general objectives for the mathematics they for instance find again14: - the acquisition of the knowledge and the essential mathematical competences to the future life and the study of other disciplines; - the application of the mathematical thought to the life of every day; - the understanding of the formative value of the mathematics and the relation of it, the nature and the human society, - the development of creativeness, competence and personality of every student through the learning of the mathematics. To completion of the nine years of obligatory education, a lot of some students continue their studies of secondary education frequenting other three or four years of school, before accessing the university education inside which the university departments almost all offer courses of advanced mathematics. Returning to the official document for the obligatory scholastic curriculum, transversally to the discussed indications, for a learning aware of the discipline in object, it furnishes some didactic suggestions on the strategies of teaching and evaluation. This last ones they articulate in relationship to those that in the past as fundamental elements were introduced for the scholastic education and therefore for a general improvement is of the methods of teaching, that those proper of the 13

The compulsory school included the primary school and the secondary school of first degree. The articulation among the two cycles however he was not extended to the whole China, in some cases it remained to discretion of the various regional authorities. The most diffused system was that of 6+3 (six years of Primary school and three of Secondary of first degree), systems existed I divided however in 5+4 and, in the zones rural nine years without any subdivision in cycles (Educational Commission, 1996). 14 Same objectives find him again in the Italian ministerial document that is conforming to that European. The problem that we set there is as these objectives you/they are reached in the single realities.


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learning. The two aspects come therefore understood as faces of the same medal, reunited trials in a hold relationship of the type yin-yang. In the routine evaluative owe therefore to be only not kept in consideration the "simple" resulted of profit of the students; on the contrary, all the aspects related to the possible cognitive development, social and emotional, which holds in consideration and valorizes the real experience and the knowledge progresses of every single student (Zhang, 2005). Through the concrete practice, the elaboration of conjectures, the communication of express meaningful regularity also through possible computer metallization, mathematical activities have to serve to the student in the autonomous elaboration of the fundamental knowledge, derived by the real experience and systematized by the teacher.

2.2.2 The Teaching of the Mathematics in the Today's Didactic Practice In accordance with the Confucian tradition of the study, of the respect of the elderly ones, of the teachers and of the formative value of the culture, the learning is held today, as in the past years, an element not renounceable for a middle Chinese that he/she sees then in the mathematical knowledge a to know badge of culture and education in special way for that that it concerns the arithmetic abilities and of mental rapid calculation, but not only. The scholastic system reflects this social demand through the definition, of about twelve weekly hours of disciplinary study (of Mathematics) and interdisciplinary (mathematics, sciences, physics, biology) for the primary school and not less than you are for the secondary inferior. On one hand, the notions, principal protagonist for the scholastic learning of once, seems entirely almost today overcome by didactic strategies more aimed to privilege the logical reasoning15 and the autonomous construction of mathematical models conceived on the base of situations and real problems; on the other hand, the calculation and the rigorous memorization of this, through the strong repetition of facts arithmeticians and formal ownership, play in every case a fundamental role in the scholastic curriculum and in the traditional daily didactic practice for teaches him of Mathematics of the various scholastic orders. Today, the ministerial indications push, in this sense to the use of formulas, rules and algorithms able to hierarchies the various problematic situations introduced in class, seeking therefore, for them meaningful algorithms of solution in terms of simplicity and flexibility of thought. General procedures and specifications that in the years, through a thick use of exercises and concrete practices, the students have to memorize and to repeat for an aware learning. As discussed previously, in the next chapter, this repetition, that also finds one natural application of his in the Chinese linguistic learning, does not reduce him to 15

The term "logical reasoning" is to intend him in a meaning "typically Chinese" that we will define in the chapter 4.

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a simple repetitive operation, it hides on the contrary to its inside an ability of non indifferent metacognizione. The education and the didactics, in past, in all the circles of learning, were held effective and productive, how much more they allowed the student to exactly reproduce how much shown, explained or simply told by the teacher in class and out of it. Today this model has been, even though partly, supplanted by a learning more aware than however, in our opinion, it sets the bases on that historical that emerge in quite a lot aspects connected to logical-philosophical interpretations of the learning of the Confucians Heritage Cultures students however. Considerations it discussed by now in literature from different authors (Biggs, 1996; Lee, 1996; Leung, 2001; Reagan, 1996). Leung, according to this vision, introduce some of the possible elements key for the Chinese mathematical didactics and therefore for the study of the salient phases of the processes of teaching/learning, it discusses, as tells the chapter II, six dichotomies that can present, in an optics of didactic comparison multicultural as distinctive elements of the western and oriental cultures: product (content) versus process, rote learning versus meaningful learning, studying hard versus pleasurable learning, extrinsic versus intrinsic motivations, whole class teaching versus individualized learning, competence of teachers: subject matter versus pedagogy. (Leung, 2001, pp.38-46) Six dualities, read, in a first approximation, in relationship to the figure of the teacher of mathematics inserted in the context class; they allow us, in our opinion, to define the "typical" the teacher's role in the Chinese school; role that, despite the different formulations of the educational systems developed in the past years, it seems today still, as mentioned in the preceding chapters, the principal actor of the didactic practice, all centered on him and on his to know. The student learns following step-by-step the instructions of the teacher that autonomously proposes the circles of learning and the problems to it contextualized and models them according to his to know (Xie, 2004). Her ministerial indications, seem to a first glance, in this sense, not acted because of one strong repartition partner-cultural that sees the transmission of the know how important patrimonies from let to manage to the "elderly" and to try to "to imitate" knowingly. The passage from her "simple" "imitation" to the autonomous meaningful knowledge is deduced however, and it strongly comes in demand to the students of all the scholastic orders, in the transferability of the contents and therefore in the conscious ability to do connection with other knowledge, with other abilities, with other meaningful experiences. Expressions of this can transversally find in the application to the students of "to divide complex problems in simpler problems", "to gather the fundamental structures (the invariants) of the various problematic situations (proposed from teaches him)", "to define algorithms fondant and procedures of solution" (what categorized the concepts and the express mathematical knowledge often in registers different semiotic) etc. On one hand, for the teacher, the transferability operated by the single a students is seen in this sense as test of acquisition of the mathematical competences, from the other one is used in the process of generalization of the disciplinary


41

contents and the acquire mathematical knowledge. Also this aspect, in accordance with Chemla (2007) he shapes, in the scholastic curriculum, in relationship to how much I strongly dictate in the preceding chapter for the history of the Chinese mathematics as a cultural element for the mathematical tradition of the Asian east south. The disciplinary suits on which the new obligatory Chinese curriculum develops him are: - Numbers and algebra: numbers, arithmetic, pattern, equations, disequalities, functions; - Space and sketch: form, greatness, position in the plan and in the space, geometric transformations in the plan and in the space; - Statistic and probability: study of data coming from the experience and from the real life, them casualness; - Exercises and applications: strategies of help for the synthesis and the use of the knowledge and the experiences. Through the auto exploration and the cooperation the students have to resolve stimulating and near problems to the real life, developing new competences, deepening their understanding of the contents of the preceding circles and appreciating the interrelations among the different aspects of the mathematics. The organization of the disciplinary contents is so articulated (Wang, 2001)16 Table 3. Organization of the disciplinary contents of the new Chinese obligatory curriculum Degree scholastic

Organization of the disciplinary contents

Elementary school (6 obligatory years) I and II Phases

Numbers and operations; Measures; Elementary Algebra; Elementary Geometry; Elementary Applications. Algebra: identity, powers and operations among powers, square roots, logarithms; Equations and disequalities: first degree, quadratic, systems of linear and quadratic equations, irrational, logarithmic: Successions and series: arithmetic, geometric; Geometry: congruence similarity,

Middle school (3 obligatory years) Phase III

16

The term "logical reasoning" is to intend him in a meaning "typically Chinese" that we will define in the chapter 4.

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2 The Chinese Written Language as Tool Table 3. (Continued)

Superior school (3 non obligatory years)

17

points and notable segments of a triangle, angles of a triangle, relationships between angles and sides, theorem of Pitagora, circle, theorem of Talete, angles to the center and the circumference, quadrilateral subscribed in a circle (quadrangle of ropes), quadrilateral circumscribed in a circle (quadrangle of shares); Probability and statistic. Functions: elementary functions and them ownership, transformations; Theory of the whole: notations, ownership of the whole, operations among whole; Trigonometry: definitions17 of the trigonometric functions, relationships among the trigonometric functions, theorem of the breasts and theorem of the cosines, trigonometric formulas, demonstration of the theorem of the shares; Vectors and Cartesian geometry: operations with the vectors, coordinate, equations of straight line, circle and parable, center of the tetrahedron, distance between points and straight lines and among parallel straight lines, equations of bisectors, equations of ellipse and hiperbol, general ownership of the conic sections, canonical equations of the plan, equations of sphere, cylinder, surfaces of rotation; Geometry in the space: concepts of point, straight line and plan in the space, mutual ownership of points and straight lines, ownership of the cube, of the parallelepiped, of the cylinder and of the sphere; Number complexes: concept of number complex, representation and conjugated; operations with the complex numbers, trigonometric form of the complex numbers, theorem of De Moivre, extraction of roots of numbers complexes, roots of the unity; Analysis: limits of successions and functions in the ended one and to the endless one, continuity of functions, rules of differentiation; Series: concept of series, convergence of series, geometric series, sum of convergent geometric series; Probability:

We need to make attention that the term "to define" in a hypothetical-deductive system has a meaning, but here you could have another of it.


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Table 3. (Continued) exchanges and combinations, elements of classical probability, fields of probability, elements of statistic with average and standard deviation

In the specific one of our job, the reading of the official documents that you/they define the curriculum scholastic Chinese for the study of the arithmetic and the algebra and the study of the characteristics of the algebraic problemsolving (Bao, 2006) proposed to the students in the classroom and through the books of text, it seems that they define the thought mathematics as a whole processes of analysis and synthesis that evolve him in the development of the phases of abstraction and generalization through a thought type "variational", as observation and recognition of the invariant elements that can give possible place mediate generalizations through the research and the cognitive fixation of algorithmic procedures memorized unified and contextualized in more circles. This trial to which all the students are submitted in the various scholastic orders has been called by some taught of Mathematics as "The trilogy of the problem-solving" (Gu-HuangMarton, 2004). In this process of metacognition "autonomous" particular mentions are not found again to mainly the use of definitions, demonstrations schemes of reasoning type hypothetical-deductive that characterize the western culture. This to which makes him particular mention is the ability to rationally read the information and the data offered by the faced problematic situation, categorized in relationship to the acquired knowledge, to choose the best formalities to represent them and to contextually use them and finally to advantageously treat them to draw corrected conclusions possible of it (Ministry of Education, 2001). The autonomy of thought application in the phase of problem algebraic solving is also vindicated strongly in the books of text that the students direct in the presentation of the situations problematic offers and in the choice of the knowledge that they has to prefer for the solution. Often the exercises and the proposed problems have an only solution but you/they can be resolved according to more different strategies: it recognizes him as correct only the best, that faster and that it requires less operations. In some books of text possibilities of alternative runs are not underlined for the solution of some exercises of algebraic and pre-algebraic die, the attention of the student is focused on considerations of technical and procedural nature. Many of the aspects here discussed in relationship to the curriculum scholastic obligatory Chinese can certainly find the way more or less emphasized, in other cultural realities as for instance that Italian. The situation that introduces him is not certain dichotomy even though in some aspects he can clearly be defined an identity cultural east Asian (Leung, 2001) and Italian.

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The following scheme puts in evidence the coordinates cultural takings in consideration for the done analysis:

Natural Language

History of Mathematics

Philosophy&Culture China

History of Demonstration

Chinese algebraic thought

Aristotelic Logic

China: Fundamental Algorithm

Meta-rules for algebraic thought, Life Etc…

Hypothetical Deductive

Example: Making Omogeneus and making equal Fig. 10. General reference schema

2.3 A Resume Chart on the Relationship Natural Language Mathematics The Chinese language:

Chinese

Recognition and Composition of lines As first activity for the recognition of a character. Relationship of order according to the rules already exposed in the paragraph 2.1.6.

Radical Use of the "parameter" to connect more characters.

Ideograms Unaware use of the Algebraic thought (Equations, principles of equivalence and equality). Logic lived with mechanisms "for association" with use of destination-rules.

2.3 A Resume Chart on the Relationship Natural Language Mathematics

45

The Italian language as possible representative of the Indo-European languages18:

Italian

18

Word Order relation on the minimal significant sense units.

Predicate Direct and inverse relations (passive and active phrase

Argumentation Bivalent Logic. “Absurdum” reasoning. Hypothetic –deductive system.

This represents better a profit schema to compare the analogies and the macro differences.

Chapter 3

The Meta-rules between Natural Language and History of Mathematics

3

3.1 The Algebra between the History and the Didactics, a Variety of Perspectives 3.1 The Algebra between the History a nd t he Didactics, a Variety of Perspectives

Before going on to outline teaching, we need a short historical–epistemological process that can draw criticism, quite basic, the key stages of the long journey and suffered from the trend on Algebra as the Arithmetic. We believe the combination of arithmetic-algebra as significant for the relationship between mathematical thinking. Considerations should be made parallel to the geometric area in which we mention in the discussion of the chapter but that remains an issue open to possible future theoretical and experimental investigations. This chapter is not only an important key to try to understand what can be the basis of epistemological obstacles related to this topic (Spagnolo, 1996, 2006) but, in light of this, attempts to "clarify" those that may arise as potential patterns of reasoning of the students involved (both Italian and Chinese) in situations of a pre-algebra and algebra. In this sense, the study of representations, preliminary epistemological and historical-epistemological, it is essential to confront the experimental contingencies, as described in Chapter 5. The presentation of the historical context related to the evolution of disciplinary and cultural offers, in our opinion, the possibility of an organic ap-deep in a particular concept and can lead emissions fundamental reflection on the genesis of the cognitive mental representation (Bagni, D'Amore, 2005). In our opinion, the history of mathematics should be "read" in this sense as an expression of culture in the process. As already widely debated in the literature, this will be more thorough analysis, more "easy" you can argue the situation analyzed in the context of teaching / learning, and therefore chose to use the general procedure in similar conditions (Spagnolo, 1998; Spagnolo et alii, 2009). What do we mean when we talk about the relationship between mathematics and culture? It should be pointed out immediately that what we want is not only the presentation of specific techniques through which certain groups of people (students but also mathematicians in history) have dealt with the mathematical knowledge, but also a critical discussion of possible correlations of these with the cultural context where they are and have been included and in which mathematical knowledge and then is processed and treated and that, in all likelihood, it was sent. This is precisely the approach that will guide us in the next chapter for the analysis of experimental results. The theoretical framework in which we are to get more and more is that socio-cultural context in which knowledge is related to activities in which players F. Spagnolo and B. Di Paola (Eds.): European and Chinese Cognitive Styles, SCI 277, pp. 47–89. springerlink.com © Springer-Verlag Berlin Heidelberg 2010

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3 The Meta-rules between Natural Language and History of Mathematics

undertake (Radford, 1997, 2003a, 2003b) and this must be considered in connection with cultural institutions of context from time to time considered. As pointed out by Radford, configuration and content knowledge of mathematics is intimately and properly defined by the culture in which it develops (Radford, 1997, p. 32). Culture in this sense is much more than an obstacle for knowledge1, this is closely rooted in its cultural place. Knowledge is a product of a specific type of human thinking and, as suggested by Wartofsky (1979) and Ilyenkov (1977), the thinking is a social practice, "a form of reflection on the world according to conceptual categories ethical, aesthetic, and other cultural categories" (D'Amore, Radford, Bagni, 2006, p.5). The thought of greek classical period, as previously expressed in the first chapter, it was conformed by eleatica distinction between being and non-being as part of a bivalent. This pattern of thought supported the Greek episteme and between its various manifestations including that of mathematical thought. The episteme of China, as discussed, albeit briefly in Chapter I, was complied from different conceptual categories such as, inter alia, directed Taoist yīn-yáng. In this vision, to Radford, the connection Mathematics-culture cannot be regarded as a pure coincidence, there is a deep connection between the two terms of the binomial: the mathematical forms are cultural reflection of the world, cultural forms to give meaning to it. " Therefore, in this sense, the cultural diversity of interpretations and suggestions can be a profound richness that is unfortunately all too often its collapse in the name of a universality that does not exist in reality and not to be found as a fact of culture. It is reductive and false. (D'Amore, Radford, Bagni, 2006, p.5). In D'Amore (2005) in this respect one speaks of Babel semiotics of class and cultural relativity to the learning of mathematics. The basic idea is not to conceive of culture as a shirt of force within which to read and interpret all the processes of teaching / learning, but to rethink the possibility that the cultures have a dimension of reinforcement. Different cultures that transmit values and attitudes different shareable class in society and able to be accepted by the students in a vision of equality (Sfard-Prusak, 2005). Another problem regards the deviance compared to a certain well-established practice teaching (cultural). As can be seen, albeit in part by the Chinese teacher, reported in the previous chapter, this is a sensitive issue in education, especially now that classes are increasingly multicultural. It is a problem that involves several players in parallel: the student placed in class, the teacher as mediator of knowledge and one who is led to question whether a single practice is deviant or not with the expectations, the family the student as an outside the micro-class society but highly influential on it etc.

1

According to the vision of Radford epistemic nature of culture, in agreement with the idea of "milieu" (as is seen in the Theory of Situations Brousseau) is often conceived as something that is opposed to the individual.

3.1 The Algebra between the History and the Didactics, a Variety of Perspectives

49

The situation that presents itself to our eyes may then schematized thus:

Fig. 1. "Subjects" active in multicultural learning situations

The binomial Mathematics-culture is probably very complex and contains within it different readings related to each other. This work, through experimental investigation, tends to develop, albeit in a first approximation, several components, with a strong emphasis on what may be the differences and similarities in respect of certain specific elements related to learning of mathematics and particularly of algebra. For the analysis of "differences" seek, Bernhard (1995) distinguishes four key moments highlighted in the past: the difference as deficit; difference as disadvantage / deprivation, no difference as a difference of substance, as a fundamental difference heterogeneity. If the first three correspond roughly to the dominant perspective in the history of mathematics in different eras, in the smooth transition and universality of Western standards and consideration of underdevelopment of the practices of other cultures, and the fourth describes or suggests the future development of the problem with an ever greater willingness to understand diversity as a fundamental heterogeneity in the knowledge game. History of Mathematics in an ethnomathematics may be this sense of great utility: it can provide important tips on how to understand diversity. Of particular interest is in this sense, the mathematical context in which we decided to move the algebra, complex milieu and always at the center of a heated debate on the historical development. Aware of the possible parallel between cognitive development and historical evolution (Piaget & Garcia, 1983), the theoretical framework in which we are moving us to consider the role of history in relation to different cultures and thus provide a valuable opportunity for a critical reading of cultural contexts of the past and, even better, this (Vygotsky, 1990).

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"Read" and "interpret" the history of algebraic thinking in education is a key as difficult as fascinating. Biunivocal establish a history-teaching is, in fact, a complex operation that requires the identification of a domain of validity and applicability of the report in reference to the mathematical content, the subject of investigation, mainly to the phenomenon of teaching / learning is investigated experimentally and thereby to regulate didactic transposition (Chevallard, 1985). Referring to our research work, the history of mathematics and in particular the evolution of algebra may be a "slow observer" interesting. Already by the end of 1800, it was thought that the mathematics education of the children could benefit from a comparison with the development of mathematical thinking in centuries, and that this comparison could be a source of inspiration for teachers and researchers (Demattè-Furinghetti, 2006). "The genesis of knowledge the individual must follow the same course of the genesis of knowledge of the breed ... it would seem that knowledge of the history of science should be an effective remedy in teaching this science" (Cajori, 1896) Several experimental studies (Harper, 1987; Sfard, 1992; Sierpinski, 1994, Spagnolo, 2006) also seem to confirm as some of the difficulties encountered by students in the acquisition of mathematical concepts and procedures can be identified as barriers experienced in the history of the discipline. In this context, a retrospective analysis of the historical development of the Algebra shows a long and difficult path of growth of this discipline, marked by the debates rage between mathematics cultures of different peoples. A careful reading of the historical picture shows off a slow and difficult than the geometry and difficult "relationship" with the arithmetic, evidenced by the constant effort to transition from computational procedures to "mathematical objects". Wanting to draw a picture of the past of the algebra the development and thus of the symbolic system used to express the concepts of this discipline in order to briefly highlight the most important steps of the algebraic thinking, a first reference theory can certainly be one GH Nesselmann and the three stages he identified and distinguished history in algebra: - Phase rhetoric (Diofanto earlier in Alexandria, 250 AD): an Algebra word, all words, not symbols. - Phase syncopated (Diofanto by the end of the sixteenth century): the abbreviations are introduced for the unknown but the calculations are performed in all natural language. - Phase symbolic (introduced by Viète, 1540-1603): the letters are used for all quantities, uncertainties or not, and "exploits" the Algebra not only to discover the value of the unknown, as in the second stage, but also to test the rules that bind the various quantities and to express those solutions. While, however, the classification of the historian said there appeared to be of particular interest for the task envisaged, on the other hand, it appears, as pointed out by more recent studies, not exhaustive. Locate it in history, in a certain, exact, the separate and distinct phases that mark the development of algebraic thinking, it seems almost impossible. Breaking, for example, between the rhetoric Algebra and syncopated or between this and the last token is not clear, well defined. One certainly has not


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supplanted the other of a sudden, the transition was slow and gradual (Gheverghese J., 2003, Spagnolo, 1995; Malisani, 2001, Spagnolo, Malisani, 2008). The proof is the analysis of the parallel development of mathematical knowledge, for example, the equations and systems of equations of first degree2 of some ancient civilizations, such as Babylonian, Greek, Egyptian, Indian, Chinese, Arab and European, and the study of some problems existing in the different books written at different times in history and came down to us through the work of several scholars. As quoted in J. Gheverghese (2000) and the volume of the Enciclopedia Italiana Treccani on the History of Science, China, India, Americas (Treccani, Vol II, 2001), in large part, the general historical works on the reserve Mathematics Greek and European tradition, the more attention often ignoring other cultural traditions such as the Mathematics Arabic, Indian, Chinese or even American pre-Columbian. Not lacking in any interesting work on Arab science and its peculiarities in this respect to a possible international context. Examples include the work of Ambrosetti (2008), Djebbar (2001), R. Rashed (2004) and Nasr (1977), works that recognize the Islamic scientific culture is direct and not mediated through the function between the Greek culture and the medieval and Renaissance Europe. Same considerations should be made for the study of mathematical traditions that have evolved in geographic areas affected by the Chinese Mathematics, like Korea, Japan, Mongolia, Tibet and Vietnam (Martzloff, 1997, pp. 105-110, Needham, 1985 Treccani, Vol II, 2001). The task that we intend to develop in this chapter certainly does not want to be a comprehensive work from a historical–epistemological point of view. The generalizations that are discussed considering the different mathematical taken into consideration, be they Greek, Babylonian, Arabic, Chinese, etc. Certainly leave some gray areas and geographical content, and vice versa, in some cases, perhaps delineate contours too well defined at not detrimental to the separation of ideas. Overall our objective is to propose an educational overview of some aspects considered fundamental to algebra and algebraic thinking, such as the study of equations and their resolution in the history of mathematics. Particularly important in this respect are the considerations contained in the chapter on the concept of variable and unknown in last parameter, load-bearing elements, as stated, to the algebraic thinking and keys to the next in the forms of reasoning brought to light by the students in experimentally 'approach to the algebraic context. During the section and then look at some examples from the history of the discipline (with specific reference to certain periods and certain geographic areas) trying to show how often certain types of processing techniques and concepts have been made possible thanks to a very specific social situation and culture that has encouraged and even led to the contemporary approach to the issues at stake on the part of scholars from different geographical areas. In this light, then the mathematical knowledge should not be considered as the result of a linear evolutionary process started in ancient Greece, it seems more appropriate to describe it as the result of fragile layers of different conceptual 2

Items of interest for analysis of algebraic thinking and then focus privileged, as mentioned, for analyzing historical epistemological of algebra. The mathematical content is central to one of the experiments discussed in this work.

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complex and constantly evolving in the most disparate “cultural traditions”. The algebraic development of the idea that you are trying to revive what was then a puzzle, a complex mosaic that we will investigate only in some respects like the study of equations and their solutions methods. In the discussion of the chapter, we have preferred to give more space to the classical Chinese mathematics as a system of reference external to our own cultural context and therefore more difficult to monitor and study, but for this very useful to reflect “internal”.

3.1.1 The Algebraic Tradition in Antiquity: An Overview From the documents in our possession, we can say that Algebra was born in the West with the ancient Babylonian Mathematics (Høyrup, 2002, 2007, Maracchia, 2001) and was constituted in a manner independent of the geometry and arithmetic, with Viète and Bombelli. The knowledge of Babylonian mathematics derives chiefly from the study of clay tablets and is based mainly on the work of Otto Neugebauer and Høyrup (Høyrup, 1982, 2002, 2007 and Neugebauer, 1974) that are considered the leading experts in the field. The Babylonian Algebra is generally un'Algebra rhetoric, verbal, even if, according to latest studies, noting for example the tablet AO8862 (1800/1600 BC), unanimously considered one of the oldest, and analysis of the writing given by the scribe, is can be noted in this people a first-level algebra that follows, in formulas, the formula solution of the equations of second degree (Neugebauer, 1935). The difficulty of classifying the methods determined Babylonians and then identify them as cases of algebraic geometry rather than intuitive, set out and solved only verbally, you can still say that whatever the type of solution, there is an algebraic substrate with which it was synthesized, how to ask the question seems kind of algebraic. This stems from the fact that on the one hand, it shows the difficulty of determining one or more unknown values that must verify certain conditions to obtain a given result, in many other cases, the nature of the resolution follows in several boards, certain transactions now standard that often do not reproduce every time the geometric figure from which descend, and then mark a first step from an algebraic to a geometric (Table BM 345689; tav. BM 13901, es.12; tav. BM 80209). It is difficult, in fact, to think that before proceedings are substantially similar, has not gained a mechanistic and hence a certain "formula" for a resolution to be applied to known values to those requested. The Babylonian nell'Algebra symbolic notation was as follows: Table 1. Symbolic notation in Babylonian Algebra Modern symbol x y x2 z xy xyz

Geometric term length width square height area volume

Size (Magnitude??) Babylonian ush sag lagab sukud asha sahar


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“Even the Egyptians, like the Babylonians, as set out and resolved the questions in a verbal or explaining why that particular method used for conducting or why it worked in that context” (Kline, 1972, Italian Edition, p.25). Egyptian mathematics is in the form of a set of simple rules are not shown, corresponding to problems arising in the daily life of people. The Algebra and their arithmetic appear, therefore, very limited3. One of the most important writings of Mathematics Egyptian culture is certainly of the Rhind papyrus (transcribed to 1650 BC) preserved in the British Museum, the papyrus is that various problems of daily life, such as the distribution of food. Problems that can be linked to resolution of equations of First degree. The collection of the problems presented in the papyrus is not easy, the unknown factor is designated by the word hau which means “heap”, addition and subtraction are indicated by the legs of a man who walks or to the symbol of the number which must operate, or move away from this, etc. In the study of issues related to algebraic language and therefore the analysis of dual Algebra-Arithmetic relatively to the phase transition between the two at the time of “openness” from a semiotic field as significant as the arithmetic algebraic (by the simple numerical resolution to the possibility to define a class of problems related to each other), we felt it was of particular interest for our work, the problem 24 of the papyrus in which he asked what was the value of the “pile” if the “bunch” and a seventh the “pile” are equal to 19. He wondered, therefore, to determine the unknown quantity in the equation of First degree(written under the modern notation): x + 1 x = 19 7 The method of resolution, called “rule of false position” of Ahmes was based reference to an unknown value probably false and execute on this value transactions listed to the left of the sign of equality, then comparing the result of this operation with the desired result and the board with the proportions to find the correct answer. This can only be defined as a method of arithmetic resolution. Through it, in fact the solution of an equation was sought for successive approximations and does not refer to any abstraction. One can now do is that the papyrus of Ahmes has been written in the style of a textbook for students of the period and that therefore, although showing the general rules for solving various types of mathematical problems arithmetic, not the present especially in general. According to this view, the papyrus is a rather advanced arithmetic. Not all agree with this vision. It is generally accepted that the Egyptians did not have a deductive structure based on axioms that establish the soundness of their rules. I do not feel the need, given the convenience, the practicality for which mathematics was used, the social context of reference could not be challenging in this regard. More speech is related to the systematization of Mathematics in history as Euclid for Western culture and the “Nine Chapters” for the Chinese. Presumably, there are reasons for socio-economic, such as having to teach more people to know the same. At this point, it is inevitable that we have to reorganize the knowledge in an organic body. This has happened in more recent eras (late 1800) in the 3

For a thorough discussion of traditional Egyptian mathematics: Giacardi, Roero, 1979.

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West with the crisis of the foundations coincides with the needs of a middle class that he was reorganizing the transmission of knowledge to wider social groups. The type of rule mentioned earlier, in the Middle Ages was called regula alchataim, word of eastern origin, or regula falsorum. The same techniques can be found in Mathematics Indian Chinese and Arabic for solving concrete problems mostly solvability of equations with a first degree unknown, and in some cases, systems of equations and linear equations of second degree (Guillemot, 1990). Dealing, in this chapter, the cultural diversity as an opportunity to learn Mathematics and the Mathematics wanting Chinese, it should be remembered over the method described above, but also that of double false position. The latter procedure “algebraic” consists of two specific values to a unknown, perform the necessary calculations to find the errors committed with the use of these particular values, and then apply the formula for linear interpolation. The method of double false position was brought to Europe by the Arabs and is found in the works of Al-Khuwarizmi (author who will discuss in more detail in this chapter) of the ninth century, the same rule is found in Chinese Mathematics in problem N.9 Chapter VII of the Jiu-zhang Suanshu under the name of “excess and defect”. The same algorithm is used, centuries later, from the Trattato d’Algibra (work of the XIV century) for the resolution of some systems of linear equations (Franci and Pancanti, 1988; Franci, 1999). In the historical-cultural puzzle that we are trying to achieve seems noteworthy in this regard the excellent computational performance that are found in ancient Chinese mathematics. Example is Chapter VIII of Jiuzhang Suanshu in which we found ways of solution of systems of equations up to five unknowns. As the Boyer, “The Chinese were especially fond of patters; hence, it is not surprising that the first record (of ancient but unknown origin) of a magic square appeared there. [...] The concern for such patterns left the author of the Nine Chapters to solve the system of simultaneous linear equations [...] by performing column operations on the matrix [...] to reduce it to [...] The second form represented the equations 36z = 99, 5y + z = 24, and 3x + 2y + z = 39 from which the values of z, y, and x are successively found with ease.” (Boyer, 1995, pp. 197) Solving linear systems highlights an interesting aspect in dealing with the relationship between mathematics-culture specifically for the Chinese culture, something that we will be more precise in the next paragraph and that it brings to the effect of techniques for calculating with numbers-chopsticks in Development of Chinese Mathematics. As Viola (2005), despite the promising debut,“The Nine Chapters on the Mathematical Art holds the first known example of a matrix as a table coefficients of the system of three linear equations with three unknowns”, not found this way of addressing simultaneous equations in other Chinese traditions, if not in modern mathematics. In an ethno-mathematics, so we are led to think that the method in question for solving linear systems is a logical consequence of the particular techniques used in calculating the sticks (to be considered in this vision as a cognitive4 artifact in the process) and that these, because of the strong 4

In a vygotskian perspective, an artifact is a mediator (Bartolini Bussi, 2002; Bartolini Bussi, Mariotti, Ferri, 2005; Vygotskij, 1974 and 1987).


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practical, subsequently inhibiting the development of an abstract Algebra which is reflected in other cultural contexts. An example can be found in Chapter VII of Jiuzhang Suanshu, called Ying put up (excess and defect), N.2 problem: "A group of people buy the same hens. If each person paid 9 wen, wen 11 remain after the purchase. If each person gave only 5 wen, there would be a shortfall of 16 wen. When you have people in the group and what is the total cost of the hens?" The problem is solved bringing this solution: order the two types of contributions made by members of the Group for the purchase of the hens in the first row. The excess and defect resulting are sorted in a row beneath the first, which contains the contributions of members. Multiply them diagonally with each other, plus the products and call the sum shih. Add up the excess and lack, and call the sum does. If a fraction appears in shih that makes you become their denominators equal. Divide shih for difference of the two contributions to obtain the total cost of the hens. I divide the difference between the contributions to give the number of people in the group. In modern algebraic terms, given spectively the excess and lack:

⎛ a a1 ⎞ ⎛ 9 6 ⎞ ⎜⎜ ⎟⎟ = ⎜⎜ ⎟⎟ ; ⎝ b b1 ⎠ ⎝11 16 ⎠

⎛ ab1 ⎜⎜ ⎝ b

a and a1 the two contributions b and b1 re-

a1b ⎞ ⎛144 66 ⎞ ⎛ ab1 + a1b ⎞ ⎛ 210 ⎞ ⎟=⎜ ⎟; ⎜ ⎟=⎜ ⎟ b1 ⎟⎠ ⎜⎝ 11 16 ⎟⎠ ⎜⎝ b + b1 ⎟⎠ ⎜⎝ 27 ⎟⎠

ab + a1b 210 Consequently, the total cost of hens is: shih = 1 = = 70 ; the 1

a − a1 a−a b + b fa 27 1 number of people in the group is: = = = 9. a − a1 a − a1 3

3

The problem is algebraically expressed in terms of system of two equations in two unknowns for x number of people, and the cost: ⎧9 x − y = 11 . The method described above would seem to present a variant of ⎨ ⎩6 x − y = −16 the method of Cramer (1750). Not found in any general formalizations for the solution of linear systems using the determinant. Other examples are found in Chapter VIII (Fangcheng) of the same text. Resume after these problems. Returning to the Trattato d'Algibra, to illustrate the method of double false position, for example, we can cite the problem 38 that, in modern symbolic language, can be presented as a system of four equations in four unknowns. The author, in fact agree with the method of double false position, by successive substitutions, consider a system of two equations in two unknowns such as: ⎧7 y = 13 + 4 which resolves in five successive steps : ⎨ ⎩4 y = 2 x + 176

(Franci & Pancanti, 1988, pp. 145-150)

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1- Adopt the false position x1 = 21 +

y1 = 40 and calculates the first equation

3 13

2- Replace these two values in the second equation by finding as a result the first member 160 and 218 + 6 the second member. Since the two members 13 should be equal to their difference is d1 = 58 + 6 . 13 3- Similarly by adopting the false position

d 2 = −(58 +

y 2 = 80 , calculate x 2 = 42 +

10 ; 13

6 . ) 13

4- Is obtained then5:

6 6 ⎤ 6 6 ⎡ y = ⎢80(58 + ) + 40(58 + )⎥ /(58 + + 58 + ) = 60 13 13 ⎦ 13 13 ⎣ 5- Substituting y = 60 in the first equation is x = 32 .

Also in Chapter VII of Jiuzhang Suanshu, the problem N.9, said: “A tank capacity of 10 tou contains a certain quantity of rice. Of the grains are added (rice not glazed) to fill the tank. When the beans are glazed one discovers that the tank contains a total of 7 common tou of rice. Find the initial quantity of common rice in the tank.”. Solution: If the initial quantity of rice in the tank is 2 tou missing sheng 26, if the initial quantity of rice is 3 tou, there are 2 more in sheng. Run 2 cross-multiply the excess tou sheng 2, and 3 between the lack tou 2 sheng and summed the two products for 10 tou. Divide the sum by the sum of excess and lack (ie 4) to get the answer: 2 and 5 tou sheng.” supplies, even in natural language, the rule of double false position shown above:

x=

(− 2) * (3) − (2) * (2) = − 10 = 2 1 tou (− 2) − (2) −4 2

Unlike the Babylonians and Egyptians, the Greeks did not want to scholars only "use" mathematics as a tool for calculation, but always tried to justify the rules used in algebraic calculations, it is precisely in this period that one finds the first “demonstration”. We cannot yet speak of Algebra’s axiomatisation, but it tends to greater awareness of the processes used (Kline, Italian Edition, 1972). 5

6

The formula refers to the process which allows to solve equations of type ax=b with x ∈ Q . d1 and d2 defined as the differences or errors obtained by considering how the unknown values of x1 and x2, can be translated into modern algebraic language as follows: a = ( d1 − d 2 ) /( x1 − x2 ) and b = ( x2 d1 − x1d 2 ) /( x1 − x2 ) . Since x=b/a is: x = ( x2 d1 − x1d 2 ) /( d1 − d 2 ) . It is assumed that 1 tou grain yield 6 sheng of rice, since 1 is equivalent to 10 tou sheng.


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Looking at the Greek writings we can see, perhaps because of their sociocultural context, a remarkable development of geometry, but not as the arithmetic and algebraic. In their algebraic equations, the unknowns are always geometric elements, such as segments, rectangles, squares, cubes... geometrical sizes often related to everyday contexts and concrete issues that were analyzed and resolved by means of geometric derivation. A label representative of this type of algebra is that of geometric algebra (term of HG Zeuthen in Van der Waerden, 1983).

3.1.2 The Elements of Euclid as a Canon of Reference for the Western Mathematical Tradition Continuing in this brief historical overview of algebraic thinking in close connection with those that may be cultural elements of Eastern and Western thought, we cannot fail to take account of what constitutes Western culture to the work of reference, with regard to not only the structure of the work in itself but also to the philosophy that you can find "behind" the text, which refers, as mentioned in the previous chapter, a type of bivalent logic axiomatic deductive: the Elements of Euclid. The best part of the mathematicians of the classical period has come down to us through his writings of Euclid, in his most famous work: thirteen books in which they are many rules of calculation. Interesting, for example, in this context is the book II (in which the author justifies some fundamental results of modern algebra, using a language quite geometric), and in particular the proposition 6 which is geometrically solved a problem, which set out algebra is in the form of second-degree equation with unknown positive coefficients. To analyze specifically the methods of solution used by Euclid for the solution of the equations of second instance can then refer to the geometric problems presented in the Book VI to propositions 28 and 29 (Clavius, Euclid, 1574): Proposition 28. On a line to construct a parallelogram equal to a given polygon, missing a parallelogram similar to a given parallelogram. It is appropriate for the given polygon is not greater than the polygon constructed on the middle of the line data, and similar to the missing polygon (Clavius, Euclid, 1574, pp. 218-219). Proposition 29. On a given straight line to construct a parallelogram equal to a given polygon, in excess of a parallelogram similar to another given. (Clavius, Euclid, 1574, pp. 220-221)

Concerning the first proposition, the wording given in the text representation has the equivalent of the geometric solution of the second degree: ax −

b 2 x =S, a c

represents the straight line, S the area of the polygon as b and c are the sides of the parallelogram given. The phrase “It is appropriate for the given polygon is not greater than the polygon constructed on the middle of the line data, and similar to

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the polygon missing” requires that the restriction is necessary so that the roots are real in the equation: S ≤

a 2c . 4b

As far as the equations of I degree and then to the method of resolution used by Euclid, an example may be the analysis of proposition 12 of Book VI of the Elements (Euclid, 1930, p. 107). Here are asked to find the fourth proportional to three segments ( AB : BC = AD : DE ). Considering as segments AB=a; BC=b; AD = 1 and DE = x expressed by the method allows to solve geometrically simple equations of First Instance of the type ax = b with a and b coefficients positive.

E D

x

1 a A

b B

C

Fig. 2. Euclidean Construction of the fourth proportional

According to the thought Bourbaki, geometrical methods used heavily by the Greeks and then by Euclid in particular, have not been helpful in developing selfAlgebra because: “[...] the progress towards the penalty must be supported in Euclid of paralysis, and some also have a back, in the development of algebraic calculations. The predominance of the geometry is blocking the investigation of autonomous algebraic notation, the elements that appear in the calculations should always be represented geometrically [...] “ (Bourbaki 1976, pp. 75). The Algebra, then, was not appropriate developments in ancient Greece if not the very late due to Diofanto of Alexandria (about 250 AD) and to Heron (100 AD) who began to treat the arithmetical and algebraic problems not “relying” the geometry and thus overcoming the design of geometric algebra. Elements of the language is impersonal and not effort or in the meta-language in philosophical discourse, the demonstration in Euclid does not use examples with specific numbers but draws general information applicable to infinite cases. This is particularly significant in relation to the text of the Chinese Jiuzhang Suanshu and its internal structure.

3.1.3 An Initial Comparison between the “Elements” of Euclid and the “Jiuzhang Suanshu” The aspect mentioned in the previous paragraph is perhaps central to the comparison between the Euclid’s Elements as a canon of culture for the West and the Jiuzhang Suanshu for Chinese thought. As we shall see in more detail below, the very nature of Jiuzhang Suanshu is profoundly different in many aspects and thus


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transversely to define the specificity of a cultural development differently. Stimuli and different needs that have produced mathematical knowledge in different contexts and different historical eras. The essential elements that will highlight later in this process of comparison relate to the situations proposed in the text, highlighted in the proceedings for resolving the individual problems that refer to the application of algorithms for resolution, the type of text used in the presentation of the problems and proposed solutions. Elements, in our opinion, for an overview of the cultural specificity of the ancient Chinese mathematics. A final comparison is then proposed the epistemological value of the “demonstration” in Chinese thought. Value that will deepen as it is clearly uneven from greek, which is considered in a Euro-centrist, excellence in rigorous mathematical thinking.

3.1.4 One Possible Comparison between Diofanto and Babylonians Returning to the historical stance, to better highlight the contribution of Diofanto to the development of Algebra and then thinking algebraically, it can be useful, in our opinion, a comparison of its problems with an “equal” of which you know a solution older, for example, the Babylonian and highlight the different approaches to solving the problems. The problem that you have chosen to analyze is the following: “Find the dimensions of a rectangle of area 96 and the semi-perimeter is 20” (in a modern way: xy = 96, x + y = 20). The Babylonian method consists in acting on the data according to the numbers given by the scribe steps: Table 2. Babylonian method for resolution of the question proposed 1 2 3 4 5

Divide by 2, the semi perimeter Raising to the square Remove the given area, 96, 100 Extract the square root

20:2=10 2 10 =100 100-96=4 2 The length is : 10+2=12; The width is : 10-2=8

What we see, by reading the steps, it is only the presence of a sequence of numerical operations. It is absent any notion of a calculation on an unknown quantity. In Diofanto, the problem of Babylonian (grade II) proposed previously stated in quite general terms: “To find two numbers whose sum and whose numbers are the data” (issue # 27 of Book I of the Arithmetica) and solved from the same numerical example was: ”Suppose that the sum of the numbers form 20 units, that their product form 96 units, the difference in numbers is 2" aritmes "(unknown): x + y = 20, xy = 96;

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x-y = 2Z. Then, as the sum of the numbers is 20, if we divide into two equal parts, each of them will be 10 units and if you add a half part of the difference of two numbers, 1 "aritmes", and the other of you subtracts the sum of the numbers is the new 20 units and the difference 2 "aritmes": x + z = 10, y = 10-z, xy-z2 = 100. But the product of the numbers must be equal to 96 units. Their product is 100 units less than 1 square "aritmes" which beat it to 96 units. L ' "aritmes" is 2 units. Consequently, the larger number will be 12 units and smaller units would be 8 and these numbers satisfy the proposition.” As you can see the solution revolves around to auxiliary unknown z. The emergence of this unknown auxiliary, "aritmes" is together with the symbolism, a real conceptual change (Ver Eecke, 1959, p. 2; Radford, 1996, p. 43). It can be argued that history of Algebra begins with Diofanto. Babylonian mathematics is light years away. The transition from Babylonian mathematics to the Greek, from this point of view, it is clear: the introduction of unknown and a special symbol appears to describe an innovation equal to that of the zero so that changed the course of history of mathematics. Diofanto marks in this sense the transition slow, gradual and difficult to Algebra rhetoric than syncopated. As mentioned previously also educationally concepts of unknown, variable and, more generally, parametric, are generally for students of the concepts are difficult to internalize, as carriers for algebraic reasoning and thereby to the algebraic thinking, and he will “develop slowly, moving from the initial report between numbers, quantities related dynamics through a formula” (Malisani, 2006, pag.161). The initial report by the passage between numbers, quantities related dynamics through a “formula”, the full awareness of the concept of variable and parameter, may highlight the cognitive differences, the obstacles, connected to a different socio-cultural background? These may follow, although some of the historical and / or cultural? What interests us here is trying to understand what patterns of reasoning, children from different cultures such as those we take in consideration, make out in the process of acquisition of mathematical concepts treated. In the coming chapters, we will try to answer these questions through consideration of a different nature.

3.1.5 The Algebraic Tradition among the Arabs and the Indians With regard to Indian Algebra, in agreement with the Malisani (Malisani, 1996) we can certainly say that the symbolism used by the people of India, although rudimentary, it is sufficient to classify them as Algebra “almost symbolic”, and certainly to a greater extent than it was of the syncopated algebra Diofanto’s. This is no longer just verbal, expressed in words and without the aid of symbolism, the unique features of the phase of Nesselmann rhetoric, but it cannot even regarded as fully belonging to the third phase identified by the historian, the symbolic, as “this people while developing good processes and having great skill and technique, [...] the various steps were not accompanied with reasons or evidence” (Malisani, 1996, p.34). Most research on the mathematics of India focuses on particular details of the mathematical discoveries of the people in question, perhaps omitting that which


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interests us more in this work: the way in which they perceived mathematical knowledge as intellectual discipline and as a body of knowledge (Keller, 2001) in view of possible trade, scientific links with China. Still highly debated topic among historians of mathematics. In a mathematical astronomy Bhaskara comment reads: “The mathematics includes the figure, the shadows, the succession of numbers, equations, the sprays (indeterminate linear equations), etc. [...] The mathematics is of two kinds: that of figure and that of quantity. The proportions, the sprays, etc., that are specific to mathematics, were named among the mathematics of quantity. Succession, the shadows and so on, [including] the mathematics of shapes. So in this way, mathematics as a whole is based only on quantity of math and mathematical figures.” According to his thinking, the basic principle of the global definition of mathematics is the distinction between “quantity” and “figures” that is between computational manipulations of known or unknown values and use of the properties of plane figures. According to this view then: Arithmetic Geometry against. The Algebra? At the same time, in a work composed in 629, Brahmagupta defines a different classification of mathematical thinking as a unique set of knowledge. Classification which will be generally accepted as a standard of mathematical thought, and that does not feature in “quantity” and “figures” but between “calculated” and “unknowns”. “Anyone who knows the twenty separate operations starting with addition, and the eight practices that end in the shadows, this is a computer [...]. [It was] familiar with the masters of the treaties, based on knowledge of the gun, zero, of negative, the positive, the unknowns, the elimination of [term] in the middle, [the reduction] to only one unknown, the products of unknowns, and the nature of the square [indeterminate equations of second degree]7.” The basic principle of Brahmagupta in the arrangement of “mathematics as a whole” is similar to the distinction between arithmetic and algebra, conceptual categories that constitute the fundamental structure of learning math. Neither author provides a detailed discussion of the proposed classification. We agree with the suggestion that the debate has certainly helped shape the medieval ideas about mathematics that are likely to have also taken into account those who were the Chinese mathematical texts. The witnesses on scientific and cultural contacts between China and India before the advent of Buddhism, around the fourth century, are only fragmentary. The most reliable evidence regarding the last decades of the Han dynasty, with the spread of Buddhism8. A character that can represent the cultural exchange between the two countries is perhaps the Buddhist pilgrim Fa Hsien that for fifteen years through northern India and traveled throughout Central Asia. The solution of indeterminate equations by the method kuttaka in qiuyishu in India and China, was a constant passion in both countries. In both contexts, the issue of indefinite born in astronomy and for the creation of calendars, with the determination of orbits of the planets. 7 8

Brahmasphutasiddhānta, cap. XII, v.1, and cap. XVIII, v.2. Over the same period seems to have been written the Jiuzhang Suanshu (Eastern Han dynasty : 25-220 d.C.).

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In this context of comparison of mathematical knowledge and knowledge is, however, in our opinion, interesting to note that, unlike the Egyptian mathematics, and later to the Chinese, Indian mathematics at least at the time of Aryabhata (born in 476 AD), a most important mathematician of ancient, method of false position, seen previously, was not used (Gheverghese J., 1987). The hypothesis that some historians of mathematics are in this regard is that perhaps the Indian mathematics, precisely because of the progress of algebraic symbolism, did not need to that algorithm. Examples of geometric solutions of linear equations to an unknown factor found in Sulbasutra of Baudhayana, algebraic solutions are reported in Bakhshali Manuscript’s of the twelfth century AD. The algebraic method used was based on inversion, a method which is five centuries after the Arab mathematics. The procedure required a reasoning backwards from a certain information and was also used for quadratic equations. With the objective to identify a possible difference in the formulation of a problem of this kind from that expressed in such Jiuzhang Suanshu, given the text and the solution, expressed in modern language, the following problem: Wonderful shiny-eyed girl, tell me, because you know the method to be applied, what number multiplied by 3, plus three quarters of its product, then divided by 7, less than a third of the result, then multiplied by the same, and then decreased to 52, whose square root is extracted before being added to 8 and then divided by 10 results in the final 2? Solution: We start from the answer (2) and then backward, If the problem says to add, subtract it, if he says to multiply, divide it, if he says to extract the square root, squared rises etc. The solution is then: [(2)(10) − 8] + 52 = 196 ; 2

3 4 (14)( )(7)( ) 2 7 = 28 ; 196 = 14 ; 3

The Manuscript of Bakhshali is the oldest evidence we have on the Indian Mathematical Association detached from any religious or metaphysical. Cultural context that has driven most of the scientific work of classical antiquity. The scientific environment is described in the text is similar in form and content to the general mathematical Jiuzhang Suanshu, older than some centuries. The latter, however, is much broader and more advanced than Bakhshali (J. Gheverghese, 1991). Indian mathematics did not direct influence on Europe, but it seems almost certain that the Arabs studied the Arithmetic and Algebra Indian indirectly: through the representatives of the elite science, accepted in the courts of the caliphs of the IX and X century. In that time were several classical works, literary and scientific publications that were translated into Arabic, from sancrito and greek, and were studied by the sages of Islam. The claim of self-algebra as a subject, start to get light in the Arab world, in Persia, in the fervor of the great school that was developed in Baghdad in the second half of the ninth century. It is in this socio-cultural context that for the first time is on the first recognition of the subject “in itself”, with problems and methods of self (Betti, 2001).


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The great development of culture and science reached its one of the highlights during the caliphate of Al-Mamun (809-833) with the founding of the "House of Knowledge9". Among the many members who made the part, the astronomer and mathematician Mohammed Ibn Musa al-Khowarizmi, a scholar who wrote more than a dozen works of Astronomy and Mathematics, the oldest of which were probably of Indian derivation. The title of his most important “Aljabr w'al muqabala” provided in modern languages the word “Algebra”. “Aljabr” had the meaning of “completion” and was one of the main methods resolutive along with “almukabala”, “balance” with the meaning which we today give to “simplification”. In his work, he systematically studied the equations of the second degree, classified according to different types and are represented graphically. While, however, the Arab Algebra seems to some progress in the development of algebraic thinking from another point of view, these people seem to make a big step backwards is the level of arithmetic (even knowing the numbers by the Indians negative, refused them) that of algebra itself. “The Arabs are not used or abbreviations [...] only a few names to name the unknown and its powers ... a step back, then, is compared to Algebra Indian to that of Diofanto” (Malisani, 1996, p.34 ). They also influenced by the Greeks, always felt the need to explain and justify the procedures used to solve algebraic problems, but they were used for this purpose, only the geometry10. As the greek people, in fact, felt the need to justify everything geometrically. The predominance of the geometry, geometric methods, it would seem, in fact, in this case, to prevail on the conduct of autonomous algebraic notation. In the texts of al-Khuwarizmi, the study is free from any immediate application and calculation of the solutions becomes progressively problem in general on the resolution method. A proof of this we tried to analyze, in a first approximation, a problem among those offered in some Arabic texts, highlighting the processes algebraic / geometric solutions, variety. Considering for example the problem: “A square and ten roots are equal to nine and thirty dirhems, (39 units), i.e. chief add ten root to square and the sum is equal to nine and thirty” (Kline, 1972, Italian Edition, pp. 226-227). Analyzing processes resolutive reported by Al-Khuwarizmi, we find two different techniques that are (stating with the first strategy I and II with the second) in order to highlight the first resolution, although logically correct, was not considered by them the solution to the problem and only the second, the geometric, was considered as optimal (today we might say that the second strategy can be considered superfluous): In modern notation, the solution is as follows: x 2 + 10 x = 39 ;

10 = 5; 2

x 2 + 10x + 25 = 39 + 25 ; ( x + 5) 2 = 64 ; x + 5 = 8 ; x=3 9

“Casa del sapere”(Italian Language), “Maison du Savoir” (France Language), the term “Sapere” has no equivalent in English. 10 As Djebbar Islamic culture is Aristotelian but remained firm on the translation of Averroes.

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3 The Meta-rules between Natural Language and History of Mathematics Table 3. Resolution procedure reported by Al-Khuwarizmi equation for second-degree

Ia

Take half of the roots

Ib

Multiply them for herself

Ic

It adds to these 39

Id

It makes the square root

Ie

This number is subtracted half of the roots

10 =5 2 5·5 = 25 25+39 = 64

64 = 8 8-5 = 3 x=3 is the root string

II. Draw a square ab to represent x ² and four sides of this square to put the rectangles c, d, e, f 2 * off each ½ unit. To complete the square, add more of the four small squares at the angles, each of which area of 6 * ¼ units (i.e., to complete the square you have to add 4 to 6 * ¼ units, ie 25 units) resulting in a square whose area measure 39 +25 units.

Fig. 3. Geometric resolution procedure set out by Al-Khuwarizmi for the equation of second degree

The greater part of the square should measure 8 units, from which you subtract 2 for 2* ½ units equal to 5 units found so that x = 3 is the solution of data. It thus shows that the solution presented was algebraically exact. This geometric reasoning is based on the fourth preposition of the second book of the Elements. Another text of the ninth century interest in dealing with the relationship between mathematics-culture and more specifically Western cultural tradition, Eastern cultural tradition, we are trying to treat, albeit roughly and not a strictly historical, is perhaps the Liber Mensuratonium of Abû Bekr. This is a text that contains many problems solved with different resolutive methods. One of these is explicitly called by Høyrup (Høyrup, 1990) method of "cut-paste geometry", a method that can be found, albeit in different ways, in the Jiuzhang Suanshu, The Nine Chapters of Mathematical Art (first century BC-I century AD ). One example is the statement and the resolution of the problem 25 said: “The area is 48 and the sum of the two sides is 14, how far each side?”.


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The problem therefore seems to require to determine the length of the sides of a rectangle that meets certain conditions. The text expressed in algebraic language is equal to the system:

⎧ xy = 48 ⎨ ⎩ x + y = 14

that is the equation: x 2 − 14 x + 48 = 0 . The author explains the solution of the question by applying the method of “cut-paste geometry” by adding and subtracting an appropriate figure given quantity. Radford and Charbonneau (Charbonneau & Radford, 2002, pp. 5-6) believe that the solution was accompanied by some drawings but not in the text. In a possible reconstruction of these designs, propose the following sequence:

Fig. 4. Resolution procedure according to the “Geometry of cut-paste”

Divide in half the 14, the result will be 7 on 7 multiplies itself and will be 49 (creation of a square of area 49), subtract 48 from it and will remain 1, which gets the root is 1, if added 1 half of 14, what will be the first major, if subtract this number from the mid-14, what will be the smallest dimension (you get the exact figure 4, the rectangle of side 8 and 6). The method of cut-paste, but with the necessary modifications to absolutely stress the concept underlying the various mathematical problems, is reflected strongly in Jiuzhang Suanshu Chinese, in the commentaries of Liu Hui (Ca. 220-280) of 263. The author in Chapter V (Shang kung, Handbook for engineering works), by dealing with the rules for calculating areas and volumes of various three-dimensional figure, refers to a particular method of “demonstration” 11: the method of “re-test” or principle of “complementarity external-internal” or “dissection and reassembly”. Essence of this method, “similar” to that of cut-paste, is based on two assumptions that constructive in some ways, in some respects purely conceptual, the difference just from that of Liber Mensuratonium while constructively resume: - Is the area of a plane figure that the volume of a sound remains the same when subjected to a rigid shift in another place; - If a plane figure or a solid is cut into sections, the sum of their areas or volumes of the sections is equal to or the amount of the original figure. 11

The term evidence is properly inserted between quotation marks because there is a problem on the epistemological demonstration schemes in different cultures. You mention this aspect at the end of this chapter. Interesting work of teaching in this sense can be Huang, 2005 and Di Paola-Spagnolo, 2009.

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The method used, which is often viewed as a method of trial and error in its simplicity of construction and explanation, in our opinion, shows something more than a simple numerical attempts to carry and cannot, therefore, be regarded as a banal arithmetical thought put into in a geometric light. Example of application of cut-paste Chinese are found in the “demonstration” of the Pythagorean theorem (Chapter IX called kou ku) which do not deal in this work12 and other different problems presented in Chapter V and IX. Problems IX 15 and IX, 16 are listed as such (we consider a translation of tense in modern algebraic language):

Fig. 5. The figure illustrates the description given by Lui Hui in Jiuzhang Suanshu, in applying the principle to demonstrate the theorem of Pythagoras. Chapter IX, issue 3.

Problem 15. Since x side of a square inscribed in a right triangle. Define a and b the lengths of the two legs and c the length of the hypotenuse of the triangle. What is the square of side x? 13

The method for finding the unknown x is shown in the following sequence which is supposedly similar to the one lost in the commentaries of Liu Hui: As shown in the figure, through the principle of cut-paste (dissection and reassembly) and try that ab = ( a + b) x and then the side of the square, x, required is: x =

12 13

ab . a+b

For an deep investigation. Chemla, 1997, 2001. Va puntualizzato che il testo del problema riportato nel Jiuzhang Suanshu non è espresso nella forma generale qui proposta per una trattazione algebrica più esplicita ma si presenta in una forma concreta ed espressa senza alcun simbolismo.


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Fig. 6. The figure shows the description in Jiuzhang Suanshu for the solution of the problem given. Chapter IX, issue 15.

Problem 16. 2r is the diameter of the circle inscribed in a right triangle. Define a and b the lengths of the two legs and c the length of the hypotenuse of the triangle. To find the diameter 2r of the circle.

The method of finding 2r can be seen in the following sequence of figures taken into account in dealing with Liu Hui in the commentaries: The figure above shows through the picture, the instructions given by the scholar, the first figure reconstructed (a) consists of four right triangles whose lengths of legs are a and b and length of the hypotenuse is c. The comparison of this with (b) and (c) states:

2ab = 2r (a + b + c) hence r =

ab . a+b+c

As Youjun Wang “These are some typical examples of the use of principle of congruency by addition and subtraction. In fact, Liu Hui also used an extension of this principle to explain the theory of limit, which Liu Hui called the method of validation with Chinese chess whose shape is cube (Wu 1982). By using this principle, the propositions pertaining to areas of figures with linear sides and to volumes of simple solids, such as column, cone, and so on could be validated […]. The principle also played a very important role in the field of solving equations. In general, this unique Chinese mathematical tradition of doing mathematics by handson manipulation or by experiments was further developed since the time of Liu Hui. It influenced the future development of Chinese mathematics, such as during the Song dynasty (960–1279) and Yuan dynasty (1279–1368).14

14

These types of verification of the proposed questions was open in the literature, considerable debate about the epistemological value of demonstrating a proposition. Address, even if in part, this argument will next paragraph.

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Fig. 7. The figure shows the description in Jiuzhang Suanshu for the solution of the problem given. Chapter IX, issue 16.

As we will see in coming chapters, we think that these assumptions resolutive methods applied to most mathematical contexts, may find themselves, albeit in a different form today, in multicultural classes with students of Chinese culture. Principles and patterns of reasoning probably conveyed through Confucian philosophical principles and language (composition of cut-paste / dissection and reassembly on the Chinese written language). But there is yet another aspect that is related to the spatial approach of algebraic formulas. As seen in the previous chapter, ideograms in the rules of association (for the Chinese culture: logical Association) follow the forms of space that might be reference to a "translation" mathematics possible algebraic simplifications, the principles of equivalence of equations, etc. ..., as evidenced by:

Fig. 8. Linguistic principles related to the possible meta-rules (see chap. 2) of the composition (Cornet, 2006)


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Similar considerations may be in the context of geometric equidecomposable. The “cut and paste” is a fundamental element to understand the approaches to solving the problems of Chinese geometric problems already presented. For us this is an open problem to be studied experimentally. Suggested that as in Euclid algebra does not develop because of the difficulties to move from III to IV size, so in the “Chapters 9” and earlier in the geometry of the “cut and paste” can be a barrier to a thought geometric “generalized” in the sense that we have given to this term in Chinese culture.

3.1.6 Significant Developments for the Symbolic Algebra Over the years, slowly and not without difficulty, the higher culture of the Arabs came with penetrating even in Europe. The Arabs of Spain and the Arabs of the Levant were largely responsible for having encouraged the rebirth of European culture that throughout the early medieval period (from about 400 to 1100) was in a situation completely stagnant. At that time it “[...] the European civilization is not worried the development of mathematics, there was no progress, and there were serious attempts to generate new knowledge. All the problems faced requiring only the use of the four operations of whole numbers [...] the calculations were performed with the help abacus [...]. It used the Roman numbering system, avoiding the zero you do not understand the meaning. The fractions were used rarely and irrational numbers did not appear at all.” (Malisani, 1996, p.48). The center of the intellectual renaissance it was Italy, and one of the scholars who began to light at that time was undoubtedly Leonardo Pisano, called Fibonacci which you trace the use of the term equation (from the Latin aequatio). We can consider the year of 1202 held on the year in which appeared the Liber abaci, the book “on which are based completely arithmetic and algebra” (Kline, 1972,Italian Edition, p.278) According to historical data in our possession, we are now able to say that from the thirteenth century, the most important discoveries that are made in Maths came mainly in the field of arithmetic and algebra. Indeed many scholars devoted increasing interest in these subjects seeking to deepen their studies. In these writings, called “Treaties of abacus”, the algebra had a geometric rather than arithmetic, and many of the issues addressed were resolved with the help of the equations. A classic abacus Treaty is the “Trattato d’Algibra” we have already mentioned before by treating the double false position. Written at the end of the fourteenth century, by an anonymous Florentine, an elementary text in which many topics are treated merchant (which characterize this type of work) not only proposed for a practical purpose, but also addressed in a more complex and abstract. The Trattato d’Algibra clearly states twenty-five rules for solving equations of first, second, third and fourth degree with coefficients always positive.

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Franci and Pancanti believe that this is one of the best treaties abacus medieval and Renaissance that they have examined. In particular, state that “... the final chapters devoted all'Algebra ... are fundamental in the reconstruction of the history of this discipline in the thirteenth and fourteenth century.” (Franci, Pancati,1988, pag.6). In Liber Abaci, Fibonacci solves several problems that can be defined as concrete, daily newspapers, mostly related to commercial transactions. They found among other things, examples of specific and indeterminate equations of first, second and third grade. The resolution method for quadratic equations follows the style Arabic and diofanteo separately in five different cases so that the coefficients are always positive. For each of these, the author determines the different solutions using geometric Euclidean reasoning. The various problems of analysis unspecified are resolved according to the application of various artifices or the method of false position (see Loria, pp. 386-391). Comparison of mathematical traditions between east and west should be clarified how “many problems in this book are closely related to chinese mathematics, e.g. there are problems on page 304, cognate to Problems 26, Volume 3 of Master Sun’s Mathematical Manual15”(Kangshen, 1999, pp.18; 415). What was built, however, until the sixteenth century were a slow and little precise Algebra: the symbols most commonly used in mathematics texts were the result of common abbreviations of words, of “pieces” of words that designate the unknown of the problem. In the Renaissance style was still common rhetorical and drew the symbolism adopted by Diofanto. The most significant change in character of algebra was introduced in connection with the symbolism of Viet. He, in a change of conception in the use of “formalized language”, language is more complete and more agile than through a symbolism more “uninhibited” gave more speed and precision, not the symbolism used to indicate only the unknown but used points also to see that generic terms be express. His “new” language was not created solely to solve problems but algebraic calculation was applied to test different rules. The boundary between the Arithmetic and Algebra is thus traced to prohibit the distinction between logistica numerosa and logistica speciosa, the first dealing specifically of numbers and the second is defined as a way to work on the species and forms of things. The algebra then becomes, in this sense, the study of general forms of representation: infinite cases expressed in a simple and concise language. Bombelli (1526-1573) and Vieté (1540-1603), following, in a first approximation, the classification of Nesselmann time, thus driving the Algebra in stage highlighted by the scholar: the symbolic.

15

The texts of Master Sun’s Mathematical Manual and the Zhang Qiujian’s Mathematical Manual, writings of the IV and V centuries are regarded as simplified versions of Jiuzhang Suanshu. In Master Zhang Qiujian’s Mathematical Manual we find some evidence on the problems Arithmetic "recreational" typical of the Western tradition: issue 38, Chapter 3.


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In his Algebra (1572) 16, Bombelli develops the theory of equations of the first four grades. Separately considers many special cases of equations of the same grade so that the coefficients are always positive. For each type of equation states (in rhetorical language) the “rule” practical resolution, performs geometric construction (as possible) to justify the validity of equality expressed in equation and to analyze the nature and multiplicity of roots. Following the traditions of medieval Arab and accept only the positive real solutions not void. The negative roots or complex were difficult to interpret in an appropriate way in relation to the problems to be solved, and the socio-cultural context not pushed in this direction. Bombelli uses geometric construction to solve algebraic problems, but the procedure is the reverse of that taken in Algebra geometric of the ancients. It does not solve the problem directly geometric solution for the analytical interpretation of arithmetic building built, but uses exactly the resolution to derive the algebraic geometric construction (Bortolotti, 1966, pp. XLIII). Bombelli can be considered one of the protagonists of history of algebra. With regard specifically to the resolution of algebraic equations and then looking for the solutions of these, which we tied to the historical development of algebraic thinking and long identified with this (Betti, 2001), it certainly reminded the work on cubic equations, by Bombelli in Algebra. The treatment, also referred to procedures for resolving Cardano, Del Ferro and Tartaglia, leads in the search for solutions, a radical with no real quantity. In a vision of algebraic thinking and thus of the same, in a continuous balance between history and teaching, which we believe should be stressed in order to highlight, as mentioned, any obstacles and difficulties of the learning process with regard to today's thought algebraic and its interactions with the arithmetic, is the trend continues, in various historical writings, at least for Western culture, the generalization of the problems addressed. Element strongly felt in the “thinking algebraically”. The proposed task is not easy. Taking a step back in history epistemological just discussed, we limit ourselves to consider as an example, in the first instance, a parallelism between the “Liber Quadratorum” by Pisano, the "Trattato d’Algibra" by anonymous Florentine and “Algebra” by Bombelli. They found, as mentioned earlier, a growing awareness of the “need to generalize” all the processes of resolution recreated and thus the possibility to apply the resolution to a variety of specific cases. The Algebra, and, according to this view, strongly western, is being “read” as a study of general types of problems: everything that can be applied to the general case you can use it to infinite cases. While Algebra was then considered a “Universal Arithmetic” a discipline that is capable of expressing in general the rules apply to arithmetic operations, on the other hand, it tried to free it, tripping from this “bond” making it less restricted.

16

The full title of the text is “Algebra opera di Rafael Bombelli da Bologna, divisa in tre libri, con la quale ciascuno da sé potrà venire in perfetta cognitione della teoria dell’Aritmetica”, published in two editions (almost identical) in 1572 and 1579 (Bombelli, 1966; Loria, 1929–1933, pp. 316–317).

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According to this view, the variable, could no longer be regarded as a generalized, it was necessary “emptied” of all meaning outside it “superior” to the idea of projecting the parameter as a means of solution for classes of problems. Like the rest of the evolutionary path, however, this phase was very slow and difficult. (Arzarello, Bazzini, Chiappini, 1994). This route is also what we try to encourage educationally, with many difficulties in students. The concept of variable in thinking is a dynamic concept that evolves through different stages and interpretations. In agreement with Malisani (2006), as already mentioned, the concept of variable carrier for algebraic reasoning and thereby to the algebraic thinking “develops slowly, passing from the initial report of the numbers included in the tables, the dynamic quantity related through a formula” (Malisani, 2006, pag.161). This was also the historical algebraic, his formalism and the stage of generalization. A teaching degree is necessary to point out how the conceptualization of the variable steps and then, more or less casually, from an initial sense of “thing that varies” (term used by Malisani, 2006) meaning essentially aimed at a first point of any possible numerical reports submitted, for a more mature observation and assimilation of any property of the variational mathematical objects in question. The third stage of conceptualization then builds to a more complex and takes into account the ability to define, in a problematic situation, which may look like abstract positional values that are (or become) the quantity dynamics, mathematical variables. At this stage, the variable takes different forms and meanings and binds strongly to the functional and parametric. Are interesting in this context, the educational work of Radford (1996, p.51, 2004b) and Gagatsis (2000). Around 1700, the algebra had reached a mature autonomy that allowed it to become detached from the geometry and especially from Arithmetic. But certainly what was missing was still a strong and resilient on which to build the future. Foundations of logic similar to those for the Euclidean Geometry. In this long period of growth, technical developments are accompanied by little to the conceptual and Algebra was clarified to his status today. Around 1770, Joseph-Louis Lagrange explicitly connects the solutions with those who will be the groups of transformations and in 1799, Paolo Ruffini first announced the nonexistence of a formula for resolving the equation radicals fifth overall grade; discovered and discussed in early dell'800 by Gauss and Abel successfully addressing some special classes of equations (the binomial equations and then, more generally, those to be called “abelian”). Were the ideas and studies to provide Galois then a theory which could decide the solvability of the equations is the equation which generally confirming the non-solvability by radicals when the level exceeds the fourth. Thus, in the first half of the 800, the theory of Galois further determined the turning point for the history of algebra; an historic socio-cultural context in which the structural aspects of scientific thought are becoming mature and predominant. As will be recognized long after the death of its author, the theory of Galois provide “also” a criterion of solvability for radicals of all algebraic equations but that non-solvability of

3.2 A Look at the Ancient Chinese Mathematics

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equations of degree higher than the fourth, his study shows the centrality of the concept of “group of substitutions”, which soon spread to many contexts. The theoretical perspective of algebra changed radically again with the definition of a new problem: finding a formula resolutive became the study of the properties of a group associated with the equation on the “Group of Galois”. The “Galois theory” became the theory of extensions of number fields and the relationship that binds these extensions automorphism groups: one that in fact today is called “Galois correspondence”. The problem originates in the radical to solve algebraic equations which both pushed the development of algebraic thinking is slowly “disappeared” transformed into something more complex. In the meantime, have changed the outlook of algebra. The reading we have given the process of growth of the algebraic thinking cannot be completely generalized to the Chinese culture especially given that those are the values of mathematics epistemological eastern China that put the roots in Jiuzhang Suanshu.

3.2 A Look at the Ancient Chinese Mathematics 3.2.1 The Historical-Cultural Understanding the history of Chinese mathematics requires some knowledge of Chinese history, very large and difficult to analyze in its entirety, since it extends for more than 4500 years. The paragraph that we are discussing is not intended to be an exhaustive discussion of mathematics in China, go beyond the objective of our history of algebra in a epistemological-cultural perspective for a “application” to didactics of disciplines. Then we are interested in some particular aspects of learning Chinese mathematician specific reference to some written records that can better describe the “traditional Chinese mathematics” seen in the relationship between mathematics and culture we are studying. References in this fundamental sense is the work of Chemla (2001, 2004), Cullen (1996, 2004), J. Gheverghese (1987), Guo and Liu Shuchun Dun (1998). The first sources that reflect an initial mathematical activity in ancient China dating from roughly the Han dynasty (206 BC - 220 AD) 17, when territorial expansion for the Chinese people, thanks to the silk trade with India and the Empire Roman18. These texts, expressed only through a discursive approach to the 17

A list of Chinese dynasties that have occurred up to 1912 (Birth of the Republic of China moved to Taiwan from 1949) is reproduced in Appendix B. 18 As mentioned earlier, the economic and cultural relations with India is made even more intense public through the acceptance of Buddhism. It seems then that Rome had not cured long to know the "far country from which the silk." The Chinese chronicles, however, recall that in 166 AD Marco Aurelio sent a mission to China, after this first meeting, the silk trade between China and the Roman Empire became more intense (Needham, Vol. I, 1985).

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problem and a presentation only in natural language, are expressly refer to calculations with numbers “chopsticks”, numbers that were, until recently, the main tools for the job math and science (as stated above shall be considered and artifacts derived from the Chinese tradition of mathematics). The research undertaken in the early decades of the last century by Vacca, Mikami and Van Hee19, studying the first editors of memories from the ancient Chinese tradition, Chinese mathematics defined as “a totally uncoordinated collection of information, sometimes imitate those of Greek or those Indian, sometimes incomprehensible in their inner meaning, sometimes in incomplete concepts, methodology and applications, and only rarely with some originality in arithmetic” (Adamo, 1968, p179). As seen previously by the few examples from the text Jiuzhang Suanshu, considered the canon of reference for the traditional Chinese mathematics, the ancient Chinese world was unaware of it certainly need deductive arguments and his mathematics, was not guaranteed by the rigor of a technique logic (in the Aristotelian sense) organized exactly as the Greeks, for example, but by practical people who distributed them in the elementary problems of economy, agriculture, etc. The result was such that it can be submitted after due consideration to a Westerner, not as a harmonious edifice of knowledge, but as a set of terms, developed, just as a “technical grammar” the spontaneous and natural, the language described by its economic relations so useful to trade. In other words, according to a vision etno-mathematics, “Mathematics is a universal activity, that is, it is a pan-cultural and panhuman activity. In all cultures, mathematical thinking has taken place, whether spontaneously or in an organized way, all human beings are spontaneously doing some mathematical thinking and are capable of learning more (D'Ambrosio, 1985), mathematics is not the property of any particular (sub) culture or cultural complex, like the Greek, European, Western, white, male, city dwellers, mathematicians, mathematics teachers etc.” (Anderson, 1990). The ancient Chinese world is not felt the need to pose problems in the abstract and deductive character much loved by the Greek mathematicians because their socio-cultural context of that was different, completely different concept of life. These general considerations are not surprising: unlike what happened in the Middle Eastern world and then in the West, the Chinese civilization, according to a vision of the Western, does not seem to have had up to the XX century no period presents an analogy with “Enlightenment rationalistic”, yet, from an educational perspective, there is no denying that the Chinese have strong provisions to mathematics, to its “rigorous”, as demonstrated in the last century and are still doing, “in line” to Western20 science and giving it its own significant contributions. Then enter the details of debate and analyze this content. Two mathematical historical documents (“classics” jing) that there have been transmitted through the written tradition and that are significant in shaping the 19

Vacca, Boll. di Bibl. e Storia, Tomo VII, 1904; Mikami, “The Development of Mathematics in China and Japan”, Lipsia, 1912, Van Hee, Articles published in various journals between 1911 and 1926, particularly in T'oung Pao; Leyda, 1923. 20 The term "line" we mean the ability to "take" from Western culture what might be considered useful for their own social group. How did the Roman Empire before and after the Catholic church for the "old Europe".


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history of Chinese mathematics specifically addressed to evolution of algebra are the Gnomone of Zhou (Zhoubu) presumably written in the time period of the first century BC - The century D.C. (Cullen, 1996, pp.148-156), and the Jiuzhang Suanshu, The Nine Chapters of Mathematical Art, text, as said, written about during the Eastern Han dynasty (25-220 AD) and as mentioned earlier, is the oldest text ever in mathematics and a fee of culture for the Chinese thought. Just because of what will be, in this paragraph, the main topic of analysis. The transmission of these references in writing was concerned, over the years, different commentaries that accompanied and handed the texts in various historical periods from the III century onwards. The first two commentaries are written and completed in 263 by Liu Hui on Jiuzhang Suanshu and another written by Zhao Shuang in the III century on Gnomone of Zhou. Another important task for the transmission of the “classics” in the history of Chinese mathematics was made by Li Chunfeng then, during the Tang dynasty (618-907) with the collection and commentary of ten “traditional math” (the Suanjing Shibu), a monumental work, an encyclopedia of traditional Chinese mathematics. It includes: Jiuzhang suanshu (Nine Chapters of the Mathematical Art), Haidao suanjing (Sea Island Mathematical Manual), Sunzi Suanjing (Master Sun's Mathematical Manual), Wucao Suanjing (Mathematical Manual of the Five Government Departments), Wujing Suanshu (Arithmetic in Five Classics), Qiujian Suanjing Zhang (Zhang Qiujian's Mathematical Manual), Xiahou Yang Suanjing (Xiahou Yang's Mathematical Manual), Zhui Shu (Method of Interpolation), and Xugu Suanjing (Continuation of Ancient Mathematics). The only other ancient mathematical texts that has come down to us is the Book of mathematical procedures established during the Han dynasty or before, and found in 186 BC . The nature of written text is profoundly different from the other two, not, indeed, that the accompanying commentaries, and there is no indication that we confirm the nature of “classic”, a distinctive feature for works written for the imperial interests. This aspect as it should be noted, as well Chemla (2007, p.93) “this is a first discriminating relevant to describe the culture, this fact distinguishes the different kinds of tracts, which were read and used in different ways.” The different writings testify to that effect, mathematical distinct cultures and therefore different cultural aspects, albeit within the same social context. For the purpose of our work, we are interested particularly in the Jiu-zhang Suanshu (meaning the classic, accompanied by his comments, specifically that of Liu Hui to which it will from time to time reference).

3.2.2 The Jiuzhang Suanshu For the treatment of a subject as vast, complex and unusual, would not have made sense in turn present some results of mathematical text. We preferred to focus on specific aspects of mathematical thinking in Chinese and then across the cultural in general, in which it was entered. The discussion thus includes considerations and examples discussed in the preceding paragraph and provides a more detailed description.

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3 The Meta-rules between Natural Language and History of Mathematics Table. 5. The Nine Chapters of Jiuzhang Suanshu (Kangshen, Crossley e Lun, 1999) Chapter 1: Land Surveying.

Chapter 2: Millet and Rice. Chapter 3: Distribution by Proportion. Chapter 4: Short Width. Chapter 5: Civil Engineering. Chapter 6: Fair Distribution of Goods. Chapter 7: Excess and Deficit.

This consists of 38 problems on land surveying. It looks first at area problems (the types of shapes for which the area is calculated includes triangles, rectangles, circles, trapeziums), at rules for the addition, subtraction, multiplication and division of fractions. The Euclidean algorithm method for finding the greatest common divisor of two numbers is also presented. In the problem number 32 an accurate approximation is given for ʌ. This chapter contains 46 problems concerning the exchange rates among twenty different types of grains, beans, and seeds. It possible to find a study of proportion and percentages and an introduction of the rule of three for solving proportion problems. Many of the treated problems apply as simple exercise to give to the reader the practice to work with the calculations with fractions. There are 20 problems which involve proportion (direct proportion, inverse proportion and compound proportion). In particular arithmetic and geometric progressions are used in some of the problems. 24 problems (the first eleven problems take the name to the chapter). Problems 12 to 18 involve the extraction of square roots, and the remaining problems involve the extraction of cube roots. Notions of limits and infinitesimals appear also in this chapter. 28 problems on construction of canals, ditches, dykes, etc. it is possible to find volumes of solids such as prisms, pyramids, tetrahedrons, wedges, cylinders and truncated cones Liu Hui, in his commentary, discusses a "method of exhaustion" that he invented to find the correct formula for the volume of a pyramid. This chapter contains 28 problems involving ratio and proportion. The problems refer to travelling, taxation, sharing etc.

20 problems that report the rule of double false position.

Chapter 8: Calculation by Square Tables.

This chapter contains 18 problems which are reduced to solving systems of simultaneous linear equations. However the method given is basically that of solving the system using the augmented matrix of coefficients. The problems involve up to six equations in six unknowns and the only difference with the modern method is that the coefficients are placed in columns rather than rows. The matrix is so reduced to triangular form, using elementary column operations as is done today in the method of Gaussian elimination, and the answer interpreted for the original problem. Negative numbers are used in the matrix and the chapter includes rules to compute with them.

Chapter 9: Right angled triangles.

In this final chapter there are 24 problems which are all based on right angled triangles. The first 13 problems are solved using an application of Pythagoras's theorem, which the Chinese knew as the Gougu rule.


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The basic idea is always that of Radford, in the interpretation of mathematical culture as forms of cultural reflection to the world, cultural forms to "give effect to it" (Eco, 1999). So, as said earlier, in this sense, the cultural diversity of interpretations and contexts may be a profound richness that, as pointed out earlier, too often, unfortunately, is its collapse in the name of a universality that does not exist in reality and does not lie as a cultural fact. It is reductive and false. Our considerations in this regard following the work carried out in recent years by Chemla (2001, 2004) Martzloff (1987) and Needham (1985) and try to explain what may be interesting, even from the academic point of view for the specific cultural context of Chinese mathematics: the nature of the problems presented in the text, Cases and comments resolutive reported, the epistemological value of the demonstration of the validity of a certain proposition, the types of contexts used in different situations and their proposed role in solving them. The main element from which we start the description of the corpus of knowledge that we want to analyze Jiuzhang Suanshu and that is the culture of which he is a witness, it's certainly the types of problems included in the text. The Chinese terms are found in the text refer to two different translations: wen which translated means "to ask a question" and that is reflected here, however, "seek", the meanings are very different. Before going into the review of some major problems related to the arithmetic / pre-algebra we consider a frame, albeit quickly, the text in its entirety. This fact includes 246 problems divided into nine sections (chapters) which address various mathematical topics, ranging from measurement of the fields with the calculation of fractions and operations on them, in the disposal of solid and plane geometry. Below is a summary table on the English translation of Chinese text (Kangshen, Crossley, and Lun, 1999). The text, articulated, such as through exposure to situations in natural language on specific examples and concrete, immediately followed by the answer, the answer is also specific, has no algebraic notation and is therefore within a tradition arithmetic-algebraic similar to that of previously treated Babylonian mathematics. There is no shortage problems related to situations so abstract, and the problem I.9 which is followed by a process for the addition of fractions: Problem I.9

Suppose further that one has

1 2 3 4 , , , , wonders what is achieved by putting 2 3 4 5

everything together. Although the wording of the problem that relates to specific numerical values, unlike other problems of the same chapter (like the questions I.8 or I.9), amounts are abstract. One cannot, therefore, speak of an algebraic thinking, structured around the concept of variable, but, in our opinion, there is a need for possible generalization, however, still strongly coupled to an arithmetic thought. Other examples of that can then be found in Chapter IX in which it addresses the issue of the “Pythagorean theorem” (kou Ku) and the Pythagorean terne (example is the problem IX, 1) (Chemla, 2007).

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Proceedings resolutive (shu) of these problems as well as of others, brought to light in the different chapters of the work, they always use, through a kind of “procedural”, the data presented in the text of the problem, expressed in magnitudes and specific numerical values. This is a key element that goes in depth for a full understanding of practical mathematics of China and perhaps also in the day. Practice occurred, albeit in part due to cultural exchange with the West, as we shall see in Chapter 5, in disciplinary didactic and in the learning phase. The distinctive characteristic of the methods of solution derive mainly represented by the text in your process as simple list of operations (Kline, 1972, Italian Edition) and use the algorithm solver as crucial steps for the "demonstration" and the widespread the proposed wording. Then considerations should be made parallel to the use of the presentation of the various questions. The text, indeed, situations in which, for the resolution process, the context described is fundamental as well as specific numerical values reported and used in the algorithmic procedure followed, and others where this is almost entirely absent (as in I.9 problem described above). In these proceedings resolving problems do not relate either to specific situations or to numerical values established a priori. The process of resolving problems I.7, I.9, I.19, I.21 are an example. They are found only technical terms, such as "numerator, denominator, multiply ..." (Chemla, 2007, pp.98-99). In the commentaries of Liu Hui, for this type of solutions we read "there doushu ye" which translated means: "this is a general procedure." Problem I.7-I.9 Parts to be collected.

Case: the numerators multiply the numbers that do not match them, it added, and takes this as a dividend. The different denominators between them are multiplied the divisor. You then divide the dividend by the divisor [...]. Problem I.19-I.21 Parts multiply.

Case: the denominators multiplied for each other are the divisor. The numerators multiplied for one another are the dividend. Dividing the dividend by the divisor. These considerations therefore show different possible levels of abstraction in the text. Even if it is correct to interpret the proceedings resolutive procedures as arithmetic, numerical, in the text, it shows a very clear, however, a continuous search algorithms to "General" can be applied to different classes of problems that are Always refer to the specific concrete situations. In this sense, the operations that are defined and that they are "used" in the application of various algorithms, are back to ordinary arithmetic operations (multiply, divide, etc.). And abstract as the use of terms: “to communicate” (a example is found in the problems I, 17th I, 18), “Procedure of the positive and negative”, “the increase and decrease”, which then define the meta-rules such as the multiply to break combine to simplify,


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standardize, match them to communicate. Meta rules can also be found for the resolution of the equations21. The terms used in the solution of the question proposed algorithms determine different nature. Another key element in the Chinese cultural tradition mathematics is, as mentioned before presenting the resolution of linear systems, the use of chopsticks and the calculation of these available for the execution of arithmetic operations. In a vygotskiana, the rods can be considered, as mentioned before, in this view, a possible artifact in the cognitive process, a primary artifact. The conventions are artifacts representing the corresponding secondary and are used for setting and transmitting the mode of action on the rods themselves. A calculation by the rods in the field of research in mathematics education requires a thorough study and a well-defined specification of the theoretical framework that will be used. The proposed topic has already been discussed in several researches (Bartolini Bussi, 2002, 2003, Mariotti, 2005) which relate to the work of Vygotsky (1974, 1987), of Bachtin (1979, 1988), of Engestroem (1990) and Wartofsky (1979). We do not want to enter the teaching on the issue, but we believe useful to analyze the historical socio-cultural context that the rods have "defined" in the history of cultural tradition and ancient Chinese mathematics22 define another aspect to the algorithm, as a procedure concrete calculation in mathematics. Educationally significant is that the use of primary artifacts requires handling (Vygotsky, 1987, p. 45) directly. The importance of the body is consistent with the recent position of cognitive science, based on the work of Lakoff and Núñez (2000), that the formation of mathematical ideas is based on sensory-motor (Mariotti, 2005 ). It is possible then that these activities of calculation with the sticks together, as we shall see in Chapter 5, the handling characteristics of the Chinese written language, and has carried vehicles today for students attending school in China (not only) a certain knowledge that could define the key capabilities in thought "mathematician"? This represents an application of research closely linked to our research work, assumptions that formalize rigorously in Chapter 5. Reconstruction of the instrument used to calculate the rods requires a complex historical analysis and is based on a variety of historical documents. Thus, outside of our considerations. Parallel assessments to be made, however, the strong character that positional algorithms with chopsticks may have conveyed. The table for calculating algebraic version was set so that certain positions were still occupied by specific types of variables (unknowns, powers, etc.), This convention can be considered a secondary artifact. The system introduced entailed the recording of “mathematical models” (Needham, 1985, p. 113). Significant models, in our opinion, for the ancient Chinese cultural tradition. Historically, the positional Chinese algebra had different consequences: while, implicitly placed the

21

The meta-rules are therefore to be understood as cultural products characterized by Philosophy, Religion of the Chinese written language etc. 22 The use of chopsticks sets only in the late Ming period (1368-1644) when they were replaced from Abacus.

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importance of the matrix23, determined in parallel, on the contrary, inhibition of the development of algebraic symbolism and thus a formalization of the concept of variable and unknown but there is in resolving algorithm with chopsticks. Taking then the statements made on algorithm as the key to the procedures for resolving mathematical procedures of China, although discussed in the first instance in the previous paragraph, it seems interesting to reconsider the text of Jiuzhang Suanshu solving linear systems using chopsticks. From reading the algorithm of resolution makes it immediately as a direct approach through the manipulation we can approach without difficulty and in an intuitive, with some properties of invariance of the equations.

3.2.3 The Rule of Fangcheng as Meta-rule for the Chinese Algebra As mentioned previously, the resolution of linear systems is introduced in Jiuzhang Suanshu in Chapter VIII, called Fangcheng name associated with the mathematical approach developed for the resolution. The commentary of Liu Hui in this chapter, but in defining a first approximation the theoretical basis of elimination of unknowns that the “algebraic algorithm” implies, proposes a definition of the term given as the principle of equality and homogeneity to test the procedure described in the various issues proposed. Since mid-century past the period of Fangcheng was appointed to constantly refer to what we call today “equation” 24. To properly define the original meaning of Fangcheng, we must examine the original meanings of the two characters and Cheng Fang. Fang can be translated as “put together”, the origin of the term seems to come from the provision of “a raft made of wood or bamboo that are assembled side by side, parallel to each other” (Chemla, 2004). Cheng is the origin, the name of a measure. The Shuowen jiezi (Interpretation of graphs, Explanation of characters) of Xu Shen gives the following definition: “10 is a form cheng, cheng 10 forming a fen, 10 fen cun do”. According Chemla, and Fang Cheng for derivation is that they have acquired the meaning of “rules of things”. The term Fangcheng, in this sense is used as a verb meaning to “look for the rules of things.” (Chemla, 2004) From this terminology seems to arise as a primary meaning of Fangcheng to “check the rules of assembling things side by side”. The expression means, namely: “to put alongside all the quantitative relations between things in order to assess the standards of the measures of each of them”. According to the description that gives Chelma “This report is prepared quantitatively in a column, in the manner of a branch of bamboo or a stick of wood, and all of these columns are then placed side by side, in the same way that a raft is mounted from the tables”. 23 24

Crucial to the concept was developed rather late, in 1683, the Japanese Kowa Seki. The word "equation" has been introduced in China with the European mathematical works.


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(Chemla, 2004) The algorithm of Fangcheng can be considered the result of more elaborate Jiuzhang Suanshu. His way of description is modern and fitting equations and the transformations that accompany the elimination of all unknowns using a positional writing. This procedure is virtually identical to the algebraic method of separation of the coefficients of modern mathematics. The algorithm is considered, as mentioned, one of the most important algorithms in traditional Chinese mathematics for the specificity of the definition of different meta rules as those of the multiply to break down, to convene meetings of simplifying, of homogenate, equal to communicate with it connected25. Rules that we believe may find strong connections with their cultural aspects of Chinese culture as the Taoist yīn-yáng and the cardinal virtues, treated in the first chapter. In the following, we intend to develop possible relationships between these, the written language and the teaching of mathematics today. An example of application of the Fangcheng you can find the problem VIII.3 proposing a system of three equations in three unknowns within which presents, in the solution, the negative amount. For complete information, please refer to the work of Chemla (2004) and Kangshen (1999) Spagnolo & Di Paola (2009) for a more thorough discussion of the algorithms proposed. Problem VIII.326: Suppose you have 2 Bing miles of high quality, 3 Bing miles of medium quality, 4 Bing miles of lower quality, and production (shi) will fill in any case 1 Dou. If (the mile) of superior quality is the sum (of miles) of medium quality, with (a mile) of average quality is the sum (of miles) of inferior quality, and (the mile) is lower quality sum (of miles) of superior quality, each time with no reason to Bing 1, then the production (shi) is always 1 Dou. It asks what produce (shi), respectively, 1 Bing miles of high quality, middle and bottom. Answer: A Bing miles of high quality produce (shi) 9 / 25 Dou. Bing miles of a medium quality produce (shi) 7 / 25 of Dou. Bing miles of a lower-quality produce (shi) 4 / 25 of Dou. Procedure: In modern notation, the solution starts putting x, y and z as their yields of wheat Bing upper, middle and bottom. The question can then be placed in

this way (since

2 x ≤ 1, 3 y ≤ 1, 4 z ≤ 1 ): 2x + y = 1 3y + z = 1 4z + x = 1

25

A thorough study of the meta-rules can be regarded as being in the work of Chemla (1996, 2004) and the French translation of Jiuzhang Suanshu (Chemla, 2004). 26 The Italian translation of the cap. VIII is Di Paola&Spagnolo 2009.

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The information contained in the text of the problem are, as in these earlier sites, initially arranged in a table of calculation 27:

⎛1 ⎜ ⎜0 ⎜4 ⎜ ⎜1 ⎝

0 2⎞ ⎟ 3 1⎟ 1 0⎟ ⎟ 1 1 ⎟⎠

The three columns contain the coefficients in order and the constant of the three equations (the first equation is represented by the third column). The first column is then multiplied by 2 and the third column subtracted from the result. The first column is then replaced by the new column obtained:

⎛0 ⎜ ⎜−1 ⎜8 ⎜ ⎜1 ⎝

0 2⎞ ⎟ 3 1⎟ 1 0⎟ ⎟ 1 1 ⎟⎠

The first column is then multiplied by 3 and the second plus the result:

⎛0 ⎜ ⎜0 ⎜ 25 ⎜ ⎜4 ⎝

0 3 1 1

2⎞ ⎟ 1⎟ 0⎟ ⎟ 1 ⎟⎠

Consequently, says the text, 25 Bing grain with a lower yield of 4 dou. A Bing was lower then the yield of 4 / 25 dou. 3 Bing grain and medium grain Bing with a lower yield of 1 dou. So an average grain Bing then has the performance of 7 / 25 Dou. 2 Bing higher grain and Bing have lower grain yield of dou 1, so a grain Bing higher then the yield of 9 / 25 Dou. The previous examples have been provided, for ease of reading, using the IndoArabic digits. The representation may also be obtained through the sticks from the calculation. An example of solving the system with the rule of Fangcheng may be this: The system is considered:

27

⎧2 x − 3 y + 8 z = 32 ⎪ ⎨− 6 x − 2 y − z = 62 ⎪3 x + 21y − 3 z = 0 ⎩

Unlike Problem 1, in the description of this problem has been appropriately chosen to set the table for the system on a more formal representation, according to a presentation reminiscent of the mathematical concept of matrix, a term is found in ch Kangshen the English translation of Jiuzhang Suanshu (Kangshen and alii, 1999)


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The elements described so far put the emphasis on two important epistemological values for the ancient Chinese mathematics culture. The first we have to recall (as we have mentioned above) is certainly that of the general as an element of discrimination between the Western European and Eastern China, and conditions predominate for the full, mature achievement algebraic thinking. The second, also a lot of it is debated in the literature demonstrating the importance of resolving the methods discussed so far. Regarding the first matter for reflection, analysis of the Jiu-zhang Suanshu and its commentaries, highlights how each problem, the relationship between the text and the procedure that follows, which, as we have seen, is expressed natural language, is something more complex than it might appear at first sight. The procedures, as mentioned, are systematically placed in the possession of more problems and should, therefore, be made in relation to more specific contexts (quantities and values issues). Through the reading is clear then, as mentioned earlier, a continuous search for a “basic algorithm”, defined for classes of problems between their equivalent. In agreement with Chemla (2007) could show that each algorithm was built not only to solve a specific problem, but the concrete class of problems that this represented. In this step, from simple calculations to think arithmetically through more or less complex, at a pre-algebraic thinking through the design of a possible change of context and numerical values, it often reveals an inability to move towards greater abstraction and hence the formulation a formalized algorithm applied to infinite possible cases. An inability to “define” then a symbolic algebra perhaps due to a lack of interest in this activity. Important in this perspective is, as mentioned, the use of chopsticks is calculated. The observation and enforcement of the provisions of the calculations, as we have seen, show the need for strict and systematic rules which established the initial setup and future development. “The development of the calculation on the area-was governed by the constraints and conditions very similar to the formalism that we follow when writing mathematical formulas” (Chemla, 2004). This formalism was to provide a dynamic and precisely because of this, any position, any numeric value, was considered in our view, within the variable of “thing that varies” (Malisani, 2006), first step, as discussed earlier, the effective "capture" of the algebraic variable. This aspect will be central later, when examining what is the practice of teaching today's school system for Chinese primary schools, will observe how the calculation with all the sticks are now charged with small children and students immigrants, although not used in schools in our classrooms, for the calculation, since children learn these skills in the family. The second aspect that we consider interesting to point at the end of this paragraph on Jiuzhang Suanshu, analyzed, albeit in a first approximation in its epistemological value, value is the demonstration of the methods discussed in the text inconclusive. Processes which, as mentioned, have for years influenced the

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Chinese mathematical tradition and beyond28, which in our opinion, can be regarded as canon of cultural reference for their mathematical thinking.

3.2.4 Argue, and Demonstrate in Conjecturing Jiuzhang Suanshu an Example through the Algebra29 The elements described so far put the emphasis on two important epistemological values for the ancient Chinese mathematics culture. The first we have to recall (as we have mentioned above) is certainly that of the general as an element of discrimination between the Western European and Eastern China, and conditions predominate for the full, mature achievement algebraic thinking. The second, also a lot of it is debated in the literature demonstrating the importance of resolving the methods discussed so far. Regarding the first matter for reflection, analysis of the Jiu-zhang Suanshu and its commentaries, highlights how each problem, the relationship between the text and the procedure that follows, which, as we have seen, is expressed natural language, is something more complex than it might appear at first sight. The procedures, as mentioned, are systematically placed in the possession of more problems and should therefore be made in relation to more specific contexts (quantities and values issues). Through the reading is clear then, as mentioned earlier, a continuous search for a “basic algorithm”, defined for classes of problems between their equivalent. In agreement with Chemla (2007) could show that each algorithm was built not only to solve a specific problem, but the concrete class of problems that this represented. In this step, from simple calculations to think arithmetically through more or less complex, at a pre-algebraic thinking through the design of a possible change of context and numerical values, it often reveals an inability to move towards greater abstraction and hence the formulation a formalized algorithm applied to infinite possible cases. An inability to “define” then a symbolic algebra perhaps due to a lack of interest in this activity. Important in this perspective is, as mentioned, the use of chopsticks is calculated. The observation and enforcement of the provisions of the calculations, as we have seen, show the need for strict and systematic rules which established the initial setup and future development. “The development of the calculation on the area-was governed by the constraints and conditions very similar to the formalism that we follow when writing mathematical formulas” (Chemla, 2004). This formalism was to provide a dynamic and precisely because of this, any position, any numeric value, was considered in our view, within the variable of “thing that varies” (Malisani, 2006), first step, as discussed earlier , the effective "capture" of the algebraic variable. This aspect will be central later, when examining what is the practice of teaching today's school system for Chinese primary schools, will observe how the calculation with all the sticks are now charged with small children 28

There are three main cultural areas that have undergone the influence of mathematics in China: Korea, Japan and Tibet. The circumstances in which this influence manifested itself, the response and the assimilation of which enjoyed differ considerably from one area to another. (Joseph, 1991). 29 Di Paola, Spagnolo, 2008; Spagnolo, Ajello, Xiaogui, 2005; Spagnolo, Ajello, 2008.


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and students immigrants, although not used in schools in our classrooms, for the calculation, since children learn these skills in the family. The second aspect that we consider interesting to point at the end of this paragraph on Jiuzhang Suanshu, analyzed, albeit in a first approximation in its epistemological value, value is the demonstration of the methods discussed in the text inconclusive. Processes which, as mentioned, have for years influenced the Chinese mathematical tradition and beyond30, which in our opinion, can be regarded as canon of cultural reference for their mathematical thinking. Problem VIII.13 There is a common well for five families. That which is lacking (in the distance of the water) to 2 ropes of A is a rope of B. That which is lacking to 3 ropes of B is like a rope of C. That which is lacking to 4 ropes of C is a rope of D. That which is lacking to 5 ropes of D is like a rope of E. That which is lacking to 6 ropes of E is like a rope of A. If each family has the rope corresponding to what lacks, all of them reach the water. The question is what is the value of the depth of the well and the length of the respective ropes.

Answer31: The depth of the well equals 7 ZHANG 2 CHI 1 CUN. The length of Jia’s rope is equals 2 ZHANG 6 CHI 5 CUN. The lengths of Yi’s rope equals 1 ZHANG 9 CHI 1 CUN. The length of Bing’s rope equals 1 ZHANG 4 CHI 8 CUN. The length of Ding’s rope equals 1 ZHANG 2 CHI 9 CUN. The length of Wu’s rope equals 7 CHI 6 CUN. Procedure: Liu: The first thing is to introduce the matrix. The problem is therefore a limit of five equations in six unknowns can be expressed in modern algebraic terms according to this matrix

⎛1 ⎜ ⎜0 ⎜0 ⎜ ⎜0 ⎜6 ⎜ ⎜1 ⎝ 30

0 0 0 5 1 1

0 0 4 1 0 1

0 3 1 0 0 1

2⎞ ⎟ 1⎟ 0⎟ ⎟ 0⎟ 0 ⎟⎟ 1 ⎟⎠

There are three main cultural areas that have undergone the influence of mathematics in China: Korea, Japan and Tibet. The circumstances in which this influence manifested itself, the response and the assimilation of which enjoyed differ considerably from one area to another. (Gheverghese, 1987) 31 This has to do with the rope for getting water from the well. The ropes associated with a family must be understood as having uniform length. Note the use of the abstract series of the celestial trunks. Jia, Yi, Bing, Ding and Wu are the frist 5 of the series of celestial trunks to which one returns for the ordinals. They are equally used as letters of the alphabet in their employment as markers for the numeration.

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The resolution that gives the author of Jiuzhang Suanshu refers to the procedure of Fangcheng, seen before, to eliminate the uncertainties, and transform the matrix in:

0 0 0 721 ⎞ ⎛ 0 ⎜ ⎟ 0 0 721 0 ⎟ ⎜ 0 ⎜ 0 0 721 0 0 ⎟ ⎜ ⎟ 0 0 ⎟ ⎜ 0 721 0 ⎜ 721 0 0 0 0 ⎟⎟ ⎜ ⎜ 76 129 148 191 1275 ⎟ ⎝ ⎠ In conclusion, we list the results by saying that the depth of wells is 721 cun respectively, whereas the rope of the Jia family has a length of 265 cun, the Yi family of a length of 191 cun, the home of Bing length 148 cun, the home Ding a length of 129 cun and finally that of the Wu family in length of 76 cun. As is highlighted in the matrix above. The answer given is a case of infinite solutions that the system admits. Marked with x, y, z, u, v the lengths of the strings t and the depth of the well you would have: x = 265k, 191k = y, z = 148k, 129k u = v = = 76k and 721k The solution reported in the text refers to the position k = 1 It is clear that just taking a pit depth equal to 721 for the length of cord Jia, Yi, Bing, Ding, Wu, fulfilling the original issue. The Jiuzhang Suanshu states then the smallest set of whole solutions that can be obtained. In agreement with Kangshen that, citing the text of commentaries on Li Huang's, speaks of “arbitrary but convenient solution” (Kangshen, 1999, p. 415), we believe that the choice of referring to a solution of this kind could be due to a strong connection with the reality of the same. This fact comes out, in this sense, not only in the choice of the situation/problem32 (the choice of the well as a meaningful situation) but also in the “choice” of the depth of the latter. 7 Zhang 2 Chi 1 Cun is a solution “acceptable” as the depth of a well that could be really feasible. Another famous riddles of indeterminate analysis is the “problem of a hundred birds”, algebraic problem is not easy33. This is, again, questions whose resolution is equivalent to that of two linear equations in the form: ax + by + cz = h x + y + z = k, which is seeking integer solutions. The first witness who finds it is in China as Shibu suanjing mentioned before. The text is presented in this form: “A cock 5 qian, a hen 3 chicks, and 3 chiks 1 qian. With 100 qian will buy 100, how many cocks, hens, and chicks has.” (Martzloff, 1997). 32 33

(D’Amore, 2000, p.285). In the history can be defined two types of classical problems of analysis unspecified: birds of a hundred problems and issues of division of victuals (R. Franci, introduction of Alcuino York’s book, 2005, p.20).

3.3 Conclusion

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In modern algebraic terms is according to the system:

⎧ x + y + z = 100 ⎨ ⎩5 x + 3 y + z / 3 = 100 and therefore as a solution: x = 4 +4 t, y = 18-7t, z = 78 +3 t In the solution must be considered that positive results are obtained only for t = 0, 1 and 2. The solution is also shown in this case only numerical expressed in words, reporting appropriately three possible answers numerical concrete, three sets of integers for cocks, hens and chicks: 4/18/78, 8/11/81, 12 / 4 / 84. How has achieved such results remains a mystery (Kangshen, 1999, p.415). We do not know if the reasoning used for the solution has been to trial and error (arithmetic) or prealgebra through a vision of the variability of the numerical cases tied to being positive integers. It should be pointed out, in our opinion that, even in this case, the problem is real, numerically solved, and the link context, however, is kind of different compared to that of Jiuzhang Suanshu. In this case, the choice of three options is not conveyed by the "convenience" as a numerical parameter was inferred from the choice of the depth of the well. The riddle of a hundred birds is one of the most popular pastime in the history of mathematics, they found almost similar in their dealings with Indian mathematics (the 8152 Proposition Ganita-Sara-Sangraha of Mahavira'se in Bakhshali Manuscript of the twelfth century), in 'Medieval Europe (with the works of Alcuin from York eighth century), and in the Arab (in the work of Abu Kamil in the 900 and that of al-Kashi in the XV century) and even in Liber Abaci.

3.3 Conclusion We can conclude that the choice of the “context well” allows a meaningful choice essential that the algorithm in the case of the problem “of a hundred birds” will not be able to identify. Returning to the mathematics of China and the process demonstration that we were discussing, in this light, we can therefore say that a proposition is considered proven if the solution algorithm was correct, if it functioned within a class of significant problems. The whole structure of Jiuzhang Suanshu, seems to call this procedure and, on the other hand, encourages the search for a generally formulated in a Algebra rhetoric. We insisted on these issues, because we consider them as important epistemological values of Chinese cultural tradition in relation to the Italian. In this sense, may be important to focus on those issues in Spagnolo&Ajello 2008 are as possible similarities and differences that the algorithm takes in the two cultures. For the Chinese culture, this constant research for a “fundamental algorithm” and the concreteness of thought, associated on the one hand to a mathematician, social, well-defined and, to a drive to the generality, may be significant elements in conjecturing and arguing in mathematics and therefore significant for Algebra.

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Table 6. Similarities and differences between the meaning algorithm that takes in two different culture (Spagnolo, Ajello, 2008)

From the Eastern point of view Procedure. Research algoIntuitive alProcedure rithms as a basic gorithm reference A paradigmatic Algorithm Algorithm: example is the rule formalized 1. Effectiveness, actually executable by an automaton. The automaton must be able to of three: tell which parts of the minimal description of the rule of three algorithm (accept the language in which the based on the algorithm is written, well-formed sentences "quantity of what are said to instructions); you have" and the 2. Finiteness of expression over the succescouple formed by sion of instructions. Cycles, conditions, the "lü of what jumps. you have" and the 3. Finiteness of the calculation: the concept "lü of what one of algorithm is usually included the condiseeks" to give rise tion of termination of the procedure for any to the "quantity of situation of the initial data within a certain what you try." domain. 4. Determinism: at every step of execution of the procedure must be defined one and only one step further. Research through analogies of algoThe condition 1 is essential. The other gives rithms applicable Deterministic rise to different types of algorithms. If mis- to homogeneous Algorithm sing the 4 will be called non-deterministic classes of probalgorithm. lems. Reference to the algorithms as a real model. Approximate algorithms, probability, NPcomplete (if there is a polynomial algorithm Probabilistic that can say if this is actually solving the Algorithm fuzzy? Algorithm problem), algorithms that are arrested after a number of steps that grows exponentially. From a Western point of view

The “Multiply to break down, put them together to simplify, to make uniform and equal so that they can communicate how these may be the fundamental points of mathematics?” Represent strategic targets in the research for invariants in the different methods of calculation. the “making equal” and “making homogeneous” may be in view of this, indications on the algebraic manipulation, but also strategies of reference that can then realize the correctness of the reasoning by the algorithm.

3.4 How to Summarize the Role of History?

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3.4 How to Summarize the Role of History? Wanting schematized, albeit in a non-exhaustive, as discussed at the historical and epistemological level we wanted to insert the following table to highlight the macro similarities and differences between Eurocentric and Chinese thought. Systematization of contents Europe

Hypothetical-deductive method of Euclidean type. The process modeling focuses on “syntactic model” in the hypotheticaldeductive sense.

China

Canon of "Chapter 9" as a system to “generalize” classes of problems in the processes of conceptual modeling. The modeling process priority “conceptual maps” and reasoning by analogy.

Argued, conjecture and demonstrate Conjecturing and arguing in semantic models. When you formalize the model, it introduces the demonstration. The “demonstration” is a tool with widespread use of bivalent logic in a hypothetical-deductive system. Research for "Fundamental Algorithms" defined for classes of problems (Chemla, 2007). Metarules to natural language, to algebra and to culture in general as tools “demonstration” and allowing “generalization”.

The table shows that the processes of modeling have been and are central to the systematization of knowledge in the subject. The modeling has been the only factor to interpret and make prediction on the phenomena of reality in the two cultures. But the process of modelling is different. Similarities and differences will be played in the choice of the algorithm as we have emphasized in Table 6? This for us is an open research problem.

Chapter 4

Common Sense and Fuzzy Logic∗ 4 Co mmon Sense and F uzzy Logic

As mentioned in Chapter 1, it is our opinion we that believe different types of logic can lead to different patterns of reasoning. Examples are the use of the logic of a bivalent and fuzzy set of argumentation in stages used in mathematical contexts and beyond. This observation may be significant, in our opinion, problem addressed and on the integration of different cultural references. With this conviction, we believe useful in discussing, albeit briefly, some significant aspects of common sense and fuzzy logic. This chapter presents considerations within epistemological nature validated by experimental investigations conducted in the classroom in Palermo. It remains open to the interpretation of the results obtained from a comparison of cognitive styles between Chinese and Italian.

4.1 Fuzzy Logic, Fuzzy Thinking and Linguistic Approach The fuzzy logic has one history of his that is tightly tied to Lofti A. Zadeh1 which initially introduces the whole fuzzy for then to pass about matters of the fuzzy logic2. The interesting problem is that the official science has welcomed with a lot of separation, the ideas of Zadeh on the fuzzy logic. The problem is in first appeal of cultural nature. From the scientific revolution actually to halves 900, the reference ∗ 1

This chapter was written by Maria Ajello and Filippo Spagnolo. Lotfi A. Zadeh was born on February 4, 1921, in Baku, Russia. He attended the American College in Teheran, Iran, and was awarded B.S. degree in Electrical Engineering by the University of Teheran in 1942. He came to the United States in 1944 and entered Massachusetts Institute of Technology where he received the M.S. degree in 1946. Until 1965, Dr. Zadeh's work had been centered on system theory and decision analysis. Since then, his research interests have shifted to the theory of fuzzy sets and its applications to artificial intelligence, linguistics, logic, decision analysis, control theory, expert systems and neural networks. Currently, his research is focused on fuzzy logic, soft computing, computing with words, and the newly developed computational theory of perceptions and precisiated natural language. Il suo libro “Fuzzy Sets” é del 1965. Bibliographical references of L.A. Zadeh and some jobs can be unloaded by the following site web: http://www.cs.berkeley.edu/People/Faculty/Homepages/zadeh.html.


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4 Common Sense and Fuzzy Logic

on the processes of reasoning and demonstrating within the scientific knowledges is almost always pasts through the ambivalent logic, Aristotle to intend us. This for different motives: • •

Although the ambivalent logic they had previously, always sure and controllable results. The tool of the ambivalent logic is rough, but sure. The structures of some natural languages that have supported this cultural operation is tightly tied to the Aristotelian logic (English, French, Italian, German, Spanish, the neo-Latin languages for instance).

The fuzzy logic has been lived from the westerners as a probabilistic logic, endless values of truth among the value of void probability 0 and the value of probability 1 (certain event). This vision is once more tied up to the ambivalent logic and, that is, when an event for instance has a certain value of probability 0.35, this he is read as the event that can have this value of probability (true) or not to have it (false).

4.2 Fuzzy Sets and Their Representations Fuzzy represents the due vagueness to the human intuition, not the probability (Zadeh). Your probability has to do with the verification him on some events, and when we have all the data a certain event or he is occurred or not. Certain whole fuzzy, those constituted by numbers, can especially be visualized by drawing graphic. In mathematical terms, these are simply the graphs of the functions of affiliation y=A(x) in a system of Cartesian coordinates. These graphs are frequently of triangular form, trapezoidal and, occasionally, to bell or of some other form. Among the definition of a whole fuzzy and his representation A(x), there is a difference. As a line, well defined poverty does not exist that you separate poor men from the others, so she does not exist "true" function of affiliation of the whole the poor people, or of anybody else together fuzzy. This does not mean, but the choice of the degree of affiliation is completely arbitrary: in the choice, considerations both theorists and empiriche as the context and the particular situation intervene of which he is looking for to give a model. Operations with fuzzy sets: • • • • •

Inclusion A⊆B “If all the elements of A are also elements of B” Degree of belonging: (A∪B)(x)=max{A(x),B(x)} (A∩B)(x)=min{A(x),B(x)} Ā(x)=1-A(x) (If x is an A, Ā(x)=0; if x it is not in A, then Ā(x)=1).

4.2 Fuzzy Sets and Their Representations

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Some graphic exemplifications:

(a)

(b) Function of union A B

AB

Function subset fuzzy (d) Complementary

(c) AB

Fig. 1.

The fuzzy logic is based on the mathematical method on the theory of the whole fuzzy that helps the cars to reason in more similar way to that of human. The fuzzy logic usually realizes him through an algorithm or a program, on a conventional digital calculator, and, as such it is exact. However, the method also has a subjective component (therefore essentially empirical and inaccurate) because it implies the translation in numerical form of the vagueness of the language and the human knowledge. The fuzzy rules. The fuzzy rules can express him with the natural language in the following way: if x is small and y is middle, then z is great. The variables x, y and z are type linguistic. With the whole fuzzy these words can be translated in numerical relationships, and to perform calculations. "A method to calculate with the words." Rules of inference: If x is A, and y it is then B, z it is C (Where: A, B and C are words that point out subsets fuzzy). This rule of inference would seem to be analogous to the rule: If x is A and y it is then B z it is C

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Premises: x is A’, y is B’, Conclusion: Z is C’. The relationship fuzzy–probability according to the point of view. 1. from the point of view of Aristotle, the logical fuzzy is seen with the tools of the probability. The measure than a subset B is contained in A (B/A). The "subsethood", as Kosko calls, concerns the degree of affiliation of an element x to a determined together fuzzy. We can interpret this "subsethood" as: 1. measure than an element x belongs to a determined together fuzzy. 2. the degree of truth of the affirmation "x belongs to the whole A.", for instance. In this case, the value of truth can be a whatever value understood among 0 and 13. Fig. 2.

What we see is an event with the maximum probability. The probability as the free will, shape him as a collateral psychological effect regarding creatures "provident", or rather predisposed to expect a certain sequence of events. It helps us in organizing our perceptions on, the memories and the most greater part of ours attended. The probability from an ordinary structure to the causal forecasts, between them conflicting, on as it will evolve the future in the next instant, day, season or millennium. It assigns a place and a weight to our future. The ability of forecast is an important element for the biological and cultural survival. More knowledge and information I lead probability. The probability as cultural constant that it makes to think about a biological substratum and this implies an evolutionary history.

The underlying figure is a circle fuzzy, in a certain measure a circle, in a certain measure an I don't hoop: More knowledge and more information it emerges the nature fuzzy of the things. Elementhood: A whole that contains his/her elements in a certain measure, that is, when it partially contains the element (A person is happy about his job in a certain measure, hardly totally happy or totally dissatisfied.)

Fig. 3. 3

Subsethood: A set that contains another set in a certain measure.

Suppose if we to want to train a car to recognize a signature, they initially recorded the signature, the speed (or better the components horizontal and vertical of the vector speed in partnership to the point of the pen). In the moment in which we want to reproduce the same signature, it is had to do it with the same speed. Insofar it will be impossible, also for the person that the first time has signed, to reproduce the same signature, with a value of equal truth to 1. we can consider valid for instance values of 0.95 or greater.

4.3 The Representative Point of View of Kosko

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(The index of implication of Régis Gras works on this aspect through the normal distribution and the probability (Gras, Suzuki, Guillet, Spagnolo, 2008)). •

A whole contains the part from the point of view of Aristotle and the whole one. Probability m/n. In the case of the figure, we have the conditional probability. A ∩ B .

•

The part contains the whole in a certain measure from the point of view fuzzy and of the part.

X

We can consider three approaches, according to Zadeh, to the fuzzy logic: 1. 2. 3.

CTP CW GCL

Compuational theory of perception Computing with words Generalized constraint language

We consider the generic relationship now (implicative): X isr R (X is the forced variable, R the relationship that he forces, isr the copula variable and r the variable whose value defines the road for which R forces X). We can have different types of constraints that allow us, therefore, in analyzing a vast range of approaches to the fuzzy logic:

Fig. 4.

• • • • • • •

possibilistic; veristic probabilistic random set pawlak set fuzzy graphic usuality

GCL is very expressive than the language of the logic of the predicates. In the computing word, the initial and terminal whole, both IDS (initial data set) and TDS (terminal data set) they are express in the natural language. The model for CW and TDS is the human mind. The approach CW represents, therefore, the approach more fuzzy next to that representable from the man.

4.3 The Representative Point of View of Kosko The language, the mathematical language of the science, especially creates artificial demarcations between the white and black, while the reason or the good sense fall through them: the reason works in light and shade. The middle point of the line is a "paradox" for the Aristotelian logic. For the logical fuzzy, it is the point where half truths are present and where A is equal to not A (equation yin-yang).

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4 Common Sense and Fuzzy Logic Table 1.

Bivalence

Polivalence

Aristotle A or not A Exact All or nothing 0 or 1 Digital elaboration Fortran

Budda, Yin/Yang A and not A Partial In a certain extent Continuity beetwen 0 and 1 Neural network (brain) Naturale Language Italian, English, …

Bits

0

Fuzzy units: Fits

½

1

Fig. 5.

In the ambivalent logic, the paradox is neither truth nor forgery, it doesnot have sense inside the language taken in examination and of the context. We consider the following concrete situation: An audience: two people. Question: Are you happy about your job? You lift the hand to answer to this question. The hand would not always have lifted equally, entirely raising, partially, etc… The following scheme analyses the possible situation with the presence of two only people, the model he can certainly widen for n people. To every vertex of the square, it corresponds a situation limit: (0 1) only the person 1 lift the hand, (1 1) they lift both the hands, etc…

(0,1)

(1,1)

person 2 ¾ -

(0,0)

(1,0)

1/3

person 1

Fig. 6.

4.3 The Representative Point of View of Kosko

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Person 1: raises the hand to 33.3 % (1/3) Person 2: raises the hand to 75 % (3/4) (1/3, ¾) A. The answer of the opposite audience is (2/3, ¼)

not A.

A and not A is equal to (1/3,¼) and not is (0, 0), that is, the empty set. The first position is that the thought fuzzy (yin and yang), and the other one is the Aristotelian position. We represent this in another scheme:

(0,1)

(1 1)

3/4 A

A e non A

. 1/4

(0 0)

A e non A

non A

1/3

2/3

(1 0)

Fig. 7.

If the answers are less fuzzy, the smallest square it widens toward the angles and not fuzzy. In addition, in this case, we have Aristotle.

Fig. 8.

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If instead the answers become more fuzzy, then the inside square tightens him toward the middle point. In the case limit, the square collapses in the middle point (all the people lift the hand to 50%). Then A and not A (yin yang) to 100%. A = A or not A = A and not A = not A (it does not distinguish him the full half glass and an empty half) Aristotle dominates in the contour, Budda in the center. The middle point is the black hole of the theory of the whole. The idea of the type of representation on exposed is of B. Kosko which consists of representing every fuzzy of X through a point in a system of Cartesian coordinates.

4.4 Some Epistemological Reflections on the Approaches to the Fuzzy Logic L.A. Zadeh recently in a conference on the artificial intelligence to Palermo4 he has redefined all the possible approaches to the logical Fuzzy. We can synthesize this way the possible approaches to the Logical Fuzzy: Tabe 2. Approach with Possibilistic Veristic words: linguis- approach approach tic approach Computing with words (CW) Computational theory of perceptions (CTP) CW-based Perceptions are possibilisexpressed as tic propositions in a natural language CW-based techniques are employed to translate propositions expressed in a natural language into what is called the generalized constraint language (GCL). X is A partial knowledge 4

Probabilistic Random approach set approach

Pawlak set approach

Approach with fuzzy graph

CWbased veristic

CW-based probabilistic

CWbased pawlak set

CWbased with fuzzy graph

X isv V partial truth

X isp P partial certainty

CWbased random set

Lotfi A. Zadeh, from computing with numbers to computing with words, from manipulation of measurements to manipulation of perceptions, Human and Machine Peception, Vol. III, Edited by V. Cantoni, V. Di Gesù, A. Setti & D. Tegolo, Palermo, Italy, 2000.

4.5 Some Experimental Observations on Common Sense and Fuzzy Logic

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The linguistic approach to the fuzzy logic seems, for the time being, one of the best approaches both for the resolutions of problems of the artificial intelligence (simulation of the human behaviors) that for the resolution of problems in the field of the technology. Scheme of approximate reasoning (Neuro-physiological Motivations)

Fuzzy Thinking Fuzzy Sets Fuzzy Logic and relative inferential systems (If … Then …) (Abduction fuzzy conjecture fuzzy) Natural Language

West

East

Polyvalent Logic

Chinese Language

Reasoning Hypothetical-deductive

Fuzzy Logic Fuzzy Thinking

Fig. 9.

4.5 Some Experimental Observations on Common Sense and Fuzzy Logic After the discussion, even though in a first approximation, on the theoretical principles subtended to the Logical Fuzzy, seem us necessary, to the goals of the analysis of our comparison, to report a experimental research work on common sense and Fuzzy Logic. In this research work it is putted in evidence the role that has the reasoning Fuzzy in an Italian set of students. The results of the research seem confirm a hypothesis related to different percentages of use of the Logical Fuzzy in the two cultures. This can engrave, in the teaching, for instance in the different approach to the "mathematical definition". An experimentally problem remains to deepen explore.”

4.5.1 Common Sense and Fuzzy Logic5 Bart Kosko’s works on fuzzy thinking have drawn great attention to some features concerning our everyday behaviour in relation to our decision-making ability or our ability to be in control of a situation. 5

This paragraph 4.5.1 was written by Maria Ajello and Filippo Spagnolo.

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The author (of the well known book: “fuzzy thinking”) has highlighted that when we are facing a decision-making problem, we usually try to obtain good solutions on the basis of global evaluations regarding the different possible problem options. When time is a critical factor, our way of thinking, analytical as it can be, is not able to help us in taking a decision quickly unless weighted “fuzzy” averages of pro and contro are used (Kosko, 1993). Talking about our behaviour, we can say that there is a “rule” whenever we associate ideas and we link a thing, an event or a process to another thing, event or another process. “If-then” clauses in natural languages, as well as in computer programming ones, help us in managing our reasoning. Kosko claims that rules on which we build our reasoning are “fuzzy” rules. This claim is widely accepted. In fact, a fuzzy rule makes a connection between fuzzy sets and presents an “IF X is A then Y is B” structure; such is the case, for example, of the following phrase: “if the weather is very hot, then I will put on light summer clothes”; if X = {days of the year}, A = {days with very hot weather}, Y = {clothes}, B = {light summer clothes}, we definitely obtain an “IF X is A then Y is B” structure (A⊂ X, A fuzzy set, B⊂ Y, B fuzzy set). Thus, despite the fact that we can have different perceptions on feeling hot or finding clothes light, still the common sense gives a common meaning to the fuzzy rule “if the weather is very hot, then I will put on light summer clothes”.

4.5.2 Fuzzy Logic and Complexity Mathematics is “the language in which is written the Book of Nature” whose characters “are triangles, circles and other geometric figures” or at least this was the position of Galileo Galilei and of many other after him. Today, if one wants to use fuzzy logic alongside with bivalent logic must consider the possibility of using “qualitative relations” besides traditional mathematical formulas as well. A qualitative relation deals with linguistic variables, in fact its values are words rather than numbers. According to this sense, Lofti Zadeh says that fuzzy logic is a methodology that makes possible calculus using words. However, Zadeh’s thought points out wideranging issues which involve way of reasoning, of conjecturing, of learning and therefore of teaching. All this leads to new inference processes, where premises and conclusions have different meaning from their ordinary sense, and where new terms, such as “fuzzify” and “defuzzify” are used, these terms mean, respectively, passing from a mathematical formula to a fuzzy rule and vice versa. New fuzzy scenarios appear, but new specifications for a fuzzy science are not fixed yet. Which attitude should take a teacher? And what can a researcher do? They can certainly wait and see what happens, but the fuzzy logic is a challenging issue that walks with another compelling contemporary issue: the complexity. Once again nature provides a common denominator. Investigating emergent behaviour of a complex system, raging from to human brain, requires much more flexible tools than classical mathematical formulas. In addition, this means that we


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must go beyond the classical two-valued logic or, as it is stated by Zadeh: “As the complexity of a system increases, our ability to make precise and yet significant statements about its behaviour diminishes until a threshold is reached beyond which precision and relevance become almost mutually exclusive characteristics.”

4.5.3 The Hypothesis Fuzzy logic is paying off in the marketplace, where there is a growing interest for products which are based on fuzzy logic as their underlying logic (usually for control systems, which are called “fuzzy expert systems”), the Japanese manufacturers were the first to embrace this technology and have been building real products around it since the early nineties. The fuzzy logic applications in reality are now innumerable and they range from control system for small domestic appliances or electronic devices (on washing machines or camcorders, for example Matsushita, Hitatchi products) to optimize planning of industrial-control applications (Kawasaki Steel industry), from efficient and stable control of car-engines (Nissan) in controlling subway systems (Sendai city subway system). These successful applications have convinced almost everyone, not only the Japanese engineers, that fuzzy systems actually “work”. To date, basically, the engineering research in this field is limited to fuzzy expert systems in which the main problem is to find good fuzzy rules. The FAT theorem demonstrates that it is possible to model any system using fuzzy rules; from a mathematical point of view, this means that a fuzzy system based on a finite number of rules can approximate, in an uniform way, any continuous system. The point is that, in most cases, finding good rules to model a real system effectively is not that easy. To solve this problem, new adaptive fuzzy systems have been recently developed: the basic idea is to build fuzzy systems that are able to model their own rules, in learning how to do it directly from data of the problem they must solve. Lots of techniques have being developed to carry out such a task, and they usually involve back-propagation neural networks determine the most appropriate rules to use, and genetic algorithms to define the type and the shape of the membership functions. Both the techniques are based on the adaptive principles and are able to deal with raw data as a starting point to build their own “knowledge” of the problem and such an adaptive fuzzy system acts like a human expert: it is able to learn from the experience and to use new data to perfect its own knowledge and ability domain. Adaptive fuzzy system developers and researchers have taken their inspiration from neural network of our brains to build neuron-fuzzy systems that are able to learn fuzzy rules, thus, adapting their own dynamic structures to the problem. All these things concern learning, and, therefore, they could be of great interest for a teacher to deal with scholastic learning issues. Thus, one wonder whether spontaneous inferences used by an individual who argues his own first approach to learn something by actually “doing” something are fuzzy rules.

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Control problems provide useful and simple examples of fuzzy logic applications in reality. Their main characteristic is readily recognised: from the input values, it is necessary to calculate the output values that can assure a satisfactory behaviour of the system that translated in fuzzy term sounds like: IF present conditions, THEN do a particular action. The proposed activity has been introduced to a 28 pupil class of the second year of a “liceo scientifico” as a computer science activity, and it deals with a classical control problem better known as the “inverted pendulum”. Of course, pupils whose age ranges from 14 to 16 have not got yet the proper tools to model the problem by means of mathematical formulas to solve it. What has been investigated, however, is their first approach to the problem in using the natural language with nobody there to guide them, but they are only given the assignment instructions. Anyway, the class have already tackled, in the previous scholastic year, an algorithm unit and the first elements of Pascal programming language, such activities are part of the planning that proposes to open again the computer science education issue that was interrupted.

4.5.4 The Assignement a Computer Science Activity (Time: 3 Hours) Phase I. The class pupils are divided into groups, each containing just two pupils; besides each group is provided with a pole, a sheet of paper and a pen (time: one hour) 1. A tries to balance the pole on his hand while B observes and writes down on the paper the actions made by A relating them to the actual positions of the pole. 2. the couple of pupils exchange their role and repeat the previous operation. 3. A and B compare what they have written and they make a list of “rules for balancing a pole on a hand palm”. Phase II. The class in now divided into groups, each containing two of the previous groups, they have the written rules, a sheet of paper and a pen. (time: two hours) 1. each group discusses the possibility to use a properly programmed robot to carry out this balancing task jotting down any draws, symbols, variables, constraints and they think it is necessary to do this correctly; 2. after they have discussed, accepted and jotted down the hypothetical conditions under which the robot should be able to execute the instructions, every member of the group tries and express all the ones he thinks right using the natural language (should he search for an algorithm?); 3. the group, once they have examined all the material that have been produced, expresses only a program for the robot.


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Note: the group must give the teacher all the material produced, even if it was considered not useful.

4.5.5 A Priori Analysis The considered activity is clearly an “open test”. To undertake an implicative analysis of the variables, assumable behaviours must be taken into consideration as well as appropriate and significant variables that must be consistent with the assumed hypothesis. As far some possible behaviours are concerned, we thought it right to consider expressions such as “the pupil does…”, whereas for other cases is it was thought that expressions, such as “the pupil does not do a certain thing” would be more revealing. Besides, it was considered to be necessary to determine the reference words of the expected answers for the two phases, in more precise terms. Phase I Point 3: making a list (one to one activity) of rules for balancing a pole Table 3.

Variable ra1 ra2 ra3 ra4

Description The pupil uses a table as a representation tool. He only tells between equilibrium and non-equilibrium positions He does not use more than four instructions He does not consider numerical variables

Phase II Point 1 and 2: expressing the robot potentialities and the initial conditions Table 4.

Variable rb1 rb2 rb3 rb4 rb5 rb6 rb7 rb8

Description The pupil draws a robot that looks like a human being He thinks that the robot is able to recognize the equilibrium state He thinks that the robot is able to move He thinks that seeing is one of the robot capabilities He thinks that the robot is able to feel (tactile perception) He specifies the way the robot receives instructions He thinks that the pole length is a possible variable that influences balance He does not specify, whether the robot is able to make numerical calculations

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Phase II Point 3: expressing the robot program Table 5.

Variable Descripion rc1 rc2 rc3 rc4 rc5 rc6 rc7 rc8 rc9

The pupil does not use a mathematical model, but he uses natural language instead He does not assign numerical values to the state variables he is considering He uses “IF-THEN” implications He does not use instructions in sequence He does not use instructions that require numerical calculations He uses fuzzy sets in the instructions He takes into consideration the angle between the pendulum and its equilibrium position He takes into consideration the movement speed to balance the pole He doe not consider the angle and/or the movement speed

How the variables are chosen. The basic issue, we are investigating is whether the pupils, under the proposed situation, is using fuzzy rules. Therefore, the expected results concern the implications between the following variables: rc1, rc2, rc3, rc4, rc5 and rc6. The occurrences of such variables will be substantially weighted, and so will be the case with the ones of the first group ra1, ra2, ra3 and ra4. Any possible “strong implication” between the three groups must be interpreted to establish, whether it is revealing about the way of reasoning.

4.5.6 Implicative Analysis Using the Chic Program6 The implicative graph containing all the items shows that the strongest implications (98) are rc4 →rb5 and rb6 →rb5, thus those who do not use instructions in sequence and those who do not specify the way the robot receives instructions consider the robot capable of tactile perception. On the other hand, the implication, hierarchical tree shows clearly the high level of implication between rc4 and rb5, but the strong implication (coloured in red) goes from rb6 to rc4 →rb5 and keeping on analysing the graph: all this block, still red coloured, implies rc5 (those who do not use mathematical models). It seems that giving up the “in sequence” structure of instructions, one also gives up any mathematical model related to strictly numerical values, and is willing to take into consideration other qualitative aspects of the problem (such as the case of tactile perception capability of the robot). Another strong (95) and extremely relevant implication is rc3 →rc6 i.e.: those who use “if-then” instructions in association with fuzzy sets, formulating, in this way, actual fuzzy rules. 6

The CHIC software is presented in the Springer’ book of the statistical implicative analysis, Gras-Suzuki-Guillet-Spagnolo, 2008.


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Besides, let us look at the occurrence values of rc3 and rc6 variables, 70% and 85% respectively, this means that we are referring to a vast majority of individuals involved in our analysis. In the similarity tree, if we look at the block in which there are rc3 and rc6 (related by means of similarity to rc9, ra4, rc5, 5b8, rc1, ra3), we can deduce that it is possible to put together those who use fuzzy rules with those who do not bind themselves to standard measures of numerical kind, or numerical calculations, or standardized mathematical models, but use only a limited number of instructions on the whole. The implications rb5 →rb8, rc2 →rb8, rc5→rb8 indicate that those who use numerical variables and numerical calculations did not specified, whether the robot was able to do numerical calculations, neither did those who consider the robot capable of tactile perception. Considering the robot able of tactile perception replaces its ability of doing numerical calculations.

4.5.7 From the Pupil Records 1. Some instructions given in the first or in the second phase : • the pole is in motion, swinging for a few centimetres only; in this case you have to compensate the pole’s movement by only moving the hand in the same pole’s direction. However, if the pole is swinging more rapidly, then you have to compensate the pole’s movement by moving your arm, and if necessary, even your body (in the first phase); • if the pole is in motion at low velocity, then compensate its movement by moving your hand at low velocity (in the first phase; • if you want to get more movement agility of your arm, then put the pole at a medium height (in the first phase); • if the pole end leans forward, then increase the angle formed between your arm and forearm and at the same time increase the inclination of your wrist forward (in the second phase); • to balance the pole do very slight, medium or fast movements, but make sure that they are always coordinated (in the second phase). 2. Some of the pupil observations on the robot potentialities: • the robot activates the given instructions as soon its tactile sensor, placed on the platform where the pole stands, detects that the pole is loosing its equilibrium position; • the robot has the ability to recognize the equilibrium position by means of its visual and tactile perceptions; • due to its visual sensors, the robot is able to recognise when the pole is standing in its upright position, perpendicular to its hand palm (or should we say platform?).

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4.5.8 A Possible Interpretation Key The previous results can be more revealing when compared with the Edward De Bono’s work on relations between creativity and lateral thinking. The following table shows synthetically the analogies and the differences between the two different way of thinking that are usually indicated as linear and parallel. Table 6.

LINEAR THINKING

PARALLEL THINKING

•

• • • • • • •

• • •

It is an intentional process • It is a mental attitude • It can be learned, put in practice, used It is selective • It is productive It chooses a path excluding • It does not select a path but the others tries to open new ones instead It selects (or it searches for?) • It produces alternative apthe best point of view proaches within the feasible solution space. It activates only if there is a • It activates in order to produce clear direction indicating a direction where to go where to go • It is stimulating It is analytical • It can jump from a place to It is consequential another When it moves step by step, • It can move freely without takeach step must be clearly jusing into consideration any tified contradictions, provided that the final conclusions are right • It does not use negation, thus, It uses negation to prevent it is possible to follow a wrong from following some paths path It explore the most probable • It explores the least probable paths paths It is expected to find an an• There could be an answer, but swer there could be many others too, besides, there could be partial solutions

The pupils that have carried out the activity have clearly used both thinking methods, but usually preferring the second one. The records, in which it is possible to see the abandoned solutions, as well as the ironic comments on the robot look and all the attempts to give a solution at any cost, even using absurd


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instructions, present different characteristics that are readily traced back to one of the two afore mentioned way of thinking. However, it must be highlighted that the activity requires pupil’s absolute freedom of using non-conventional schemes that are usually used during the normal scholastic activities instead. Usually a pupil, during any scholastic activity, tends to follow the linear way of thinking rather than the lateral one and that is also due to an “expected” logic capability that is recognized and appreciated only when its way of reasoning was based on the linear thinking requirements, while it is usually misunderstood and not enough appreciated if it was based on the lateral thinking ones instead. We can assume that besides the well-known relation: linear thinking ↔ ordinary logic (the bivalent one) there is also another one of the following kind: parallel thinking ↔ fuzzy logic and fuzzy rules.

4.5.9 Conclusions Pupils, during the proposed activity, did not use their mathematical and computer science knowledge and freely used natural language to express their models, in this way, they ended up using actual fuzzy rules to describe the control instructions for some expected actions. The lack of sequential schemes has also encouraged fuzzy thinking and creativity. This, in a nutshell, was the main result of the experience. However, it is necessary to comment further this experience to reveal its own specific characteristics: • The way the activity was carried out can be easily lead back to the so-called “a-didactic” Brousoau’s situations rather than to any conventional didactical approach (drills and class tests and so on). The pupil can talk with the classmates and can express his own opinion as well as face the proposed problem using the approach he thinks is the best. He knows that the process and the method he uses to get the solution will not be evaluated, but what really matters is a convenient solution to the problem is found. • There are no objectives referring to “logic capabilities”, and everyone knows the programmatic choices that were made. It created an atmosphere in which everyone had the feeling of being free to use irrespective of the kind of thinking he liked, linear or parallel. • In this work, speaking with the class about the fundamental choices, it was preferred to use terms, such as “different mathematics” rather than simply “mathematics" also it was abandoned the idea of knowing the “reality” in favour of knowing the “realities”. • And finally, uncertainty, indetermination, incompleteness are the terms that get to be familiar to pupils since their first year of school, facing, for example, games and problems which have no solutions or more than one, or situations that require a probabilistic approach, or open tests and so on, all situations that can easily happen during laboratory activities that are integrated in the ordinary scholastic work.

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4.5.10 Open Problems • Investigate the implications that reasoning using fuzzy rules can have on the possibility of formulating conjectures (is it possible to speak of fuzzy conjecturing?) • Investigate the pupil knowledge of “precision and significance” used in their reasoning processes • Investigate how the students use quantitative and qualitative relations in their formulating conjecture processes • Investigate the possible implications that the habit of using “fuzzy enquires”, in natural language, can have on their learning approach.

Appendix 1 Basic Fuzzy Glossary Bivalent logic and fuzzy logic: The bivalent logic is what in the Western thinking is called logic par excellence, this kind of logic goes back to Aristotelian times and it is based on the concept of true (value 1) or false (value 0) clauses. Within this logic is valid the non-contradiction principle (it is not possible that the same clause is true and false at the same time) and the law of the excluded middle (a clause can only be true or false, there is not a third possibility). Upon this logic, it is based on the formal demonstration process. The term fuzzy logic has two meanings, the first: it indicates a multi-valent logic in which the concept of truth of a certain clause can vary in a continuous way from values between completely false and completely true [0,1], introducing in this way the concept of “partial” truth. In this context, the Law of the excluded middle is not valid at 100% and so is case for the non-contradiction Principle, (there is no contradiction between A and not A, but only degrees of contradiction). This logic goes back to the beginning of the last century. The second meaning was created by Lofti Zadeh in the sixties, and it indicates the reasoning (and the calculus) using fuzzy sets and fuzzy rules. Un indicatore significativo nella didattica della matematica é l’approccio alle definizioni che può essere giustificato secondo una logica bivalente. Difatti “la lingua cinese cerca di appoggiarsi su delle basi già conosciute, di cercare delle similarità tra il nuovo oggetto e quelli di cui si è già appropriato, a creare delle categorie più generali nelle quali potrà inseguito classificare gli oggetti. La sua logica è, estremamente vicina ai processi scientifici moderni!” (p.53, E. Cornet) Una definizione è dunque qualche cosa che fissa il significato di una parola, che traccia dei limiti precisi a ciò che può voler dire. Ma i cinesi non hanno il bisogno di definire, di limitare. Che bisogno c’è di definire un carattere? Perché precisare il significato delle parole di una lingua il cui fondamento é l’essenza stessa del senso?

Appendix 2

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Fuzzy rules: Conditional relations of the following kind: IF X is A, THEN Y is B, where A and B are fuzzy sets are called fuzzy rules. "if there is poor visibility, then car headlights must illuminate farther". Every rule determines a “fuzzy patch” (AXB product) within the state space – the set of all possible input and output combinations - . The larger the A and B sets, the more uncertain the fuzzy patch; on the contrary, more certain knowledge implies smaller fuzzy patches, i.e. more precise rules. From a mathematical point of view, every fuzzy rule just operates as an associative memory that links the fuzzy B output to the input A, besides inputs similar to A activate outputs similar to B. Fuzzy system: A fuzzy rule-based system that associates input to outputs is called fuzzy system. Every input activates all the rules of the system with different degrees of intensity, just like in an associative memory. The more precise the correspondence to the “if part” of the fuzzy rule, the more activated will be the related “then part” of it. The fuzzy system sums all these “then parts” up determining a weighted average (usually called “centroid”, also you have to considered that several method as to determine the output of the system exist), such a value is the output of the system. The adaptive systems are able to learn their own rules directly from the problem data, such an adaptive fuzzy system acts like a human expert: it is able to learn from experience and to use new data to perfect its own knowledge and ability domain. FAT theorem: It is the Fuzzy Approximation Theorem that demonstrates that a fuzzy system based on a finite number of rules can approximate any continuous system in a uniform way (Borel – measurable)

Appendix 2 Conjecturing and arguing between the philosophical texts of Mao and a text of Chinese contemporary pedagogy A further reinforcement of the discussion can be found in the experiment on the comparative argument and conjecturing the writings of Mao (1924–1949) and a text by Professor teaching Chinese M. Sun (2005). Mao's writings are compared with the Hegelian dialectic. As we considered cultural reference that is expressed in the literature Taoism–Confucianism–Buddhism.

In the Oriental culture 1.

Taoism. The Tao and the Way, spontaneity (tzŭ jan), inaction in the sense of non-artificiality or conformity to nature, simplicity, vacuity, tranquillity, and illumination, all dedicated to the search for “long life and enduring vision” Dagobert (1972)

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2.

3.

Confucianism. Confucianism concentrates everything on humanity as the highest good, the superior man as the ideal being, and the cultivation of life as the supreme duty of man. Confucius taught the doctrine of the “chung”, or of conforming oneself to the principles of one’s own nature, and of the “shu” or of the application of the principles in relation to other men, thus like the doctrine of the Just Means (chung yung), that is, finding “the way of our moral being and to be in harmony with the universe”. Dagobert (1972) Confucius encouraged constant and continuous dedication to learning, which had to last all of one’s life and that still today constitutes one of the main elements of the growing oriental economy which is inspired directly or indirectly by Confucius. Buddhism

Interpreting Taoism from a western point of view as “naturalism” and Confucianism as “humanism” turns out to be very reductive, but it gives us a general idea of these two great oriental philosophies. Both the schools conceive reality as an incessant mutation and in continuous transformation. Only in 400–200B.C. did the school of Yin Yang highlights the fact that these elements of the Yin and the Yang, fundamental principles, always in contrast, but complementary, represent a common foundation. Taoist metaphysics and Confucian ethics are mixed together, while the theory of Yin Yang works as a connector, tying together, in this way, nature and man in a clear relationship of macrocosm and microcosm. To manage these changes, it is necessary to follow the Tao, the Way, the lesser line of resistance, the most profound resonance which can be obtained with the changes in our life. From the oriental point of view, the metaphysics of Yin and of Yang allow the declination of the possible nuances existing between the two opposing situations. The visual representation also gives a possible interpretation of it.

Fig. 10.

The counter positioning of the two opposites never happens in a clean way. It always has, as can be seen in the picture, an infinity of other opposing states. The interpretation of some writings of Mao Tse-tung on dialectics7 and on argumentative language. 7

Here, we refer to the official works regarding articles written in different periods of the Chinese devolution: Vol. I (1924–1927, First and second revolutionary civil war), Vol. II e Vol. III (1937–1945, War of resistance against Japan), Vol. IV (1945–1949, Third revolutionary civil war). The analysis was conducted taking considering only the argumentative indicators regarding tools: dialectic and classificatory.

Appendix 2

111

The parting hypothesis is that the argumentative language used by Mao follows the Chinese tradition in a much more marked way in that it cannot have been influenced by the argumentative systems of Marxism–Leninism with a Hegelian origin. Table 7.

Influence of fuzzy reasoning (interpreted as probabilistic reasoning8) and classificatory reasoning. Vol. I: pp. 29; p. 31: “… I made a classification…” p. 32: “…there were only 300400…” p.39: “the number can fluctuate between … and …” (also in vol. II pg. 459 and vol. IV pp 151, 212,381, 433) p.115 absolute egalitarianism does not exist and passes to an analysis of concrete cases of the possible nuances Vol. IV. p.392: “Having in mind the amount. This means that we must pay attention to the quantitative aspect of a situation or a problem and do a fundamental quantitative analysis. Each quality is shown in a specific quantity and without quantity, there cannot be quality. Even today, many of our associates have still not understood that they must pay attention to the quantitative aspect of things – to the fundamental statistics, to the principle percentages, and to the quantitative limits which determine the quality of things. They do not have the “amount” in mind and it turns out that they cannot

8

Hegelian Dialectics - Dialectics in Mao through the Yin and the Yang

Hegel, and successive interpretations of the Hegelian right and left, consider dialectics as the synthesis of opposite: a. position of a concept; b. suppression of this concept through its opposite; c. synthesis of two precedents to preserve that which is affirmative in their solution and in their transition. Mao (Vol. I, pg. 316): a. First phase of knowledge: phase of perception (and of impressions). External facts make a series of impressions tied to an approximate exterior link. In this phase, man yet cannot form profound concepts and not treat logical conclusions (examples of social practice) b. The following social practice causes numerous repetitions of things that bring out perceptions and impressions in men, and then it produces, in the human mind abrupt change (a jump) in the knowledge process and the concept is found. The concept no longer reflects, the phenomenal aspect, a single aspect and the external links of the things, their set and their internal link. The difference between concept and perception is not only quantitative, but also quantitative. Proceeding in this direction and making use of the methods of judgment and of reduction, one can arrive at a “logical” conclusion

For an analysis of this position, see Kosko (1995) and Spagnolo (2003).

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Table 7. (Continued)

avoid committing errors. … Also, in all these cases, it is necessary to determine the quantitative limits.” Observations: 1. The classificatory system passes through the percentages and not through the “relationships”. 2. The use of intervals of approximation very similar to the variables of fuzzy interval. 3. The always declared will to analyze nuances.

c. The true task of knowledge is to arrive, by means of perception, at thought, at the gradual understanding of the internal contradictions of things objectively existing, of the laws which regulate these things, of the internal link between one process and another, arriving, that is, at logical knowledge. Logical knowledge makes a great step forward, embraces the set, the essence, the internal link of things bringing one to the discovery of the internal contradictions of the world around him, and can thus seize the development in all of its entirety, with the internal link of all of its aspects

A “classificatory” type of list of things to follow or not to follow” (i.e. p. 111, vol. I; p. 147 vol. I on the classification of social classes in China in that historical period; p. 14-15, vol. II, where the argumentation begins with some questions on one proposition and its opposite, the answers to which are still of the classificatory list type and still on p. 246 of Vol. IV “Directive on the correction of errors in class groups and on unity with the average agricultural labourer”). In particular on p.57 of Vol. III in describing a very common stereotype in the party on “descriptive classification”: “ … in their articles and in their discourses, when they write or give a report, first they use the Chinese numbers in upper case, then the Chinese numbers in lower–case, then the cyclical characters, etc…”

Observations: 1. In the second phase of knowledge, Mao quoted Confucius: “There is always a relationship with human perception”. This cross-reference to the history of Chinese philosophy recurs in all of these four volumes. “Logical” knowledge must be understood here in the sense of Chinese culture. The reference is to the Chinese language; 80% of the Chinese characters are of the associative type which is the way to construct new characters. In the Chinese historical–philosophical tradition, there is no reference to Logic as it is understood in the west (Needam, 1985) (Spagnolo, 2005). When Mao speaks of “embracing the entirety”, “gather the essence”, etc… it is referred to the characteristic of this type of character. Some examples of characters by association (ideograms) are covered in paragraph 4.2.2. The character, which in this case is an ideogram (by association), is seen in its entirety. 2. The sense of “embrace the entirety” etc.. could also derive from the meaning that is given to the word “show” in the

Appendix 2

113 Table 7. (Continued)

Observations: 1. It would seem to be a relational classificatory type of reasoning, but it is all reduced to a descriptive classification. 2. In any case, Mao used the descriptive classification, in vol. III on p. 57 he ridiculed this way of reasoning which had become a stereotype for the party managers seeing that it did not bring new knowledge: “…it is none other than a Chinese pharmacy, it is an article that does not have precise content.”

history of Chinese mathematics. In the Canon of the “9 chapters”, Chemla takes into consideration the research of a “fundamental algorithm” for classes of problems and such a “fundamental algorithm” would represent the analogous operation of the hypothetical–deductive system of western culture 3. In the third instance of the dialectics presented by Mao “becoming” in the Hegelian sense is never taken into consideration. In the tract “On contradiction” (I Vol., p. 335) he also makes some mathematical examples when he analyses the contradictions in terms of “* e -, differential and integral, etc…” That is, he highlights that which is already found inserted in Ying and Yang and, that is, the presence of opposites in the most disparate of situations. 4. Thus we can outline the dialectics presented in the works of Mao: a. the first two include the internal contradictions (Ying and Yang; b. Overcoming. Almost generalization in the sense of the research of the fundamental algorithm

Perhaps, this is the greatest difference with respect to the western way of seeing. In Chinese philosophy, nothing is cleanly divided into black and white, not even the interpretating colours in Ying and Yang circle. Nisbett (2003) maintains that “The dialectic thought which developed in China is neither the Platonic understanding of dialectics (as art of discussion) nor the dialectics of Hegel or of Marx, in which they look for conflictual solutions (for example the interests of different social groups). Oriental dialectics accept the contradictions, since only through them, in their way of viewing things, does the truth become recognisable.” Nisbett, together with Peng, looks to analyze oriental dialectics through the following principles: 1. 2. 3.

principle of mutation: reality is a process subjected to constant mutation; principle of contradiction: because the mutation is constant, so also is the contradiction; principle of holism: because everything changes continuously and is continuously found in contradiction, nothing, in human life as in nature, can be understood independently from any other. Everything is connected.

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This has brought a tolerance of paradoxes that is absolutely not present in western culture. Some argumentative elements of the text of Prof. M. Sun. What is the Tizhi? It is none other than “logical” interpretation within the formation of the characters by “association” (Ideograms) and it represents that which Nisbett maintains is the holistic approach of the Chinese culture. Therefore, the “Tizhi” “…not include only one factor, but many factors …”, the framework is “…related to tizhi education, for instance, people, educational events, physical matter, activities of different levels and different kinds of education, educational organization, and educational regulation and so on.”, “…they form a unity…”.(Sun, p.143) In the same way, the character by association then represents a oneness, even if it comes from other different characters, the “tizhi” represents this element of conceptual organization of knowledge applied to the organization and the management of the public school. The conceptual maps present in the book (ex. p. 158-161) are to be considered as diagrams of this type and the “relationships” indicated with the arrows as references. The “relationship” does not have the same meaning that it can have in western culture. When one thinks of “relationship”, in western culture, it refers to the “relationships of equivalency” in the Aristotelian classificatory sense. The lists of relationships (ex. p. 48, 84, 125, etc…) represent, perhaps, more than not “definitions” of the listing type. In paragraph 2.1.5, after having analyzed five types of definitions9: “stipulative definition, lexical definition, precising definition, theoretical definition, and persuasive definition” (p.34) he concludes “…in my research, we cannot use all five types of definition to define the concept of tizhi education” (p.35), intending to give it a definition in chapter 6. However, the “definition” exhibited in chapter 6 is strictly tied to conceptual maps that are connected to each other. Then, the expression referring to the method used by Prof. Sun with regard to the approach “conceptual, analytical, technical from the philosophy of education”, becomes a bit clearer. The analytic stands for “critical analysis”, in the sense expressed above, of the possible inferences between concepts or conceptual fields: “the study of tizhi is also the study of a concept” (p. 33). The complete framework of reference together with the definition of the “tizhi” understood as “logical system” for managing these conceptual fields with regard to the educative phenomena represents the “philosophy of education”. In paragraph 1.3.2.2 “Generating categories” this is the sense of the discussion. Categories can be generated by this construction in conceptual maps. Sun on pg. 155 states: “We should use the holistic approach to study tizhi education, so as to avoid unilateral ideas in understanding tizhi education. Consequently, we should establish a framework for tizhi education which should include basic factors and tizhi sub-education.” Once more, this “holistic” system of 9

There are reported by Copi I. M. and Coen C. (1998), Introduction to Logic, tenth edition, New Jersey: Prentice Hall, Inc.

Appendix 2

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knowledge which is presumably also tied to the system of the conceptual maps is highlighted. All this is in the direction of the construction of a “tizhi model of education” (p.21). It also explains the fact that the construction of a model is none other than the systemization in conceptual maps. Again, on pg. 264: “…my conceptual model is more specific and adaptive than the starting point model. Because the starting point model of tizhi education consists of only two factors and three tizhi sub-education, if we put this model into practice, say, at the state level and school level, the reform will result only in two-factor reform, the education organization reform and the education regulation reform …” “Deduction” is understood as “…an inference in which I claim the conclusion follows necessarily from the premise” (Sun, p. 21), that is, “inference”. The inference is presumably tied to the abductive process10. As a matter of fact, in chapter nine regarding the conclusions, Sun states “The research reported here is the result of conducting a process of adduction at different levels of education …” (p.263). The term “dialectic”, after having been presented in the introduction, is never taken up again in the course of the treatment of the book; and it is not by chance to take up again the conception that the Chinese has dialectics in their interpretation of Marxism–Leninism (Nisbett). Where is the difference with the past? There is a very successful attempt to be able to see the future, with all the possible cognitive ties to western culture, but with an always attentive glance at the past. On the other hand, the cognitive style induced from “natural language” is almost always unknowingly (or sometimes knowingly) present. Education and school in a globalized world. By the end of the XX century and the beginning of the XXI century several International organization, such as OECD (Organization for Economic Co-operation and Development, 1999) and PISA (Programme for International Student Assessment) decided to monitor the results reached by the students in different disciplines in almost all countries of the world. The results of the PISA tell us that, of the students at the end of obligatory school (on average 10 years of study), at the beginning of the XXI century, 40% manage to solve only very simple tasks. “These students identify, with difficulty, the principle theme of a text and cannot manage to establish an evident connection between pieces of knowledge which are used in daily life.” (Bottani, p.379) At this point, wonders what the role of the school could be in a globalized world if students, who have attended 9 or 10 years of school, do not understand what they read. Therefore they are not able to know how to use the information contained in the text. “In general, the level of instruction of the new generations has improved for all, but the gap in the possibility of access to higher level forms of instruction between 10

Here, we could refer to abduction in Peirce’s sense, but presumably the inference used is none other than a fuzzy implication (Kosko, 1995; Spagnolo, 2003, 2005; Gras & Spagnolo, 2004).

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holders of diplomas of a technical–professional nature and holders of diplomas of general culture has remained intact.” (Bottani, p.380) Another disturbing fact is that a high percentage of 15 year old students in many countries does not have any desire to go to school, prefer, to stop attending it. Open educative problems 1. Students would like to be taken seriously by the teachers, even following courses which are difficult and structured in a rigorous manner; particularly mathematics and sciences. 2. The epistemological and psychological constructivism of the 1900s has offered the theoretical basis in founding scholastic learning (and not only) that mental categories reside in human activity and in biological organization has been acquired by and brought to socio-constructivist theories. The conclusion is that the teachers are no longer the only fonts of knowledge for the students. 3. Managing to reconcile “… two distinct universes: that of the students, who possess and use a range of different languages, and that of the teachers, to whom the task of imposing codified, stereotyped language is entrusted that still serves as a Measure to confirm the order of legitimate discussion, or rather the authorized discursive practice of the various systems of power. The generational exchange of the body of teachers will not automatically regulate this phase displacement, but it will take 20 or 30 years to accomplish it: in primis, because probably the difference between the linguistic behaviours of the teachers and those of the students could remain more or less unaltered, in that all discursive practices evolve and secondly, because the teachers, recruited and trained by scholastic institutions, cannot do other than use the codes of communication legitimated by the institutions and the strategies of socially recognized communication of the institutions of power which manage and maintain the school.” (Bottani, p.381) 4. Passage from a disciplinary canon from a medieval mould to a new canon which takes into account complex and multidimensional cognitive tools and also under the linguistic aspect. This could bring us to a redefinition of the architecture of scholastic systems that are currently based on 1800s.11 5. To what extent is the school project inspired by the enlightenment, or rather the desire to create a society of free and equal men due to the education extended to everyone still valid? 6. Can instruction of the young generations prolonged in time (no sooner than 30–35 years old can one get a PhD) allow a country to be competitive in the global market? Is this competition necessary? Is it necessary to think of other development models? 7. What has the managerial transformation12 of the school meant? Can scholastic autonomy, in an exacerbated regime, guarantees instruction for everyone, even for the weakest?

11

The organization of the public school had, in the 1800s, with the establishment of national states, the point of reference. The “religious” school, at least in old Europe, was transformed into the state public school. 12 In China, for several years, they have gone in this direction, see paragraph 4 (Sun, 2005).

Conclusions

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Conclusions The problem today is that in a “truly” globalized culture, whether economically or in its means of communication or in its cultural exchanges, it is necessary to keep the contributions of the different cultures in just consideration. “Complementarity” could be a solution which the man belonging to the “Earth” to survive, today more than ever, needs to know the different cultures in terms of ever more diversified “knowledge”13 which can allow a greater adaptation of the human species to future situations. The different approaches to research in the didactics of mathematics, to the different Philosophies of Education, I believe is a road that can absolutely be travelled over by our scientific community. What are the open problems of research? Surely, they are many and all are tied to the difficulty of being able to interpret the phenomena of teaching/learning. However, we can already identify some with regard to: 1. the theoretical reflections can be concerned with: a. a deeper study of the structure of natural language (also with the help of linguists); b. a deeper study of the use of epistemology and the history of mathematics for the interpretation of the phenomena of teaching/learning; c. a study of different tools of interpretation of these so complex phenomena both from the linguistic–communicative (semiotic) point of view and the logical–linguistic (fuzzy logic) one; d. a study of the relationship of neuro–physiological research and mathematics. 2. The applicative reflections can be concerned with: e. the study of particularly functional situations/problems to the problems of multiculturalness. In the sense, they can highlight the greatest number of solution strategies, reasoning diagrams, processes …; f. the qualitative and quantitative study of the socialization processes of the individual cognitive paths of the pupils (analysis of the validation phase of adidactic situations); g. the study of didactic innovations as sources and stimuli for the creation of didactic situations which are appropriate for multiculturalness.

13

The term “knowledge” is inserted here in its instrumental meaning.

Chapter 5

The Experimental Epistemology as a Tool to Observe and Preview Teaching/Learning Phenomena

5.1 The Experimental Context In previous chapters, we discussed some general considerations of epistemological nature, and in this chapter, we introduce two groups of investigations of experimental behaviors in the years and in the spirit of the experimental epistemology of mathematics. The first group refer to an experiment conducted with Palermo and Nanchino students of Superior Secondary School on the comparison of the schemes of reasoning regarding the logical–linguistic paradoxes in this two cultures. The analysis of two cases is introduced, then one of which had already been analyzed for about a year in 1984/85. The second in Palermo group starts investigations with reference to experimental behaviors in multicultural classes where Italian and Chinese from the primary school to the secondary superior students study together. The mathematical content defined for the experimentation has been related to a passage from the arithmetic thought to that of algebraic. The analysis of the results is discussed according to investigation types: quantitative and qualitative. The concerns of quantitative analysis refers to the jobs published in the Necklace "Studies in Compuational Intelligence" n.127 of the Springer (R.Gras-E.-Suzuki-F.Guillet-F-Spagnolo, 2008). The concerns of qualitative analysis makes reference to study of cases, single interviews, analysis of video, indicative semantic for the study of the phases –– the students' argumentative. The choice of the mathematical contents of the two groups of experience has not been casual, in both the cases the hypothesis of departure has always been the same one: The structure of composition of the language there written it favors through the phase of coding and decoding the acquisition of particular competences of area logical mathematics. In the following section, the passages are analyzed by the arithmetic thought to the algebraic thought in numerous experimental contexts, while section 5.2 are analyzes the aspect of logical argumentative in Italian and Chinese classes with study of cases.

5.1.1 Choice of the Problematic Situations in Accordance with the Theoretical Framework and the Hypotheses of Research As seen by the preceding chapters in the analysis of the difficulty theoretical context inside which our experimental investigation is inserted, the structuring of the experimental run of the work has owed take in account of manifold variable in game, F. Spagnolo and B. Di Paola (Eds.): European and Chinese Cognitive Styles, SCI 277, pp. 119–200. springerlink.com © Springer-Verlag Berlin Heidelberg 2010

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different points of view offered by the separate ones sprout of reflection discussed in precedence and necessary for one how much more possible corrected analysis of that that can present as a didactic situation multicultural reported to the algebra. Investigations give type historical epistemological on the Chinese mathematical tradition related to the algebra and to the study of the equations as heart of the discipline and generally to the Chinese culture (linguistic aspects, philosophical, etc.) we have drawn, even though in way certainly not exhaustive, some important observations on the possible one "development" cognitive of the algebraic language in the Italian and Chinese students attending the Italian school. To depart really from these reflections it is planned, therefore the experimental research with the objective to analyze in two typologies of students involved in some of the processes of reasoning shown in the different scholastic orders (primary school, inferior and superior secondary school) in relationship to a peculiar mathematical context what that algebraic and pre-algebraic and to the delicate phase of the passage from the arithmetic thought to that algebraic not necessarily formalized. The experimentations have been effected, in relationship to the declared purpose, in different institutes in Palermo; different for partner cultural context, family, typology of study and scholastic degree. The following chart brings, for every single experimental investigation the context, the objective and the sample of reference.1 Experimental analysis

Typology/Context

Aim

Target

Sudoku Magic Box

Situation a didactics, representative register type view, language natural.

Varying as unknown of position and relationshipfunctional inside a chart.1

95 students of the primary school in Palermo (age 7-10) + qualitative analysis with couples of students Chinese/Italian.

Fermat problem

Situation/problem, register of introduction: language na-turale, arithmetic/prealgebraic.

Verification of possible difficulties of the al-light ones in the passage from the numerical reasoning, arithmetician for attempts and errors, to that algebraic

140 Italian students attending the superior secondary school in Palermo (age 14-16).

1

Chinese students of the same age and degree of instruction, some of which not attending the Italian school.

In this chapter the schools considerate in the experimental work will be encoded as follows: 1. Istituto Pisco Pedagogico “Margherita”: IPPM; 2. ITC “Salvemini”: ITCS; 3. Circolo Didattico Ferrara: DDF; 4. Circolo Didattico G.Costa: CDC; 5. Istituto D’Arte di Piazza Turba: IAPT; 6. Liceo Scientifico “Galilei”: LSG. 2 We refer to the relation-functional aspect of the algebraic variable, (Malisani, 2006).

5.1

The Experimental Context

121 and pre-algebraic. Difference and analogies in the schemes of reasonchin brought ahead by the students of among-diction cultual italian and Chinese in the evocation of decisive strategies and generalization of arithmetic die pre-algebraic and/or in the phases of conceptualization and reasoning of these.

Questionnaire variable and parameter in different Semiotic contex

Questionnaire with questions to open answer and dam, regal-stro of presentation: natural language, prealgebraic, formalized algebraic, parametric.

Study of the relazionalefunctional aspect of the go-riabile in the problem-solving, relationship among with-text of introduce-zione and strategy of resolution. in aswithout of suitable mastery of the algebraic language.

Italian and Chinese students attending the superior secondary school in Palermo (age 17). IPPM and ITC "Salvemini"

The sequence

Situation/problem, register representtivo type view, natural language.

Phases of deducezione in the process of generalization: - 1. definition of a deep-mental algorithm; - 2. treatment and conversion algebraic

37 students of different etnic (4 Chinese and 33 Italians), age 13. IPPM

The grid of numbers

Situation/problem, register representtivo type view, natural language.

To mediate in a class multicultural of the School Inferior (age 10-13) with al-light possible Chinese algebraic knowledge on systems of

Couple of Chinese and Italian students

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5 The Experimental Epistemology as a Tool to Observe and Preview reference different and possible schemes of reasoning eterogeneic.

As said before the mathematical content and the didactic value of this are chosen not to purely investigate specifically on students attending institutes of scientific die, opting instead for an investigation more transversal through the involvement of students of a plurality of addresses scholastic character hips less "mathematical". The primary objective of the job is not in fact that to investigate the algebraic abilities formalized of the students on the contrary the phase of transition of these from the arithmetic thought, in contexts not standard and not necessarily algebraic formalized. The methodological choice has involved, however, some difficulties type organizational in the retrieval of the Chinese students. In total 270 students have been involved, around of which around 30 Chinese. The results here discussed are to consider themselves therefore only a spring board of throwing for future investigations more deepened in relationship to the sample and the scholastic context. Future investigations would be able in fact to directly be 3 conducted in China . For how much it specifically concerns really to the students of Chinese ethnic, to the goals of the experimentation, their products have been "classified" according to two macros different keys of reading: insertion in the Italian scholastic system and therefore possible preceding attending of the school of origin and linguistic ability for the writing ideographic and alphabetical (in the native or family scholastic tradition). The job of research here discussed is inserted, as confirmed more times, in a project of experimentation on the amplest teaching-learning of the mathematics, brought ahead by the G.R.I.M. of Palermo in the last 20 years and specifically turned, to the problem list study also treated in circle multicultural, considering a possible reading of the relationships mathematics-natural language-culture in a Chinese cultural optics. As says, the hypotheses of the job of research, you also introduce in precedence, you are been defined in relationship to a whole series of questions of research reported to the proposed mathematical content, analyzed by a didactic point of view for the different scholastic orders (the algebra and the complex phase of transition of this from the arithmetic through the primary and secondary school), and to a whole series of possible transversal reflections type epistemological-cultural on the Chinese and Italian students from us involved in the experimental investigation: The students, as are they activated and among them connected the conceptions of unknown and of relationship-functional? Does it occur the passage from a conception to the other? If yes, as does the phase of transition develop him? Is it possible to analyze in this sense of the differences of behavior between Italian students and reported Chinese to epistemological-cultural aspects? The relationship syntaxsemantics inside the algebraic code underlines different meanings for the two 3

A first work of research of the GRIM in Palermo has been conducted in this sense, in 2005, (Ajello, Spagnolo, Xiaogui, 2005; Spagnolo, Ajello, 2008).

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typologies of students? How is this used in the problem-solving? How does the process of translation happen from the algebraic language to that natural and vice versa? Which the schemes logical deduced uses you from the students culturally different in the process of test and generalization for the approach to the algebra in the passage from the arithmetic register? The hypotheses of research have been therefore you define in this way: H1: The differences and the verifiable analogies in the traditions Italian cultural and Chinese, find again him in the analysis of the schemes of reasoning showing in situations of teaching/learning of the algebra and in general of the mathematics. H2: In absence of a formalized algebraic context: - (H2.1): the students evoke decisive strategies of arithmetic die and/or "pre - algebraic", you direct, for the Chinese students, to the research of a fundamental algorithm and defined by an analogical thought recursive; to the deductive reasoning, for the Italian students. - (H2.2): The procedures in natural language and/or in arithmetic language for tests and errors, prevail as decisive strategies in absence of an appropriate mastery of the algebraic language. - (H2.3): In the phase of generalization, the Chinese students underline-no a reasoning "variational-procedural" that it conducts them to the definition of a destination-thought (what it loses meaning however unhooking itself from the concrete context). The process of generalization of the Italian students evolves him through the phases of reasoning, conjecture and demonstration, according to a deductive logic. - (H2.4): In a situation problem4 defined according to a register type view, the pupils they succeed in gathering more easily the aspects of "thing that varies", unknown and relationship-functional of the variable. The strategies of resolution hierarchizing in various conceptions can stop the development of it however. H3: In purely algebraic contexts, the Chinese students underline a good syntactic control of the algebraic formalism. H4: The structure of composition of the language written Chinese favors through the phase of coding and decoding the acquisition of particular aspects of the concepts of unknown, variable and parameter. The experimental investigation has been separated in three phases realized according to temporal scanning and didactic situations different and specifications to the plurality of interpretations that you/they can give him to the "to algebraically think" For how much it concerns to the approach methodological/experimental there it has referred to the Theory of the Situations of G. Brousseau (1997). For each of the six structure experimental activities and courses in class he is made therefore 4

The term situation problem refers to a situation of learning conceived so that the students cannot resolve the matter in examination for simple repetition or mere application of knowledge or acquired competences but such that it requires some formulation of hypothesis and conjectures. The students are appraised in their personal acquisitions. (D’Amore, 2000, p.285).

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to define a possible analysis a priori of the behaviors from the students; connected behaviors tightly to the typology of the define didactic situations, to the proposed mathematical content and to some aspects related to the cultural epistemology of the students. With the term "analysis a-priori", we intend the analysis of those that they can present as "representations epistemological", "representations historicalepistemological", "behaviors defined by hypothesis from the students", correct and not, for the resolution of the didactic situations experimentally defined. The cognitive runs of the algebra and the mathematical concepts of this as the study of the equations, becomes, as previously says also, keys of reading of the experimental contingencies and they are able, really in virtue of their nature, to make light on possible analogies and differences in the schemes of reasoning used by the students of different cultures. As brought in Spagnolo (1996), the previous analysis of a didactic situation allows in general then of "to individualize him/it "space of the events"5 on the particular didactic situation in comparison to the professional knowledge of the teacher researcher in a determined historical period; to individualize, through the space of the possible events, the "good problem"6 and therefore one "situation didactic fundamental" for the class of problems to which the didactic situation is; to individualize some variable of the situation problem and of the didactic variable7; to individualize some hypotheses of research in didactics type more general in comparison to those analyzable from a first analysis of the situation problem." The a-priori analysis represents therefore the element for this type of experimental investigation and it specifically keeps in mind some epistemology of the student and the spread out discipline with its history and its evolution. In the first phase of experimentation, chosen a situation a didactics8 particularly "family" to the Chinese cultural context (for form and decisive approach), he/she is wanted to analyze, in a representative register type view, the use from the Italian students and Chinese, even though in a first approssimation, of the conception of 5

For "space of the events" he intends the whole the possible correct decisive strategies and not, defined by hypothesis in a determined historical period from one determined community of teachers 6 The "good problem" it is that that, in comparison to the knowledge taken in examination, it allows the best formulation in ergonomic terms. 7 The "varying of the didactic situation" they are all the possible variable that intervene, the "varying didactic" they are those that allow a change of the behaviors of the students. The didactic variable are therefore an under together of the variable of the didactic situation. 8 We defines situation a didactics the part of the didactic situation in which the intention of the teacher is not explicit towards the students. The students know that the proposed problem has been select to make to acquire him new knowledge or ability and these they are justified from the inside logic of the situation. For observe this to know them they don't have to appeal to reasons for didactic nature. The teacher through a whole conditions that allows the students to appropriate some situation, allows a devolution of the situation. An action therefore through which to accept to the students the responsibility of learning (a didactics) or of a problem and ax himself the consequences of this transference. This last aspect properly differentiates the situation a didactics from that didactics. (Brousseau, 1981, 1997; Spagnolo, 1998).

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variable as unknown and relationship-functional. This first activity of research has been therefore finalized to a possible falsification or validation of the hypotheses H2.3, H2.4 and H4, reported to the cultural-linguistic context and to some typical procedures "algebraic" reported to the epistemology cultural and disciplinary Chinese. The strategies of resolution hypothesized come therefore "you support" from the historical context algebraic and pre-algebraic epistemological discussed in precedence and in literature. In accord with Ericsson et 1993 and Van Someren et 1994, with the finality to find again, in the phases argumentative it discussed from the students for the solution of the situation a didactics proposal and the sharing of this in class, possible procedures of solution "typical" of the cultures he/she took in examination, the problematic situation (proposed to 95 students of the primary school in Palermo9), has been followed by a mixed interview semi-structured to two couples of students (students Italian Chinese-students). Defined then the analysis a priori of the possible behaviors shown by the students in the resolution of the experimental situation realized, he is, using the software of statistic Chic inferential (Classification Hiérarchique Implicative et Cohésitive), to quantitatively analyze the contingencies experiment them. The statistic indexes used in this sense have been the index of implication of R. Gras and the index of similarity of Lermann (Gras, 1979; Lermann, 1981, 2004)10. In the second phase, with the objective to underline possible analogies and verifiable differences in the two "cultural traditions" in the schemes of reasoning used in formalized contexts or algebraically seed-formalized, they are submitted attending to around 140 Italian students the superior secondary school in Palermo in different scholastic addresses, two tools of investigation created ad hoc11, through which then, in a second experimentation, is tried to define a job of critical comparison (qualitative and quantitative) with Chinese students of the same age and degree of education, some of which not attending the Italian school12. The experimental job, in this phase has tried to specifically falsify the hypotheses H1, H2.1, H2.2, H2.3 and H3 related to the abilities manipulative of the students on 9

The experimental job has taken I also sprout from a job of research realized for the thesis of 2006 dott.ssa Ferdico (http://dipmat.math.unipa.it/~grim/), Di Paola, 2007; Di PaolaSpagnolo, 2008. 10 In the following paragraph, the statistic indexes are brought used in the quantitative analysis of the picked data and a brief description of these in relationship to the statistic implicative. 11 The tools of investigation used have also been realized contextually in relationship to jobs of research brought ahead by the GRIM on the same matter. The first one specifically refers to a research conducted on the students of Palermo of three different addresses of superior secondary school and in a first year of university (Di Paola, 2006). The second makes reference to an article published in 2007 and submitted to a sample of investigation of 42 Italian students of the superior secondary school (Di Paola, Manno, Scimone, Sortino, Spagnolo, 2007). 12 In total, in this phase of job, 16 Chinese students have been involved. To 7 of these (two of which external to the Italian scholastic system) the first tool of investigation has been administered. To the remainders 9 (three of which external to the Italian scholastic system) it is administered to the second tool of investigation.

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the formalized algebraic language (and on use that they do for the procedure of resolution of the various questions of it to them proposed), to their ability of abstraction, to the relationship syntax-semantics (referent-sign-sense and in the meaning of the triangle of Frege as I compare among zeichen-bedeutung-sinn) inside the algebraic code and therefore transversally, to the application and the control of the algebraic and pre-algebraic thought on more environments of presentation (numerical, geometric, algebraic pure). Also defined in this case, the analysis a priori of the possible behaviors of the students, he is made to quantitatively analyze the experimental contingencies putting in evidence as additional variable (Gras et alii, 2008)13, those that, as says, you/they could be, varying discriminated among the two cultures for the algebraic thought (Natural language, formalized language, aspect report her functional of the variable in reference to the idea of unknown and thing that varies etc.). In relationship to the results of the first two investigations he/she is made then necessary a third phase of the job of research. In this last phase, I/you/they have been submitted to Italian students and Chinese of the inferior and superior secondary school, two problematic situations of pre-algebraic die that through activity of autonomous job (the first one) and mixed group (Chinese-Italian students together) they were able take on a shape not only himself/herself/themselves meaningful for the proposed disciplinary objective, but also able "to underline" some aspects key of the phase of mediation of the to know mathematical among the students culturally different involved different (in total 37 students). Phase of mediation that is tried to study (qualitatively and quantitatively) through the analysis of some of the meaningful phases of Metacognition shown in the phase of validation of the job. This third phase wants to be therefore a further moment of verification of the four hypotheses of research. H1, H2, H3 AND H4. Contextual to the definition and management of the experimental run, he is made to pick up even though in non formal way (seen clean refusal of the interviewed ones) a series of transversal reflections on the phenomenon of teaching/learning in situations of multiculturalism, observations furnished by parents of Chinese students and Italian teachers that teach or you/they have taught in fine-cultural classes in the primary and secondary school of first and according to degree in Palermo. This last investigation, even though formally brought not in this job of thesis, is shown very useful for a general organization of the treated didactic problem list and it owes therefore to consider himself/herself/themselves a spring board of throwing for future investigations. Does the palermitan teacher manifest a knowledge of those that you/they can present as didactic problem list for the teaching to Chinese students?

13

We reports in the specific one to the jobs of research published in the volume in the third section "A methodological answer in various application frameworks", Malisani, Scimone, Spagnolo, pp. 247-276, 2008.

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Can The presentation of a research of this type help them in the definition of didactic practices and innovative methodologies for the integration of the knowledge?

5.1.2 The First Experimental Investigation: "Sudoku Magic Box" The first experimental phase has been defined, as says precedent-mind, beginning from a situation to didactics14 (reported below) built beginning from the rules of the sudoku and the magic square subsequently simplified for the primary school through a registered representative, pictographic, with the use of some images of animals. The Principal objective of this first experimental investigation has been to falsify, even though in the first approximation, the conception of variable as unknown of position and relationship function inside a chart, regulated by laws of composition it expressed in natural language. This first activity of research has been therefore finalized, as says, to a possible validation or falsification of the hypotheses: H2.3: In the phase of generalization, the Chinese students underline a reasoning "variational-procedural" that it conducts them to the definition of a destination thought (what it loses meaning however unhooking himself/herself/itself from the concrete context). The process of generalization of the Italian students evolves him through the phases of reasoning, conjecture and demonstration, according to a deductive logic. H2.4: In a situation problem, defined according to a registered type of view, the pupils succeed in gathering more easily the aspects of "thing that varies", unknown and relationship-functional of the variable. The strategies of resolution, hierarchizing her various conceptions, you/they can stop the development of it however. H4: The structure and composition of the language written Chinese favors through the phase of coding and decoding the acquisition of particular aspects of the concepts of unknown, variable and parameter. Fig. 1.

14

You define situation to didactics the part of the didactic situation in which the intention of the teacher is not explicit towards the students. The students know that the proposed problem has been for him to acquire him new knowledge or ability, and these they are justified from the inside logic of the situation. For bulding this to know them they don't have to appeal to reasons for didactic nature. The teacher through a whole condition that allows the students to appropriate some situation, allows a devolution of the situation. An action, therefore, through which to accept the students the responsibility of learning (a didactics) or of a problem and ax himself the consequences of this transference. This last aspect properly differentiates the situation a didactics from that didactics (Brousseau, 1981, 1997; Spagnolo, 1998, 2009).

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“Sudoku Magic Box” Animals to insert: 1. flashes 2. dog 3. cat 4. lizard 5. fly 6. cow 7. fish 8. spider 9. mouse . Rules: The symbol - flashes - Dog - Cat - Cat - Lizard - Spider

means “to eat”. fish ; Cat; Fish; Mouse; Fly fly

1) You need to insert in the chart all the animals; 2) every animal is able you insert once only. Fig. 1. (Continued)

The experimentation has been conducted on 95 students of a primary school in Palermo (The II, III, and IV classes) and he/she has been accompanied by a qualitative investigator with an interview semi-strutturata to a couple of students (Italian and Chinese students) verbalized the solution of the "game" and the strategies of resolution (Ericsson et alii, 1993 and Van Someren et alii, 1994). He/she must be specified immediately before the administration of the rules the chart of the, to didactics proposal, to falsify, the possible meaningful variable of the didactic situation, if the Chinese students were able to read and understand the text furnished in Italian language. All Chinese students have underlined good linguist abilities i.e., the understanding on the expression of the rules of the "game". The same has been inserted in the Italian scholastic context since the first year of school and don't result to have frequented any course of studies in their country of origin. Here in the definition of the hypothesis of experimental research in H4, we hold both necessary to specify besides the good abilities of writing ideographs in all the Chinese children involved in the experimentation. Ability of this is verified in entry for all the classes of the primary school in which the experimentation has been developed. The ability of writing has been considered for the analysis of the experimental contingencies, as one of the meaningful variable for the

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interpretation of the schemes of reasoning of the Chinese students, for the job of research on the algebraic thought. The ability to read and write the Chinese ideographic language is in evidence from all the students of Chinese ethnic who are experimentally involved, really in operation of the peculiar linguistic structure for a pre-algebraic thought (recognition of the "radical" as possibility of reading and writing in the relationship part/all "thing that varies" inside a single character, definition of a parametric structure for the generalization of a character to level semantic/syntactic) has been considered in this sense as one of the discriminating elements of the two typologies of culturally, different students. For each of the experiences of research in the various classes coin-turned in the investigation, the principal three phases of job are followed substantially: Phase I: Explanation of the rules and presentation of the animals. In this phase, he/she is made use, besides the oral verbalization, of a placard showing the scheme "prey-raider" and is the symbols introduced which is representative of the action "to eat." Phase II: Administration of the situation a didactics and observation of the strategies of solution. Divide the couples in the class into small subgroups of three students, his/her, you/they have tried to resolve the situation to didactics, verbalizing among them, the schemes of reasoning used. A mobile television camera has been kept to taken us the whole experimental situation. Contextually, through a grade of evaluation is brought for every group of students, some particularly meaningful and reported semantic indicators have been transcribed, in a first approximation to the analysis a priori built in precedence and brought the following. Table 1. Grade of evaluation used for the analysis of the semantic indicators

Gesture

Language

Description

Locution Illocution Epistemological and cultural references Phase III: Verbalization of the strategies of resolution. Finished the phase of "game" and validated, for all the students, to complete the execution of the chart Sudoku Magic Playpen and his oneness of solution; you find particular differences in the used decisive scheme and discussed inside some groups of Chinese students (attending the D.D.S. "G. Costa", IV class) in comparison to those used by all the Italian children of the same class, he is made to isolate a couple of Chinese students and a couple of Italian students of the same class with the objective to make to verbalize them and to mediate the strategies resolute different. The seed-structured interview has been conducted through the questions, stimulus brought following and transcribed to handle the most salient phases.

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Some of the meaningful sentences of the teacher interviewer and some students been transcribed in the underlined cursive character. 1. What have you done for reaching the solution? Would you know how to explain him/it to your companion? 2. Which rules you have followed for positioning the various animals? Have you effected some changes of position for the various animals? 3. Have you noticed some most important rules of others? If yes, have you used them? 4. Can you insert the papers you imagine in a different way to get the three solutions for the chart? The questions have only been mailed to the students as I stimulate to the discussion and therefore impute to subsequently examine post-hisses analogies and differences in the conception of the cell/animals of the chart in terms of variable as unknown (of position in the a beautiful one) and relationship functional (in relationship to the rules of the situation of learning and the different positions of the animals in the chart). How are these two conceptions activated in the Chinese and Italian students? How is it underlined (if it is underlined) the passage from one to the other in the situation of problematic proposal? Do referable conceptual differences exist to cultural aspects? The teacher's role as an experimenter has been alone as that of an external observer and facilitator. 5.1.2.1 A Priori Analysis We hypothesize that the pupils can apply to one or more than the following decisive strategies: S1. It interprets the blank cells of the chart following a correspondence "biunivoque"in relationship to the "hostile Animals". It interprets therefore the symbol in a different meaning from that proposal (x/y and z/x) and it proceeds for attempts and errors departing from the first animal quoted in the chart preypredatory: the Cat; S2. He/she reads the rules of the "game" and it defines the animal more quoted between the preys and the raiders: the Cat. It positions therefore the paper image Cat in 2A; S3. It recognizes in the cell 1B, a cell "meaningful" and without the appeal to the reading of the rules it positions the paper image of the Mouse. It proceeds for exclusion on the chart prey-raider; S4. It recognizes in the animal Cow, a non-influential animal (an animal Joker) and it decides to position him/it for last; S5. It recognizes in the animal Cow, a non-influential animal (an animal Joker) and paper positions at random as before it in a select cell;

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S6. It cleans separation between preys and raiders. Reasoning type combinatory on the possible relationships among the preys; S7. It cleans separation between preys and raiders. Reasoning type combinatory on the possible relationships among the raiders; S8. He/she reads the chart of the preys and the raiders using the symbol to his/her inside in an unitary vision of the rules; S9. It inserts at random the cardboards showing the various animals. It doesn't show any reasoning of solution; S10. Strategy-type probabilistic on the variable of position and choice of the position of a paper image in relationship to the rules of the "game" read in an unitary vision; S11. In accordance with the strategy S1, the paper image of the Cow is only inserted at the end of the "game." For how much it concerns then to the behaviors related to the argumentative hypnotized phases in the phase of verbalization inside the various groups in the third phase and in the validation of the situation to didactics for the whole class the varying behaviors are hypothesized as follows: B1. It justifies the personal reasoning of solution through the analysis of the various local rules for every single animal in the paper; B2. It justifies the personal reasoning of solution through the analysis of the various rules read in way holistic for every single animal paper; B3. It justifies not the personal reasoning of solution through a reading of the rules, but with concrete examples on particular positions of insert animal in the papers; B4. He/she reads the rules of the "game" in a serial way and subsequently it connects her among them, it deduces in operation of these local global considerations; B5. Law justifies, even though with possible errors, the solution, departing from an unification of the rules for every animal; B6. You position the various papers you imagine; it proceeds in the oral reasoning in way "algorithmic" not considering, the cases previously developed in the following footsteps; B7. It deduces the insertion of the various paper images reconsidering every step of the various insertions and justifying her various papers imagine you put B8. To feel the oneness of the solution again applies to the reading of the rules of the "game" and it justifies the various insertions in serial manner B9. To feel the oneness of the solution underlines a reasoning type procedural algorithm. 5.1.2.2 Quantitative and Qualitative Analyses of the Data Following tables are the graphs gotten by the realization of a Excel-built matrix according to the underlying chart or transposes in the case of the similarity of Lermann:

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5 The Experimental Epistemology as a Tool to Observe and Preview Table 2. Example of matrix in Excel realized previously for the analysis

Student 1 Student 2 Student 3 Student 4 …

Strategy S1

Strategy S2

….

Argumentatio

B1

Argumentatio B2

0

1

…

1

0

1

0

…

0

1

1

1

…

1

1

0

0

…

0

0

…

…

…

…

…

From the graph of similarity, two meaningful results emerge, the first one is related to the Italian students and he/she understands all the scholastic orders of the primary school (from II to IV classes), the second, clearly separated by this last, it brings the similarities of the behaviors of the involved Chinese students instead. The first general consideration is therefore that for which, also being the sample of Chinese experimentation, small in comparison to the Italian counterpart (only 13 students on 95), it defines in an evident way, to our opinion, a class of behaviors distinguished by that typically used by the Italian children in the various involved schools, behaviors and schemes of reasoning that besides you/they can consider, for both the cultures represented in the experimental investigation, stable for almost all the course of study of the primary school, fairly finding again himself/herself/itself you distribute in everybody and for the classes of the elementary school. To define better the meaningful variable that has experimentally been underlined among the behaviors of the involved students, brings the following graph implicative with a level of significativity for 99% implication. Implicative graph From the implicative graph, we observe interesting connections among some of the anticipated strategies in the analysis to both for how much it specifically concerns to the research of the solution of the situation a didactics proposal that for the argumentative phase of this. We bring through the cloud yellow, the grouping that represents, in percentage, some of the typical strategies of solution for the involved Chinese students: (S4, S8, S3, S11, B5), (S8, B2), (B6, S10), (B9, S4) among them you implicate between 95 and 99%. The strategies inserted in the celestial clouds represent the behaviors put in evidence among Italians instead (S1, S9), (B7, S3), (B1, B8). The data, picked relatively to this first phase of experimentation, seem to confirm a type of pragmatic, concrete, behavior from the Chinese students shown by a reasoning-type algorithmic procedure tightly connected to the holistic thought on the coding and decoding of the proposed chart and the rules of "game" you introduce (evident in the use of the strategies S3 and S10).

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B1

B9

S1

S9

B8

S4

B5

B2

S2

B6

S8

S6

B3

S10

B4

B7

S3

S11

Fig. 2. Implicative graph situation didactics situation Sudoku Magic Box

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For the Italian students, it seems to be able to confirm, even though in first approximation, a thought type logical deductive used on the reading of the rules of the game, you analyze in serial way in relationship to each of the paper–animal ones and therefore to the positioning of the same inside the chart. This scheme of reasoning still results more evident in the research of the oneness of the solution of the proposed situation. If for a certain verse in fact, during the phase of verbalization inside the various groups of students, the Italian students have manifested, for the proposed purpose, a continuous need to bring back to the reading of the rules of the "game" and to a continuous verification of group of the choices sorts in precedence in the various insertions of positioning of the papers (B7, B8), the Chinese children have manifested a certain difficulty in produce a reasoning-type argumentative that was not typical of constructiveprocedural (B6, B9) on the chart. They have then justified among them the certainty of the oneness of the solution unifying her/it with the "correctness" of this. Some examples of reasoning verified during the interview, following semistructured to the situation a didactics they have been to this intention: "I am sure, I have always followed the same thought… as a chain of animals all united ones among them"; "This is the solution of the game, cannot be us of it another"). The strategies of solution observed in the class in the Italian various groups of job and Chinese strongly seem different in relationship to the passage from the local thought for the various paper–animal (thought that all the children have certainly brought ahead in the "game") to that global, necessary for the correct compilation of the chart. The Chinese students, seem to underline a holistic type thought report, her functional on the variability of compilation of the single cells. These are, in fact, shown more uninhibited in to parallel check their reasoning type local on the single variability of the position of the paper image (it same variable in relationship to the rules proposed among prey-raider) with the definition of a general decisive procedure able to report all the present cells. Unlike the Italian students, none of the 13 Chinese children have in fact reasoned locally for tests and errors. All, in a more correct way have looked for, in the phase of conjecture, to define a scheme of relationship among preys and raiders underlined, even though in the first approximation also on the chart their proposal (S8). You phase of conjecture of the Italian students has almost been common for the totality on a reasoning for tests and errors on the single variable (cells, animal, papers) you organize on logical sequences. The scheme of reasoning observed in the class in the various groups of "game" of the Chinese students subsequently confirmed and strongly in the comparison realized with the couple of children interviewed by us, it seems near even though with the due differences, to that from them used for the realization and the reading of the formal diagrammatic writing. The Chinese students also, in this case, in fact, are metabolizing the concept of "variability" as initial relationship among "expressions" (paper-animals) different and subsequently as "expression" dynamics reported through one "formula" to others "expressions" also the dynamics, gather according to an approach type procedure the idea of variable as one orchestrate for individualizing and to compare relationships and structures connected among them.

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As the preceding chapter also tells, in the Chinese language writing every character shows a plurality of interpretation to its inside; possible different readings that come out of the analysis of the existing functional relationships inside a composed character, among the various parts that compose it and the text in which it can be inserted.

Fig. 3. Scheme showing a possible thought report functional on the chart, Sudoku Magic Playpen

The strategy of insertion of the "paper Cow", has been then the element of departure for the third experimental phase of the situation "Sudoku Magic Playpen". This in fact has been the first consideration debated among the couples of Chinese and Italian students attending the IV year of the primary school "G. Costa" from interviewed. The justification brought by the Italian couple on their use of the strategy S5 has been so expressive: "The Cow is not hostile of any animal and therefore could be where I wanted. Then I have realized, also Mark, that the Cow could not be here (he points out the position 2A) because there was the cat and we have moved her… at the end she is here (he points out the position 2C)" (Strategy S5) The mediation of a different strategy of solution, discussed with the Chinese students, and therefore the verbalization of their strategy of solution (S4), has brought the Italian children to the individualization of a possible strategy "faster and more profit." As underlined by the words of one of the two Italian students: "there we could made to forget some Cow and to put it at the end"; "the other boxes, the other animals follow some rules, this no". (a passage is observed by the strategy S5 to that S4). Departing from these considerations, the description and the mediate discussion of different schemes of reasoning (algorithmic and logical deductive with a reasoning you global type on the insertion of the various paper-animal) put in evidence by the two couples of students, it has allowed us to discuss together with them, possible different decisive interpretations in the approach to the chart. If from a side, in fact, every single cell can be considered, as variable in the meaning of "unknown" (potentially all the images can be placed in that cell, without any limitation: S5, S9, B4) from the other, as said, the choice of the correct image comes out of the possibility to read the binomial cell-animal in a

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relationship type function through a global reading of the rules of the situation to didactics and of the position of the considered cell. At the end of the run of resolution, this second strategy is revealed to the eyes of all the students as a winning strategy. The experimental contingencies have, in fact, confirmed a good boldness from almost all the students in the passage from an aspect of the variable to the other, and also thanks to the structuring of the situation problem defined according to a register type view. A few cases of failure seem to derive from a fixity of thought (type local, expressed by the strategy S9) that has stopped the evolution of it. The hypotheses of research defined for this didactic situation therefore seem to be validated, though partly. The situation a-didactic it doesn't have in fact permission to specifically investigate on the aspect of the generalization, form typical of the algebraic thought. In this sense, what has been debated by his/her children has only been a possible definition of a general algorithm of resolution for all the images of the animals introduced in the situation of learning. The other meaningful aspect, discussed in the previous chapters in discovery to experimental level of the observation of the phase of verbalization of the Chinese students for the reasoning of the correctness of the solution is the "game", that he/she has been able to notice in all the cases, the continuous need from the Chinese students to refer, in the phase argumentative of compilation of the chart, to the process of repetition and therefore to the memorization of the rules of the "game" (strategy B2). The strategy underlined, as previously, stated is within the research, also from students of the Chinese culture, attending orders scholastic superior, experimentally involved by us. In accord with Marton, "In the process of repetition, it is not to simple repetition. Because each time The repeats, The would have burdens new conceives of understanding, that is to say The Khans understand better." (Marton, Dell’Alba and Tse, 1996, p. 81). The same behavior doesn't find again in the students of Palermo. This aspect results, is our opinion, therefore to a cultural element strongly for the phase of reasoning (and not only) and it needs investigations more specifically deepened revolts to the phase of acquisition of the algebraic thought. 5.1.2.3 Second Experimental Investigation: "The Problem of Fermat", "Varying Questionnaire and Parameter in Different Semiotic Context " In relationship to the results of the first experimental investigation, it is made necessary in more aspects, as stated, a second phase of research. As confirmed in the beginning of this chapter, with the objective to underline possible analogies and verifiable differences in the "traditions cultural" Italian and Chinese in the schemes of reasoning used in way that specifies in formalized contexts or algebraically seed-formalized, they are submitted to around 140 Italian students attending the superior secondary school in Palermo in different scholastic addresses, two tools of investigation created to hoc (brought following), through which then, in a second moment, he/she is tried to define a job of critical

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comparison (qualitative and quantitative) with the Chinese students of the same age and degree of education, some of whom are not attending the Italian school. In this sense, the choice to involve the two Chinese boys not attending the Italian scholastic context has been dictated by the possibility to falsify possible differences in the behavior cognitive of the free students from every type of contract didactic scholastic-cultural Italian and to compare them with peer of Italian and Chinese nationality attending the palermitan school. The experimental job, in this phase of research that it has tried to specifically falsify the hypotheses H1, H2.1, H2.2, H2.3 and H3 related to the abilities manipulative of the students on the formalized algebraic language (and use that they do for the procedure of resolution of it), to their ability of abstraction, to the relationship syntaxsemantics (sign-referent-sense) inside the algebraic code and therefore transversally, to the application and the control of the algebraic and pre-algebraic thought on more environments of presentation (numerical, geometric, algebraic). “Fermat problem”

5 = 12 + 2 2 Look at these numerical example: 13 = 2 2 + 3 2

17 = 12 + 4 2 As you know, 5, 13 and 17 are first numbers. According to you it is possible to write all the first numbers (except 2) in this form? It deduces your answer. Are you able to find a general expression of these numbers through a formula? It deduces your answer. “questionnaire variable and parameter in different semiotic contexts” QUESTIONAIRE 0 Exercise 1: In an field, in one year, 10 kgs of lemons are produced. to) Thing translates the chart under brought? b) According to you, in it, must thing be replaced with the place of the symbol "* "? 12 month

24 month

10 Kg.

*

Exercise 2: You read the following chart:

1 year 10 kg.

2 years 20 kg.

3 years 30 kg.

P = Production What does P=10t Kg represent the formula? Fig. 4.

5 years 50 kg.

t years 10 kg.

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QUESTIONAIRE 1 Exercise 3: In a triangle, an angle is of 57° while the difference of the others two are of 83°. It finds the ampleness of the three angles. N.B. he/she Remembers that the sum of the inside angles of a triangle is 180° Exercise 4: The perimeter of the triangle ABC brought nearby is 180 cms. In general AC measures 20 cms while BC is smaller of the general AB of 46 cms.. it Determines the sides of the triangle ABC. Exercise 5: Two friends, Giacomo and Andrew, met in school to exchange their CDs; Giacomo said to Andrew: >. Andrew answers to the friend saying: . How much CD had each one? Exercise 6: To find the ages of the two Red brothers: Charles and Mario, knowing that: - adding five to the age of Mario gets a quantity equal to the triple one of the difference between the age of Charles and five; - adding five to the age of Charles and subtrahend five to the age of Mario, the two quantities so gotten they are the same. Exercise 7:

⎧x

y

+ =2 Solve this equation system: ⎪ ⎨2 3 ⎪⎩ x + y = 2

Exercise 8: 12 − 3x ⎧ 2 ⎨ ⎪⎩ y = 2 − x

Solve this equation system: ⎪ y =

QUESTIONAIRE 2 Exercise 9: The exercises N.3 and N.4 have the same decisive method? Do they synthesize the same matter? It motivates the answer. Exercise 10: The exercises N.5 and N.6 have the same decisive method? Do they synthesize the same matter? It motivates the answer. Exercise 11: The exercises N.7 and N.8 have the same decisive method? Do they synthesize the same matter? It motivates the answer. Exercise 12: You succeed in describing some problematic situation using the following parametric system Fig. 4. (Continued)

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⎧x = a ⎪ ⎨y − z = b ⎪x + y + z = c ⎩ QUESTIONAIRE 3 Exercise 13:

⎧x 2 = Solve this equation system: ⎪ ⎨y 3 ⎪2 x + 2 y = 80 ⎩ Exercise 14: The perimeter of a rectangle is 80 dm. and the relationship among the two sides is

2 . Calculates the measures of the sides and the area of the rectangle. 3

Exercise 15: Determines the ages of Giovanna and Licia knowing that: 1. the relationship among their ages is

2 ; 3

2. the double sum of the respective ages is equal to 80 years. Exercise 16:

⎧x

According to you, do the sistem reported below, ⎪ ⎨y

=a

is a modeliza-

⎪2( x + y ) = b ⎩

tion of the exercise N.13, N.14 and N.15? Argument your answer. Fig. 4. (Continued)

The defined hypotheses are articulate as follows:

H1: The differences and the verifiable analogies in the traditions Italian cultuali and Chinese, find again him in the analysis of the schemes of reasoning evidenziabili in situations of teaching/learning of the algebra and in general of the mathematics. H2.1: the students evoke decisive strategies of arithmetic die and/or "pre-algebraic", you direct, for the Chinese students, to the research of a fundamental algorithm and defined by an analogical thought recursive, to the deductive reasoning, for the Italian students. H2.2: The procedures in natural language and/or in arithmetic language for tests and errors, prevail as decisive strategies in absence of an appropriate mastery of the algebraic language.

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H2.3: In the phase of generalization, the Chinese students underline a reasoning "variational-procedural" that it conducts them to the definition of a destination-thought (what it loses meaning however unhooking himself/herself/itself from the concrete context). The process of generalization of the Italian students evolves him through the phases of reasoning, conjecture and demonstration, according to a deductive logic. H3: In purely algebraic contexts, the Chinese students underline a good syntactic control of the algebraic formalism. The experimental investigation on the boys of Italian behavior together with other researchers of the group of research of the G.R.I.M. of Palermo in the academic year 2006/2007 have been brought ahead in different public institutes of superior secondary education, different for typology of study and number of times of lesson of Mathematics and Sciences. For how much it specifically concerns to the situation problem (D’Amore, 2000, p.258) related to the "problem of Fermat" the students of the first ones, seconds and third classes of the Institute “Galieo Galilei” have been involved (High Scientific school LSG), Institute “Psico Socio Pedagogico Finocchiaro Aprile", those of the Institute "E. Medi" (Technical Professional) and in a second moment, the students of the Institute IPPM. The second experimental investigation ("varying questionnaire and parameter in different semiotic context") has been brought then before near the fourth grade classes of the “IAPT, ITCS, IPPM, LSG of Palermo.” As reported many times, the choice is to purely investigate not only students attending institutes of die scientific, but also opting therefore for an investigation more general through the involvement of addresses of study to character less "scientific", he/she has been dictated by the same mathematical content and his/her didactic value, necessarily sees also the significance of the possible results in the definition of the phase of transition from the arithmetic thought to that of algebraic in contexts algebraically formalized. This type of methodological choice has allowed us to analyze in parallel the different approaches to the algebra in the various didactic contexts, besides among them very different. The choice of the classes has been effected to the purpose to falsify a difference of verbalization among the students, with the purpose to appraise if a determined behavior has the tendency to disappear or to persist at the time of studies. With such purpose, we bring in the text (in underlined cursive character) some of the meaningful sentences that is found again him in the protocols of the students.

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Every student has been able to understand the natural language of expression of both the tools of investigation and the mathematical objective of solution of these. As it regards that the Chinese students not attending the Italian scholastic system, contacted in a second moment of experimentation for the investigation of research, must be specified since immediately their discreet linguistic ability both on the understanding and on the expression of the express text in Italian for the formulation of the questions of the two tools of investigation. The translation of the texts in Chinese language, realizes however, they are not made necessary. The experience of research has followed therefore two principal phases: Phase I: Administration and analysis of the results of the problematic situation "the problem of Fermat." The questionnaire (brought in precedence) that it has been distributed in the three classes involved together to a table with the first 500 numbers and a table with the first 500 square numbers. The time of disposition of the students for the solution of the question has been fixed as 120 minutes of autonomous job. The choice of the tool of investigation, particularly meaningful for form and content15 that he/she has been dictated in relationship to the possibility to put in evidence, through the phases verifiable argomentative in writing on the protocols of the boys, the way according to which students of different age and cultural tradition can be able to algebraically formalize an express problem of theory of the numbers in natural language. In accord with Kieran and Chalouh (1983, p.179), the question is, in this sense consider him a good milieu of pre-algebraic study realized through a context numerical arithmetician. The resolution of the problematic situation is in fact able to underline only not possible difficulties of the students in the passage from the numerical reasoning, arithmetician for attempts and errors, to that algebraic and pre-algebraic (and therefore to underline if the notion of variable as "unknown" and "thing that varies" interferes or less with the interpretation of the relational-functional aspect of this) but to show then, its our opinion, possible differences and analogies in the schemes of reasoning brought ahead by the students of tradition Italian culture and Chinese in the evocation of decisive strategies of arithmetic die pre-algebraic and/or in the phases of conceptualization and reasoning of these. The phase of "generalization" required by the question in natural language it is able in fact to furnish us a key of reading more deepened for the process of reasoning, conjecture and "demonstration" used by the students in the passage to the symbolic language. 15

The first formulation of the theorem finds again him in a letter of Pierre de Fermat (16011665) to Merin Mersenne (1588-1648) December 25 th 1640. After the formulation of Fermat, the first demonstration of the theorem goes up again to Leonhard Euler (17071783). Other demonstration finds again him in Carl Friedrich Gauss (1777-1855) in its Disquisitiones Arithmeticae (1801).

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The considerations that we discuss in this chapter therefore take in examination through the experimental results, the reflections that we have discussed to level historical-epistemological in the preceding chapters. The comparison of the behaviors of the Italian students attending with two uprooted boys of Chinese etnia from the didactic contract of the italian16, cultural tradition, has furnished us further to sprout out reflection on the possibility to find again and to discuss stable schemes of reasoning reported to the culture of origin and not mediated by the teacher. In accord with Hanna (1990) the questions that we have therefore places and that they also represent a spring board of throwing for future experimental investigations on the same matter of research that they have been therefore: "What is the personal natures of proof17 " "Or why macaws students' personal justifications different from the paradigmatic mathematical proof? ", "What might warranting mean in classroom practice? ", "It is possible to give to-cultural "definition" of argumentation and proving? ". To be specific one therefore can have a particular problematic situation as that expressed by a problem of theory of the numbers in an aritmetic/pre-algebraic context be an milieu stimulus to answer to these questions? II phase: Administration and analysis of the results of the "varying questionnaire and parameter in different semiotic context" In relationship to the contingencies theoritial/experimental discussed in literature on the didactic problem list of the passage from the arithmetic thought to that algebraic in environments not multicultural, initially fixed the attention on those that could present for the Italian students, naïve knowledges to the algebraic and pre-algebraic thought and held these necessary conditions for a first approach to the algebraic thought, he/she has contextually decided, of "to compare her" with those possibly underlined by the Chinese boys. In this sense, after a first research on the italian18 students, the investigation is extended palermitan to scholastic contexts multiculturali of the IPPM and of the ITCS that they introduced students of Chinese etnia to their inside (9 boys of Chinese etnia have been involved distinguished by those participants to the experimentation of the The phase). Seen the objective of the research, in the experimental job there is particularly reported to a first concept of "function" or "functional relationship", concept to 16

Seven Chinese pupils have been examined that frequent the first three classes of the institute Partner pedagogic Psico "Margherita" of Palermo. 17 Intending, specifically for the proposed question, the term proof as it verifies, test and not formal demonstration. In accord with Cooney et alii, 1996; Hanna, 1990, 2001). 18 On the research experimental behavior together to other researchers of the group of research of the G.R.I.M. of Palermo in the academic year 2007/2008 have been published an article of research on the notebooks of research of the G.R.I.M. (Di Paola, 2007). The experimental investigation here discussed takes I sprout from this job of research. Experience is developed in collaboration with the Prof. Messina (scientific High school Galileo Galilei), the Prof. Catania (IAPT), the Prof. Scillieri (Commercial Technical Institute Slavemini).

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consider itself, as previously stated also, necessary condition to a first approach aware to the pre-algebraic thought and to the algebraic language. The concept of function or functional relationship he/she has been from us intended, in line with the first experimentation (developed in a context of different presentation by this), in terms of correspondence or at least ability of "to know how to read" a chart putting in evidence a lacking datum in it. Operation cognitive to this is validate through a first reading of the same chart and a definition of the initial relationships among the numbers included in the following and it Metacognition on the dynamic quantities reported through a formula subtended to the same chart. (Malisani, 2006). With this finality, the tool of investigation used has been structured in three different parts (questionnaires) among them complementary and to consider itself, in a first approximation, exhaustive for the proposed investigation. In the first preliminary questionnaire (what we have called questionnaire 0) we have asked students to reflect on two express problematic situations in natural language that you/they introduced in a chart of values to their inside. The solution of the proposed questions asked for the ability to report a quantity that varied in discreet way formalizing the gotten result. The thought activated by the students in the proposed assignment, studied to back in relationship to the behavior adopted by the same on the remainder of exercises, can be defined him therefore as a thought type pre-functional that does reference to a simultaneously think on whole families of numbers rather than on any quantities it specifies (in demand thought instead in the first question of the questionnaire), as well as to the mutual relationships among families of numbers. (Arzarello et alii, 1994). The various problematic situations administered in the different questionnaires The III have been formulated according to contexts different semiotic but isomorfus, denominated geometric, algebraic and arithmetic/natural-daily and express according to different languages. The present experimentation proposed him in fact to study from a point of view didactic-cultural-epistemological reported to the cultures in examination, the aspect relational-functional cognitive of the variable in the problem-solving and, sees the formulation of good part of the proposed questions. How can then the natural language and/or the arithmetic language you/they can prevail as systems of reference in absence of suitable mastery of the algebraic language. For an analysis to be more deepened, according to as I dictate in precedence, in terms of objectivity and clarity of the results we have placed side by side then to the exercises above described some questions stimulus (you introduce on the questionnaire II) that you could make reference to the free thought of the students and to refer therefore to the analysis of the phase of Metacognition of the same. In this part of the questionnaire, we have asked the boys to autonomously reflect and in more aware way on the exercises proposed in the questionnaire The (in terms of context of presentation of the problem, formalized language and decisive procedure to be adopted) and to have therefore as reverted, as our hypothesis, a greater awareness in to face the third questionnaire created to hoc according to this motivation.

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In this senses, besides the questionnaire II, introducing a series of exercise/questions stimulus allowed one greater attention of ours schemes of reasoning of the students through the reading of their observations, justification and motivations to the answer. The experimental data have quantitatively been analyzed beginning from the definition of an analysis a priori of the behaviors ipotizable from the students in relationship to the analysis epistemological previously discussed and to the present references in literature on the treated problem list. In accord with Malisani (2006), the mathematical content used in the questionnaires III has been that of linear equation in two variable. Disciplinary content that we have discussed to level historical-epistemological and that its representation, as discussed the chapter II, the heart of the historical development of the algebra for all the cultures. We underline the intention the presence in the same questionnaires of two questions related to the concept of parameter. The choice to insert two problems of the kind has not been casual. According to precedence in relationship to the concept of correlated parameter to the algebraic thought, the proposed exercises reduced him, in a first analysis, to appraise the mature awareness in terms of algebraic thought acquired by the students and to appraise therefore, in way more objective and less ambiguous possible, the full attainment, if ever conquered, of the same algebraic thought. As said the concept of parameter it assumes then a central role for the Chinese culture and for the thought categorization observed in the preceding experimentations. The analysis of the decisive strategies of the students reported to these particular and complex problem is list "read" in parallel with the others reported to the use of the variable with the different meanings that can give (Malisani, 2006), it has allowed us, to our opinion, to underline in a first wisecrack, the relationship between varying-parameter in terms of algebraic thought and the contestualization of this in the involved Chinese and Italian pupils.. The tool of investigation has been verified in two separate moments of experimentation proposed in two lists and the following phases of job in class. In the first phase of experimentations we have administered the questionnaires 0, The II and only in the second phase we have brought in class the third questionnaire. For both the batteries of test, we have allowed to the students to resolve (in maximum 40 minutes) the exercises, underlining the various passages. Also in this case the following comparison of the behaviors of the Chinese and Italian students attending, with three boys of Chinese ethnic not attending has been thought with the idea to seek further you sprout of reflection on the possibility to underline the stable schemes of reasoning reported to the culture of origin and not mediated by the teacher on the mathematical content and on the procedures of resolution. The smallness of the sample of investigation of Chinese students has been caused, also in this case, from the select elevated scholastic degree for the investigation. I am in fact the Chinese boys that result affiliate in the IVs classes of the it drains secondary palermitan.

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5.1.2.4 A Priori Analysis: “Fermat Problem” You can hypothesizes that the pupils can apply to one or more than the following decisive strategies:

S0: The student doesn't understand the text of the problem; S1: The student proceeds for attempts and numerical errors; S2: The student only calculates 10 select numerical cases random and doesn't justify the procedure; S3: The student proceeds in methodic way (according to an analogical thought recursive) in the choice of the numerical cases; S4: The student proceeds not with justified arithmetic calculations; S5: The student proceeds in arithmetic way amiss considering composed numbers and first numbers introduced in the charts; S6: The student proceeds in arithmetic way opportunely considering first numbers and squares introduced in the charts; S7: The student calculates the sum among square numbers in way random and doesn't deduce the solution; S8: The student proceeds calculating the difference with following square numbers and tries a generalization in natural language; S9: The student tries a generalization of the decisive procedure used without an explicit use of the formalized algebraic language (he/she only characterizes some numerical relationships); S10: The student tries a generalization of the used decisive procedure and considers at random some numbers, select opportunely among the greatest. S11: The student tries a generalization of the used decisive procedure and considers at random some numbers, select among the first numbers. S12: The student tries a generalization of the used decisive procedure using an opportune algebraic formalism. S13: The student tries a generalization of the used decisive procedure and puts in evidence a recognition of a structure unifying for some numbers; S14: The student underlines a possible formalized parametric writing; S15. The student changes decisive procedure, abandoning that pseudo algebraic and he returns to the natural language and/or to the arithmetic calculation. S16: The student deduces the procedure of resolution followed through possible reasonings, conjecture and theoretical demonstrations; S17: The student deduces the procedure of resolution followed through specific numerical examples;

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5.1.2.5 Quantitative and Quantitative Analysis: “Fermat Problem” From the chart of the frequencies it is immediately deduced that the tallest percentages are recorded for the strategies of resolution pre-algebraic resolutions (S1, S2 and S6). They used the strategies that are purely algebraic formalized S12 and S14.. Meaningful that the second data which the 95% of the students justify the procedure of resolution according to the strategy S16, furnishing the possible reasonings, conjecture and demonstrations of theoretical nature. A specific analysis for the Chinese students underlines that none of the involved students has used this procedure, preferring to deduce the decisive procedure through the formal-procedural languages instead, assisted from specific numerical factors of various type (S17). This experimental datum, strongly cultural for the Chinese etnia to our opinion, must be analyzed therefore separately in operation of the observations historical-epistemological and didactic discussed in precedence. As in fact also underlined by the Chinese teacher, from us involved in the analysis of the experimental19 data, "the Chinese students have seen some relationships and therefore you/they have not heard the necessity to justify her with theoretical facts and them reasonings, also because you/they have made some practical tests, of the constructions and so all is seemed them clear." The graphic implicative (realized, as in the preceding case with the Chic software) underlines with 90, 95 and 99%, three well defined and separate groupings individualized by the experimental variable of the analysis a priori. The grey cloud describes the grouping corresponding to the type of strategy used by the few pupils that have not underlined neither algebraic thought nor arithmetic nor (strategy that we call "Procedure natural simpleton", represented by the varying S2s, S7, S4, S0); in the celestial cloud we point out then the grouping type natural-arithmetician (S1, S15, S16) used by the Italian students, students that typically manifest then the strategies brought inside the green cloud, grouping of strategies afferente to a pre-algebraic procedure. Inside the yellow cloud, the selected arithmetic/pre-algebraic grouping is typically inserted, in the experimentation, from the Chinese students for the resolution of the proposed question (S11, S17, S13, S9, S6, S8, S3). The strategy S8 is underlined in the green and yellow intersection of the clouds and represents therefore a strategy of solution common to which however he arrives, in the two cases, according to a type of different arithmetician-algebraic thought.

19

As previously says it deals with a Chinese teacher of Mathematics, retired, that established him to Palermo, perhaps without a regular permission he/she has been from us consulted more votes, even though in parallel and anonymous way entirely (seen the clean refusal to release any form of formal collaboration is oral that writing) in the phase of planning and analysis of the experimental data. Its contribution is revealed precious for ours greater clarity of possible cultural heritages on the Chinese students involved in the experimentation and an analysis of some particularly meaningful protocols. The cultural mediation has happened in Italian language and always through of our specific questions.

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S16

S10

S7

S9

S0 S13 Saff

S15

S4 S17 S14

S2

S12 S11

S1

S3 S8

S6

Fig. 5. Graph implicative situazione sperimentale problema di Fermat

We hold necessary to underline, since immediately, as from the analysis of the graph is deduced among the Chinese and Italian students a behavior of solution, for the question proposed that, also being both the correct decisive procedures in form and expression (little students reach the definition of the in demand formula in the text however), as a rule it differs in more aspects, to specifically report you to the activation of the conceptions of unknown and relationship-functional and to the schemes logical deduced uses you in the process of test and generalization for the approach to the algebra, in the passage from the arithmetic register. The graph of similarity, for a certain verse it underlines, as said, a clean separation among the behaviors of the students of the two represented cultures, from each other they don't notice a clean distinctions in the choice of the strategies of resolution used by the students of Chinese etnia attending the Italian school and those not attending. This datum, meaningful to our opinion, must be investigated in deeper way however, involving a vast sample of students. The research here discussed, is not able whether to be, in this sense, a first experimental verification on possible analogies and different cognitive shown by free students from every type of scholastic didactic contract and therefore cultural, and their peers attending the school of Palermo. Returning to the graph implicative, it is interesting to notice that is typically the yellow cloud raffigurante the select pre-algebraic arithmetic grouping, in our experimentation, from the Chinese students, is the grouping cloud celestial green cloud (typical, as said for the Italian students) strongly introduces a cultural connotation to our opinion for the considered etnies.

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Before, the represented variable refer to a thought type procedural algorithmic underlined through an initial approach to the problem for attempts and errors (according to an analogical thought recursive: S6-S8-S9) through which in succession (S11), the students define, in the final part of the protocol, a possible relationship unifying the various numerical cases (S9-S13) without however to produce a formalization type symbolic and expressing this instead in natural language again deduced by numerical examples (S17). Meaningful it is for instance a sentence given in a Chinese protocol "the formula of the built numbers is: to choose the odd ones and from these to remove one and to choose those equal to the multiples of four. I have helped with some arrows on the chart of the first numbers." This strategy of resolution allows to evoke the dependence among the different numerical variable, and even though in a first approximation it underlines a strong conception of the relational-functional aspect among the various numerical attempts. However, aspect that not vien out through an algebraic writing unifying the various numerical attempts but only in the natural expression of the law of composition of the first numbers that you/they can write him as sum of two squares. Meaningful sentences brought in the Chinese protocols are for instance: - "I have thought about having to find a formula and… those that I have found are all odd…can to write not him really multiples of four but it is added or it escapes one. This is the general expression of these numbers. It needs then to build them." - "All it takes is adding to one 4, 8, 12, 16. They are written so 5, 9, 13, 17. One there are always, the other ones are equal. Perhaps this is the formula that serves us. I have not had a lot of time." - "The two charts have not helped me to find the first numbers that are highest of squares… (a lot of cancellations of calculations arithmeticians are underlined). I/you/they have not succeeded in finding a general formula. I am sure of 5, 9, 13, 17, 21. 19 are not good because if I do 19 - 12 have 18 that it is not a square, 19 - 22 = 15 the same is not all right, 19 - 32 that it comes 10, 19 - 42 = 3. [skipping some calculations to border of the protocol,19 cannot be written. With 21 him, the formula works … I can write to write that the first one I lead the square it is a square. On the charts I have not found a formula, they have served only me to connect the numbers that I have found." - "All it takes is escaping to the numbers of the chart one 4, 9, 16 8, 12 etc. and to see if an equal number comes (cancelled and replaced) to the square. I have not understood however a general formula. I have not had a lot of time." More deeply, the use of the term "formula" is the meaning of "procedure", "algorithm", "research of invariant". Meaning this deeply different from the Italian students that it seems recognize at the end datum a characterization of algebraic nature.

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In accord with Kieran on the mature expression of the algebraic thought "[…] in addition to seeing the general in the particular, one must also be able to express it algebraically" (Kieran, 1989, p. 165), the behavior manifested by the Chinese students is not considered to be algebraically think aware. The research and the definition of the procedure generalizing through a procedural approach as that it needs further close examinations on the same nature of algebraic thought and on the phase of transition of this from the arithmetic thought that how rooted semantic environment is strongly seen in their thoughts. Skipping then the grouping underlined by the grey cloud, few meaningful for the study here discussed, for how much it concerns to the other schemes of reasoning quantitatively underlined for the Italian students we can consider the procedure natural-arithmetician (green cloud) as the method for attempts and errors in natural language or in seed-formalized language. The Italian pupils that this strategy generally applies this strategy consider different numerical cases of first and square numbers (S6) and they proceed according to the application of the text with simple arithmetic calculations (S8) select in way random. After various numerical attempts (not always correct) some try the first approach to the use of the algebraic symbolism (S12) and they try to write "a general law for all the first numbers that you satisfy the asked rule20 " (S14). Not all reach the jump cognitive among the arithmetic thought for tests and errors and that algebraic however. Those people who manifest a first pre-algebraic thought, after few reasonings of "test arithmetic" abandon the formalized language (S10) and they try of "to justify" the reasoning followed with considerations of theoretical nature. They omit however to repeat the definitions of "first number" and "square number." The algebra, in this sense, as language of mathematics, rather than to present himself/herself/themselves to the students as a bridge toward the following studies an impossible wall reveals him that hinders the acquisition of a thought "superior". The principal obstacle is what he finds again him in the passage from the procedural aspect seen not in this case by the students as a point of arrival for their reasoning of test but of discovery of certain regularities, to that structural of the algebraic language (Arzarello et alii, 1994). In the phase of "test" of the symbolic writing, with the objective to falsify the affirmation it posts in the problem (according to you it is possible to write all the first numbers (except 2) in this form?), are underlined in the protocols different two typical schemes of reasoning, that underline as for these students the to formalize and to define therefore then one "general formulation" you mean to falsify that a certain ownership results are true considering at random select great numbers. The two different schemes of reasoning can easily reassume him bringing some meaningful sentences are introduced in the protocols of the boys: - "To feel the truth of the rule respect to the set problem I have reasoned this way: I consider as initial examples the examples introduced in the text of the problem and to show that all the first numbers can decompose in the sum of 20

Brought by the protocol of a student of the scientific High school Galileo Galilei, III year.

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two square numbers I have to do it for of the very great numbers I have not succeeded however to trovar many great numbers. 9 13 is small. 181". The student follows a trial type inductive on the generises. Some algebraic symbolism is not underlined. - "I have tried for various cases of first numbers that I have chosen at random among the greatest that I have found they follow the rule of the problem: 449 =202 that ago 400 more 9 that it is the square of 3; 457=192 + 16 (what 4 are to the second); 181=100 (what 0 is to the second) more 81 that it is 92; 53 can also be written, 61 also and so street. I think that I am always worth. I have tried to look for a general law for all the first numbers that I have chosen but if I call these numbers x and y I have that p=x2+y2 and we have not studied the equations of according to degree…. it formulates her however it works for all the numbers prime…with the numbers it works, it comes better me". (You student in this case initially underlines a reasoning type arithmetic abductive, according to Peirce, that pushes her/it to a first symbolic writing. After few reasonings of "test arithmetic" it abandons the formalized language and he/she returns to manifest an arithmetic thought, for her to check semantically). In accord (Malisani, 2006; Matz, 1980; Wagner, 1985) and also in this case the point of transition among the two "thoughts", considering that that is the phase of breakup between the simple numerical calculation and the to think of algebraic terms through the introduction of the concept of variable is of obstacle for the reasoning of the student with the presence of the 0% of the Chinese students, some Italian boy of the high school Scientific (however a low percentage) and a solo of the Institute Technical Professional "E. Medi" they reach the formulation of the parametric writing p=4K+1. The run argumentative underlined in all the protocols is variegated and articulates through a gradual passage from the arithmetic strategies (S2, S4, S7, S8) to the general formulation type algebraic (S12, S14, S16). As I visit in the graph implicative and underlined in precedence, the strategy S8 is underlined in the green and yellow intersection of the clouds and represents therefore a strategy of common solution, to which however the Chinese and Italian students arrive according to a type of different arithmetic/pre-algebraic thought. If in fact the Chinese students it seems that they use it in the phase argomentativa of the solution, justifying this procedure with a "ransom glues" recursive ("The same I have for example done for 20 removing and putting the numbers to the square and it doesn't work… Uniting you to them you acknowledge the rules, and you find the solution that you want"), Italians use him/it in way random in the phase of initial conjecture of the problem. As says, also in the preceding chapter, reporting us to the Jiuzhang Suanshu, the method used by the student that it is assimilated to our opinion to the technique of the Fangcheng, Chinese it underlines, even though in a different circle from that of the Jiuzhang Suanshu, something more of a simple progress for numerical attempts and it cannot be considered then in the case of the students of Chinese etnia as expression of a banal to persist nl arithmetic thought. The principal meaning of the Fangcheng is, as says chapter II, that of "to look for the norms

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of the things assembling side by beside". The expression means, more precisely: "to put side by beside the whole the quantitative relationships among the things to appraise the norms of the measures of every of them (Chemla, 2004). it is one of the most important algorithms in the Chinese mathematical tradition for the specificity of the definition of different destination rules as those to multiply for disaggregating, to simplify for gathering them, to homogenize, eguagliare to communicate it how to connect. To our opinion they find strong connections with the cultural aspects proper of the Chinese culture as the Taoist yìn-yáng and the cardinal virtues, treated in the first chapter and with linguistic aspects of composition of the characters, as seen in the chapter V. Her strategies of resolution used by the Italian students, read in parallel with those typical Chinese they allow of validare, even though in a first approximation, the hypotheses of research defined in the experimental job. A different conceptualization to think for cases. The experimental contingencies seem in fact to confirm as all the students, in an express problematic context in arithmetician-natural language, evokes decisive strategies of arithmetic die pre-algebraic and/or in absence of an appropriate mastery of the algebraic language. If however these, Chinese students involved in the experimentation, they seem direct the more from an analogical thought recursive finalized to the research of a fundamental algorithm unifying, a thought type theoretical/deductive (defined according to logical schemes of reasoning different) for tests and errors it seems to prevail as scheme of fixed reasoning for the Italian students that manifest a greater "need" to translate their numerical conjectures through a more formal language. If then in the phase of reasoning and generalization the Chinese students underline a reasoning type "variational-procedural" that it only conducts them to the definition of a reported destination thought to the recognition and the definition of an inside numerical structure to the in demand first numbers, the process of generalization of the Italian students it evolves him through the phases of reasoning, conjecture and demonstration, according to a logic more than type deductive/abductive. The tool of investigation, for his/her nature a little algebraically formalized in the context of presentation, has not allowed us to falsify the hypothesis H3 from which we have departed for the definition of the second experimental situation realized beginning from the "varying questionnaire and parameter in different semiotic context." 5.1.2.6 A Priori Analysis: “Questionnaire Variable and Parameter in Different Semiotic Context” You hypothesizes that the pupils can apply to one or more than the following decisive strategies CN1a: The chart translates the annual growth of the production. The student reads the chart and spontaneously brings back her/it to the calculation of the lacking datum. In the phase of justification it underlines a dynamic vision of the problem list; CN1b: The chart translates the annual production of lemons. In the phase of justification, the student underlines a static vision of the problem list;

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CN2a: The formula is the translation of the introduced chart. The chart represents the text. The student translates the relationship brought in natural language in terms of production and growth; CN2b: The student translates the formula remaining hooked to the symbolic introduced without making to appear the idea of function,; A31: Reported strategy to the study of a planned linear system and opportunely resolved: ⎧α + β + γ = 180° α + β + γ = 180° or α > β ; ⎪α = β + 83° ⎨ ⎧β − γ = 83° ⎨ ⎪γ = 57° β γ + = 123 ° ⎩ ⎩

β + γ = 180° − 57° = 123° so

A32:

α = 57° ; β + γ = 123° ; β − γ = 83° then: β=

β + γ + (β − γ )

= ... 2 ; β + γ − (β − γ ) = ... γ= 2 Ar31: Strategy for attempts and errors. Systematic strategy; Ar32: Strategy for attempts and errors. Casual strategy; A41: Reported strategy to the study of a planned linear system and opportunely resolved (defined as in the exercise N.3); A42: Defined as in the exercise N.3; Ar41: Strategy for attempts and errors. Systematic strategy; Ar42: Strategy for attempts and errors. Casual strategy; A51: Reported strategy to the study of a planned linear system and opportunely resolved: ⎧3( y − 5) = x + 5 x=number of Cd of G.; y=number of CD of A: ⎨ ⎩y + 5 = x −5 52: Reported strategy to the study of a planned linear system and resolved dynamically not formalizing the problem: ⎧x + 5 = 3y x=number of Cd of G.; y=number of CD of A: ⎨ ⎩y + 5 = x Ar51: Strategy for attempts and errors. Systematic strategy; Ar52: Strategy for attempts and errors. Casual strategy; A61: Reported strategy to the study of a planned linear system and opportunely resolved (defined as in the exercise N.5); A62: Reported strategy to the study of a planned linear system and resolved dynamically not formalizing the problem (defined as in the exercise N.5); Ar61: Strategy for attempts and errors. Systematic strategy; Ar62: Strategy for attempts and errors. Casual strategy; A71: Reported strategy to the study of a planned linear system and opportunely resolved with one of the studied methods; Ar71: Strategy for attempts and errors. Systematic strategy; Ar72: Strategy for attempts and errors. Casual strategy;

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A81: Reported strategy to the study of a planned linear system and opportunely resolved with one of the studied methods; Ar81: Strategy for attempts and errors. Systematic strategy; Ar82: Strategy for attempts and errors. Casual strategy; A9/10/111: The student recognizes the analogy among the two proposed questions and correctly interprets the matter; A9/10/112: The student only recognizes the analogy in terms of decisive method; A9/10/13: The student only recognizes the analogy in terms of structure, according to a static vision of the problem list; A9/10/114: The student doesn't recognize any analogy except a generic problems affiliation of mathematics; A9/10/115: The student doesn't recognize the analogy justifying his/her answer to every single detail of the proposed exercises,; A9/10/116: The student doesn't recognize any analogy, manifesting a sectorial knowledge, tied up to the context and the used language; Ap121: The student answers to the question not taking back, for the appointed purpose, an already present problem in the questionnaire. He/she could show a high maturity in to introduce one of them new correctly (conceivable an aware algebraic thought); Ap122: The student answers to the question taking back, for the appointed purpose, an already present problem in the questionnaire; Ap123: The student doesn't understand the exercise. Syntactically he/she faces him/it (incorrectly) but he/she doesn't notice the analogies with the preceding exercises; A131: Reported strategy to the study of a planned linear system and opportunely resolved with one of the studied methods; A132: Strategy type proportional through a conversion from the algebraic system:

x : y = 2:3 2x : 2 y = 4 : 6 2( x + y ) : 2 y = 10 : 6 80 : 2 y = 10 : 6 48 = 24 ⎯⎯ → x = 16 y= 2 Arl31: Strategy for attempts and errors. Systematic strategy; Ar132: Strategy for attempts and errors. Systematic strategy; A141: Reported strategy to the study of a planned linear system and opportunely resolved according to one of the studied methods; A142: Strategy type proportional through a conversion from the natural language (as the strategy A132) Ar141: Strategy for attempts and errors. Systematic strategy; Ar142: Strategy for attempts and errors. Casual strategy;

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A151: Reported strategy to the study of a planned linear system and opportunely resolved according to one of the studied methods; A152: Strategy type proportional through a conversion from the natural language (as the strategy A132) Ar151: Strategy for attempts and errors. Systematic strategy; Ar152: Strategy for attempts and errors. Casual strategy; Ap161: The student recognizes in the parametric system the tool for modeling the proposed situations; Ap162: The student only recognizes the analogy in terms of decisive method. It doesn't go over simple observations type procedural; Ap163: The student doesn't recognize the analogy, it limits him to a syntactic reading of the system; Ap164: The student doesn't manifest a mature awareness of the role of the parameter. It doesn't recognize in the proposed system the possibility to describe analogous problematic situations; 5.1.2.7 Quantitative and Qualitative Analysis: “Questionnaire Variable and Parameter in Different Semiotic Context”

Following, we bring the grafis gotten by the realization of the usual experimental matrix Excel and of his transposed in the case of the similarity of Lermann realized in accord with the analysis previously introduced in the preceding paragraph. The strategies from us individualize they are shown valid and exhaustive, the behaviors hypothesized him are verified in fact, even though partly; any student, as it was easy to hypothesize, he has used the strategies A32, A42, A52, A62, A132, A142, A152. As in the preceding experimentations, in this case also the analysis of the results has been gotten through simple applications of descriptive statistic and the use of the software of inferential statistic with CHIC. Through the analysis implicativa and of the similarity, the used software has allowed us to underline meaningful relationships among the various items introduced in the questionnaires, could do therefore possible inferences in relationship to the significance of the various contexts of presentation used. The grafis parallelly produced by the Chic beds to a qualitative analysis of the protocols have, in fact, permission, to our opinion, to put in evidence meaningful conceptual knots related to the algebraic thought of the involved students. Possible inferences also thanks to the interrelations among the various pars of the tool of investigation (Questionnaire 0, The, II and III).

Implicative graph From the chart of the frequencies it is immediately deduced that the percentages related to the variable retentions Necessary Condition to the pre-algebraic thought they are relatively high for all the involved students. The students who exhibit some difficulties in this sense are some of those who attendings the institute of art that you/they don't underline, in the three tools of investigation, neither a

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pre-algebraic thought neither nor an arithmetician. The scarce results almost exclusively reached with reasonings for tests and errors on the express questions in natural language, are probably to impute to a scarce interest manifested for the experimentation. In general, the strategies used more commonly are recorded, in percentage, among those of arithmetic/pre-algebraic die. In the questions N.3, N.4, N.5, N.14, N.15 these seems to in fact prevail for almost all the students, an initial resolution in natural language and/or for tests and errors follows, for the almost all of the students, from the first algebraic formalization through equations and systems of equations. If then the students of the scientific high school seem more skilled in to algebraically formalize the text of the various problems expressed in natural language (even though manifesting some errors), a disembodiment of the experimental data in relationship to the various subgroups of students attending addresses scholastic different, show as the percentage of strategies type algebraic you lower notably for the addresses of the institutes Magistral and of art. For these exercises, among Italians, most greater difficulties find him in the difficulty relationship syntax-semantics (sign-referente-sense and in the meaning of the triangle of Frege as I compare among zeichen-bedeutung-sinn) inside the passage to the algebraic code and therefore transversally, to the application and the control of the algebraic and pre-algebraic thought on more environments of presentation ("daily-numerical" and "geometric"). For the Italian students of the Commercial Technical institute of the Magistral one and the institute of art, the conception of predominant variable to resolve the expressed questions in natural language is mainly that of unknown. The pupils calculate some particular solutions required by the problem through reasonings for tests and errors or through the resolution of equations among them independently in the impossibility to form an only representative system. The problem of the ties don't be set imposed by the context in which the expression is considered. In the students of the Scientific High school it seems to prevail a first relational-functional aspect of the variable. The solutions of the various exercises constitute for them a whole range of values that you/they are gotten varying one of them and calculating the other beginning from the linear dependence among the variables. Not always, however, this type of scheme of reasoning is correct in the procedure of resolution. A typical error is that which is manifested by quite a lot students in the question N.5 with the strategy A52 (hooped in red in the implicative graph). reported Strategy to the study of a planned linear system and resolved dynamically not formalizing the problem: x=number of Cd of G.; y=number of CD of A:

⎧x + 5 = 3 y ⎨ ⎩y + 5 = x

The questions N.7, N.8 and N.13, introduced in contexts purely formalized, they have been faced (even though in some cases in incorrect way) mainly according to strategies of algebraic type (the marked strategies with Ai). Only two Italian students of the Magistral address have tried to resolve the questions N.7 and N.8

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for tests and errors, abandoning, however, this in the operational procedure of an algebraic form in the questionnaire III. These last therefore they result in percentage the strategies more used both from the Chinese students and from Italians. In this context two meaningful sentences are brought in the protocol of two Chinese students attending the teacher's college "Margherita" IPPM. "The exercises N.7 and N.8 are equal. They are only writings in a different way. With simple operations on the equations they are written equally. I have resolved only the according to whether it is easier to be escaped in the system. (A111), the procedure of solution is the same one." "The systems of the questionnaire are equal. I am good in Algebra, I immediately am aware of it. They resolves equally". (A111) To the institute Margherita, none of the companions of the same class has produced this observation. All the Italian students that have resolved the various formalized linear systems, introduced in the tool of investigation, you/they have used, as method of resolution the method of substitution. Five out of students seven (among which the not attending) you/they have opted for that of difference. While the remainding two have worked for substitution. The percentages of resolution of the questions N.12 and N.16 are relatively low (19% for the Ap122; 10% for the Ap163) and they show therefore, above all for the Italian students (what you/they have almost ever answered to the questions) an absence of mature algebraic thought. The algebra, in this sense, as for the preceding experimentation, as language proper of the mathematics, rather than to present himself/herself/themselves to the students as a bridge toward the following studies an impassable wall reveals him that hinders the acquisition of a thought "superior" that ago reference to a thought type holistic report her. All the students, in accord with quite a lot of the results discussed in literature, they don't always seem able to conceive the possibility to hardly express through a symbolic expression a whole family of problematic situations as those first resolved. In the few cases of answer, the behaviors analyzed for the Italian students underline as the students, in the question N.12, after a first vain attempt of algebraic manipulation (Bp123) or they abandon the exercise not answering to the application or, in the case of few students of the Scientific High school, they bring a possible analogy with one of the already present problems in the questionnaire (N.3, N.4). In the question N.16 these same limit them to deduce their answer with considerations type syntactic (Ap163). Any Italian student underlines the strategy Ap161. Typical sentences that manifest the uneasiness of the Italian students toward the question exercise are for instance: - "I have not understood what I have to find […] I Invent a words problem" (Italian student, Institute Technical Commercial, Es. 12); - "The result cannot come because there are no numbers. It has to be a number" (Italian student, Scientific High school, Es. 12); - "The x is not a number, I have not understood what they represent to, b and c" (Italian student, Institute Technical Commercial, Es. 12);

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- Bringing the text of a problem unhooked by the proposed parametric system the student declares, "I have invented a problem, but there are too many symbols" (Italian student, IPPM, Es. 12) The writing x+y+z=c comes in this case "translated" as perimeter of a triangular geometric figure and, even though in indirect way and it doesn't render explicit, ago therefore reference to the question N.4 - "The first equation is the same one if I put a=2/3 but the second it is different" (Italian student, Teacher's college, Es. 16); - "They cannot be equal because in the first one is had to resolve a system and this is equal, it is also a system but the other two not are even if I always have the relationship among two things is always 2/3. " (Italian student, Scientific High school, Es. 16); - "every formula is assestante. The system if I put a=2/3 (b I don't know him/it) it resembles to the problem 15. The first one certainly no because the second equation is different "(Italian student, Scientific High school, Es. 16); The last three quotations confirm then a difficulty, by now enough documented in literature, related to the understanding of the polisemia of the writing as possibility to consider the invariance of the denotation in comparison to the algebraic sense of the expressions, agreements as the esplicitazione of the way with which the denoted one can be gotten through the application of rules computationals (Arzarello and alii, 1994). As confirmed before also, in the chapter III, an interesting example in such sense has been developed by Chevallard in 1989, 1990, 1991. More deep considerations must be made then for the resolution of the question N.12 from the three Chinese boys not attending the Italian scholastic institution. For these in fact the problem list it posts in the questionnaire II it doesn't avowedly have some sense. As brought in one of the protocols of these, judged too much "misleading" and general in his/her formulation, "for the system written in the exercise I cannot invent a situation this way. In the other questions I know what you are doing, here no. What am I to, b and c? […] This problem doesn't have sense. You had to give me some numbers"21. Relatively to the question N.16, another boy not frequentante declares "the exercise doesn't have sense. In the exercise 14 have to calculate the sides of the figure, in 15 the ages. Then the 16 type is the same. It is true, they are all of the same one […] it is easy, but here you have to say what you want to know. I cannot resolve it. I think that they are years or sides or numbers "103 it Recognizes therefore a certain equivalence among the representations but it manifests a difficulty in accepting the abstract parametric writing. All the Chinese students attending, from their song, they easily enough recognize in the formulation of the parametric system brought in the exercise N. 16 a structure unifying ("it has the same form and they represent the same formulas of resolution of the problems N.13, N1.4 and N.15"; "I would Say that I am the same category all it takes is resolving one of it that then they do all") various groupings of questions (Ap161), and unlike their contemporaries not attending, in virtue 21

The brought text has been polished up by grammatical and orthographic errors that find again him in the protocol.

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perhaps of a possible didactic contract, them conceptualize implicitly semantically declaring a concrete value for the parameters to, b and used c and correctly resolving the exercise in algebraic way. Always for the exercise N.16 finds again him in a Chinese protocol: "to resolve the system I have put some equal numbers to those precedents. I don't know if it is correct but I have used the same scheme of 13 and 15. This way I find age that I already know because I have found her to mind." It inserts in the protocol a=2/3 and b=80. In the exercise N.13 a strategy is underlined for subtraction, in the exercise 15 are only brought the solutions of the question without any procedure. For the exercise N.12s solutions of the are not found again attending Chinese students attending even if for the exercises N.9, N.10 and N.11 they clearly underlines to have understood the key of reading and decoding of the questionnaire in classes of problems. Meaningful examples of the reasonings are as follows: A111-A91: "… they have the same number to couples and this is the same of the N.4 and the N.3" A91-A101-A111: "The exercises are to two to two peers even if different. I have resolved together them, they resolve with the same calculations", "They are equal. They internally vary but they are equal in the solution" A91-A101: "you, are equal if it is understood that they are the same" "I have immediately understood that the same thing is worth for the problem of the CD and the age, all it takes is following the rules". From the references logically deduced by you (also read in parallel with those Italian afferent to the varying A93-4/A103-4/A113-4s), it seems to deduce him for the Italian students a possible facility in the reading of the questionnaire in terms of categorizzazione and classification of the proposed exercises. It seems that through a job on the numerical and procedural relationships, the students of Chinese etnia determine more easily the classes of meaningful situations unified by the symbolic writings and define her/it "perfection" of these in terms of simplicity and generality. Passing therefore from a first approach to the concept of variable as thing that varies, concretely gathers the aspect of unknown and functional relationship and they deduce him/it, in the parametric writing only in terms of procedural. Is it possible to find again in this scheme of reasoning a further epistemologico-cultural reference to the Jiuzhang Suanshu? In accord with Chemla (2004) on the structure subtended to the Jiuzhang Suanshu and the research of a "fundamental algorithm of resolution" in partnership to a thought olistico concrete pragmatic, believes really of him. The contingency of reasoning for the question N.16 among the involved Chinese students, who attending and can bring back besides not him, according to us, a cultural aspect of the learning of the Mathematics; shared aspect is that an Italian teacher that teaches mathematics to Palermo in a class multicultural with Chinese students, both from the retired Chinese teacher, with which, as says, we have discussed these experimental results. The reference to the concrete one, to the categorization, to the hierarchization of procedures of solution and memorization of these in the resolution of a problematic situation, discussed to level historicalepistemologico in the chapter II, seems to find himself/herself/ themselves therefore, even though according to different meanings from those discussed in the Jiuzhang Suanshu, also experimentally in this phase.

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This aspect certainly needs following close examinations in relationship to the didactic contract to which the Chinese students are "subjects" in Italy, to the Chinese scholastic circle and therefore to the "mathematical tradition" brought ahead in the last years in the country of origin of these students not attending but it is able, according to us to be considered a first interesting result. Even though the questions N.6 and N.15, with the due differences between the one and the other, they underline on the Italian protocols, identical percentages about for the strategies type for arithmetician and algebraic (A61-A151-B151 and Ar61-Ar151) almost always contextual. Almost at the totality of the students that in fact begin the procedure of resolution with a research of possible references numerical solvers ("I have tried with 18 and 23 but it doesn't work, then with 5 and 15 etc".), they almost immediately replace, in a more correct way, with a formulation of the variable, the meaning of unknown, ("I have tried to call x the ages that I had to find". Then I have understood that I had to call an unknown for Mario and one for Charles and I have used M and C. Ho read therefore the data of the problem and I have written some mathematical formulas. It resembles to a system but I am not sure, I have lost in the calculations"). The students that manifest this passage of strategy underline since immediately a scheme of pre-algebraic reasoning with the individualization of the unknown one and the definition of formal writing. The sentence brought by the student makes reference to the difficulty relationship existing in the arithmetic and algebra among procedural aspect and that structural of the mathematical language (Arzarello et alii, 1994) and it sets the accent on that introduces him for her the phase of breakup between the simple exploratory numerical calculation and the to weigh in algebraic terms through the introduction of the concept of variable and the nominalization of this (Malisani, 2006). concept this, as seen, variegated in overall its facets and really for this difficult to "to check." The meaningful result related to the two considered questions and to the change in the decisive strategy that must also be interpreted in operation of the particular formulation of the data that really in virtue of their exposure type serial it could communicate this passage. Passage that doesn't result instead evident in the Chinese protocols, which is difficult to classify in many cases, for one it doesn't render explicit declaration of the strategy of solution. In these they are underlined in fact only few simple numerical cases and the in demand result. In eight protocols on nine some reasoning is not brought. The only meaningful protocol in this sense brings the sentence "I have calculated to mind the in demand numbers. All it takes is reading together well the two rules and to find the two hidden numbers" Same reasoning for the question N.15. Same considerations for the questions N.3, N.4, N.5, N.14. The exercise 14 underline a variegated situation instead. Four of the students attending resolve him/it re-writing the system of the question n.13 and expressly declare the correspondence among the two expressions. (These students it would seem that leaflet therefore a good x syntactic-semantic control on the expression "relationship, , perimeter of recy tangul 2 x + 2 y ”), the lasts students report the remaining solution. These first

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quantitative data seem to confirm though to a first approximation the research hypotheses set for this second experimental investigation: H1: The differences and the verifiable analogies in the Italiani and Chinese cultural tradition, find again him in the analysis of the schemes of reasoning evidenziabili in situations of teaching/learning of the algebra and in general of the mathematics. H2.2: The procedures in natural language and/or in arithmetic language for tests and errors, prevail as decisive strategies in absence of an appropriate mastery of the algebraic language. H3: In purely algebraic contexts, the Chinese students underline a good syntactic control of the algebraic formalism.

The general hypothesis from which the present job has departed, and therefore the verification leading, put in evidence in the research in didactics from more subjects (teachers, researchers and same students) of the "good algebraic abilities"22 of the Chinese students in relationship to the inherent difficulties "to algebraically think" shown by now by different searches in didactics of the Mathematics (Arzarello, Bazzini, Chiappini, 1994; Matz, 1980; Malara-Navarra, 2001; Radford, 2003a, 2006; Sfard, 2002) has been partly confirmed. None of the nine involved students has manifested some errors type algebraic in the exercises algebraically formalized (N.7, N.8 and N.13). The boldness in to stir inside the formalized linguistic register has put not in evidence of a banal syntactic control that in many cases it is not found in the Italian students (if not to the Scientific High School). This, however, an evident behavior of algebraic thought? Can we conclude that the students that have manifested these good abilities manipolative have the algebraic thought? The Chinese students that "they are formidable indeed in Algebra, they quickly enough resolve equations and systems and often in fast way and technique"23 and what "they resolve the exercises […] in way meccanic "102 has almost reached indeed the algebraic thought? Or are they perhaps only good solvers, good repeaters of techniques of calculation done opportunely by memorizing and do they apply according to a control type syntactic more semantic? A parallel analysis of the grafis of non parametric statistic (implication and similarity) can help us in this sense. For a greater legibility of the text, we propose following the graph implicative previously introduced. A first analysis of the variable in game puts in evidence from one side the implication (brought in the green cloud with a level of implication 99%) among the varying CN2b, considered necessary condition type static on the question II and a series of incorrect algebraic strategies (besides the varying A95s, A105);

22

23

The fear uses he/she has been quoted of it really from a teacher of inferior secondary school to describe a his/her a Chinese student. Sentences brought by a teacher of mathematics of the superior secondary school of Palermo that he/she works in classes multiculturali with Chinese students, from us interviewed.

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CN2b

A105

B61 A101 Ar62 A104 B151

B131

Bp123 A151

A114

A95 A52

CN1a

Ap163

B71

A141 A61 CN2a

Ar141

A41

Br151

A91

B81

Br51

Ap161

A131

Ar61

A92

Ar41

A31 Ar151

A71

A81 A102

Br31 Ap122 A112 Br141 Br41

Fig. 6. Implicative graph experimental situation "varying questionnaire and parameter in different semiotic context "

from the other, brought in the orange cloud, the implication among the Necessary Condition CN1a and some of the correct algebraic strategies hypothesized in the analysis a priori (red cloud). For the same meaning of implication has therefore a result that confirms the initial hypothesis of the job of structuring of the tool of investigation related to the planned Necessary Condition: - a Necessary condition type static, tied up to the single case and without any possibility of generalization from the student, it seems to bring him, a knowledge in wrong Algebra, "false". The student, also elaborating and underlining a first idea of functional relationship seems not to succeed in reading the formula in an autonomous way. It doesn't boss this and therefore the present variable in it. Typically The strategies of algebraic resolutions can result, as in this case, wrong both for calculations and procedure. - a Condition Necessary dynamics would seem to bring a maturity and aware of algebraic reading instead and therefore a manipulation conscious of a formula, of a formalized algebraic expression. The implications B123 and A163, A104, A114 (celestial cloud) they put in relief the acquisition however, the acceptance in the Italian students of one before, ingenue, naive approximation of the idea of parameter and parametric thought (as before ability to recognize classes of problems) and therefore not a full attainment of the algebraic thought.

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For the Chinese students the exclusive use of the strategy Ap161 (hooped in yellow) it underlines one ability of theirs of it recognizes in the parametric system a tool of modeling; this scheme of reasoning is not found again however in the exercise N.12 and doesn't manifest therefore even in this case a mature algebraic thought. The experimental result, even though partly, it confirms us from a side the good ability of syntactic control of the Chinese students from the other one them it doesn't mature acquisition of the reported algebraic thought to a simultaneous control of the syntactic aspect - difficult semantic/procedural-relational for all the students of the superior secondary school involved in the experimental investigation. Schematizing the different layers of algebraic thought manifested by the Chinese and Italian students we could say that also not underlining a mature algebraic thought, the Chinese students show him, mostly skilled in comparison to the Italian contemporaries, as "solvers" in formalized algebraic contexts. This is underlined as a strong result that, if for a certain verse, confirmation the "good algebraic abilities" of the Chinese students, it doesn't even manifest in them, a full mature algebraic thought that would be manifested, as says, through a correct, aware, approach to the parameter in the exercises N.12 and n.16. Their syntactic control inside the symbolic writing is able to conduct them to recognize, even though partly, classes of problems expressed by an only symbolic writing (Ap161) it seems to fade away, however, in the phase of breakup between the simple numerical calculation and the to think of algebraic terms formalized through the introduction of the concept of variable formalized functional relationship. In the point of transition among the two "thoughts", the Chinese boys, attending and not, also manifesting a thought report procedural among the variable in game and therefore the limitations of these, seem that they prefer to remain in concrete arithmetic contexts almost always developed through mental calculations. Them, as also confirmed from more taught Italian that teach in classes of inferior and superior secondary school to Chinese and Italian students, "in technical environments of job, with symbols and numbers they stir better in comparison to their Italian contemporaries. And not because they don't understand the Italian because it is perhaps a language to them more congenial and they can go fast." In contexts algebraically formalized not, in accord with one strong historical tradition of theirs (even though discussed partly in the preceding chapters) they underline a reasoning type arithmetician (even though helped by the strong memorization of "made arithmeticians" that they help them in the definition of a thought olistico type report) and they strongly remain anchored to this. As says it is our opinion to believe that their ability symbolic manipulative on the written language ideographic strongly helps them in the formalized algebraic syntactic control. The environment of symbolic reference seems in fact, with the due differences of content, to them relative from the procedural point of view. The search confirms then, for the Italian context, the reported problem list to the passage to semantically give register semantic arithmetician to that algebraic

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different. Relatively to the formalized symbolic language it underlines once more as for the students this writing (equations or systems of equations in this case) you don't activate in many cases forms of productive thought, doesn't absolutely come considered as I model interpretative of a problem (exercise N.12) better still of a class of problems (exercise N.16). This ability that, as says, it is reached by all the involved Chinese students. As it finally regards the questions set in the questionnaire II, the graph underlines entirely in a way 99% reasonable implications among the A91-A101; A92A102-A112; A95-A105 (brown clouds). The quantitative datum underlines a scheme of fixed reasoning, that is common to the involved students, in the approach to the various proposed problematic situations. The strategies A91/A101/A111s mainly refer to the Chinese students, where the others to the Italian contemporaries. As resulted meaningful related to the methodology of experimentation, we hold meaningful to underline as for the most greater part of the students involved in the experimental phase the developed possession the questionnaire II, has been an important moment of attention. The autonomous reflection in the questions N.9, N.10 and N.11 has brought the students, in a more evident way, to face the questionnaire III in more critical and aware way. The Metacognition seems therefore to have favored in the boys a greater awareness, at least in terms of attended procedures and verified. Beginning from the effected analysis possible profiles of pupils emerge "clearly typical" that they face the three structured questionnaires. These can be schematize him this way: Table 3. Profiles typical of students for the experimental situation "varying questionnaire and parameter in different semiotic context"

NatIng (Natural naife)

RelazProc (Relational procedural)

I outline correspondent to the pupil that exhibits a procedure in natural language or it resolves the various problematic situations with reasonings for tests and casual errors. This profile is typically characterized by the presence of the experimental variable you Plough and Bri and it doesn't even underline a pre-algebraic thought. I outline correspondent to the pupil that applies a strategy for attempts and errors in natural language and/or in language seed-formalized not in contexts purely formalized and it underlines good abilities manipulative in symbolic contexts. Him therefore, generally, in contexts not really algebraically formalized, it assigns more concrete values in the arithmetic register to one of the variable in game and it finds the values corresponding of the other varying showing some solutions that falsify the problematic situations. It necessarily recognizes a functional relationship among the variable in game but not

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5 The Experimental Epistemology as a Tool to Observe and Preview Table 3. (Continued)

the apparent in symbolic language remaining often in an arithmetician-procedural circle. On the contrary succeeds to gather the essential aspects of the algebraic syntax and through this it underlines a thought categorization. I outline correspondent to the pupil that uses the pseudoalgebraic procedure of calculating for the questions in language arithmetic natural daily paper some particular solutions required by the problem through reasonings for tests and errors or through the resolution of equations among them independent in the impossibility to form an only representative system. The problem of the ties is not set imposed by the context in which the expression is considered. I outline correspondent to the pupil that evidence a first attempt of algebraic procedure (correct and not) formalized or seeds formalized in all the contexts of presentation. He underlines her/it "necessity" to translate express the problem in natural language in an equation of first degree in two unknown and it tries to consider in implicit or explicit way how it can represent a functional relationship. Often in this run of growth him it limits however him to an aspect of variable as unknown. In the questions algebraically formalized apparently a good syntactic control not accompanied by a thought categorization and general of the errors N.12 and N.16 it doesn't recognize in the parametric systems a modeling of the develops exercises.

PsAlg (Pseudo algebraic)

PreAlg (prealgebricaic formal)

0

22

V1 rI

tcI V

sp m I V8 A

A

A

V5

28 IV

sA lg P

17

sp mI

tcI V

sp m I V6

sp m I V9

tc A

A

A

A

A

V1

V4

V2

sp mI A

V3

sp mI

19

sp mI

sp mI A

A

A

10

3

4

tcI V A

tcI V A

11

tcI V

tcI V A

A

8

18 IV

tcI V

tcI V

tc A

A

A

2

ro c

9

laz P

tcI V A

Re

4

33

cs pm IV A

V2

cs pm IV

ct cI

tcI V A

A

A

cs pm IV

es t4

es t2 A

A

A

es t3

es t1 A

A

6

cs pm IV A

21

7 IV

tcI V

tc A

A

26

lg A re

tcI V A

P

27

5

31

tcI V

tcI V A

A

tcI V A

30

32

24

23

29 IV tc A

tcI V A

tcI V A

tcI V A

1

tcI V

tcI V

tcI V A

A

IV

tcI V

tc A

A

A

2

16

3

1

In the hierarchical graph of similarity brought following, and related to only the only classes with situations of multiculturalism (ITCS and IPPM), three macros are observed groupings of meaningful variable that implicate: to the left the profile PreAlg underlined in red color, follows to the right from the profile RelazProc (marked in blue) inside which almost exclusively has the Chinese students (a substantial difference is not deduced in the schemes of reasoning related to you address him of frequented studies), finally a grouping Pseuo Algebrico (PsAlg) (marked in black) and among this last ones a group of students that to the 95% similarity level they don't reenter in some macros grouping a small whole variable, he notices belong to the grouping.

Árv ore coesitiva : C:\Us ers\Benedetto\Desktop\PROVA1.csv

PreAlg

RelazProc

PsAlg

Fig. 7. I plant with trees similarity experimental situation "varying questionnaire and parameter in different semiotic context". ITC "Salvemini" and Institute Psico Pedagocico "Margherita".

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The brought graph, even though with due limitations of the case (very narrow sample of investigation, addresses of studies a little specific deliberately to the mathematical level etc.) can represent therefore the didactic situation of the various classes that introduces students to the inside in situation of multiculturalism. The didactic implications of this study can be, in a first approximation, to our opinion, meaningful for the analysis of the phase of teaching/learning of the algebra and the processes of reasoning shown in relationship to a peculiar mathematical context what that algebraic and pre-algebraic, and to the delicate phase of the passage from the arithmetic thought to that algebraic not necessarily formalized. Further investigations in relationship to the experimental sample and the definition of the additional variable are necessary, however, for a more deeply research more deepened. The hypotheses of research defined for this didactic situation seem validate, even though partly. 5.1.2.8 Third Experimental Investigation: "The Sequence", "The Grid of Numbers"

In the definition of the hypotheses of search fixed for the experimental job, the hypothesis H2.3 is opportunely defined relative to the phase of generalization used in the definition of a decisive algebraic thought for a problematic situation. The proposed objective was, in fact, that to try to falsify the general hypothesis according to which in the phase of reasoning and algebraic or pre-algebraic generalization, the Chinese students underline a reasoning type "variational" tightly tied up to the mathematical context and opportunely formalized in the definition of a fundamental algorithm able categorizzare various numerical cases (but not only, as seen even though partly in the first experimental situation) separate through a reasoning type analogical procedural invariant. In parallel with the process of generalization of the Italian students, that is hypothesized to be came itself through the phases of reasoning, conjecture and demonstration (Ajello, Spagnolo, Xiaogui, 2005; Spagnolo, Ajello, 2008), according to a deductive logic, he can be a further aspect key in the study of the cultural aspects of "thing that varies", unknown and relationship-functional of the variable. Tied up Concept tightly to the phase of generalization and algebraic thought. In the problematic situations discussed earlier, this aspect has been almost always unexplored and it has therefore required of further close examinations. The didactic situation "the sequence" (brought following) has been defined really with this objective. A further investigation experimental call "The grid of numbers" (brought following) has been defined then in relationship to the possibility to mediate in a class multiculturale of the Inferior Secondary School with possible Chinese students algebraic knowledge contextualized according to systems of reference heterogeneous different and possible schemes of reasoning. The job of search has been in this case, opportunely duct through a job of group (on a couple of students: Italian-student, Chinese-student) from us qualitatively analyzed in relationship to an analysis a priori of the possible behaviors.

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“The sequence” Look at these four images of black dots

1° Figure

2° Figure

3° Figure

a) Can you succeed in drawing the following one? What procedure have you followed? It deduces to the answer; b) How many black dots it will have the sixth configuration? What reasoning have you followed? It deduces to the answer; c) The 1° figure has 3 black dots, does the 2° figure have 8 of them. If a figure has 80 dots which it shows up it is? It explains the reasoning that you have followed; d) Can you find a figure of 143 prepared points equally? and) you Succeed in finding a general expression that can represent the different figures? “The grid of numbers” You insert in the squares the numbers from 1 to 12. The sum of the numbers inserted in the grey squares has to be the double one of that of the numbers inserted in the white squares.

Fig. 8.

We intend to specify that for both the situations of learning proposed in class, each of the Chinese students has been able to understand the natural language of expression and the mathematical objective of solution. All the students of Chinese etnia experimentally involved, in fact, have been inserted in the Italian scholastic context since the first year of school, they don't result to have frequented any course of studies in their country of origin. The phase of reasoning has put in

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evidence of a good linguistic ability both on the understanding and on the autonomous expression. As regards the experimental situation "The sequence" it has been conducted as for the preceding experimentations, a statistic analysis implicative (with the Chic software) on the experimental protocols. They have been involved, 37 students of different etnia (4 Chinese and 33 Italians) who are attending a first class in the Superior Secondary School (IPPM). The investigation is developed to the beginning of the year to falsify the knowledge at entry to the superior secondary school. For the same nature of the situation of learning, it is shown essential to falsify the knowledges prerequisite of the whole class on the concept of in recursive way, pattern sequential and possible strategies of generalization. Knowledge discussed can be falsify through an exploratory investigation that has been conducted at the beginning from the presentation of the underlying figures: "Observes these sequences of four configurations. Do you succeed, for each, to calculate the following configuration? " a) --------------b) --------------c) --------------d) What does it mean for you to generalize? Do you know how to make me an example? Fig. 9.

For each of the three sequences, the verification at entry has been, after an autonomous exploratory phase and a job of mediation on the definition of the figure in demand (the following one) in small groups, verbalized and criticized subsequently in collegial order of class. The negotiation of the various strategies of generalization of the sequences has served then there for the definition of the analysis a priori of the didactic situation " The sequence" that we introduce, in the wrong most meaningful strategies and not, in the paragraph 6.5.1. The collective discussions on the decisive trials of the various numericalgeometric cases result in this phase important because able to force every student to the autonomous reflection on his own mental trials, to the verbalization of his own thoughts and his own strategies. The phases of job of group in the class have

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allowed the students to listen to the companions in the different phases argomentative not only exalting the aspects cognitivi but also those metacognitivis and metalinguistic. The use of this initial experimental situation prerequisite results for our ducklings meaningful and not banal for a correct approach to "The sequence". The varying of behavior used as meaningful indicators for the analysis of the schemes of reasoning of the Chinese and Italian students on their protocols of the didactic situation de "The sequence" they have been in fact you contextually define to the behavior from them underlined in the analysis of the prerequisitis and therefore to the strategies of solution, autonomous reasoning and verbalization of the group. Them, in accord also with numerous searches published early from similar problematic situations (Radford, 2000; Neria and Amit, 2004; Zazkis and Liljedahl, 2002; Malara and Navarra, 2003), they have held therefore in consideration some key aspects in the definition of the process of generalization: - recognition and use of a single variation; - recognition and use of possible inside relationships to the same pattern; - quantification of the variation; - recognition and verbalization of a possible relationship between position and pattern (description type numerical disposition); - recognition and verbalization of a possible relationship between position and pattern (description type view). The collective validation phase of the strategies of solution of the three single sequences as conclusive elements of the didactic situation of verification, have made reference for all the students to a generative thought type analogical sequential that, however, is distinguished in the phase of conjecture and reasoning used by the Chinese and Italian students. The distinction of thought has been the key of time for the mediation of the decisive procedures and therefore the conceptualization of general possible schemes of different reasoning. The last question on the term "generalization" has been thought for communicating the question "and" of the didactic situation de ("The sequence") subsequently them proposal. In the phase of generalization of the three sequences, what has mainly been discussed by the Chinese students, has been the necessity to define a possible least variation recursive inside the various sequences and therefore a possible unique rule of inside and external composition (according to the relationship inside pattern - external pattern) to each of the three compositions. If this exists. It regulates that has been almost always expressed on linguistic registers that were supported to considerations type figurative and numerical. Almost totality of the Italian students, opposite, in a first moment of exploration effected in the various jobs of group, has opted instead for the possible definition and collective discussion of how many more possible solutions of composition could recognize in the single sequences. Therefore the equivalence of these is recognized to the goals of the solution and the definition of the following figure.

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The examples of the sequences have been, in this sense, elements of discussion. The third sequence has not put in evidence of any meaningful strategy. The mediation of the strategies is introduced to the class as a monument of growth. The choice of such methodology of investigation is shown, to our opinion, remarkable for a first socialization of the cognitive styles of learning and therefore a mediation of the "knowledge" in the phase of validation of the situation (Brousseau, 1997). The schemes of reasoning shown in this first investigation prerequisite to the following didactic situation are found again then, almost totally, also of it "The sequence". this last experimental investigation is shown profit, in this sense, in a first approximation and in relationship to the environment multicultural from us investigated, for the study of the algebraic thought in the phase of generalization of the numerical calculation for tests and errors, defined as arithmetic thought, and contextually in the formulation of an expression symbolic representative a functional relationship in abstract form. The activity de "the grate of numbers" has been thought with the objective to investigate, according to a non formal approach as that of a situation of learning verbalized in the dialogue among peer performed by the outside by the teacher researcher (Ericsson et alii, 1993), those that could present as you model mental proper of the thought algebraic built footstep after footstep in an arithmetic environment through initial forms of "algebraic stammering" (Malara, Navarra, 2001). To think therefore the arithmetic algebraically. In other words, through a tool of investigation pre-structured he is passed, gradually and through an analysis of "game" among equal, to experimentally falsify in the students a possible prealgebraic thought in a rent interlacement with the arithmetic and his/her meanings. The experimental investigation therefore departing from the mediate compilation among equal (classmates of the same culture) of the grade of numbers (context arithmetician, pre-algebraic) he/she is evolved, in a context multicultural (two Chinese students and two Italians), through of the questions stimulus, to the research of a general symbolic language defined by the possibility to identify some inside rules of relationship. To the beginning therefore the exploration of the tool of investigation and the compilation of this as solution of the situation of "game", he/she is developed in an arithmetic environment, to slowly widen toward the algebra and the naïve discovery of the use of the letters and the equations. The possibility to define more solutions has allowed to explore then the "flexibility" of thought of the students. The experimental investigation has been conducted, as says, with four students attending the school of Average "G. Garibaldi". Two Chinese and two Italian affiliate regularly in third Average. This third experimental phase wants therefore to be, in line with the tools of investigation proposed, a further moment of verification of the four hypotheses of research. H1, H2, H3 AND H4. The experience of research has followed therefore two principal phases: The phase: Administration and analysis of the results of the problematic situation "The sequence", II phase: Administration and analysis of the results of the problematic situation "You grate of numbers."

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5.1.2.9 A Priori Analysis “The Sequence”

You can hypothesizes that the pupils can apply to one or more than the following decisive strategies:

Sa0: The student correctly draws the following figure but he doesn't render explicit any reasoning on the followed procedure; Sa1: The student draws the following figure and explains the decisive procedure through simple numerical reasonings on the sequence of alignment of base and superior of the black dots represented in the three brought figures; It notices an unitary increase in both the alienations (1,2,3 in the alignment of base and 2,3,4 in that superior) and it hypothesizes a geometric figure that has for base (4 black dots) and superior alignment five. It reconstructs the figure supporting itself to the pictorial register; Sa2: The student draws the following figure, he recognizes in the first three traced figures a regularity in the number of points lined up that he repeats him in vertical. The 2° figure underlines a to be repeated twice of three lined up points, the 3° figure it underlines etc four alignments of four points. The last horizontal alignment is always lacking of a black dot; Sa3: The student draws the following figure and deduces the solution reporting to the possible structuring of a square form around the black dots. The square is always incomplete of a dot in the angle in low to the left; Sa4: The student draws the following figure recognizing to every footstep an addition of a turned upside-down L always composed by black dots of greater dimensions of an unity in horizontal and vertical; Sb1: The student declares the number of dots of the sixth figure but he doesn't render explicit any reasoning on the followed procedure; Sb2: The student declares the number of dots of the sixth figure also drawing the preceding cases; Sb3: The student is to declare the number of dots of the sixth figure test to define a regularity among the introduced numerical cases: n1=3; n2=8; n3 = 15 but it doesn't underline any reasoning; Sb4: The student in to declare the number of dots of the sixth figure test to define a regularity among the introduced numerical cases: n1=3; n2=8; n3 = 15 and the positions these (1° figure, 2° figure, 3° figure). Test to formalize a possible correspondence but it doesn't underline any solution; Sb5: The student in to declare the number of dots of the sixth figure test to define a regularity among the introduced numerical cases: n1=3; n2=8; n3 = 15 and the positions these (1° figure, 2° figure, 3° figure). to correctly Formalize a correspondence and it deduces it in natural language; Sb6: The student is to declare the number of dots of the sixth figure motivating his choice in geometric terms (on the quadratic form of the configuration of little balls); Sb7: The student is to declare the number of dots of the sixth figure opportunely makes petition to an algebraic expression writing; Sc1: The student doesn't answer and doesn't underline any reasoning;

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Sc2: The student defines the position of the in demand figure in the proposed succession also drawing the preceding cases; Sc3: The student defines the position of the in demand figure in the proposed succession and deduces the solution supporting itself to numerical relationships contestualized to the typology of the proposed configuration (80 dots are equivalent to a square of 81 dots less a dot. 81 are the square of 9); Sc4: The student confuses the number of the dots with the position of the succession; Sd1: The student doesn't answer and doesn't underline any reasoning; Sd2: The student wanting to define the figure in demand in the proposed succession starts to also draw the preceding cases but it loses him after few footsteps; Sd3: The student defines the figure in demand in the proposed succession trying to also draw the preceding cases; Sd4: The student defines the figure in demand in the proposed succession and deduces the solution supporting itself to numerical relationships contestualized to the typology of the proposed configuration (143 dots are equivalent to a square of 144 dots less a dot. 144 are the square of 12); Sd5: The student confuses the number of the dots with the position of the succession; Se1: The student correctly formalizes the in demand general expression; Se1: The student formalizes but in incorrect way the in demand general expression; Se1: The student doesn't formalize in any language the in demand general expression. 5.1.2.10 Quantitative and Qualitative Analysis of “The Sequence”

In general, the strategies used by the students seem to primarily be type numerical-arithmetical. Through the reading of the experimental results three types of processes of generalization are identified founded mainly contextually on strategies of similarity (properly numerical, show up her and type pragmatic). The students that proceed in their reasonings with strategies type pure arithmetician, underline (for the questions to, b and c) strategies of similarity for tests and errors (on the various represented figures) not contestualized on any patterns of reference of the model of succession. Not underlining therefore not even a thought type pre-algebraic, they don't effect any generalization. They don't answer neither to the question d neither nor to the last one. The representative aspect is not held in consideration and the variable, in this sense, it doesn't assume any meaning if not a generator of linear sequences of numbers. The students that manifest a scheme of reasoning type show up her for the phase of generalization, they underline strategies of solution founded on possible linear sequences, read in relationship among them according to an approach of analogical perceptive thought. The variable, in this sense, he is conceptualized only not by the students in the preceding meaning but above all inside a context of functional relationship. The students that underline reasonings of generalization type pragmatic (contextually underlining strategies type numerical and show up deduces her in a

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continuous conversion semiotics of registers) they succeed in gathering the aspects key of the sequence in terms of ownership and inside relationships (to the single pattern and in the passage from the one to the other). Your generalization clearly manifests him in these cases through the conceptualization of a numerical structure unifying. In accord with (Warren, 1996, 2000), the few students that don't even manifest a naive thought of generalization (and what therefore they don't succeed in even not defining the eighth in-demand figure from the text) they underline a serious difficulty in the phase of conversion of the succession through the various approaches of thought previously defined. As it regards the situation multiculturale, a specific analysis of the picked data it underlines as, unlike quite a lot of Italian students, none of the four involved Chinese students, has expressed in symbolic formal language the in demand general expression. Perhaps intending, as in the case of the reported experimental situation to the problem of Fermat, the wording "the general expression" as research of invariant and definition and verbalization of a rule of unique composition. The protocols bring in fact, for the question d, an express description in natural language of the inside relationships the various figures brought in the text of the didactic situation and therefore of the relative figures required in the various questions of the tool of investigation. The research of the invariant possible seems to be for the Chinese students the key of time for a possible interpretation and generalization of the succession of the figures. This aspect results, according to us, meaningful also in relationship to the abilities of categorization of the Chinese students on the reading and writing of the characters linguistic writings. As seen in the chapter V, the possibility to define the semantic and syntactic aspect of a character and to interpret it therefore in relationship to his inside structure and his regularity of compositions, he is strongly connected to the decoding of this in relationship to the fundamental lines composed and to the research of a possible key of interpretation (the radical) that it assumes the form of invariant in her "generalization" syntactic/semantic of the same character. Meaningful it is for instance a sentence brought in the protocol of a Chinese student for the question and: "the makeup is that to always build a square and to remove the angle in low. All the numbers that have this rule are of this sequence. If I for example change the form of some dots, I have other numbers of different sequences", "I can represent all the figures that have numbers of points similar to the squares. Difference has to be always one. I choose any number of points, as I want, and I raise it to the square and I remove one. For instance that following is then 99 120." This type of scheme of express reasoning in a complex register what strongly supports him to that pictographic, few proper to a formalization generalizing but enough stable among the students of Chinese etnia, he is revealed however fallacious in the approach to a more abstract thought as in the case of a student that in the solution of the question d confuses the structure of the figure with the construction of this (phase of conversion from the arithmetic context to that show up her Sd5): "142 are not a square, I cannot build this figure". Her strategies that are

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glimpse in the Chinese protocols make greater reference in percentage to the varying Sa3s, Sb4, Sb6, Sc2, Sc3, and Sc4 type pragmatic. An only student underlines (through of the lines of pencil drawn on the introduced figures) initially a strategy Sa2 that then he abandons for the Sa3. As in the case of the problematic situation related to the problem of Fermat, a Chinese student brings a writing type algebraic "seed-abstract" strongly contextualized to the proposed exercise "he formulates it is: .2-1" and it allows to glimpse a possible general translation of the problem. Unlike the Italian students, for the four Chinese students from us involved in the experimental investigation, the passage to the algebra and therefore to the mass in formula of the problem it doesn't seem to be held necessary. The picked data, in consideration of the strong limitations of the sample, they must also be reconsidered if however, in relationship to a possible environment of presentation of the problematic situation, more complex and articulated that naturally pushes to the mass in algebraic formula. The experimental results can consider in this sense a possible first approach to the comparison of the results in the searches of Cai on the Chinese thought (Cai and Stephen, 2002). For the Italian students, the situation introduces him variegated even if don't miss the points of contacts with the Chinese students. Some, approaching themselves to the problem, they underline some strategies for tests and errors that actually maintain to the resolution of the questioned (what besides they don't resolve because "too complicated to draw"), they are limited by this reasoning type numerical place and they remain anchored to a numerical thought not generalizing (Sa1, Sa2, Sb2, Sb3, Sc1, Sc2, Sd1, Se1). Others, initially underline also the strategies Sa2, Sa3 and Sb2, they almost immediately manifest a thought type pragmatic and they pass therefore, through quite a lot of numerical calculations of discovery and verification (graphically visualized also) to a symbolic formalization (Sa3 Sb6 Sb7 Sc3 Sd4 Se1). The result seems to confirm the experimental contingencies discussed in the preceding experimentation on the "Problem of Fermat". The implicative graph and of similarity (realized, as in the preceding cases with the Chic software) also underlining interesting groupings of variable, they are relatively results express train according to low percentages (70%, 67% and 65%) due perhaps to a sample of investigation few "representative". For this motivation they are not brought following. This first investigation related to the third experimental phase seems to be, in line with the tools of investigation proposed, and with the limitations of the case on students' sample involved, a further moment of validation of the four hypotheses of research. H1, H2, H3 AND H4. In accord with Radford (2001, 2008), the proposed investigation specifically needs however further reported close examinations to the passage to the algebra and therefore to the study of the phase of transition from the arithmetic, from the function deictic of the language (in this written case) to that generative, central for a full understanding of the phenomenon and possible analogies and differences among the involved students. In this context of job of thesis he has not been possible to define an analysis semiotics it specifies some situation, that instead he holds necessary. This, therefore, shapes him as an open problem.

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5.1.2.11 A Priori Analysis of “The Grid of Numbers”

You can hypothesizes that the pupils can apply to one or more than the following correct decisive strategies and not:

S0: I know: The student declares not to be able to insert in the cells the number with odd sum; S1: The student declares to want to insert in the cells the number that are all peers; or all odd ones to do yes that they are of equal sum; S2: The student divides the equal numbers from the odd ones; S3: the student conjectures possible combinations (select at random) inside the cells; S4: The student conjectures possible combinations among the numbers equal to the inside of the cells; S4: The student conjectures possible combinations among the odd numbers inside the cells; S5: The student declares to assemble only his attention on the white cells and only later on the grigies; S6: The student declares to want to insert in the white cells the tallest numbers. The more lower part in the external cells; S7: The student divided the grate in subunit of triple declares to want to work on small unities to report at the end; S8: The student conjectures possible combinations among the numbers inserted in "subparts" of the grate and the whole grate of numbers; S9: The student declares the constant sum of the numbers inserted in the grate; S10: The student declares a sum constant for the numerical combinations of the inside cells; S11: The student conjectures possible combinations of possible numbers (among those first discussed) to position in the white cells with constant sum; S12: The student conjectures possible combinations of possible numbers (among those first discussed) to position in the cells grids with constant sum; S13: The student correctly deduces possible generalizations of the grate through the approach to the variable. S14: The student conceptualizes the possibility to define more reported solutions to the limitations of the variable. 5.1.2.12 Qualitative Analysis of “The Grid of Numbers”

The typology of investigation discussed in this second investigation of the third and last experimental phase of the job of research, has been select, as says, to favor, as last objective of the job of thesis, a first approach to a proposal of integration of the knowledge in game and the mediation of the experiences through different perspectives among students of Chinese and Italian etnia. Phase of proposed mediation even though partly also in the preceding experimentations but in this more evident case. The proposal of a particular situation as that realized in this last investigation has, as says top of this chapter different motivations. As before consideration,

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certainly, deals with a problematic situation that is able to our opinion well to suit for the considered scholastic level (Inferior Secondary School, class III) and really in operation of this, to allow us on the matter of research, a parallel analysis of possible different phases of conjecture, reasoning and reported verbalization to the arithmetician-algebraic thought, ipotizzabili in the free discussion among equal of students of different etnie. Of other song than the experimental situation, as after all that just discussed in the preceding paragraph, is able, to our opinion, to help us in the study of some aspects of the period of transition between the arithmetic language and that algebraic and pre-algebraic analyzing possible analogies and differences in the schemes of reasoning used for the conceptualization of the idea of variable as unknown and relationship-functional. The methodology of investigation continues with, as says, for a couple of Chinese and Italian students attending the same class24, can schematize him in this way: 1 - presentation to the two couples of students of the proposed activity: reading and understanding of the text and the rule of composition of the grate. 2 - explanation of the methodology of the "game": - every couple (team) singly works (Chinese student - Chinese student, Italian student - Italian student;)25; - every couple, through a comparison and a mediation of strategies inside his own team, it has to compile the proposed grate and to get ready to subsequently deduce it with the other team, putting in evidence: criterions of compilation, possible solutions, possible regularities among the inserted numbers, possible ownerships of the operations used etc.26; 3 - finished the phase of compilation of every team, he proceeds to a verbalization of the strategies of solution; 4 - every team is compared with the couple "opposing". In this phase it pushes him to a possible reinterpretare of possible different strategies and therefore a critics reading of these in comparison to the proper ones; 5 - you verbalize and discussed the various strategies the teacher experimenter, through of the questions stimulus, proposes to both the couples, a possible rilettura the schemes of compilation in pre-algebraic terms: he does therefore gradually resorted to the introduction of the letters as a tool of exploratory investigation on the numerical regularities of the grate. Regularity previously found by the two couples and mediate among them; 6 - They observes possible difficulties in the acceptance of the new writing and in the semantic-syntactic control of this in relationship to the numerical grate numerically compiled in precedence; 24

Four have been involved some students that have participated in the investigation experimental precedent ("The sequence"). 25 This methodological choice has been forced by the refusal of the Chinese children in to work since the first phase of the game to mixed group. 26 In accord with the Theory of Brousseau, the researcher experimenter has assisted both the groups, his/her role is limited to that of external observer and " facilitator".

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7 - in this stadium of exploration of the phase of transition among the arithmetic thought that algebraic in the phase of introduction of the letters, a possible aware reflection is favored from the students on the equivalence of writings what: a+a = 2a; a+2a = 3a (odd number); 2(a+n)=2a+2n that they allow the transformation of the numerical writings involved in others more expressive in the form of equation. The validity of the use algebraic formulas every time comes verified on some numerical grates previously realized (the letter assumes, in this sense, the idea of unknown indeterminate); 8 - contextually, the students are invited to deduce multiplicity of solutions and generality of thought on the compilation through of the applications stimulus contemplated: "It is normal to get numerous equivalent representations but "different" among them as for instance…" "the solutions that you have gotten are unique therefore? You reflect and compared you among you"; 9 - The manifold activities of numerical algebraic verification on the letters, try to favor a possible devoluzione to the pupils of the choice to represent all the numbers hidden through a symbolic representation that holds in consideration the rules of the "game" and the ownerships numerical discoveries previously. Contextually, it is tried to devolve through the mediate verbalization among the couples a progressive social construction of the meanings; 10 - as last phase of discovery of the grate proposes him to the students to work in the built algebraic environment and to again feel the validity of the symbolic writing through the appeal to different numerical cases. Each of the phases above described (from the third one in then) has been, as says, mediate orally among the couples of students (Chinese/Italian). This has allowed us, even though in a first hypothesis, to analyze those that could present as stable schemes of reasoning and to compare them with the results underlined in the preceding experimental investigations in relationship to the hypotheses of research. In the impossibility to propose the transcript detailed of the experimental situation (is the Chinese students that those Italians have refused to be recorded audio/video) we bring some of the meaningful behaviors annotated for the two cultural typologies of couples of students. We point out with C1 and C2 the Chinese couple and I1 and I2 to the Italian. In cursive outlined, some of the sentences brought by the students27 In cursive we transcribe someone of the meaningful sentences brought in the salient phases by the students and transcribed by one of the teachers experimenters involved in the research. The teacher's role experimenter has only been that of mediator and external observer.

27

For a complete and correct evaluation of the behaviors of all the students, to the Chinese students he/she has been asked to only express him in Italian language. This aspect doesn't is not result to them difficulty it approves the good knowledge of the language.

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Table 4. The table brings some of the discussed meaningful sentences from the students in the various phases of the didactic situation "The grid" Chinese couple The first step of compilation of the chart is underlined in the introduction (from C1, without mediation with C2) of the numbers 1 and 2 in the cells aloft. 3 as sum of these in the cell q9.

Italian couple The student I2 oversimplifying the set problem starts to insert at random select numbers from 1 to 12 (S3), checking with regularity the external and inside sum of the grate.

C2 underlines the error and reflects in silence. Then it declares to want to separate the two squares of cells (S5); C1 doesn't seem of accord but it doesn't propose an alternative strategy. It reflects in silence.

The student I1 doesn't propose any strategy; The student I1, seen the difficulties of calculation of the companion it proposes to reread the text of the problem and he writes on the sheet reporting the grate the sign "x2" deducing to the companion the decision in terms of call to the rule on the whole grate "these added have to be the double one of the other sum."

C2 proposes then to divide the grate in two parts lists of triangular form (q1,q2,q3,q4, q9, q10) and (q5, q6, q7, q8, q11, q12) and it deduces this choice with C1 in terms of simplicity of calculation. I2, taking back his strategy for tests and er(S8) C1 accepts this strategy of solution and rors, he asks experimenter to the teacher if it proposes to use the greatest numbers (S6); is possible to repeat the numbers in the grate and he/she explains the application with the ida immediately to write 2 as sum of 1 and 1. I2 mentions to a strategy type casual and it inserts 3 and 4 in the cells q2 and q9. I1 invites him to the reasoning. "It takes logic! " C2 writes in succession in the cells q3,q4, q10, the numbers 4,5,6. it Deduces with C1 the correctness of the choice recalling the rule of the "game". C1 arranges and proposes to make the spread out thing for the triple q5,q6 and q11.

The sheet that reports the grid is changed; I1: "If the sum has to be the double one, then…they (the outside) added they have to be the double one of the other … correct?" I1 conjectures possible incorrect configurations and not (S0, S1)

I2 doesn't deduce. C2 declares to want first to complete the first underlined triangular configuration I1 after long conjecture, you orally verbal(q1,q2,q3,q4,q9, q10) and only later to izes with the companion, on the ownerships of grouping of the addition ("I remember pass to the other (S8); that this could be done") it recalls the associative ownership of this and it deduces to I2 the possibility to separate the grate in four ternes (S7). I1 inserts the numbers 1 (q7), 3 (q8) and 4 (on q12, that then straps autonomously with 2) "I have read badly the signal that I was me fate."

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5 The Experimental Epistemology as a Tool to Observe and Preview Table 4. (Continued) It proceeds writing 6 in the cell q10 and he/ she asks to I2 to find two numbers that give 6. You straps in: "No, they have to give 12." I2 favors the application and they tries, for attempts and errors on couples of selected numbers at random within 12. They find the numbers 5 and 7 and they write them in q3 and q4.

They underline the same strategy for the cell q9 (what originally had been compiled with 4) inside which insert, of commune accord, the number 10. After an oral examination of the couples that gives as resulted 20, they hypothesize the choice "12 and 8" that however they don't write; C2 inserts therefore the numbers 1, 2 and I2 planning a reasoning on the possibility to 3 in the cells q1, q2, q9 and it verbalizes only define in the white cells equal numthe solution with C1 recalling the rules of bers, it proposes to consider the number 8 the "game"; as possible number to insert in the cell q11. I1 and I2 write all the possible couples that give 16 as resulted. Planning a reasoning type logical deductive "We know that 12 more 4 give 8, 11 more 9 give 10, if we insert in the cells q1, q2, q9 and q5, q6 and q11…?" I1 and I2 arrange on the correctness of the solution (S11-S12); The teacher experimenter asks to render explicit the choice of the number 8, contextualized from the Italian couple, to the possibility to have to the inside (in the white cells) only equal numbers.

C1 proposes to pass to the configuration (q5, q6, q7, q8, q11, q12) and it mentally tries (it underlines an use of gestures (index and ring finger of the hand) to mark the cells and the relationships among these, divided in triple (q5, q6 and q11), Both the students don't answer to the ques(q7, q8 and q12). tion and they justify the reasoning as a posC2 follows in silence the gestures of the sible uniformity of equal and odd numbers companion; (S2); C2 after an invitation of the teacher experimenter to verbalize his thoughts and to share them with the companion, interrupts the silence and it declares not to be sure that the strategy of the separation in two parts (and then in others two) is correct in the following global reading of the grade. "The rule says that this has to be true for all the numbers that we put in the grade". "we can see them as union?"

To the application of the teacher experimenter to change some numerical positions, I1 and I2 they declare the availability to find other numbers that, organized in triple, involves as those marked (S7-S14);

5.1



C2 proceeds to a verification of the rule The teacher experimenter invites them to refor the cells (q1,q2,q3,q4, q9, q10): "here read the rule of the "game"; it works… I now have to follow the same reasoning in the other part of the square"; C1 confirms the necessity to still find the cells hidden second a reasoning in numerical triple (S7). C2 declares to want to again insert the spread out numbers not accepting the fact that in the delivery the possibility renders explicit him to insert all the numbers from 1 to 12; C1 after a long break of observation underlines a regularity in the cells already compiled: "it looks, every time I add one in diagonal". C2 confirms this scheme of reasoning and together they decides the blank cells to be compiled with the numbers: q5=7, q11=8, q6=9, q7=10, q12=11 q8=12; (S12-S13) Contextually to the observation of the teacher experimenter to falsify the correctness of the solution, C1 and C2 they proceed to the sum of the cells and to the verification of the rule of the "game" (S9S10); C2, discovering some regularities in the grate declares "it is true, if it is worth here (q1,q2,q3,q4, q9, q10) and here (q5, q6, q7, q8, q11, q12) it is also worth in the sum"; To the application of the why of this rule gimmick and therefore a possible general verification of this affirmation, C1 and C2 remain in silence.

I1 and I2, recognized the difference of the rule required by the "game" (I1: "Us this we know, the rest we have told him/it from the rule of the addition") they underline the possibility to add all the numbers of the grate. Both in the external part that interns (S9-S10). The application of possible other possible corrected configurations is accepted both from I1 that I2. The teacher experimenter asks them to discover possible constants of the "game" (the external sum, that inside of the grate and the gotten one adding all the inserted numbers); Proceeding as in the preceding case, I1 and I2 they realize other two grates and they ascertain the constant sum for the three greatness (S13-S14). They hypothesize possible other corrected configurations, with sum constant; Declared expressly the various strategies of composition and therefore the different solutions of the "game" you underline in precedence, the teacher experimenter proposes to the couple to prepare a report to introduce opposing to the team putting in evidence:

C1 declares then: "I have put together them, you the you can separate and to unite, the same comes, if you separate them and pi you unite them it usually criterions of compilation, possible different comes easier." solutions, possible regularities among the inserted numbers, and possible ownerships C1 and C2 returns then in silence; of the used operations. The students discuss on thing to deduce to the other team. They arrange in to introduce the necessity to follow the rule it dates for the compilation of the grate in his entirety and to have found four possible solutions.

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After quite a lot of time, in silence, (they are hypothesized some possible mental calculations on the numbers inserted in the cells) C2 orally verifies with the companion the ownership used for the operation of addition (but it doesn't orally recall in this sense any mathematical knowledge previous). C1 nods (S9-S10); C2 discovers besides the possibility to change among them (to couples) the cells "so much the result would not change". C1, initially a little convinced for a possible violation of the rule of composition it continues with in precedence (the +1 rule), apparent after tests type numerical a favorable attitude. "So much the rule him preserve, contrarily"; To the application of the teacher experimenter to change other numerical positions among the cells; C1 and C2 are manifested very hesitant because this would violating the +1 composition rule in diagonal. The teacher experimenter invites them to reread the rule of the "game" and to the possibility to modify the grate varying the positions of the inserted numbers. C2 separately verifies and in writing that the rule of the "game" continuous to be worth in general. It observes that he/she is true for the triangular configurations (q3, q4,q5,q6, q10, q11) and (q7, q8, q1, q2, q12, q9) but not for the various groupings of three cells previously individualized. Strongly confirming the bond to the separation of composition, follows before, it doesn't accept to violate it. After quite a lot time of silence, C1 deduces the possibility of change in the configuration rereading the rule of the "game". "here it says that the sum of all the numbers of the cells has to be the double one, this is the rule that we have to follow"… "it regulates her principal it is that of the game, the other no, we have given her us"… "this is the formula that we have to follow";

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C1 and C2 deduce possible different numerical configurations only gotten rotating the cells already inserted (S12-S13). In every shaped numerical case, C2 verifies both the rules and deduces the possibility to find again her both or less. For this whole phase of research and reasoning, C2 has a preference for a verification you type part-everybody. It proceeds then to a first local verification of the correctness of the rules for "to generalize her/it" trying her/it on the complete grate of numbers. The application of possible other possible corrected configurations is denied both from C1 to C2. The teacher experimenter asks them to discover possible constants of the "game" (the external sum, that inside of the grate and the gotten one adding all the inserted numbers). C2 after a break of silence, declares that the sum of the numbers has to constantly be always equal to 78 ("the sum of the first 12 numbers"). It Seems to recall a consolidated arithmetic fate. C1 arranges nodding. Calculating her other sums, they discover and they discuss among them the constancy of these. The teacher experimenter informally asks to think about possible different configurations from the realize ones. The application is again denied both from C1 that C2; Declared expressly the various strategies of composition and therefore the different solutions of the "game" you underline in precedence, the teacher experimenter proposes to the couple to prepare a "report" to introduce opposing to the team putting in evidence: criterions of compilation, possible different solutions, possible regularities among the inserted numbers, and possible ownerships of the used operations.

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The students discuss on thing to deduce to the other team. They arrange on to introduce the necessity to follow the rule and to divide the grate in more ways to falsify her/it "generality" of the solution. They arrange then in to underline the discovery of the regularity of the sum gimmick and of the numerical regularities verified as first step for the didactic situation. Comparison among the couples "culturally different" The teacher experimenter underlining some differences of compilation proposes to both the couples a possible verbalization of the schemes of reasoning used for the solution of the didactic situation. In operation to criticize solution together of possible analogies and differences of compilation, convenience of these, best strategy for the research more solutions, possibility of definition of numerical ownership subtended etc to the application of the text. After a long phase of comparison and mediation of the knowledges numerical-relational you individualize inside the chart, in hold connection with the numerical strategies underlined by the two teams, the teacher experimenter proposes to both the couples, a possible rilettura of the schemes of compilation in pre-algebraic terms. He applies therefore gradually to the contextual introduction of the letters as tool of exploratory investigation of the numerical regularities (correct and not) of the grate. Arithmetic truth discussed by the two couples and you face, through the debate with the students, in a change of register type pre-algebraic. Matters of discussion have for instance been: - symbolic writing of equal number and odd number underlined by the Italian team in the research of different possibilities of definition of other configurations; - "rule-Chinese of +1": symbolic writing: (to), (a+1), (a+2). Contestualized from the Italian students to the not parity of the number written as (a+1) and "generalized" from C2: - A2: "the rule that you/they have found them, we could not write because a+1 is odd" - C1: "but a+1 is always worth. We also have 8 in the low square (in q11) and there it is worth" - equivalence of writings: a+a=2a; a+2a=3a (odd number); 2(a+n)=2a+2n (always Contestualized in circle concrete numerical arithmetician on the configurations of the two teams): - possibility to point out all the cells with letters separate and following redefinition of these in accord with the rule of composition (read in the form of single triple, triangular configurations and whole grate after an initial difficulty from all the students in to accept the new writing (difficult semantic-syntactic control of this). - I1: "we can use the letters to, b, c, d,… that then we can replace with the numbers as in Geometry with the segments" (I1 underlines a thought type pre-algebraic legacy to the test of the variable on an arithmetician-numerical context. The idea of variable is in this phase that of "unknown" - C1. "but if we point out with the symbol small q and great Q the squares (it points out q1e q2) also the other ones we are able to write her so, we multiply two for the sum of the symbols Q " (C1 seems to underline with this affirmation an idea of variable and symbolic writing report her functional between q and Q) - I1: "… correct… opposing quadrates has been better… so we avoid there so many letters, that it then comes difficult" (I1 assimilates the scheme of reasoning of C1 and does him/it really)

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- algebraic translation of the observation of both the teams of the constancy of the total sum (Si+Se=78), of the external sum and that inside. The validity of the single symbolic expressions has been of time in time verified on some of the numerical grates previously realized (the letter assumes in this sense the idea of unknown indeterminate); Contextually to the observations algebraic sorts and to different realized grates and discussed among the couples, in the students the possibility was born of almost spontaneously to conceive the manifold solutions for the situation didactic proposal through "an only" literal grate that, as it declares C2, "holds together all the hidden numbers, the rules of the game and the ownerships numerical discoveries that we have discussed together. In this then one can decide that number to put in thing base he wants". This possibility is recognized, even though with some initial resistances, from the Italian students. In the contextual reconsideration of the impossibility to initially build different grates from those realized C1 it is expressed this way: "We have been wrong because we could put in the squares other numbers. In the letters that we are writing you/they can be us so many cases…. but the grate is not unique because he/she can turn and then I can put so many numbers to the place of the letters. Inserting the numbers, we owe remember us the rule for the game"

This last investigation related to the third experimental phase seems to be, in line with the tools of investigation proposed, and, also in this case, with the limitations of the case on students' sample involved, a further moment of validation of the four hypotheses of research. H1, H2, H3 and H4. For the Chinese students, in the analysis of the phase of transition from the arithmetic/pre-algebraic thought it emerges once more, also in this case, a concrete behavior strongly type algorithmic-procedural driven since suffered by a facility of thought type report her functional on the grate (H2). this aspect is underlined not only in the compilation by the same grate since the first step (type pure numerical arithmetician), but also in the semantic control of the symbolic writing gotten in the second phase of the situation of learning through the mediation with the Italian students. These, "overcoming" a reasoning for tests and errors type casual that instead it is the origin of the Italian investigation in the phase of reasoning and conjecture (H2), they immediately define an algorithm solver that drives them in the insertion of the in demand numerical values (H2). The passage from the "false" algorithm to that communicated by the situation is the key for the configuration of possible different numerical representations. The semantic control type report it arithmetician it is however, in this case, from obstacle for an ampler generalization of the grate. For them, the algorithmic-procedural approach seems so meaningful in terms of compilation and verification of the procedure to initially limit them in the research of further schematizations and therefore in the autonomous algebraic development. I hinder that is to overcome through the mediation with the Italian couple that instead, initially proceeding for tests and errors, according to a reasoning type logical deductive on the single insertions (H2), it comes to a greater numerical generality of the grate. Generality that comes so discovery, in the second phase of the "game", also from the Chinese couple. The semantic control of the algebraic writing is initially, however, of obstacle for Italians; their idea of variable is that of unknown. The overcoming of this

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comes out of the mediation with the Chinese team that underlines, as in the preceding experimentations, a greater ability on the algebraic symbolic register and the manipulation of this (H3). All the students come, at the end of the didactic situation, to the algebraic schema of the grate and the reading of this in terms of variable is relationshipfunctional. The hypothesis H4 on the structure of composition of the written language and as this, in a first approximation can favor through the phase of coding and decoding the acquisition of particular aspects of the concepts of unknown, variable and parameter it is, even though partly, to our opinion, validate in the didactic situation. The behavior of the Chinese couple in the reading of the grate, through a parallel investigation on the numerical composition part / whole and subsequently in the acquisition of the grate as tool of representation of manifold configurations, it recalls, to our opinion the operations cognitive discussed in the chapter V on the reading and writing of a character written Chinese, in its autonomous composition and in the insertion in a written text. This aspect remains a problem open of the thesis, however, and also needs greater close examinations in different mathematical circles. We can reassume the results of the paragraph 5.1: Context AlgebraicFormalized

Pre-algebraic

Argomentation/ Generalization

Use of the parameter as attainment of the algebraic thought.

Chinese students In formalized algebraic contexts, even if don't underline always an aware algebraic thought, they macaws mostly able in comparison to the Italian contemporaries as good "solvers". The experimental contingency seems to confirm this to all the scholastic levels. Good capacity of manipulation on algebraic formulates. In pre-algebraic contexts they tightly underline to arithmetic reasoning schemes connected with to strong memorization of "arithmetical facts" that help them in the definition of holistic thought as to relational thought. The reasoning and the phases of argumentations in the process of generalization (“to think for cases between Arithmetic and Algebra”) are defined through an algorithmicprocedural thought. The conceives of parameter as possibility to express different problems through "similar" writings. The role dl context as significant to the solution of the problem.

Italians students Don't underline a mature algebraic thought. The idea of variable is limited to the idea of "thing that varies" (Malisani 2006).

They underline difficulty in the individualization of the unknown one and the control of this to semantic and syntactic level. The Arithmetic Thought, predominant, it results of obstacle to the Algebraic thought. Reasonings in natural language, difficulty in the phases of treatment and conversion of the registers semiotici (D’Amore, 2000, 2003). Absence of awareness on the role and use of the parameter as possible tool of generalization.

5.2 From Natural Languages to Logical Linguistic Aspects

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5.2 Fro m Natural La ng uages to Logical Ling uistic A spects

5.2 From Natural Languages to Logical Linguistic Aspects28: Cognitive Styles of European and Chinese Students 5.2 Fro m Natural La ng uages to Logical Ling uistic A spects

In relation to what we discussed in the chapter 4, the passage from the analysis of language to the analysis of the logical-linguistic aspects is not difficult; in the historic Chinese tradition the main reference is that of the “School of Nouns”, 370-310 B.C. It is exactly in this period that the logical-linguistic paradoxes were proposed which are taken into consideration in this work: 1.

2.

“the distinction between that which you receive more and that which you receive less is the minimum of reception and distinction: that which in all beings is entirely received and entirely distinct corresponds to the maximum of reception and distinction” “a white horse is not a horse“

In agreement with the authors cited in this paragraph, the “School of Nouns” and the “Dialectics” are given a central role in the elaboration of scientific thought in China. Kosko’s hypothesis of fuzzy logic as a logic of reference of unknowing Chinese thought (at least until the end of the 1900s) represents one of the main references for this and other works (Ajello, Spagnolo 2002). We analyse the first paradox using the following diagram. It is an attempt to give greater strength to Kosko’s hypothesis. Table 5. The Paradox

From the point of view of Fuzzy Logic

From the point of view of Bivalent Logic

The distinction between that which you receive more and that which you receive less is the minimum of reception and distinction: that which in all beings is entirely received and entirely distinct corresponds to the maximum of reception and distinction.

A set A and the set not-A have in fuzzy logic an intersection which varies from a minimum to a maximum that depends on the possibility of receiving A and not-A and distinguishing A and not-A.

A proposition of this kind does not enter in the Aristotelian syllogisms and is not found even in Hegel’s dialectic.

28 29

From the point of view of Classical Chinese Culture It is part of the classical diagram of cohabitation of opposites as in the case of yin and yang29.

This paragraph was written by Maria Ajello, Filippo Spagnolo and Zhang Xiaogui “The symbol yin-yang is the emblem of nuance, it represents a world of opposites”, it also represents the instrument of fundamental reference of Chinese thought in all fields of knowledge both humanistic and scientific. We can compare it to the organisation in categories of Aristotelian logic.

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5.2.1 Some Reflections on “Arguing, Conjecturing and Demonstrating” in Chinese Culture with Relation to Occidental Culture This paragraph analyses in a schematic way some substantial differences found in the history of Chinese thought and in the history of western thought. As we said before, in the comparative analysis of science in pre-modern China and the west, Geoffrey E.R. Lloyd (2001, pag.574) says that, “The aspirations of ancient Greek tradition represented by Euclid, which proposed deducing all mathematics from a single set of indemonstrable but evident axioms were not shared by the Chinese at least until the modern age. In China, as a matter of fact, the goal was not axiomatic-deductive demonstration, but gathering unifying principles from all of mathematics.” The analogies with the work of Fibonacci are very strong and as always are about concrete problems, analysing them by classes of problems and also with the intent of constructing a didactic work. In the history of education, in general, the comments on the Nine Chapters and the Liber Abaci have also represented a stimulus for investigating new mathematical paths. The following table analyses some differences in reasoning patterns in a holistic vision. Table 6.

Occident 1200 algebra: no formalisation

Orient 200 B.C. algebra: no formalisation

Paradigm of geometry, equations

Positional system, matrices (system of the rods) Aprioristic formulas that hide the Solving equations by means of algeprocesses, favouring, with the re- braic manipulations with the stratesult, determinism gies: 1) making equals, 2) making homogeneous, and 3) research for fundamental algorithms Reductio ad absurdum in a potenExisting infinity of operations tial infinite

5.2.2 The Algorithm as Fundamental Element of Arguing and Demonstrating? In Chinese mathematical thinking the main reference is the algorithm. We just discussed this aspect analysing the historical–epistemological development of the Chinese mathematical thought. It plays a central role in the Canon of mathematics and also represents a tool for algebraic demonstrations. In the solving of a problem which foresees the rule of three (substantially it has to do with the uniqueness of the fourth proportional), for example, the initial data are considered as conditions


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(if… then) “if I have a certain quantity of silk then I have spent a certain amount of money”, and also the solution of the question “What quantity of silk can I buy if I have a different amount of money than previously?” Can express, in the same way (repeating the previous condition), “if I have a certain quantity of silk (which I don’t yet know) then I have spent a certain amount of money (which I know)”. Thus, the variable is identified and with the process of reduction to unity (by means of the properties of the proportions, in our way of proceeding), the unknown value is obtained. The process for the solution is standard and is therefore an algorithm. Demonstrating the validity of that reasoning means demonstrating the correctness of the procedure (use of the properties of the operations) in the steps of the algorithm. The algorithm is a combination of an iteration and of chosen ‘conditionals’. The chosen conditional is a first interesting element of the pattern of reasoning: 1. 2. 3.

Iteration. Conditionals (If…then…). Assignment of variables.

Human thought is based on heuristics and not on algorithms as understood in the formalised western sense. Human decision makers formulate their decisions on subjective heuristics. These heuristics are founded on personal experiences, on (abductive ?) extrapolations and on probabilistic (fuzzy?) evaluations of the costs and benefits with the goal of arriving at the least risky decision possible in the presence of the scarcity of available objective data.

5.2.3 What Are the Stable Reasoning Patterns in Chinese Culture? Each argument is concluded with phrases of this type “from here the result”. The algorithm is seen as an instrument for demonstrating the precision of an argument. If there are successive divisions, in geometry for example, the algorithm is declared correct only when it is demonstrated that in the process followed the quantity not yet dealt with tends toward zero (recall Archimedes’ method of exhaustion). One stable reasoning pattern is the following: “Making homogeneous and making equal”: (from the commentaries of Liu Hui, 263 B.C. (Chemla, 2001, pg. 142)). “Multiplying to separate them, simplifying to unite them, making homogeneous and making equal so that they can communicate: how could these not be the fundamental points of mathematics?” The demonstration is not only the correctness of the reasoning. “Making equal” and “making homogeneous” which represent concrete indications on algebraic manipulations enter into play, but also strategies of reference for then being able to concretise the correctness of the reasoning through the algorithm. An interesting example of the “Making homogeneous and making equal” is that of the rule of three (from the commentaries of Li Chunfeng, 656 B.C. (Chemla, 2001, pg. 142)). This algorithm once again is an operation which ‘makes equal’ and

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‘makes homogeneous’ (in the reduction to unity). So, the rule of three, as a fundamental algorithm, is the parallel in western culture of the postulate. The fundamental algorithm can combine itself several times always arriving at a sure argument. As Liu Hui observes, applying such algorithms, the values should not change and this guarantees the truth. Therefore, particular attention must be given to the examination of the algorithm on the classes of problems to be able to highlight its correctness. One strategic objective of the Chinese was that of correlating the different processes of calculation employed in diverse areas of mathematics for demonstrating their unity (research on invariants). Needham (1985) maintains that after 1700 the two cultures fused, while P. Engelfriet (2001) maintains that this process was longer and perhaps is still being carried out. Table 7. Scientific technological revolution.

Occident

Orient

1600 scientific revolution: bivalent logic Tool of bivalent logic: a priori knowledge of the possible scientificmathematical and technological modelling activities. XXI century scientific revolution: fuzzy logic? Tool of fuzzy logic: a posteriori knowledge of possible scientificmathematical and technological modelling activities. Logic of analogies? Logic of correlations?

Inferences

How one knows

How to confront, today, the question from the viewpoint of the science of complexity

1) Inductive 2) Deductive 3) Adductive

Categorical diagrams (Aristotle) Manipulations of algebraic formulas out of context for constructing abstract modelling activities to foresee phenomena in a deterministic way

Semiotics? Systemic approach?

Semiotic inferences. 1. If…then… 2. Adductive 3. Making equal 4. Making homogeneous 5. Algorithm - Interaction - Conditionals (If…then…) 6. Assignment of variables

Consciousness by way of one’s whole body: modern neurophysiological theories, manipulations of algebraic formulas always referred to a context (as in tradition) Embodiment?

The computerbased demonstration (for example the theorem of the four colours). The technological applications of fuzzy systems. Possible tool of unification of the consciousnesses: neuro-physiolocial, consciousness by means of one’s own body, overcoming of the divisions of the mind – body heritage of Cartesian philosophy, holism.


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In this table, a diagram of comprehensive reference on some significant differences between the two cultures is presented with respect to the cognitive instruments of the deducing. Naturally, such a table is still a work tool to be perfected and with interesting open problems to be discussed again. There exist, effectively, many analogies with western thought at least with respect to the recent developments of the neurosciences. Presumably, the most important reference is that of acquisition by means of Models and the Hierarchization of Models which correspond to Deducing for organising arguments. Such Hierarchization is strictly tied to linguistic organisation: Good order depends entirely on the correctness of the language which according to the point of view of the authors is in agreement with the fuzzy approach: Table 8. Indications for correct argumentations according to Chinese thinking Correct designation and correct predication: these are the practical indications which are indicated by numerous Chinese intellectuals30

Interpretation according to the Fuzzy thinking of the passages by topic Correct Fuzzy relationship, conforms to concrete situations: translation of the linguistic rules in inferences fuzzy sets on fuzzy sets.

5.2.4 How Were the Situations/Problems Chosen? Each situation foresees a possible reasoning pattern, but does not exclude others (the questionnaire is in appendix 1). In questions 1 and 4, the term “prove” is used deliberately and in 3 “demonstrate” is used. This is because the first and the fourth questions lend themselves more to processes which induce proofs to empirical attempts. The third, instead, foresees reasoning by deduction, in any case, whatever the technique may be (by means of the representation of the possible cases or not) to arrive at the solution. The fifth necessitates reasoning by “exclusion of cases” which comes closest to a reasoning to the impossible. Question 2 is a paradox of the traditional Chinese culture which dates back to the “School of the Nouns” (370-310 B.C.), that plays on the linguistic ambiguity relative to qualities and it lends itself very well to analysing the different oriental (Chinese) and occidental points of view. The argumentations for confronting questions such as the ones proposed are closer to natural reasoning than to mathematical demonstration. The analysis of the different discursive forms and the different levels of organisation of the argumentations produced by the young people in the sample chosen offer, however, 30

In reference to Tao, ancient Chinese philosophy, there are some analyses of a Complex of very close ideas which could follow the following pattern: 1. Order 2. Totality 3. Responsibility 4. Efficacy

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the possibility of distinguishing and comparing different reasoning patterns and making some instructive observations about different behaviours. The only implicit bond to which the argumentations are subjected because they are considered acceptable is the pertinence. 5.2.4.1 The a Priori Analysis of the Situations. We Report the a Priori Analysis of the Hypothesized Behaviours

1a Does not answer 1b Does inductive reasoning but does not arrive at the solution 1c Places a trimino at the centre and resolves inductively and expresses it in NL 1d Does proofs ( 22, 23, …) and attempts at reasonings, but does not arrive at the general case 1e Uses the linguistic register of arithmetic deducing (ex 2nx2n – 1 =22n-1, and multiple of 3?) 1f Does inductive reasonings and arrives at the solution by with processes different from case 1c (in LN) 2a Does not answer 2b Demonstrates that it is false using the language of sets also with graphic representations 2c Manages to deduce both by demonstrating that the proposition is true and by demonstrating that it is false without however posing the problem that he is in the presence of a paradox. (LN) 2d Tries to demonstrate that it is true 2e Demonstrates that it is true not separating the quality from the object (fuzzy behaviour) 2f Understands that he is in the presence of a paradox 3a Does not answer 3b Does not answer correctly, not distinguishing premises and consequences 3c Answers correctly but does not give reasons for his answer 3d Answers correctly and gives reasons for his answer using the language of sets also with graphic representations 3e Answers correctly and gives reasons for his answer using tables (use of matrices in the patterns of reasoning) 3f Answers correctly and gives reasons for his answer using patterns of reasoning of the Aristotelian syllogism (proportional calculation, uses the language of logic of the 1st order) 4a Does not answer 4b Gives a wrong answer or rather does incorrect reasoning 4c Makes attempts (tries in various cases) because he does not know how to interpret the quantifier “at least” 4d Answers correctly and gives reasons with combinatorial reasoning using tables 4e Answers correctly using division by distribution 5a Does not answer 5b Gives a wrong answer and does not give reasons 5c Answers correctly but does not give reasons


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5d Answers correctly and gives a solution by means of reasoning to the impossible using, however, NL. Without any graphic help. 5e Answers correctly using combinatorial reasoning (considers all the possible combinations of names, surnames and ages excluding the false ones) 5f Answers correctly using matrices (possibly with relationship indicators: arrows) without linguistic argumentation 5.2.4.2 Presentation of the Experimental Work in the Italian Classes

Five situations/problems have been formulated with the primary objective of identifying different patterns of reasoning. This work was carried out at the state Scientific High School “S. Cannizzaro” in Palermo. The classes involved were: one third year (16 - year - olds) and one second year (15 - year - olds). The young people of the third year had already confronted the question of Aristotelian syllogisms and therefore they expressed themselves more suitably, where the students of the second year were, in any case, able to solve the questions (the percentage of the questions solved was very similar). Everyone was able to use the language of set theory correctly. The methodology followed and the analytical tools used: The questionnaire was distributed to the young people of the two classes by the same teacher (Prof. Ajello) and they were given the same information and the same clarifications of the questions. The time available to them was 90 minutes. The protocols were collected and analysed on the basis of previously formulated a priori analysis and the data were drawn up with the CHIC, for the implicative analysis, and with the SPSS for the factorial analysis. 5.2.4.3 The Quantitative Analysis: Implicative and Factorial 5.2.4.3.1 The Factorial Analysis Fig. 10. The variables relative to the “missing” and “incorrect” answers were eliminated from this analysis just as in the implicative analysis and this was done to better highlight the patterns of reasoning. The information is 31.86% important with respect to the number of variables (29). Two factors can be identified. The first has to do with the group of variables on the right (2b, 4c, 2e, 3d, 1c, 5f) which refers to the richest reasoning (inductive,

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deductive, Aristotelian, etc.) and more elaborate reached by a certain group of the young people. Analogously, the same variables are in a circular implication in the implicative analysis (implicative graph follows). The other group is represented by the remaining variables (except 2c) and corresponds to correct behaviours but is not always explained in an exhaustive way. A separate discussion regards variable 2c which corresponds to the ability to deduce contemporaneously with opposite arguments that arrive at opposite conclusions (the proposition of question 2 can be, at the same time, true and false, a logical-linguistic paradox of Chinese culture). 5.2.4.4 Implicative Analysis and of the Similarities

The following graphics are more meaningful, other graphics have only confirmed how much is already deduced from the following two reported. 1 a

3 b

4 a

5 a

5 b

2 a

3 a

4 b

5 c

3 c

1 b

1 d

2 f

5 e

3 f

4 d

1 c

2 e

4 c

3 d

2 c

2 b

Arbre de similarité : C:\Documenti\marilina\il pensiero cinese\schemi.csv

Fig. 11. Similarity tree

Fig. 12. Implicative graph

5 f

1 e

1 f

3 e

4 e

5 d

5 d


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5.2.4.5 General Considerations on Factorial Analysis, Implicative Analysis and of the Similarities

From the circumstances one deduces that: 3f and 2b were the most frequent responses and they both correspond to a pattern of reasoning ascribable to the rules of deduction of Aristotelian syllogisms. From the graph of the similarity tree, one notes that the correct, but not explained, answers are grouped either with the attempts or with the incorrect or missing answers. From the implicative graph, one notes that the same answers, above, are in relationship among themselves but never with others of other types. It is thus appropriate to exclude from the graphs answers 1a, 2a, 3a, 4a, 5a, 3b, 4b, 5b. Looking at the implicative graph, the most important considerations are the following: * 2f Æ 3f whoever recognises a paradox in the proposition of the 2nd question responds to the 3rd question with Aristotelian reasoning patterns. * from answer 3c (answers correctly using tables and matrices in the reasoning patterns) three implications branch 3c Æ 5d Æ 2c ; 3c Æ 4c ; 3c Æ 1f Æ 2c. One possible interpretation: who makes use of tables or matrices is able to do inductive reasoning and confront a paradox even with contrasting patterns and is able to produce proofs for the impossible and correctly use the meaning of the division by distribution. * 1c is another importance crux, in fact there are two significant chains of implications: 1cÆ 2e Æ 3d Æ 2c e 1cÆ 2e Æ 3d Æ 5f Æ 2b literally: whoever solves the 1st question, using induction correctly (which appears the most difficult from the circumstances), manages to leave the traditional patterns of deduction and in question 2 proves that the proposition is true interpreting it like “a white horse is not just any horse”, but he is also the one who correctly uses inclusion of the sets to show the structure of an inference graphically and still uses matrices with arrows that indicate the relationships. Summary of the results

Among the possible reasoning patterns that the students of the sample used, the most difficult to support and was the tied to induction while the most often correctly used ones were the ‘chains’ of deduction also with graphic representations of the inferential structures. This, basically, is a quite foreseeable result because of the set-up of the study of mathematics in schools, but perhaps a part of all disciplines has always favoured inferences and therefore deductive reasoning, neglecting the importance of training the students to utilise alternatively induction – deduction – induction – deduction. Perhaps the second result, which involves the different ways of reasoning, is more interesting: the ability to utilise graphic representations, combinatory methods, accepting, without much surprise, the possibility of encountering a paradox,

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encourages the correct use of syllogisms, preparing the way for a conscious use of mathematics demonstration. Giving a look at the characteristics of parallel thinking (pt) and serial thinking (st), one can compare the ability to use multiple ways of thinking, even accepting contradictions to pt, while the ability of the conscious use of syllogisms can be compared to st. So, the alternating pt – st – pt – st – pt can correspond to moments of creativity in which one result is seen from many different points of view and moments of systematisation in which one reconstructs and explains a result.

5.2.5 Interviews with Two Chinese: Qualitative Considerations Two interviews were carried out on the sketch of the situations/problems done in class. Tong (born in Canton 1954) went to Chinese schools until the upper secondary experimental school, but did not, however, complete his studies. He moved to Palermo in 1978 and obtained his Italian middle school diploma in 1985. Currently he manages a Chinese restaurant in Palermo. Jouzou (born in Palermo in 1986?) is currently attending the last year of an experimental study course European High School. He has studied Latin, Greek, philosophy, mathematics, etc., and he considers himself to be culturally Italian. His awareness of the Chinese language and culture has been by way of his parents who are both Chinese. Tong is Jouzou’s father. We report the work carried out in the interviews: Tong Question1: Arithmetic approach “23 x 23 = 64 64 – 1 = 63 and it is a multiple of three”, “therefore it is possible to cover the chess board with triminos”; Practical attempts; Despite the proposition to place the first trimino at the centre, reasoning by induction did not “click”. Question 2: “Which context? Without the context it is not possible to decide, there are always more meanings if there is not a subtext”. Telling of a story as a reference (a parable?). For each situation there is an appropriate story which comes from one’s personal experience. Question 3: Request for greater analysis of the text! “You must measure with the condition” (models as references which must be measured, compared) Question 4: He looks for the reference (a model) in the things he knows; that he has learnt; applicative aspect: “to count many boxes in a warehouse, one proceeds like this: they are arranged according to a known structure, for example a pyramid, and


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then it is enough to count the elements of the first row and to know how many levels have been built to arrive at the total number. Reference to previous experience. “In this case there is a minimum of 1 and a maximum of 12: Look at the formulas and apply the one that works!” Jouzou Question 1: “I’ll make some attempts”. With 2 and 4 he manages to cover the chess board and looks for a way to generalise the mechanism with 8 (iteration of the process). The mechanism for induction, instead, is not clear. He looks for analogies with other arguments. Question 2: He recognises it as a paradox. “a horse can be any colour, while a white horse cannot be brown or red; if one considers white-horse as a single entity then the property is true; white is quality, horse is form. It depends on the point of view from which one starts: true in reality and false in abstraction.” “The real absolute does not exist; everything is refutable”. “Deducing is making reference to historic facts, even one’s own personal history”. “One argues to be right about someone, one always tries to argue more strongly: you block the attack and restart!” Justify facts Æ rules Æ historic facts as model in a set of possible models (more experience more models). Resolve problems Æ rules Æ formulas as model in a set of models (more mathematical awareness more models). Accepting a demonstration Æ looking for its confirmation in a real model (amongst the possible models). Question 4: “Missing data?” Analysis of the text, organisation of the data to arrive at formulas. The model, in general: previous outlines or ideas, stories, mathematical model = literal formulas. “In what sense try?” “In this case it is enough to understand what is possible!” He does not manage to answer the question even with further clarifications of the text.

5.2.6 The Experience in the Chinese Classes The experience was conducted in September 2003 in some classes of the upper secondary schools of Nanjing and with some students of the 1st year of university. The questionnaire was the same one given to the Italian students. The a priori analysis of the behaviours was shared by Doctor Zhang Xiaogui and translated into Chinese. The data collected were tabulated following the same scheme used in paragraph 3.1. Important results were not discovered with the implicative analysis of the variables.

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My exam subjects were 12th grade high school students and first year university students. Students chosen were representative. There were 105 students, 65 high school and 40 university, who joined the exam. The questionnaires received back included 52 from the 12th grade and 30 from the university students. 5.2.6.1 Factorial Analysis

The variables relative to the “missing” and “incorrect” answers were eliminated from this analysis as in the implicative analysis and this to better highlight the reasoning patterns. The information is 25.3 %, important with respect to the number of variables (20). A small number of variables appear because the answers were concentrated only on some variables. The most significant result is that relative to the variable 2c which differentiates itself form all the others as in the case of the Italian sample. This analysis did not give other relevant information. 5.2.6.2 Supplementary Variables

To be able to better analyse both the Italian and Chinese data, we have introduced supplementary variables. These variables, which we called “europ” e “fuzzy”, characterise different patterns of reasoning:

Fig. 13.

Table 9.

“europ” Reasoning patterns typical of western scholastic9 education and especially in Europe. 1c, 2b, 2f, 3f, 5d, 5e

“fuzzy” Reasoning patterns typical of classical Chinese9 mathematics education. 1e, 1f, 2c, 2e, 3d, 3e, 4d, 5e, 5f

These variables were selected on the basis of epistemological and historic epistemological analysis, but also on the basis of preceding experimental works (Spagnolo (2003), Ajello-Spagnolo (2005)). This foresees that in the initial table with the addition of the supplementary variables (profiles of student type) it is done there. At this point, the variables will be the students and the two profiles “europ” and “fuzzy”. This allows a new vision of the situation.


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5.2.6.3 Considerations on the Analysis of the Data Relative to the Chinese Sample Both in the implicative - analysis of the variables and in the factorial analysis, one observes a well-defined division of students who imply or get close to the two supplementary variables in a distinct way (Ajello, Spagnolo, 2005). For this reason, we can, without doubt, confirm that these two patterns of reasoning are present today contemporaneously both in occidental and in Chinese schools. This is an important result that is not completely identifiable a priori with only epistemological analysis (which instead has clear references to education, in general). Fuzzy reasonings are always present even in western culture in agreement with a wide body of literature which confirms the fact that spontaneous conceptions on deducing and demonstrating are not always tied to bivalent logic. Thus, the attitudes which refer to bivalent logic are well present, in the answers of the Chinese young people. However, on the other hand, the presence of the deductive type of reasoning in the Chinese school and of the fuzzy type in the Italian school causes us to take into consideration the role of scholastic culture. Further experimental works are in progress to be able to better understand the role of the patterns of spontaneous reasoning in the two cultures.

5.2.7 General Conclusions and Future Perspectives Following is a comparative analysis between the different behaviours which emerged from the enquiry of the questionnaire in class in Italy and China and the interviews. The different points of view include all the casuistic of the answers and are reinforced by the historic-epistemological analysis of mathematics in the two cultures with particular reference to deducing and demonstrating. Three important results can be identified: 1. Deductions both for demonstrating that proposition 2 (the Chinese paradox, variable 2c) is true and for demonstrating that it is false are present in high percentages in the two samples examined (I) and (C). This brings to light that fuzzy reasoning is present in equal measure in the two scholastic populations (Ajello-Spagnolo (2002)). 2. With respect to question 4 (which refers to the so-called “the pigeon hole principle”), most of the young people of the Chinese sample did not answer while the percentage of the young people in the Italian sample that did not answer is insignificant. The question is not referable to patterns of preorganised reasoning (it does not have reference models). From the point of view of classical Chinese mathematics education, this constitutes a problem in the moment in which one looks for the reference to a pre-established model for confronting an “analogous” situation. 3. While numerous similarities come across between the two samples examined in a situation of stated didactic contract (I) and (C), the answers in the protocols of the interviews (PC) turn out to be more varied where one sees greater cultural influence in the absence of a didactic contract.

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5 The Experimental Epistemology as a Tool to Observe and Preview Table 10.

Questions

1

2

3

4

5

Prevalent behaviours in the protocols of the interviews (PC) Heuristic approach for attempts and errors. Research of an algorithm as a tool of formalised demonstration. Request of a concrete context to analyse the adequacy of the proposition in hand. Measure of the conformity of the affirmation at hand with the premises. More care of the analysis of the text. Use of tables or matrices. Organisation of the data for the research for conformity with a model (diagrams, previous idea and analogous situations) Reasoning of the combinatorial type with representations by tables. Analysis of all possible case to encourage the renewal of the model.

Prevalent behaviours in the experimental results in Italy (I) Inductive reasoning: finite chain of conjunctions.

Prevalent behaviours in the experimental results in China (C) Experiment and induction. Proof and intuition.

Use of Venn’s dia- Proof of the truth and grams for deduction falsity of the proposi(the proposition thus tions. ends up false).

Use of Venn’s dia- Correct us of the reasograms for deduction ing patterns of the sylloand a correct inter- gisms. pretation of the syllogisms. Deductive processes in N.L.

Organisation of the data for the analysis of all the possible cases. Use of division by distribution. The pigeon hole principle.

The problem is not recognised as referable to a known pattern of reasoning. It is not solved.

Use of contrapositives and therefore of reasoning to the impossible in N.L. and with the help of double entry tables.

Use of counter contrapositives and therefore reasoning to the impossible in N.L.

Appendix 1

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The initial hypothesis needs, therefore, to be reformulated in the light of the role that didactic contract plays in confronting the situations/problem proposed. In the preceding work, Ajello-Spagnolo (2003) where the investigation of the Chinese scholastic sample was not present, the differences between the patterns of reasoning in the two cultures were more marked. Other developments:

1.

The analysis must continue with other experimental works through interviews of Italian young people and Chinese young people (who live in China) outside of the didactic contract. Conjecturing and demonstrating (here, we refer to mathematical demonstration) have not been confronted in the experimental phase. We are thinking of preparing a series of problem situations for a careful investigation.

2.

Open problems:

1. 2.

Up to what point does the didactic contract also impose reasoning patterns thus intervening in the deep-rooted logical-linguistic structure? What is the range of action of the didactic contract in the case of didactic situations in multi-cultural environments?

Appendix 1 Questionnaire on the Abilities “To Argue” Problematic Situation 1.

A chessboard of 2n x 2n cells is given. You remove a cell in one of the four angles, for example:

Is it possible to cover the whole chessboard with pieces of this type? Suggestion. To reason by induction, you put the piece from three in the centre:

How does it proceed? 1a) Solution 1b) It motivates the proposed solution 2.

"A white horse is not a horse." Are we able to declare this proposition true or false?

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2a) Solution 2b) It motivates the proposed solution

3.

The premises are given: "All the adults can vote. Sabrina is of age. All those people who have a driver’s licence are of age." It considers the following affirmations: a. Sabrina can take the licence b. Who is not of age doesn't have a licence c. Who doesn't have a licence is not of age d. Sabrina can vote It shows that three of them are true and only one is false. 3a) Solution 3b) It motivates the proposed solution

4. In a class there are 30 pupils. In the dictation, all have made at least one mistake. Alex has made 13 mistakes and all the others have done less than he did. Prove that there is at least one group of three pupils that has made the same number of mistakes. 4a) Solution 4b) It motivates the proposed solution

5. Mario, Benedetto and Giovanna are the first names of three young people aged 14, 16 and 17. Rossi, Bianchi and Verdi are their last name’s. The order of the first names can correspond or not correspond to that of the last names, and it is not known to whom the ages belong. Knowing that: the girl Rossi is three years older than Giovanna and the young Verdi is 16 years older, find the complete name of every pupil and his age. 5a) Solution 5b) It motivates the proposed solution

Chapter 6 Strategy and Tactics in the Chinese and European Culture: Chess and Weich'i∗ 6 Strategy and Tactics in the C hinese a nd Europea n C ulture: Chess and Weic h'i

In previous chapters we observed that hypothetic-deductive reasoning may represent in western culture a fundamental reference for “strategic” choices. It is concerned with maths and also with other cultural contexts including economy. What for Chinese Culture? We stress that in the Chinese classical classification of the most important arts Wei-ch'ì (an ancient Chinese strategy game, known also by the Japanese name “Go”) is considered a very important art. We believe that wei-ch’i plays an important role for arguing, conjecturing and proofing in Chinese thinking. We glimpse in some experimental work significant differences, confirmed by historical–epistemological references (Chemla, 2007), but we are not in a position to say more on this topic.

6.1 Introduction Didactics is interested in activities improving the concrete development of logic and metacognitive skills effectively. While recognizing the primary role of the subjects that historically represent the main topics of the growth process of students, we believe that other activities can authoritatively add to the traditional ones, especially if they are structured in games. It supports education in a multitasking way, typical of subjects without a strong epistemological framework. Playing is very important in the learning processes, in particular with children. It is considered an issue of great educational value, as pointed out by several important scientists like Piaget and Vygotskji. Using strategy games as an educational tool is frequently adopted by several educational agencies over the world. The builder-child in playing becomes a situation simulator, a builder of strategies, an evaluator of risks and advantages connected to the various possibilities. Chess are a “cognitive gym-hall”: And not by chance the major part of studies concerning cognitive processes in playing are focused on chess1. This is referred to the motivation towards a strategic project chosen independently, and to use logic and meta-cognitive skills, especially in cause–effect relationships, and to apply divergent thinking inside a convergent thinking framework. ∗

This chapter was written by Giuliano D'Eredità and Filippo Spagnolo. We report the specific references of the chapter at the end of it. 1 Roberto Trinchero and Mariella Piscopo (University of Torino) Gli scacchi: un gioco per crescere, in print, T. Ito, H. Matsubara, R. Grimbergen, The Use of Memory and Causal Chunking in the Game of Shogi, http:// minerva.cs.uec.ac.jp/~ito/kokusai/3.pdf F. Spagnolo and B. Di Paola (Eds.): European and Chinese Cognitive Styles, SCI 277, pp. 201–218. springerlink.com © Springer-Verlag Berlin Heidelberg 2010

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6 Strategy and Tactics in the Chinese and European Culture: Chess and Weich'i

In particular, considering first the most diffused strategy game over the world, chess, various studies by psychologists and scientists proof that it is a powerful tool for improving several competences and skills in students. In particular, there are clear benefits for: • • • • • • •

Focusing Visualizing, with particular respect to pattern recognition Hypothetic-deductive thinking Thinking abstractly Creative thinking Evaluation Decision making

It is not the aim of this chapter to consider in detail such items, just emphasize that playing chess enhances the ability to use different resources in a certain situation. Furthermore, there are other important aspects in the personal formation of students, like the consideration and acceptance of their limits, of possible defeats, and the sense of legality and respect for rules. These are fundamental topics in didactics, and we found them in chess and in other strategy games, when the player faces the difficulties using his own skills, without any external help, within definite rules. It was stressed in literature2 that chess mastery manifests itself not only in the logic-analytic capacity to visit a tree (more or less deep and/or branched), but also in recognizing already-known structures, such as chunk, template and patterns3, and it allows a quicker and safer evaluation of position. In particular, pattern recognition not only suggests what to do, but also orients the choice in a strong way. This is pointed out as a content-oriented selective procedure4. We know that Human thinking processes are a very complicated issue. There are a lot of studies about it, especially in Cognitive Psychology and Neurosciences, in continue evolution. We believe that learning conjecturing and arguing are not independent on the social, cultural and specific educational context. In the modern world, where the exchange of information has increased quite a lot, and where one must deal with continuous problematic relative to the integration of cultures, we hold it useful and constructive to propose a reflection on the different approach to the problematic of choice and decisions, which different sectors embrace and place themselves as common denominator for the setting up itself of some 2

3

4

De Groot A.D., Thought and choice in chess, The Hauge, Mouton Publishers, 1965; Binet, A. Mnemonic Virtuosity: a study of chess players, Genetic Psychology Monographs,74, 1966; Di Sario P. Apprendere e applicare Strategie seguendo un modello cognitivo per il Gioco degli Scacchi, Tesi di laurea A.A: 2001-2002 Università di Bologna, Diderjean; Ferrari, Marmeche Nel cervello di un grande maestro- Mente e Cervello marzo-aprile 2004; J. Levitt, Il Genio negli Scacchi, Milano, Messaggerie Scacchistiche, 1998. The chunk is the minimal recognizable unit, in chess typically from 1 to 7 pieces in a given position, The template is a configuration with 1 or more chunks.. The pattern is a given configuration, intending also attributes and logical relations among components, and also more abstract generalizations. Saariluoma, Chess and content-oriented psychology of thinking Psicológica (2001), 22, 143-164.

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human activities. Strategy and tactics are some of the generalisations of behaviours and of choices which can be very useful for the comprehension of the same. Games of strategy, in particular chess in the occidental world and wei-ch’i (go) in the orient are paradigmatic environments for the mentioned themes. In these (as probably also in other games) the abstraction is such that it consents to an analysis which is independent enough from contingent bonds. In particular, the examination of the different approaches to the themes of the strategic and tactical type builds interest both for the understanding of determined behaviours, also on a macroscopic scale, and in the environment of Research in Didactics of the Sciences. In this field, these deeper studies have value as a tool for the understanding of teaching/learning phenomena and for possible aimed didactic proposals, which can also consider the use itself of the games of strategy.

6.2 Strategy and Tactics “Tactics is knowing what to do when you have something to do; strategy is knowing what to do when you don’t have anything to do.” (Tartakower). Strategy and tactics are moments of the activity in a determinate context, and, in general, are deeply interconnected, even if the respective phases of intervention are normally quite separate one from the other. One strategy is the formulation, even in an implicit form, of a plan of action which, even long term, is taken as a reference for the coordination of the activities addressed to the reaching of a predetermined goal. The sectors within which one can speak about strategy can be the most disparate, and the strategy is formulated, also through different phases of actuation, in that area where, to reach the objective, there is not only a single choice and the outcome is, in general, uncertain. The word ‘strategy’ is derived from the work with which in ancient Greek was meant ‘general’ (στρατηγός). The first necessary option is precisely the determination of the goals, i.e., the explicit identification of the objectives on the basis of an evaluation of the situation. The evaluation phase is perhaps the most delicate and depends on the data at hand, on the decision-making ability of the subject in terms of aptitude, on experience, and on knowledge. This often poses some important bonds on the successive strategic planning operations. The true and real strategic operations are based on objective considerations and also on psychological considerations. The explicit identification of the objectives, or better yet, more in general, the choice of the aspects of the situations worth consideration and/or passably important developments, depends on the situation and on the decision-making subject, in the terms expressed above. It is interesting to underline that the decision-making subject can be a person, a group of people, a machine, a group of machines, and even an animal or group of them. The possibility has often been discussed that machines or animals can have a true and real strategic ability, in any case that is outside of our aims here. One important aspect of strategy is the frequent placing in hierarchy of the objectives, i.e., the fact that often useful or indispensable partial or intermediate objectives for the successive goals are often defined. This frequently implies the establishing of a chain of implications which, in its details, often resembles tactical type operations. It is often stated by the experts in various sectors that “it is better to have an incorrect strategy than to not have one at all”, in

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the same way, it is highlighted that those subjects are destined to fail which have an absolutely unequivocal strategy and are devoid of flexibility and adaptation to reality, meaning both the evaluation and the decision in progress and the fact that the initial conditions very rarely are always the same. We must walk a thin line between flexibility and consistency5 Strategy has its natural complementary action in tactics. Tactics comprise the methods used for achieving established objectives. Tactics are the means, real or logical, used to obtain a goal, be it partial or total. A tactical operation has the goal of realising a single action within the strategy, or also for gathering possibilities offered by an adversary or from the physical or logical environment in which it is found. Strategy refers to operations done to reach a long-term objective and is put into effect on a wide geographical scale (more in general contexts). Tactics, instead, refer to actions done to reach a short-term objective and generally is put into effect on a reduced geographical scale (more in general contexts). In tactical operations, by means of concrete actions, one aims at obtaining an advantage, recovering from a disadvantage, or maintaining the status quo which is held to be satisfactory. An advantageous tactical operation which stands out can not only convince one to decisively change his formulated strategic plans if necessary. Not only, but also the evaluations asserted in the strategic phase can turn out to be erroneous, or much less evidently absolute, following the realisation of a tactical operation. This can also happen in Physics or in Science, in general, where determinate mathematical or applicative difficulties can convince one to abandon a previously formulated modellisation. In particular, in science, a theory provides a set of relationships between properties and quantity of the real world, while the models determine the levels of accuracy, select the details and areas of interest, the variables to consider, and specify initial conditions and constraints. Theories and models, together, allow us to make predictions, and the consequences of results and deepening lead us back to refine the model or to reject it. This may cause to question the whole theory, or reconsider it as a special case of a broader theory; in mathematical logic, we say a theory model of any structure that respects the axioms, is this sense of the word that does the so-called Theory of models. The first meaning is local and covers classes of problems and has no pretensions of generalisation. The second meaning concerns with the organisation of knowledge in theories. In mathematics, a theory concerns, in particular, the “Theory of models”. In the twentieth century, this had different interpretations as that named “Abstracts Models”, or also “Syntactic Models”. In any case it concerned a formal, axiomatic systematization, with a well-defined and sufficiently rigorous language. Both meanings concern also with different ways to represent real phenomena. First meaning is “tactic”, analysing local situations. The second one is “strategic”, analysing broader problem classes, and making long-term predictions. Given the complexity of science at the end of the century, it requires a systemic approach for modelling in different fields (Morin, J. De Le Moigne). 5

Kasparov, Gli scacchi e la vita, Mondadori 2007, pag. 38.

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It requires a general theory that encompasses both the formal and the modelling of complex systems. It seems that modelling is the only tool to interpret the reality at the beginning of the twenty-first century. 6.3 Notes on the Co nceptions of Strategy a nd Tactics

6.3 Notes on the Conceptions of Strategy and Tactics in the Orient and the Occident In advance, we notice that the terms East and West are obviously very general, and show almost entire hemispheres, with many regions and ethnic groups. It should be noted that by West we mean western thought, in some features that distinguish it from the eastern. We refer to the culturally Western identity, distinct from the culturally Eastern one. In particular, we consider, for present purposes, a general framework that can refer to terns Confucius / Tao / Buddha for the South East Asian (Chinese) thought and Socrates / Plato / Aristotle to the West (Italian). For the first treatise on strategy, one can certainly go back to Sun Zu, a general who lived in China (VI-V century B.C.), entitled The Art of War (Sūnzǐ Bīngfǎ). The text is explicitly in reference to war, but it is still considered one of the basic texts for the learning of strategy in every field of human action. Sun Zu, besides establishing several fundamental principles of a strategic character as well as political, morale, and practical, often stops himself to underline how success can depend completely on the quality of the execution of an operation (tactics). He goes beyond this and supplies some useful indications for understanding that success, seen as the obtaining of a goal, brings with it losses and, more generally, that the actions undertaken have some consequences; with his own words Obtaining one hundred victories out of one hundred battles is not the epitome of ability. Beating the enemy without having to do battle, that is the epitome of triumph.6 Make visible a rational layout where the risk must absolutely be minimised. “Therefore, the victories obtained by the Masters of the Art of War do not distinguish themselves either for the use of force or for their audacity. Their successes in war do no depend on good luck. Because to win it is enough not to commit errors. “Don’t commit errors” means placing oneself in a condition to win with certainty. In this way, an enemy already beaten is subdued… in that way, a victorious army wins first, and then goes to battle. An army destined for defeat first goes to battle and then hopes to win”. In short Total control. It is very interesting to note how audacity is considered a virtue which is not strictly basic for a strategy. In this, we note a contrast with the aggressive conceptions which are encountered in the western world, above all in the field of economics, where strictly connected to the concept of risk. This is fanatic in a world where innovation is decisive and coming in second in the development of a sector is like not finishing at all. Audacity, the attack, the aggressiveness contain in themselves the advantage often being able to orientate things on one’s own preferred terrain. The classical western approach to a conflict can be summarized as follows (by D. Lai) “the Greeks developed what has been called the Western way of war a collision of soldiers on an open plain in a magnify cent display of courage, skill, physical 6

Sun Zu, L’arte della Guerra, p. III.

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prowess, honour, and fair play, and a concomitant repugnance for decoy, ambush, sneak attacks, and the involvement of non-combatants”7. It is interesting to underline how in the first chapter “Evaluation”, Sun Zu traces the five guiding principles for the evaluation of the real situation: The first of the fundamental elements is the Tao, the second is the Heavens (climate), the third is the ground, the fourth is the command, and the fifth is the doctrine (Rule, organisation). Not by chance, the first place is reserved for the element “Tao”, the word, difficult to translate8, here it is placed to indicate that people, army, and sovereign have the very same intent, that the Chiefs have great moral strength, that they are united in reaching their goal, and that the reciprocal trust and esteem are total. This is present in a very similar way in Von Clausewitz9. This has a lot to do with motivation, and also finds an easy comparison in the didactic sphere. It implies a clear definition of the goal, and in general, the profuse spending of the respective resources and abilities in a synergetic way. Also, generalising about more limited group or individual activities, commitment and Concentration are in the first place. Nevertheless, and this is very interesting, Sun Zu does not hesitate to declare that in some cases it is necessary to disobey the orders of the sovereign…the ninth is: there can be circumstances in which the sovereign’s orders must not be obeyed”. In synthesis, the decisions must be taken by who is competent and has the elements to do so, and not on the basis of pure hierarchy, if this is necessary for the supreme common good; a concept also present in Von Clausewitz, but criticised in a strongly hierarchic vision of the military and government organisation. In any case, the strategic skills have had a decisive role in the history of the human race: is assumed that after a genetic modification modern Homo sapiens has acquired "... The ability to devise and implement action plans in the long term, a feat which the Neanderthal could not ever enjoy. The distinction was small, but ultimately had profound consequences: Our species survived and prospered, while Neanderthals died out.”10.

6.4 Historical Games of Strategy in the Orient and Occident: Chess, Wei-ch’i and the Different Conceptions of Strategy and Tactics 4 Historica l Games of Strategy in the Orie nt and Occident

The world is a game of Go, whose rules were unnecessarily complicated (Chinese proverb). Life is Chess (Bobby Fischer) A game of strategy is typically a board game or a videogame in which the ability of the player to take decisions has a great impact in determining the result. Many games include this element to a greater or lesser degree, making it difficult to establish a 7

D. Lai, Learning from the stones: a go approach to mastering China’s strategic concept, Shi. E-book da www.asiaing.com 8 “ Via, Principio, Modo, Legge immutabile, Cammino, Via Diretta, Stile di vita …”. 9 C. Von Clausewitz, della Guerra, 1832. 10 T. Wynn-F.L.Coolidge, Un incontro di menti nell’età della Pietra, Le Scienze, n.485, gennaio 2009.

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demarcation. It is therefore more appropriate to speak of degree of strategy of a game, rather than of the fact that it is or is not a game of strategy. A game of strategy is a game in which the rules are well defined and clear and are known by the players. Von Neumann demonstrated that chess, in that it is a game of complete information, is described by a matrix (endless, the possible games are of the order of 10 to the 50th) which contain a saddle point, and therefore, the game is solvable, i.e., there exists a perfect match where both of the players, if they were able to evaluate everything, play their best (how this is defined is not known, the chess player’s experience makes him inclined to a draw). The same is true for wei-ch’i, with the difference that there are many more possible matches (10 to the 172nd). In chess, the goal of the game is to capture the adversary’s King, the checkmate. And so, the game, above all in its theoretical beginnings (XVI–XVII century A.D.), was heavily orientated towards, favouring concrete operations for attacking the adversary King, exquisitely tactical, even though recognising that there were some configurations the reaching of which was a sufficient basis for victory. Only with the theorists at the end of the XIX century (Tarrasch, Steinitz)11 did the conception move ahead, in an explicit way, that it was not possible to play only for the checkmate, but that it was rendered necessary to aim at the acquisition of little advantages of a ‘positional’ type, which after became the basic strategic elements for the evaluation of a position. Chess strategy is a complex of player’s activities which bring one to consider peculiar aspects of the position, to establish priorities, and to carry out some forecasts on the advancement of the match, to plan a series of operations. Tactics in chess are concrete operations, normally aimed at altering the existing position by means of a sequence of one or more moves, which are often obligatory. A single error in tactics, always if the adversary plays without errors, can be fatal for the player (see appendix I). The ability to calculate, of a strictly deductive type, in a given position, becomes the balance keeper of a match. With the passage of time, the mastery of the player is no longer represented by the mere ability to calculate the possible moves, but also by the selective ability of orientation of his thought and of his attention by means of the recognition of visual or abstract patterns12. This has rendered fascinating, starting with the 60s-70s of the XX century, the competition between human players and artificial players, the latter of these which today are practically unbeatable on the tactical plane. From the second half of the XX century, the concept of initiative and dynamic game has taken the upper hand over a static conception based only on the classical strategic canons. Wei-ch’i, in the Occident better known by its Japanese name ‘go’, probably began in China about 4000 years ago and its complete development dates from the VII to the V centuries B.C. It was introduced in Japan and Korea around 700 A.D. It spread amongst the imperial Chinese functionaries and in a period of continuous war also represented, besides being a philosophical type discipline, an important exercise in military strategy. The aim of the game is the control of the territory on the goban (board/checkered board of the game) and represents an elevated form of abstraction of thought, at which point, in traditional China, it rises to 11 12

Steinitz, The Chess Instructor, 1889. cfr. De Groot, Saariluoma, Gobet etc.

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a second discipline amongst the four held to be basic for the instruction of a person of elevated rank13. The knowledge of wei-ch’i is a necessary condition for holding high government positions. It summarises, in itself, several fundamental characteristics of Chinese culture, in line with the tradition of the I-King, with some symbolic meaning tied to spatial and temporal representations of the Universe. Different from chess, where often the actions are completely aimed at reaching an identified objective and probably the adversary will impede it at any cost, in weich’i it is implicit that the choice of an area of influence or of operation brings with it an analogous adversarial action and this is part of a conception that aims at avoiding the absolute identification of an element in favour of a dynamic vision and of interchange between the various elements themselves. In traditional wei-ch’i, the players were, more than adversaries, two parts of a cause for creating something intrinsically valid. The basic strategic rules in wei-ch’i are, in a schematic form: 1. If you have a weak group of stones, reinforce it. 2. If your opponent has a weak group of stones, plan to attack. 3. If you can make a move that has a broad territorial effect, make it. 4. If the opponent can a make a move that has a broad territorial effect, try to prevent, destroy it or reduce it with appropriate measures14. Everything is based on the techniques of encirclement or counter-encirclement of the adversary. “According to his model of rational behaviour, rationality consists not in the optimum (in wei-ch’i theoretically optimal strategy exists as a principle, but is and will remain unknown even for the best player, because of the quantity of possible variants which one should take into account in calculating it) but in the satisfying15. Chess and wei-ch’i are profoundly different, and not by chance the areas of its diffusion were, for a long time, clearly distinct between the Occident and the Orient. Obviously, at an elevated level of thought processes, we find in both some common denominators in the strategic and tactical elements. But it is quite obvious a connection with some basic rules of social life. The so-called “basic virtues” of western world, rising from the double helix of Western civilization (Greek-Roman tradition and Judeo-Christian religion) are almost always focused on individuals as a part of social environment and also a san isolated subject. We can define this approach as the “perfect isolated” harmonised at the” Right in the middle” as we find in Aristotle16. The Eastern approach to Ethics is quite different, and is focused more on social organisation than on the individual, and this is connected with Chinese Philosophical traditions that have not a similar in Western world. Tao, as we mentioned earlier, is one of the most high and deep concept the human mind ever produced, and it is intrinsically unexplained, and for a long time incomprehensible to the Western world. Tao direct applications are found in various sectors like Calligraphy, Tai Chi 13

See appendix II. http://senseis.xmp.net/ 15 Boorman, Gli scacchi di Mao, pag. 201. 16 B. Di Paola, Pensiero Aritmetico e pensiero Algebrico in ambienti multiculturali: il caso cinese, PhD thesis with parts in english langiage, http://math.unipa.it/~grim/Tesi_it.htm 14

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Chuan, Medicine, and in Wei-ch’i itself. This is a strongly cultural item for Chinese people. We refer to three parts braid composed by Tao, Confucianism, and Buddhism.17 Coming back to Chess and wei-ch’i, the chess player uses patterns as tools for conjecturing and arguing in a typically deductive framework, in which moreover valuation assessments are not always sharp but can be fuzzy (e.g. “white is slightly better”). Tactics is fundamental in chess; it deals with concrete developments that require an exhaustive analysis at the most efficient level. Instead, the big strategic frameworks conceived by the greatest theoretician of chess are very general and abstract theories. Even after the criticism of the classical principles by hypermodern (Reti, Tartakower, Nimzowitch18), strategic principles fail to have an absolute validity, so much so that are considered also like epistemological obstacles19, and today are used as available elements to evaluate a position in a nondogmatic way. In wei-ch’i, according to Chinese thinking tradition, to consider or to select only a feature, even in an exhaustive way, is not useful but is extremely dangerous. Nothing makes sense except in context. In the following “Ten Commandments” (“meta-rules”) we find always the reference to the context, and it is recommended to avoid to focus too much on particular Go Ten Commandments (Otake Hideo, 9º dan20) 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

“Gluttony does not lead to victory.” “To penetrate the opponent’s zone gently and simply.” “If you attack your opponent, pay attention to your shoulders” “Abandon the easy gain, and fight for the initiative”. “ Let the little falls, concentrate on large ”. “If you are in danger, abandon something”. “ Be careful, do not wander randomly on goban”. “If necessary, blow by blow”. “ If your opponent is strong, protect yourself”. “ If your group is isolated in the middle of an opponent’s zone, choose the peaceful way”.

In Chess, the game can be decided by a single move, and in the most evident case checkmate will be or will be not (Bivalent logic), and in a more general way, in general the presence of more or less important pieces aligned addresses clearly the game towards the final target by displaying pieces towards vital points and by destroying opponent’s defence. This approach is related with western way to combat a fight, or a conflict, or any situation21. Instead, in Wei-ch’i success is represented by a 17

Ibidem. See Nimzowitch, Il mio sistema, Mursia 1975-1989 and R. Reti, Masters of the chessboard, Whittlesey House 1932.” 19 S. Bartolotta, Un approccio euristico alla strategia, alla storia della strategia ed alla didattica degli scacchi: gli assiomi strategici cone concezioni ed ostacoli, Quaderni di Ricerca in Didattica n.7 ,1997 G.R.I.M. (Department of Mathematics, University of Palermo, Italy). 20 From Wikipedia, “Go”. 21 See D. Lai, Learning from the stones: a go approach to mastering China’s strategic concept, Shi. e-book da www.asiaing.com 18

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gradual series22, somehow more fuzzy23. The target is consider not to defeat opponent completely, but to maximise your benefits. It is not a duel, but an economic competition to obtain an asset of little value24. It is interesting to stress that in Chess a little advantage must be converted in a tangible advantage to win the game, while in Wei-ch’i it is sufficient to retain an imperceptible advantage to the end. The following table, within the limits of any schematisation, may be useful to focus some interesting items Discipline Purpose of the game Checkmate Chess (capture of the Adversary King).

Wei-ch’i or Go

22

Control of the largest area possible of the territory on the goban.

Social function

Strategic elements

Function of tactics

None in particular – an exception is the Soviet Union from 1925 to about the 80s.

Identification of partial objectives and evaluation of the position. Optimisation of the action of one’s own pieces and limitation of the adversaries. Recognition of visual and abstract patterns. Concept of control, in the sense of a continuous evaluation of the total situation. Choice of that which can be held to be satisfactory. Strategic items: defence of own groups, attack to opponent’s groups, territorial gain.

Determining. A single tactical action, well carried out, becomes the main and often conclusive one of the match.

Very important – second place amongst the traditional arts and held to be necessary for the education and the instruction of functionaries and dignitaries in imperial China and imperial Japan.

Boorman, op.cit, pag. 46. See. B. Kosko, Il Fuzzy Pensiero, Baldini&Casoldi, Milano, 1995 24 Boorman, op.cit, pag. 46. 23

The knowledge of the tactical themes is very important, but a local victory risks being lost. Tactics are not tied to strategy. The concept that a tactical success leads to a strategic success is alien to the spirit of Wei ch’i.

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6.5 Connection to Didactics and Open Problems We maintain that several thematic explained above can have connections to didactics. The different approaches to strategy and tactics in different cultures supply, in their entirety, some formidable tools for confronting the most disparate thematic of life and are very interesting in a didactic context. We limit ourselves to giving some schematic notes on these possible connections, providing that a more specific analysis of some of these requires dedicated research, with quantitative and qualitative methods. 1. First point, the full awareness that without dedication, unity of intent, clear definition of the goal, attention and motivation it is rare that one reaches an objective whatever it is. This point is of a general character, but precisely for this it is often neglected in Didactics. Demotivation and too bureaucracy are the probable reasons of it. 2. The aptitude of Evaluation: the considering of the elements with an elevated degree of objectivity and selectivity of one’s attention allows the formulating of strategies, and sometimes also methods and tools capable of the solution of problems and of the definition of a situation. That in line with a modern vision of didactics which aims at an increase of competences, i.e., of the use of one’s knowledge and abilities in different contexts and situations; this particularly in Italy, where there is a proven lack in this direction. 3. Hypothetic-deductive thinking, also at a high deepening level, it is an important item in strategy games and useful in all fields. Hypothetic-deductive thinking is found in the western Culture since from Euclide’s Elements ((Model of Aristotle Logic, Spagnolo (2005)). Hence, in the Western culture, strategic thinking may be addressed by Aristotle Logic (inherent to Natural Language), while in the Eastern Culture, as we mentioned earlier, by wei-ch’i “meta-rules”. 4. Visual or abstract patterns recognition, and continuous adaptation to the reality is the normal practice in chess and in wei-ch’i, typical of a high order competences. A strategy adopted in a repetitive way, without adaptation to reality, can be proofed wrong by practice. This is related with Epistemological Obstacles Theory in Education25. This feature is found in both cultures; chess and wei-ch’i support it. 5. Education to strategic and tactical features recognition could be important in social fields. It could become, as mentioned earlier, an useful tool for understanding different cultural approaches. We can imagine a more Hypothetic-deductive approach by western students, and a sharp separation between strategy and tactics. In eastern culture, this separation is not so sharp (see also appendix 2 on Chinese language). Probably this approach depends also on Weich’ì meta-rules. These meta-rules maybe clarify the differences between strategy and tactics in a more subtle way with respect to western approach.26 25 26

Bartolotta, op.cit. See Spagnolo F. et alii , Reasoning patterns and logical-linguistic questions in European and Chinese cultures: Cultural differences in scholastic and non-scholastic environments, Mediterranean Journal for Mathematics Education, Cyprus Mathematical Society (ISSN 1450-1104), Vol. 4, N. 2, pag. 27-65, 2005.

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6. The self-concept being separated from others as a reference for decisions, and constant search for best solutions are useful topics in Education.

6.6 Conclusions There is no doubt that the different conceptions of strategy and tactics between the Orient and the Occident have deep cultural roots and highlight a different approach to reality. The same course of history was influenced by these tactical and strategic conceptions and today different experts see, in the economic development of China and in its behaviour in the world market, a strategic approach similar to some concepts of wei-ch’i (see Appendix III). In any case, this lies outside of our aims of deepening the study of said connections. Also, the flourishing of chess in the occident and Wei-ch’i in the Orient, even if in different epochs, is not casual considering the nature of the two games. However, both can give great richness of themes and suggestions in the sphere of didactics and in the training of people. The use itself of games of strategy, particularly chess, in a scholastic environment has been the subject of scientific research and even of government interventions (currently the most notable example is in France, in the municipality of Cannes). Certainly, cultural diversities emerge more at the moment in which the necessity of a multicultural integration is posed in a class where there are students present who come from different nations and these thematic absolutely cannot be undervalued at any level.

References Bartolotta, S.: Un approccio euristico alla strategia, alla storia della strategia ed alla didattica degli scacchi: gli assiomi strategici cone concezioni ed ostacoli, Quaderni di Ricerca in Didattica. G.R.I.M., Department of Mathematics, vol. 7. University of Palermo, Italy (1997) Boorman, S.A., di Mao, G.s., Editrice, L.: The Protracted Game. A Wei-ch’i Interpretation of Maoist Strategy (2004/1969) Ciancarini, P.: I giocatori artificiali, Milano, Mursia (1992) Cardellino, C.: Giocatori non biologici in azione, Proto (2002) De Groot, A.D., Gobet, F.: Perception and memory in chess, Assen, Van Gorcum (1996) De Groot, A.D.: Thought and choice in chess. Mouton Publishers, The Hauge (1965) Di Paola, B.: Pensiero Aritmetico e pensiero Algebrico in ambienti multiculturali: il caso cinese, Tesi di Dottorato, in corso di stampa Gobet, F., Simon, H.A.: Templates in chess memory: A mechanism for recalling several boards. Cognitive Psychology, 31 (1996) Kasparov, G.: Gli scacchi, la vita, Mondadori (2007) Kosko, B.: Il Fuzzy Pensiero, Baldini&Casoldi, Milano (1995) (Fuzzy thinking: the newScience of fuzzy logic, 1993) Lai, D.: Learning from the stones: a go approach to mastering China’s strategic concept, Shi. e-book da, http://www.asiaing.com Levitt, J.: Il Genio negli Scacchi, Milano, Messaggerie Scacchistiche (1998)

Appendix A

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Nash, J.: Non cooperative games, PHD Tesis, Princeton (1950) Needham, J.: Scienza e Civiltà in Cina (Original title: Science and Civilisation in China), Einaudi, vol. I, II. Cambridge University Press, Cambridge (1959/1981) Nimzowitch, Il mio sistema, Mursia (1975-1989) Reti, R.: Masters of the chessboard, Whittlesey House (1932) Saariluoma: Chess and content-oriented psychology of thinking. Psicológica 22, 143–164 (2001) Spagnolo, F.: Obstacles Epistémologiques: Le Postulat d’Eudoxe-Archimede. In: A multidimensional approach to learning in Mathematics and Science. Intercollege press and Departement of education, Nicosia, Cyprus (1999) Spagnolo, F., et al.: Reasoning patterns and logical-linguistic questions in European and Chinese cultures: Cultural differences in scholastic and non-scholastic environments. Mediterranean Journal for Mathematics Education, Cyprus Mathematical Society 4(2), 27–65 (2005) Spagnolo, F.: Alcune idee sulla Filosofia dell’Educazione Matematica tra oriente ed occidente. Quaderni di Ricerca in Didattica (Sezione Matematica) 18, 86–126 (2008), http://math.unipa.it/~grim/quaderno18.htm Spagnolo, F., Ajello, M.: Schemi di ragionamento in culture differenti: i paradossi logicolinguistici nella cultura europea e cinese. Quaderni di Ricerca in Didattica (Sezione Matematica) 18, 163–182 (2008), http://math.unipa.it/~grim/quaderno18.htm Spagnolo, F.: Philosophy of Mathematics Education among east and west. Philosophy of Mathematics Education Journal 23 (October 2008), ISSN 1465-2978, http://people.exeter.ac.uk/PErnest/pome23/index.htm Spagnolo, F.: Fuzzy logic, Fuzzy Thinking and the teaching/learning of mathematics in multicultural situations. In: International Conference on Mathematics Education into the 21st Century, General conference, Brno (Ceck Republic), September 2003, pp. 17–28 (2003) Steinitz, W.: The Chess Instructor (1889) Zu, S.: L’arte della guerra, Newton Compton (1994-2008) Von Clausewitz, C.: Della Guerra (1832) Von Neumann, J., Morgenstern, O.: Theory of Games and economic behavior. Princeton University Press, Princeton (1944) Wynn, T., Coolidge, F.L.: Un incontro di menti nell’età della Pietra. Le Scienze 485 (gennaio 2009) Yasuyuki, M.: An Asian Paradigm For Business Strategy. The Ishi Press, Tokyo (1995)

Appendix A Strategy and Tactics: – Examples from Real Games Frequently in Chess, a tactical blow arrives suddenly, when strategic forecasts do not provide it in an evident way. In the following position, Black has a plus-pawn, does not have particular weaknesses, his pieces are active, and normally White has to fight to draw the game. But by a sudden tactical blow White wins:

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White

Black

1. Rf4 + (check) 2. Rh4 + 3. g4 checkmate

Kh5 g:h4

Instead in wei-ch’i , a strategy that points just to a single target, as Black’s playing in the following example, it is not correct, in fact Black played like a beginner and has a loosing position.

Appendix B 27

B.1 Strategy and Tactics in Chinese Language strategy ∾n. 27

战略[戰-] zhànlüè; 略 lüè; 策 cè; 方策 fāngcè; 谋略[謀-] móulüè

About word we referred to software “Welin”, that allow to search pinyn characters and english words (in both directions). - Chinese–English dictionary, The commercial press, 1982, Beijng. - Dictionnaire Français-Chinois e Chinois-Français, La presse commerciale et Larousse, 2000.

Appendix B

215

Consider in detail single words: lüè, short, rough, sketchy . to walk slowly. to surround zhàn, war: fortune, old chinese weapon Other word: yàolüè n. outline; summary important plan, (strategy, plan, scheme, contour, profile, silhouette, sketch).

略夂囗战

占要略 ①

戈

②

Different words meaning tactics: [ ] shù; cèlüè; [ ] quánmóu; [-- ] tactics ∾n. xùsùxué shù : art, method, tactics. n. tactics; strategy. s.v. tactful | Nà wèi lǐngdǎo de dáfù hěn . The cèlüè reply of that leader is very tactful. [ ] n. intrigue; tactics; schemes quánmóu xùsùxué [-- ] n. lg. tactics (The word more similar to the western meaning)

术術

策略

权谋權謀

序素学學

策略权谋權謀序素学學〈〉

∼

Analyses of the parts: n. preface, introductory; initial xù sù* b.f. plain; simple; quiet xué* [ ] v. study; learn; b.f. learning; knowledge

序素 ① 学學 ①

①

First, we stress that the distinction between strategy and tactics is not so sharp in the Chinese culture with respect to the western one. It seems that often the word “strategy” is used in both senses. Probably these differences are due to wei-ch’I meta-rules. This is coherent with the classification of basic disciplines in the traditional Chinese culture: - Music, discipline related to hearing and also to sophisticated processes that regulate vibrations. It is the practice of the inner qì. It also concerns with embodiment referred to the musical rhythms. Today we know that it is related to natural language and numbers learning. - Wei-ch’i (Relationship between partial modelling and holistic view with a strategic intention). - Writing (Holistic representation, algebraic structure , variable, parameter28). - Painting. (It is considered a quite direct and subjective discipline, accessible to all). Western classification, by Marziano Capella, philosopher of late Latin age (IV-V century A.D.) is: Trivio (artes sermocinales): Grammar, Rhetoric and Dialectics Quadrivio (artes reales): Arithmetic, Geometry, Astronomy and Music. 28

This topics are also in: : Spagnolo (2003), Spagnolo et alii (2005), Spagnolo (2008), Spagnolo & Ajello (2008), Di Paola & Spagnolo , Di Paola (Doctoral Thesis, in press).

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We do not find any ranking among them, as we found in Chinese Culture. There is no trace of chess or other strategy games. Geometry is the most important referred to the modelling of Aristotelic logic. This is the deep distinction between Chinese and European cultures.

Appendix C C.1 Similiarities to the Economy There are a lot of studies concerning with Games Theory and economic behavior, starting from the fundamental works by Von Neumann e Morgenstern29and by John Nash30. A more esplicit similarity between strategy games and economic phenomena has been suggested. About Chess and Wei-ch’i, the most common similarity concerns with war and diplomacy.31 About Economy, it is easier to formulate conjectures about wei-ch’i than Chess.32 It is because in chess, the aim, the checkmate, is absolute (but partial objectives are also important, see par. 4), comparable with monopoly33. Instead, in wei-ch’i, the target is to control the largest area possibile of the lands, and it is necessary to diversify operations. This is more similar to a competitive economic scenario, in which it is impossible to prevent others gain a slice of market. Nevertheless, the processes adopted by the chessplayer (also referred to fantasy, risk, and search for the iniziative) are considered possibile paradigms for economic behaviour34. Hence, the similarity between Chess and Economy is more abstract, and implies strategical concepts of a higher order, like superprotection.35 The similarity between wei-ch’i and Economy considers goban as an economic system, and single zones as markets. A fully controlled territory is considered as a monopoly, as in the following picture36.

29

J. Von Neumann e O. Morgenstern “Theory of games and economic behaviour”, Princeton University Press, 1944. 30 J. Nash, Non cooperative games, PHD thesis, Princeton 1950. 31 See D. Lai, Learning from the stones: a go approach to mastering China’s strategic concept, Shi. e-book www.asiaing.com 32 See Miura Yasuyuki, Go: An Asian Paradigm For Business Strategy. E Parton, Dieter.. 33 See G. Amato, M.Parton,F.Scozzari, Il gioco del go tra matematica ed economia, Series of Seminars, Pescara 2008, http://www.sci.unich.it/~scozzari/go/corso_di_go.html 34 G. Kasparov, op.cit. 35 Super protection is a concept introduced by Nimzowitch. It concerns with the control of an important point. If the point is controlled by a single piece, the piece is unable to move. Instead, if the point is controlled by two or more pieces, everyone of the pieces can move, because there is at least another piece controlling the point. 36 See G. Amato, M. Parton, F. Scozzari, op.cit.

Appendix C

217

Potential territory (moyo) is a wide market still open for competition, but strongly influenced by their investments, as in the following example

Tactics are used to conquer a single market, strategy is used to choose in which markets to invest to improve interactions among economic activities37. Where do the resources of Africa? What links with China? A possible interpretation using wei-ch’i concepts. The map we present is taken from an article by Margherita Paolini “Il gran safari e le sue ombre” in Limes (Rivista Italiana di Geopolitica published by l’Espresso, n.3 2006) (pp.51-64). As we mentioned early, we can try to interpret some features of the map by wei-ch’i. In particular, Chinese activities’ basis is not located to face directly other basis, but to occupy empty zones (see. Sun Zu, throw in the blanks), ad simultaneously trying to set up a future encirclement of the other bases. In wei-ch’i to play only towards the centre it is dangerous because the centre is more difficult and will become hard a safe territory. These concepts are valid not only in a geometrical sense but also in a logical sense as well. From this point of view, it is

37

Ibidem.

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interesting to quote some general wei-ch’i proverbs, that maybe can be compared to the Chinese business strategy (from Wikipedia, Go): •“Go is an exchange game: conquer territories and make exchanges”. • “Territories are not on the walls against which the opponent goes”. •“There are no points in the centre”.

In considerations to the discussion made in the text, we give a possible interpretation of the map that represents the use and allocation of resources in Africa, using the wei-ch'i of the possible keys for reading. In particular, we note that the foundations of China's activities are not directed to openly confront the areas influence of others, but to fill in the blanks (see Sun Tzu, introduced in gaps), while seeking to be able to configure, even in perspective future, a siege to the bases of others. In wei-ch'i game directed only the middle is dangerous because the centre is much more difficult to protect, well hardly become safe territory. These concepts do not apply only geometrically and / or geographically, but specially in the economic field must be understood even from a logical point of view. In this context, we quote some sayings of general wei-ch'i, taken from Wikipedia, entry Go, which seems well suited to the Chinese business strategy: • "The Go is a game of trade: they are territories and trade." • "You are the territories on the walls against which the opponent goes." • "There are points in the center."

Chapter 7

Rhythm and Natural Language in the Chinese and European Culture* 7 Rhyt hm and Nat ural Langua ge in t he Chinese a nd Europea n Culture

Appendix 2 of Chapter 6 has been taken into account the classification of the arts in the two main cultures. In Chinese culture, the music was made in the first place: “Music, discipline related to hearing and also to sophisticated processes that regulate vibrations. It is the practice of the inner qì. It concerns also with embodiment referred to the musical rhythms. Today we know that it is related to natural language and numbers learning”. In European culture, the music is part of the quadrivium of the arts and is considered one of the sciences together with arithmetic, geometry and astronomy. In this chapter we will try to grope a reading of these differences. Rhythm, natural language and mathematics have some relationships that have come today highlighted by neuroscience and epistemological relationship. Some of the epistemological relationships will be highlighted in order to create openings in the reading of cultural issues.

7.1 Rhythm and Natural Language in the West Rhythm, element of life, is a force that has characteristic features of periods and repeat. It is present in matter and spirit, in the category of time and space. In the world of the infinitely small is rhythmic animation of molecular motion, in the world of the infinitely great is rhythmic motion of the stars of the planets. Rhythmic phenomena are the biological growth in the plant world, the phenomena rhythmic breathing, circus-bearers, walkers in the animal kingdom, including man, and are also the rhythmic vibration of the body sound, finally, music and dance, and poetry standing on the rhythm.(R.Allorto, 1986, p.5). However, it is difficult to formulate a comprehensive definition of the concept of rhythm (in greek rhytmos, from the verb reo, scrolls), the Swiss pedagogue Edgar Willems says it has counted more than four hundred different definitions of rhythm which would be proposed by not less than two hundred scholars (Willems, 1954, p.53). One that best sums up the concept was formulated by Plato to that "The rhythm is the order of movement" (Allorto, 1986, p.6-7). Even the spoken language, without the accents, pauses, acceleration and delay, is a monotonous succession of syllables and without expression, and therefore without any real meaning. What gives meaning to a speech is the rhythm that animates him: fellow, calm, punctuated, gay. Element, common to singing and dancing, the rhythm is the physical life, energy, which results *

This chapter was written by Daniela Galante & Filippo Spagnolo. We report the specific references of the chapter at the end of it.


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7 Rhythm and Natural Language in the Chinese and European Culture

in moving that is material and ideal. Already in ancient Greece, where poetry and music were complemented and the sister with the dance, the feet (which were groupings of metric syllables long — and short ∪ variously arranged according to a quantitative assessment) is to animate life rhythm. During the classical period Metrics and rhythm were closely united. In medieval Gregorian chant, an expression of contemplation and prayer, for one voice and rhetorical pattern, using a free rhythm does not follow prescribed meters. With the advent of polyphony comes the notation and the mensural metric reappears in Western music during the Renaissance on the staff when the arms are drawn delineating the measure. The entire western music from Monteverdi to contemporary, sacred and profane, which is then inserted into a metric system, is indicated by the sequence of measures. Rhythm and meter are two independent but historically linked to the West. The rhythm allows the music to manifest itself over time, the equivalent of the space for visual arts and through its elements, the musical rhythm gives a certain "form" in time, as well as the arts, through the concepts of length, height , depth, manage to give "shape" of space. The elements that make up the structure of the measured rhythm are: the pulse, the movement, the measure or beat and time, the organization of the durations. (Becatti, 1997, pp.53-69). 1. The pulse: It is a regular succession of beats, called time, the same duration, such as those produced by the metronome, and is the essential element and coordinator of any musical performance. It may actually be scanned with the instruments or scanned in mind. The duration of sound is calculated in a number of beats, or time: one, two, three, four or more times, not only, but also in its fractions: half time, fourth time and so on. In this way, a set of players can play in perfect synchrony, referring to a pre-pulse. 2. The movement: The different speeds of pulsation are called movements that are expressive of the elements of rhythm: a pattern performed by slow movement takes on a character opposite to the performing with a quick movement. In order of increasing speed, the different movements can be divided into three groups: slow movements: Largo - Adagio - Grave - Lento - Larghetto; moderate movements: Moderato - Andante - Andantino - Sostenuto - Mosso, rapid movements: Allegro - Presto - Vivace. The movement is indicated at the beginning of the composition. It is good to remember that every good musical performance is always full of light, just for perceptible changes in movement delays, suspensions, accelerations, as in the spoken language, that give it its elastic and expressive performance that breaks and times of enthusiasm and hesitation. Only in exceptional cases, the pulse rate maintains a strict timing. During execution, the original motion may be amended by writing next to the staff one of the following: slowing, delaying, diminishing, less driven, falling, dying, when it will slow the initial movement: forging, pressing, accelerate, most moved, when it intends to accelerate it. To return to the initial motion is used referring to time. 3. The measure or beat. The pulse becomes measured rhythm putting, with a certain order, the accents on some times. Accents are a primary element of rhythm in spoken language. Talking does not pronounce the syllables in a uniform, but to

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give some more strength, with the accent and some are not. Thus, every word has its own unique rhythm that contributes to its understanding. Shift the focus from being a syllable to another means to alter the rhythm of the word and, in most cases, even make it incomprehensible, while in music you can move the focus using the procedure of syncope and it serves to create tension1. A rhythm track when they say has an accent every 2 times, it says when a ternary accent every 3 Quaternary times and has an accent when every time the 4 beats the original is now divided into numerous "rhythmic cells" equal to that in first case of 2, the second 3 and third 4 times. These elementary rhythmic cells are called measures or bars. Each of them starts at the accent, also known as accent or strong time. They are indicated by two vertical arms that enclose the relevant times, 2, 3, 4, depending on the type of rhythm, and the strong accent on the first ever same time of each measure. The relation with the verbal language is very strong because the number of times (or beats) and disposition of the accents of each bar are related with the number of syllables and the accents of different words. In the chant, the strong accent of the measure and the words are always the same. In particular, the beat of the binary rhythm coincides with the disyllabic word; the beat of the ternary rhythm is the equivalent of the trisyllabic word and quaternary beat rhythm is identical to the quadrisyllabic word. (picture 1).

Picture 1. 1

There is talk of syncope when a sound starts on a weak time, extended time, that immediately follows it. The effect of the syncope is a rhythmic movement of the accent. It is realized as a lag in the regularity of the bar resulting in a load of dynamic contrast and tension. Western repertoire, both sacred and profane, vocal and instrumental, which has significant moments, syncopated composition.

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4. Time. It is a claim that arises at the beginning of each song, after the key, to determine number of times the measure or beat. It consists of a fraction whose numerator determines the number of times, and the denominator their individual value. Moreover, a composer makes music in the verses, chooses the rhythm depending on the metric of the same verses (i.e., must–examine the sequence accents strengths and weaknesses to match them the most appropriate time rhythmic, binary, ternary, quaternary, etc.); without this operation, it is impossible to proceed in the composition with the development of melodic line and harmony, followed by the form, even in this close relationship with the form of verses to compose music. 5. The organization of durations. The pulse, the motion and the measure used to measure the lengths of the sounds that give life to the rhythm of the composition and then to all the musical discourse. Using the various music figure, you can create infinite patterns.

7.2 Rhythm and Natural Language in China If in Western music the development of language rhythm is always connected to the metric of the language and poetry, both classical quantitative and the modern emphasized, in China, the relationship between rhythm and natural language has different characteristics. China is one of the oldest civilizations that are perpetuated uninterrupted to us and its culture has its character and continuity of specific units. One of the main factors of this area is the writing system that has a continuity of characters ranging from the second millennium BC until today. Writing non-alphabetic character, but consists of independent characters, each with its own semantic and phonetic, with real meaning. Some characters are pictograms, ideograms others that represent abstract concepts through symbolic representations, while others have origins phonetic (See the chapter 2). Most of the characters, however, are formed by the combination of two elements, a signifier that indicates the general category of the word, and one that is phonetically rather to indicate, on the other with more or less exact, its phonetic pronunciation. The presence of such a particular form of writing was necessarily influence the thinking, culture and music of China. Of course we must think of a different conception of the music from the western one, where the nature of music is related to the sound, as an expression of transcendent power; the idea that music contains and exceeds the sound still is reflected in practice of traditional instruments such as the tendency of players to continue psaltery achieve vibrato on a string even after the cessation of all sound hearing. Music over the sound (at least beyond the mere experience of hearing), based on the complex and articulated, determine the relationship between music and the order of the universe: the organization and structure that occur in music the same ones that govern all human activities and therefore represent only the faithful mirror the harmony of the cosmos. This view says that human activity is the consequence of nature, and leads to harmony with nature which pervades all Chinese thought.

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At the root of Chinese thought, it is the man and the cosmos in a close re-lation rhythm so that both breathe and pulsate in synchrony pervaded by the same energy. The composer in the West for centuries imitated nature in Chinese culture, however, the musician should not imitate anything because the music is nature that is manifested through the expression of human sound. This natural quality that permeates the music of China, traditional is also evidenced by the prevalence of voice: nothing could be more natural of voice, without a place of mediation (as are the musical instruments) in nature and culture: (Zumthor, 1992). Fundamental phenomenon of human sound, the voice is "primordial sound" object material expression of that desire that will not or can find fulfillment. Even before the language has started and would be in words to convey messages in the form set out in the minutes, the voice has always been home, as there is potential for signification and vibrates as indistinct flow of vitality, driven confused want to say, to express, to exist. Its nature is essentially physical, corporeal, bears no relationship to the life and death, with the breath and the sound is emanating from the bodies governing the use and survival. (Zumthor, 1992, p.23) The "vocal instrument" can basically be considered under a twofold aspect, which is summarized in the definitions of spoken and sung. More precisely, we mean the ratio of Voice (code interindividual) and Word (individual act of use of the Code). They are distinguished as two systems of social functioning: orality and voice, where orality means the sense of the voice as the vehicle of the message and for voice everything is expressed in the operation of voice irrespective of message. Language and writing are so distant from each other to exclude mutual revelations. For research in the field, those who performed were overwhelmed by the richness of an archipelago where each island is a world. Instrumental and vocal music are now separated, and the study of the first shows that they had their origin in a text of which had sound only be important, and that definitely was emptied of all meaning. The true relationship between sound and meaning, between oral and written, is expressed in China in music (Picard-Rostagno, 1998, p.55). Listening to the vocal repertoire proves that there is no universal way to sing the Chinese that transcends genres, eras and regions. Non-system in the Chinese language is an intrinsic character of this force to be consistently musical. To mark this choice has been a long tradition that has linked the destinies of poetry and music. But the relationship is not mutual, and not all music has its origin in the Chinese language. If (Demiéville, 1962, p.20), this does not mean that has known only using the “five notes” (do-re-mi-sol-la). From talking to sing, certainly all possible intermediate stages have been developed. The primacy of writing the word did not stop reading changes appeared as an aberration, or at least as a borderline case in which the sound dissolves into silence.

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The classic Chinese identified with the same term (nian) and a "read" that the "read aloud". The modern Chinese saying "look" (kan) a book indicated the silent reading. The Confucian sacred books, and symbol of the predominance of literary culture, were recited. The psalmody of the choir Buddhists (fan bai) produces an effect similar to the overall flow and ebb of the sea (haichao); the work is both ballads know no changes in the recitative and song; certain forms of ballad as the "fast book" (kuaishu) performed with a simple accompaniment of claquettes, certain songs of north-east cannot be compared only with the spoken rhythm (rap) of American blacks. Even when the poetry he invented forms (fu), which “are not sung”, it is oral, and pays attention to the game on the pitch or timbre, and even the rhythmic pulse. Ideogrammic Chinese writing does not indicate the pronunciation, which can vary without the written notes. The transmission of the meaning of a text is thus ensured regardless of the language. Conversely, the orality makes sound vulnerable to disruption or change minimum; fortunately, the immensity of the country, the breakdown in languages and dialects in the maintenance that has been facilitated by their common writing have allowed some communities to maintain a spoken much music, from Canton to Amoy to Shanghai, without doubt, much less in the North. The inability of Chinese writing to fix the sound positive aspect has an objective the importance of orality, and music. But please do not text-lines, shapes, styles, and witness of the authentic: we can still hear an alternate songs, improvised in the rice fields or in the collective farms, such as those that Marcel Granet (1982) revealed in the “Fee poems of” we can find writers able to recite from memory in the old style. Chinese is a language of monosyllables, where the words do not mark either gender, or number, or time, or form, or the case: the application of the lead needed will make poetry less, and these ambiguities are total beauty. Paul Demiéville has shown masterfully: (Demiéville, 1962, p.12). Yet the regular verses are not the only the strong syllables (each syllable in his note), although frequent it is not an absolute rule. On the one hand, the juxtaposition of a new text on an existing melody can impose the use of syllables empty (our "tra-la-la"), of this rule is in contradiction between the tones with respect language. The latter, symbolized by different absolute heights, have in reality as the main feature of movements, without absolute height or subdivision determined the range of heights in the intervals. Chinese writing, usually so full of meaning, has its magical characters, the fu; the voice has its songs and its charms, like free by the sense and therefore much closer to the sacred. The breath, the sigh, the roar, the muttered, rice, whistle are other possibilities of the human voice. Each of them has its own use, its effectiveness. The language and words do not belong only to the speaker. There is then the multiplicity of voice.

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In Chinese music, the rhythm is entrusted to the percussion instruments: skin tight, bronze plates, bells, sticks and boxes of wood, stone, clay jars, and set of tools beaten fought with each other, rubbed or scraped is immense. Scores in the Tang era, pai isochronous is the use of a percussion, and corresponds to the concept of the bar. In the notations gongche later, in which a rhythm appears, pai is the time. Ban, opposite yan, is the first time (pai) a cycle in one (liushuiban), two (one ban and one yan) or four (one ban and three yan). The claquettes may also mark the break phrases, and then would correspond to commas and periods, if the texts know Chinese punctuation. But more than the measure, in the Western sense of the return of the time, the claquettes mark the time, so variable and diverse. Also, here, the word ban is used: time floating or scattered (sanban), slow time (manban), Quick Time (kuaiban). The claquettes are played by both the operator can, in many ballads and balladsinger, both from the main drum (Picard-Rostagno, p.62). The concept of time: In Chinese thought, nothing spontaneous marked the time, which is neither smooth nor divided, nor irreversible: you can write and rewrite. It is up to build a human model for the provisional ritual. The man acts on this model and consequently acts on time and to fix it groped through the alternation of the drum and claquettes, bell and drum, this will not prevent the mute. The consequences are immensely musical: the time of the ritual, the music, first of all requires an opening in the form of a prelude to invite the spirits, which cannot take place until the area is held sacred. It is impressive to note how music, like the story, does not give lead again to a linear course. The gradual acceleration of the time, practiced with great frequency, changing constantly isochronous time, marked from the option of Claquette and drum. Time lost (sanban), fluid (liushuiban), slow (manban), medium (zhongban), fast (kuaiban) one another and overlap, and the interest is constantly returned to the art of transition and of its preparation. As for the instruments, they overlap and contrast with the pleasure they sound different densities, the placidity of a three-stringed lute and the mouth of a virtuous of the lute p'ip'a or dizi flute. Impulse rhythm, time, phrase, gesture coexist so well, and diversified mixed at the same time, to the point that a transcript in values (minim, crotchet, quavers...) with arms of the bar, which is always possible, results always false. For Taoists, the universe is inhabited by Breath and all things are the result of greater or lesser condensation of energy. The Breath lighter (Yang) came to form the sky while the heavier Breath (Yin) descended to form the earth. Breath being the energy and movement can only be the sound which pervades the world and man, so man, who keeps himself in a portion of the sound of Breath universal, be in harmony with the world and resonate in sympathy with the universe using the music, sounds and voice. In Chinese music, essentially melodic, there is neither harmony nor the polyphony in the Western sense. Each piece is characterized by a melody relatively fixed, the notation indicates a melodic line that the performers must meet. But in a complex instrument, the execution of the melody, even in unison, makes us feel simultaneously several line

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melodic slightly different due to the particularities of the instruments and produces what we call heterophony very frequently in the accompaniment of songs or the songs performed by a complex instrument such. Music is the essence of the universe, the nature of beings. The union with the substance, the agreement with this kind, this is the harmony (Ruan Ji, 1987, p.73). Division and harmony, ritual and music, form a pair of complementary bodies as diverse as inseparable and the shadow and light, male and female, sun and moon, Yin and Yang. Unlike what happened in the West, the musical language Chinese, with its pentatonic scale, has never felt the need of a centre of gravity around which the melodies are rotating and the lack of sensitivity is that the range semitone of the inside of the pentatonic scale has also opened the door to a system devoid of musical sound complex architectures, which is why the musical heritage very structured and complex, for centuries has been handed down orally (as has happened in the West during the first thousand years of Christianity) and the notation has been limited to the practice of tablatures for simplifying the implementation of the instrumentalists. The lack of written documents, especially with regard to popular music performed in the endless territory of China, makes it difficult to analyze the vast assets and more challenging comparisons with the West. Within this concept, precisely because the Chinese language is monosyllabic, rhythmic in dimension, as has been handed down for centuries, with the distinctive feature the immutability not felt the need to develop a practice and hence a mensural notation articulated and complex, the rhythm is a "dimension" of the sound and order of movement (of the matrix Platonic) should not be understood in the Western sense of the term, i.e. rigid, synchronic, but on the contrary, flexible, formalized within of rites and ceremonies to enable man to get in condition to be able in part by cosmic echo portion of the "sky" that in itself is enclosed. In this sense, the rhythm and more generally all the music of China has always had an ethical value, and this even before Plato in the West support a similar argument ethics (IV sec. BC). Among the western rhythm and Chinese rhythm, the only possible comparison is with the game, but in its richest: Rules defined underlying subtle, inventive, interactions between allies, but also the game between the elements without which a machine does not work: “the idea of the game approaches, in an unstable equilibrium and paradoxical, the two groups of concepts formed by the concepts of freedom, open, empty space on one side - and from those of adjustment, contact, closing the other systemic” (Henriot, 1990, p.1391). Long associated with archery, it is the first of the attributes of the letters before the checkered, long before the calligraphy and painting. In a "young ones" (you), there are traces of "swimming", "float", "fun", "err", and "wander", being nomads, all musical features. Play, joy, pleasure, it also says yule, where yu is written with the woman key and where le uses the same character, pronounced differently of "music" (yue).

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7.3 Conclusions We try to summarize in two tables, some of the results of this chapter: European Culture Rhythm and syllables Rhythm and Metrics Metrics such as succession of syllables Scale: modal, tonal, dodecaphonic

Chinese Culture čie (rhythm), pan-yen (measure) the rhythm track (measures 2 and 4 times) is very frequent, while the ternary (measures 3 times) is very rare in traditional classical music. Scale: pentatonic

Rhythm as an order of movement (Plato)

The music is nature which is manifested through the sound and the rhythm a "look" of the sound

Middle Age: Gregorian with free rhythm and meter without quarrels prestability

Language monosyllables Singing syllabic

Misic, text, italian Melodramma

Music lyrics: Italian Opera Music lyrics: melodic compositions and polyphonic harmony without the Western

Measured rhythm: is based on the provision of accents. Free rate (no provisions in the accents) and measured

Free rhythm (no provisions in the accents) and measured

References Abbiati, M.: La lingua cinese, Cafoscarina, Venezia (1992) Allorto, R., Perrotti, P.B.: L’educazione ritmica, Ricordi, Milano (1986) Becatti, R.: Nella Musica, Fabbri Editori, Milano, vol. I-II (1997) Berlioz, H.: 21° soirée (1852), in Les Soirées de l’orchestre; ried. Paris, Gründ (1968) Demiéville, P.: Antologie de la poésie chinoise classique, Paris, Gallimard (1962) Galante, D.: Aspetti didattici dello studio delle trasformazioni geometriche: l’Offerta Musicale di J. S. Bach. Quaderni di Ricerca in Didattica (G.R.I.M), Palermo 8, 1–25 (1999), http://math.unipa.it/~grim/quaderno8.htm Galante, D.: I suoni armonici, le trasformazioni geometriche e i processi compositivi in J. S. Bach: una proposta didattica, nei Proceeding del CIEAEM 57 Changes in society: a challenge for Mathematics Education, Piazza Armerina (2005)

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Galante, D.: I suoni armonici e le isometrie nella musica tonale: l’Offerta Musicale d i J. S. Bach, Progetto Alice. anno III, Casa Editrice Pagine S.r.l. Roma, vol. VI(18), pp. 459– 492 (2005) Galante, D.: Le matrici come espressione delle isometri e nel procedimento compositivo di Pierre Boulez: la serialità integrale in Structures I per due pianoforti. Proposte didattiche. Quaderni di ricerca in didattica, a cura del G.R.I.M (2006), http://math.unipa.it/~grim/quad16_galante_06.pdf Galante, D.: Mathematics, Physics and Music, to interpretate didactic situation in secondary school. Quaderni di ricerca in didattica, a cura del G.R.I.M., Palermo 17(Suppl. 4), 1– 203 (2007), http://math.unipa.it/~grim/quad_17_suppl_4.htm Granet, M.: Il pensiero cinese, Adelphi, Milano (1971) Granet, M.: Fêtes et chansons anciennes de la Chine, Paris, 1919 ; ried. Albin Michel (1982) Henriot, J.: Jeu, in Sylvain Auroux. Les notions philosophiques, dictionnaire, Paris, PUF (1990) Humeau, S.: Le musiche che guariscono. Teoria ed applicazioni pratiche dellla musicoterapia. IPSA, Palermo (1990) van Khê, T.: voce Cina in DEUMM (Dizionario Enciclopedico Universale della Musica e dei Msicisti) UTET, Il Lessico, vol. I Martin, F.: Note sur l’histoire de la série des quatre tons. Extrême-Orient Extrême Occident, 12 (1990) Picard, F., Restagno, E.: La musica cinese, La tradizioni e il linguaggio contemporaneo. EDT, Torino (1998) Ji, R.: Studio sulla musica (Yue lun), intorno al 250. Ripubblicato in Ji Liankang, Ji Kang, Saggio. La musica non conosce tristezza né gioia (Ji Kang: Sheng wu ai/e lun), Pechino, Renmin yinyue (1987) Tentons l’expérience, «Annales», l’editoriale, Paris, Armand Colin (Nov.-Dic.1989) Lao-Tzu: Il libro del Tao, Newton Compton, Milano (1995) Zumthor, P.: Nella Prefazione a Corrado Bologna, Flatus Vocis, Il Mulino, Bologna (1992) Willems, E.: Le rythme musical, Parigi (1954)

Chapter 8

Conclusions 8 Conclusions

8.1 The Conclusions of the Experimental Work In a school such as the intercultural today, to analyze the cognitive styles of students through a socio-cultural approach, highlighting potential similarities and differences in patterns of thinking, attitudes, beliefs and conceptions of the students of various ethnic groups, with respect to the acquisition of specific mathematical concepts, is certainly a complex operation but is now a "need" teachers and students themselves for a possible integration of knowledge at stake and the mediation of knowledge. The national and international research in mathematics education in recent years has shown sensitivity to the problems treated. We define possible "comparisons" between strategies of teaching / learning used in different countries (mainly for primary schools) in the approach to key content in mathematics such as arithmetic and Algebra (Cai, 2002;- Fan Wong, 2004; Leung, 2000), the mathematical content for the privileged treatment of this text. The literature has also highlighted proposals on education capable of fostering the integration of pupils of different ethnic groups in the classes of the host through the establishment of intermediary cultural environments (D'Amore, 2000, Sparks et al, 2003; Favilli et alii, 2004, Sparks, 2006°, 2006b); Tillema, 2005; Skovsmose, 1994; Spagnolo et alii, 2008). Over the past ten years, we have also witnessed an increase in more and more research on "comparison" between the mathematical performance of students. Particularly, those developed are aimed at students of U.S. and Asian countries such as China, Japan and Korea (Becker, 1992, Cai & Silver, 1995; Husen, 1967; Lapointe, Mead, & Askew, 1992; Robitaille & Garden, 1989, Song & Ginsburg, 1987, Stevenson & Lee, 1990; Steven-son & Stigler, 1992, Stevenson, Lee, Chen, Lummis, Stigler, Liu, & Fang, 1990, Stigler, Lee, & Stevenson, 1990 S. An, 2008; Z. Wu, 2008; J. Cao et al, 2008; Yeping Li & Chen Xi & Gerald Kulm, 2009) few are the work of comparison with Italy and specifically, works that take into account inter-environments within the classroom Italian. Our most recent work, in entering this field of study, aims to analyze the theoretical framework, experimental epistemology of mathematics, some of the possible similarities and differences in the forms of reasoning logical argument put out by Chinese and Italian students, attending Italian schools in all school orders. The generalization of the results is ensured by the method of experimental epistemology and through a careful a priori analysis that takes into account the cultural processes (epistemological representations of Mathematical and Natural Language, historical–epistemological representations and behaviors expected), analyzes the phenomenon of teaching/learning both quantitatively and qualitatively. This analysis is conducted in a continuous balancing Micro-didactic / F. Spagnolo and B. Di Paola (Eds.): European and Chinese Cognitive Styles, SCI 277, pp. 229–242. springerlink.com © Springer-Verlag Berlin Heidelberg 2010

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macro-didactic. Essentially, a detailed micro analysis may allow us to hypothesize inferences on the macro.

8.1.1 Topics Such as Research Agreements? In the text, it is referring primarily to two issues related to each other and developed through experimental investigations related to theory/methodology. The first refers to an experimental work (Ajello-Spagnolo-Xiaogui, 2005; Spagnolo et alii, 2008) conducted in parallel with Palermo and Nanjing (China), the second refers to a doctoral thesis developed in a multicultural classroom Palermo. The first research topic, not referring to a specific mathematical content, aims to present situations/problems typical of mathematical thought with particular attention to logical and linguistic problems and argumentation of the students. The most significant reference of the research is the survey on patterns of reasoning in relation to Eastern Europeans with those instruments of historical epistemology. Equally important, for the study of contingencies, is the experimental analysis on the use of epistemic logic with respect to the underlying natural languages of the students used (Chinese and Italian). The second theme of research, aiming primarily to the cognitive processes, was brought to light by the students of different educational system (primary and secondary level) in relation to a specific mathematical context in which the transition from arithmetic thinking to algebraic thinking is proposed to study, through different experimental investigations (Di Paola&Spagnolo, 2006; Di Paola&Spagnolo, 2008a; Di Paola&Spagnolo, 2008b; Di Paola&Spagnolo, 2009), analyzed qualitatively and quantitatively (Gras et alii, 2008), the “behaviour” of Chinese and Italian students, through the use of linguistic and historical–epistemological tools. Particularly, significant for the work of comparison1 is the use by pupils of different orders with different educational and ethnic group, incorporated in the same class group, the concept of variable as unknown and report functional and parametric (Arzarello, Bazzini, Chiappini, 1994; Malisani & Spagnolo, 2008; Matz, 1980; Navarra, 2003; Radford, 2003, 2006; Sfard, 1991; Wagner, 1985). In this sense, it was suggested that the Chinese written language Natural may, in its function of its internal structure, composition, encourage Chinese students in certain key skills for algebraic thinking (such as the concept of variable and the process of generalization within the linguistic structure can be accessed through a definition of a complex system of type "parameter" related to the role of "radical"). In Chapter 5, the second of the two unifying themes was as follows: H1: The differences and similarities in the history of Eastern culture2 and Western Dental also have an equivalent in the differences and similarities between the patterns of reasoning found today in a teaching / learning of mathematics. 1

The term comparation is understood throughout the thesis work within the given all' ICME-10 (International Commission Mathematics Education). 2 The set of values, behavior patterns, and also of materials characterized the distinctive way of life of a social group. In a perspective that takes into account the communication processes, natural language is then understood as a product of a certain culture because it can spread. At the same time, the culture can be influenced by natural language as a bearer of meanings historically traced.

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This was then varied in the two research themes, through under hypothesis more specific on conjecturing and arguing on the one hand, and the specificity of thought arithmetic / algebraic on the other. In order to falsify the hypothesis of research, both issues were used the following references paradigmatic: - Historical analysis and historical–epistemological mathematical thinking, as regards the study of differences in the forms of reasoning (arguing, conjecturing and demonstrate) and the algebraic tradition in antiquity. The latter, read an overview of possible through a "comparison" between the Elements of Euclid, as a canon reference to the Western tradition Mathematics and Jiuzhang Suanshu for the Chinese. The references were mainly theoretical in this sense the work of Chemla (2001, 2004, 2007), Needham (1985), Cullen (2004) and Granet (1988); - Language analysis and neuroscience on alphabetic and ideographic languages to a comparison of the cognitive aspects related to the natural language of origin of students and thus may have impacts on learning of mathematics (Tan et alii, 2001). Study of meta-rules for settlement of the Chinese written language in relation to some cultural aspects of their ethnic Chinese as the Taoist yin-yang and aspects that can find patterns in algebraic reasoning and pre-algebra; - Experimental analysis of situations / problem through the approach of the Theory of Didactic Situations (Brousseau, 1997, Spagnolo, 1998, 2009); - Analysis of cases through the individual interview method. For a detailed analysis of the comparative study between Chinese thought and Italian, in situations of teaching/learning in multicultural perspective, in a first approximation, the studies of D'Ambrosio (1985, 1992, 2002) Gheverghese J. (2000), Bishop (1988) and Nisbett (2001, 2007). Study of meta-rules for settlement of the Chinese written language in relation to some cultural aspects of their ethnic Chinese as the Taoist yin-yang and aspects that can find patterns in algebraic reasoning and pre-algebra; - Experimental analysis of situations/problem through the approach of the Theory of Didactic Situations (Brousseau, 1997, Spagnolo, 1998, 2009); - Analysis of cases through the individual interview method. For a detailed analysis of the comparative study between Chinese thought and Italian, in situations of teaching/learning in multicultural perspective, it is reported, in a first approssimazio-ne, the studies of D'Ambrosio (1985, 1992, 2002) Gheverghese J. (2000), Bishop (1988) and Nisbett (2001, 2007). 8.2 The Ex perimental Episte mology of Mathe matics

8.2 The Experimental Epistemology of Mathematics: Concluding Observations on the Experimental Investigations Discussed 8.2 The Ex perimental Episte mology of Mathe matics

Research in Teaching of Mathematics, by its nature, is never conclusive in the sense that it cannot proceed for trial. It must be satisfied with evidence. So, it is

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necessary to define what counts, that is affecting such insights, such behavior, in which students, under what conditions and which therefore the evidence in relation to such significant variables. Research in education does not provide correct answers, but then the instruments, tracks, opportunities for thinking and methods. In this sense, the very nature of research is provided by the second process which takes place and is through a series of special features such as: the descriptive power, the explanatory text, the definition of the purpose, the predictive power, and the rigor and specificity of the method, the falsify and replicability of research, multiple sources of evidence. Certainly, it is not always possible to meet all these requirements; cross condition in general, however, is a thorough study of the processes of thought of the stages of teaching / learning contextualized to disciplinary foundational content. As mentioned several times, the research work involved here has tried, albeit in a first approximation, to discuss some experimental evidence relating to a disciplinary field of inquiry in our view significant, such as algebra and pre-algebraic, "read" in a ethno-mathematics through a critical discussion of possible references epistemological–cultural significance for the stages of teaching/learning of mathematical content Treaty. As mentioned several times in the course of treatment to assess the cognitive styles through a socio-cultural approach, emphasizing what may be the patterns of reasoning, attitudes, beliefs and concepts about the acquisition of a particular concept, a specific knowledge, is certainly a complex task, more so in our case, given the reference system analyzed (cultural system Western/Italian - cultural system east / Chinese) but may be the key to teaching a more attentive to the different skills and therefore to each other; "need" now heard by all players in the school.

8.2.1 The First Experimental Investigation Regarding the first issue, the data collected with respect to this phase of the research project (Ajello-Spagnolo-Xiaogui, 2005; Spagnolo et alii, 2008) seem to confirm the Chinese students to conduct a highly pragmatic, concrete, put in Light from a discourse of algorithmic procedures closely related to a holistic thinking on the encoding and decoding of the information presented in various situations / problems. These, in the process of mathematical reasoning and conjecturing, generally show a heuristic trial and error, to find a "basic algorithm" as a tool demonstration3 for Chinese students, the argument and the organization of the reasoning are hierarchy of models of reasoning (and sub-models as sets and subsets) that seem to refer to implications Fuzzy (Gras & Spagnolo, 2004). The typical reasoning shown by students is Italian rather than hypotheticaldeductive type through a chain of finite conjunctions contextualized to a bivalent logic. The two different views have been comforted with the appropriate historical epistemological analysis of mathematics in the two cultures with particular reference to this subject and to demonstrate. 3

One of the tools used in the history of Chinese Mathematics to demonstrate conformity of a sequence of steps with respect to a class of concrete problems is the algorithm. The algorithm is referred to as iteration of two choices conditional (If.. Then) with a starting point for any.

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The most significant results can be summarized, as mentioned above in paragraph 5.2 ms with the following table: Questions Prevalent behav- Prevalent behave-

1

2

3

4

5

iours in the protocols of the interviews (PC) Heuristic approach for attempts and errors. Research of an algorithm as a tool of formalized demonstration. Request of a concrete context to analyze the adequacy of the proposition in hand. Measure of the conformity of the affirmation at hand with the premises. More care of the analysis of the text. Use of tables or matrices. Organization of the data for the research for conformity with a model (diagrams, previous idea, analogous situations) Reasoning of the combinatorial type with representations by tables. Analysis of all possible cases to encourage the renewal of the model.

iours in the experimental results in Italy (I) Inductive reasoning: finite chain of conjunctions.

Prevalent behaviours in the experimental results in China (C) Experiment and induction. Proof and intuition.

Use of Venn’s diaProof of the truth and grams for deduction falsity of the (the proposition thus propositions. ends up false).

Use of Venn’s diagrams for deduction and a correct interpretation of the syllogisms. Deductive processes in N.L.

Correct us of the reasoing patterns of the syllogisms.

Organization of the data for the analysis of all the possible cases. Use of division by distribution. The pigeon hole principle.

The problem is not recognised as referable to a known pattern of reasoning. It is not solved.

Use of contrapositives and therefore of reasoning to the impossible in N.L. and with the help of double entry tables.

Use of counter contrapositives and therefore reasoning to the impossible in N.L.

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8.3 The Second Experimental Investigation Backbone for the research work on the second experimental investigation, contextualized through a selection method such as that in the theory of Didactic Situations Brousseau (Brousseau, 1997, Spagnolo, 1998, 2009), was the firm belief and importance of evidence the integration of knowledge through a possible socialization of cognitive styles in practice teaching and the stages of learning, particularly under conditions of multiculturalism (validation phase of a-didactic situation). This (see chap. 5), concerning the discussion of the experimental contingencies, was then the cornerstone for the understanding of learning brought to light on algebraic thinking and integration of knowledge in the transition from arithmetic. In this point of view, it was considered necessary to draw a brief historical–epistemological process that can draw criticism, quite basic, the key stages of the long journey and suffered from the trend as Algebra Arithmetic and to focus attention on the concept of equation as the heart and of algebra as significant for the experimental investigation. Context the emphasis was on Jiuzhang Suanshu considered representative text of ancient Chinese mathematics culture. It has tried, albeit in a first approximation to find some classic meta-rules that seem to be as fundamental today to the algebraic thinking, in some ways for the Chinese written language. The latter reference epistemological treated in chapters 2 and 3 in a continuous balancing mathematicslanguage (the cognitive aspects of encoding and decoding of alphabetic and ideographic writing) was central in defining the assumptions of the research work and thus the definition of variables experimentally controllable behavior. The "translation" of the mathematical skills related to learning the language shows the students, as discussed in the handling of work, through a parallel learning of maths / language (contextual memory in the process and techniques of composition), the "discovery" of a first rough idea of variables and symbolic equation. As discussed in Chapter V, though not explicit in boy language writing is an equation, he realized, so implied, may acquire algebraic skills that may in some respects be of help in the process of internalization of formal algebraic thinking. Needham as reported, “Such equations constitute a semiconscious mental foundation for whoever is acquiring familiarity with the language.” (Needham, 1985, pp. 35-36, vol. I). Particularly significant is the sixth and final category of classification of Chinese characters (xíng), which is a hybrid category in which the central element that plays a shēng role of classification is that of “radical”. The “key”, the radical, according to a mathematical reading shows, from an unknown hand (a sign-specific but "indefinite"), which takes its meaning in relation to linguistic context in which it is inserted ("word → ask," heart → sentiment "etc.), on the other hand, however, could assume the meaning of the generalized sign, sign capable of conveying the character in which it is inserted and allow the reader to identify him. Further observation in the work discussed in combination of algebra-written Chinese language is that which comes out from the analysis of functional relationships existing within a compound, among the various component parts. In this sense, for what interests us specifically, the "key" assume the role of parameter. A diagram of the above can be described by the following image that represents the syntactic–semantic links between different characters from their "associates":

形声

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Fig. 1. Syntactic-semantic links between different characters with each other "associated"

The dual language–mathematics is considered throughout the thesis work as meaningful combination for the study proposed in relation to Chinese culture and has been taken into account for the experimental analysis. As mentioned in section 5.1, the results of these experiments are as follows: Context

AlgebraicFormalized

Pre-algebraic

Argumentation/ Generalization

Using the parameter as the achievement of algebraic thinking.

Chinese Students

Italian Students

In formalized algebraic contexts, even if do not underline always an aware algebraic thought, they are mostly able in comparison to the Italian contemporaries as good “solvers”. The experimental contingency seems to confirm this to all the scholastic levels. Good capacity of manipulation on algebraic formula. In pre-algebraic contexts, they tightly underline a arithmetic reasoning schemes connected with a strong memorization of “arithmetical facts” that help them in the definition of a holistic thought as a relational thought. The reasoning and the phases of argumentations in the process of generalization (“to think for cases between Arithmetic and Algebra”) are defined through an algorithmic-procedural thought. The idea of parameter as possibility to express different problems through “similar” writings. The role dl context as significant to the solution of the problem.

Did not show mature algebraic thinking. The idea of variable is limited to the idea of "thing that varies" (Malisani 2006).

Highlight difficulties in identifying and monitoring unknown this level semantic and syntactic. Thought Arithmetic, predominantly, is an obstacle to thought Algebraic. Arguments in natural language, difficulties in the stages of processing and conversion of semiotic registers (D’Amore, 2000, 2008). Lack of awareness on the role and use of the parameter as a possible way of generalization.

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The five experimental investigations are "Sudoku Magic Box", "Problem of Fermat", "Questionnaire and variable parameter in different semiotic contexts", "The sequence" and "The grid of numbers” and they are profoundly different in the context of presentation skills pre-algebraic and algebraic applications, methodology and sample survey may, in our view, define, at least with respect to the data collected, significant both for the proposed disciplinary objective and targeted primarily to the cognitive processes brought to light by students of different school orders in relation to a specific mathematical context which the transition from arithmetic thinking to algebraic thinking, both for the phase of cultural mediation in the knowledge game. These key aspects for the process of institutionalization of mathematical knowledge between culturally different students involved. Wanting schematized the results discussed in the sixth chapter to the individual experimental situations, it would seem, on the data collected, that the Chinese students, in agreement with what emphasized by several international studies, confirm a type of behavior very pragmatic, concrete, shown by an argument of algorithmic procedures closely related to a holistic thinking on the encoding and decoding of the information presented in various situations / problems. These, in the process of mathematical reasoning and conjecturing, raised in various teaching situations, seem to generally show a heuristic trial and error (like arithmetic), aimed, unlike the Italian students, in search of a "basic algorithm" as a tool demonstration. The same kind of arithmetic reasoning is instead used by the Italian students in a demonstration but no sense of numeric exploration and conjecturing that often limits the transition to Algebra. For Chinese students, the argument and organization of reasoning are generally for ordering patterns of reasoning that does not seem logical to refer to the type of divalent but through extensive use of the idea of "variability" as the initial report of "expressions" different and later as an "expression" dynamic related through a "formula" to other "expressions" are also dynamic (examples of this issue may find themselves in all trials). The idea can be discussed here, in our opinion, one of the significant aspects of algebraic thinking. The typical reasoning shown by students in Italy seems to be to mold the Aristotelian-Euclidean, of hypothetical-deductive through a chain of finite conjunctions contextualised to a bivalent logic. Then depict the different layers of thought expressed by algebraic Chinese and Italian students in the various trials (starting from the situation described for the primary school that on the context parameter to upper secondary school), it seems that the Chinese students, while highlighting a algebraic thinking always conscious (test is the cognitive behavior in environments that are not precisely formalized algebraically: experimental situations "Fermat Problem", "Questionnaire and variable parameter in different semiotic contexts", "The sequence"), you show, the more skillful compared with peers in Italy, as "resolvers" formalized in algebraic contexts and show greater ease compared to peers in Italian seize possible aspects of the report (expressed verbally, graphically and in tabular form) between variables in play. This trial appears to meet contingency at all school levels. A parallel reading of graphs CHIC reported in this thesis shows, in our opinion significant.

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As said, in a not completely formalized algebra, pre-algebra contexts such as "Fermat problem" of "Questionnaire and variable parameter in different semiotic contexts" in some item of "The sequence" and "The grid of numbers", in agreement with a strong historical tradition, show a kind of arithmetical reasoning (closely related to a strong memory of "arithmetic facts" that help in defining a kind of holistic thinking relational) and remain strongly rooted in this. The arguments in the process of generalization, as a natural evolution from a first thought for trial and error, refer to a thought of algorithmic-procedural. Good manipulative skills on algebraic symbolism (in some item of the “Questionnaire and the variable parameter in different semiotic contexts” and even in different experimental situations in "The sequence" and "The grid of numbers") may, as mentioned, on the contrary, result from the ideographic written language, as discussed in previous chapters, seem to help them into heavily formalized algebraic syntax checking. The environment of symbolic reference appears, in fact, due to the differences in content, in their family from a procedural point of view. The tree similarity reported to the experimental situation “Questionnaire and the variable parameter in different semiotic contexts” seems to synthesize all these aspects well.

PsAlg

Graf.9. Siilarity tree experimental situation “Questionnaire and the variable parameter in different semiotic contexts”. ITS and IPPM”

The research seems to confirm then, for the Italian context and partly in China, issues pertaining to the transition from arithmetic register to algebraic, semantically different, regarding the symbolic language formalized stresses once again like this for the students write (or equations systems of equations, areas of content used in the experimental work of research) is not active, in many cases, forms of productive thinking, should not be regarded as absolutely interpretative model of a problem or better yet a class of problems. This capacity that, as mentioned, is more easily seen and reached by Chinese students involved in class Palermo. The idea of argument, reasoning behind this (in a formal way in some item of the “Questionnaire and the variable parameter in different semiotic contexts” and in an informal situation in “The grid of numbers”), shows a different behavior of the two types of students, perhaps dictated by the peculiarities of language differences on that for Chinese students, according to the research of neuroscience discussed in previous chapters, is manifested in a continuous balance between a serial thinking, local, one global, holistic and capable of operating categorizations of cognitive and possible generalizations. As mentioned in Chapter 7, this aspect was analyzed qualitatively and quantitatively, in a broader framework for learning

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algebraic thinking and the idea of variable as "heart" of Algebra, interesting for the study of phase transition from arithmetic thinking, for trial and error, and the relational algebra (capable of driving in almost natural generalization of the process as "capability of noticing something general in the particular," Love, 1986, Mason, 1996) allows us to hypothesize in linguistic structure of Chinese ways of "help" in overcoming specific barriers to algebraic thinking (discussed in Chapter 2 in relation to a literary context), constructive components of the Chinese ideographic writing which could, in years, easier “access” to Algebra while not providing the students a mature algebraic thinking, perhaps hampered by strong epistemological factors, which are found on both students attending classes multicultural Palermo that their peers not attending. A comparison between parallel behavior explained by Chinese and Italian students involved in the five trials shows that there is greater similarity between the behavior of pupils in two different ethnic group where the degree of formality is lower or even nil. One could hypothesize that the differences are more evident as you choose one or other symbolic ways. This, in our view, could depend, as mentioned, in a language. The research of neurophysiology says that thought parallel (emphasized by the Chinese students on all school orders) is mainly played on Fuzzy behavior from a neuron, at the stage of awareness of self (Boncinelli, 2002) there is then a alternation between thinking and parallel thinking serial. The thought is parallel to the fuzzy thinking and thought to thought divalent serial (usually found among the Italian students, see Chapter 4). This is an open problem and requires further experimental investigations related to a reference sample broader in that it can take into account the ethnic Chinese students entered school in Palermo, but also directed research on Chinese students included in their original context. The research team of Palermo has already started working on this last area of research through international cooperation on "Classroom Teaching for All Students Re-search Working Group" which involves partner countries such as China (Beijing Normal University, Tianjing Normal University, Nanjing Normal University, East China Normal University, Hang Zhou Normal University), the United States of America (US-California State University, Long Beach, National University in California, Louisiana University, Grand Valley State University, Montclair University, Indiana University), Korea (Seoul National University) and Malaysia (University of Malaysia). 8.4 An Open Pro blem on t he "Sig nifica nce" of the Sa me in Both C ultures

8.4 Conclusions and Open Problems Related to "Strategy and Tactics" As already discussed in Chapter 6, the reference to the Wei-ch'i has enabled us to make inferences about the role, including teaching, strategy and tactics in the two cultures. This topic is closely connected with what is discussed in Chapter 4 on the logic underlying the differences in reasoning patterns of the two cultures. The game of chess and the Western Wei-ch'i (or Go) allowed us to highlight these issues and discuss a possible schematic didactic references.

8.4 Conclusions and Open Problems Related to "Strategy and Tactics"

239

The table that follows, the limit of any diagram may be helpful to focus those elements that can then be used for any educational issues: Discipline Purpose of the game Checkmate Chess (capture of the Adversary King).

Social function None in particular – an exception is the Soviet Union from 1925 to about the 80s

Control of the largest area possible of the territory on the goban.

Very important – second place among the traditional arts and held to be necessary for the education and the instruction of functionaries and dignitaries in imperial China and imperial Japan.

Wei-ch’i or Go

Strategic elements

Function of tactics

Identification of partial objectives and evaluation of the position. Optimization of the action of one’s own pieces and limitation of the adversaries. Recognition of visual and abstract patterns. Concept of control, in the sense of a continuous evaluation of the total situation. Choice of that which can be held to be satisfactory. Strategic items: defence of own groups, attack to opponent’s groups, territorial gain.

Determining. A single tactical action, well carried out, becomes the main and often conclusive one of the match.

The knowledge of the tactical themes is very important, but a local victory risks being lost. Tactics are not tied to strategy. The concept that a tactical success leads to a strategic success is alien to the spirit of Wei ch’i.

8.4.1 Links with Education We maintain that several thematic explained above can have connections to didactics. The different approaches to strategy and tactics in different cultures supply, in their entirety, some formidable tools for confronting the most disparate thematic of life and are very interesting in a didactic context. We limit ourselves to giving some schematic notes on these possible connections, providing that a more specific analysis of some of these requires dedicated research, with quantitative and qualitative methods.

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1. First point, the full awareness that without dedication, unity of intent, clear definition of the goal, attention and motivation it is rare that one reaches an objective whatever it is. This point is of a general character, but precisely for this it is often neglected in Didactics. Demotivation and too bureaucracy are the probable reasons of it. 2. The aptitude of Evaluation: the considering of the elements with an elevated degree of objectivity and selectivity of one’s attention allows the formulating of strategies, and sometimes also methods and tools capable of the solution of problems and of the definition of a situation. That in line with a modern vision of didactics which aims at an increase of competences, that is, of the use of one’s knowledge and abilities in different contexts and situations; this particularly in Italy, where there is a proven lack in this direction. 3. Hypothetic-deductive thinking, also at a high deepening level, it is an important item in strategy games and useful in all fields. Hypothetic-deductive thinking is found in the western Culture since from Euclide’s Elements (Model of Aristotle Logic, Ajello-Spagnolo-Xiaogui (2005)). Hence, in the Western culture, strategic thinking may be addressed by Aristotle Logic (inherent to Natural Language), while in the Eastern Culture, as we mentioned earlier, by wei-ch’i “meta-rules”. 4. Visual or abstract patterns recognition, and continuous adaptation to the reality is the normal practice in chess and in wei-ch’i, typical of a high order competences. A strategy adopted in a repetitive way, without adaptation to reality, can be proofed wrong by practice. This is related with Epistemological Obstacles Theory in Education4. This feature is found in both cultures; chess and wei-ch’i support it. 5. Education to strategic and tactical features recognition could be important in social fields. It could become, as mentioned earlier, an useful tool to understand different cultural approaches. We can imagine a more Hypotheticdeductive approach by western students, and a sharp separation between strategy and tactics. In eastern culture, this separation is not so sharp (see also appendix 2 on Chinese language). Probably, this approach depends also on Wei-ch’ì meta-rules. These meta-rules may clarify the differences between strategy and tactics in a more subtle way with respect to western approach. 6. The self-concept being separated from others as a reference for decisions and constant search for best solutions are useful topics in Education.

8.5 Conclusions and Open Problems Related to Music 8.5.1 The References to Mathematics, the Reasoning and the Conceptualisation We can reassume in this conclusive chart some of the meaningful results that we hold can be tools of observations for multicultural classes with Chinese students. We think that these results can be generalized in different cultural contexts. We hold in this sense that experimental investigations directly performed in China and in Italy 4

Bartolotta, op.cit.

8.5 Conclusions and Open Problems Related to Music

241

they would allow further close examinations on the Natural Language and on the phases argumentative schemes that from this can derive (you see paragraph 8.4).

Natural Language Logic

European Alphabetic writing Lateral thinking, Bivalent logic prevails: 1. Bivalent logic (about 80%) 2. Fuzzy Logic (about 20%)

History of Argumentation and demonstration

Aristotle, Plato, Hegel and the hypothetical-deductive reasoning. Euclid as a paradigm of Mathematics until 1800 and as an argumentative siystem.

Arithmetic/ Algebraic Though

The variable as unknown and which varies (70%). Variable in the relational sense (30%).

Strategy

The hypothetical-deductive system as a strategic system. Choosing the winning strategy in relation to defined predominantly bivalent logic. Using patterns of recognition (20%) according to fuzzy logic. Hypothetical-deductive reasoning (logic bivalent) (80%).

Music

Rhythm and syllables Rhythm and Metrics Metrics such as succession of syllables Scale: modal, tonal, dodecaphonic

Chinese Ideographic writing categorized with meta-rules. Relational associative thinking. Fuzzy logic prevails: 1. Bivalent Logic (about 20%) 2. Fuzzy Logic (about 80%) Tao, Confucius, Buddha. Fundamental algorithm for classes of problems (9 chapters). Meta-rules. Mao: classification ranges of percentages. The variable as unknown and which varies (30%). Variable in the relational sense (70%). Parameter in the written language system. Choosing the winning strategy in relation to defined within the system-Confucian philosophy (there is no overall winner) and adjusted by fuzzy logic on the target system ying / yang Using patterns of recognition (80%) according to fuzzy logic. Hypothetical-deductive reasoning (logic bivalent) (20%). čie (rhythm), pan-yen (measure) the rhythm track (measures 2 and 4 times) is very frequent, while the ternary (measures 3 times) is very rare in traditional classical music. Scale: pentatonic

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Rhythm as an order of movement (Plato)

The music is nature which is manifested through the sound and the rhythm a "look" of the sound

Middle Age: Gregorian with free rhythm and meter without quarrels prestability Music and text: Italian Melodramma

Language monosyllables Singing syllabic

Measured rhythm: is based on the provision of accents. Free rate (no provisions in the accents) and measured

Music lyrics: Italian Opera Music lyrics: melodic compositions and polyphonic harmony without the Western Free rhythm (no provisions in the accents) and measured

The cultural classification found in Chapter 1 (Table 1) is it still valid? May it still be a point of reference in comparing socio-cultural? We think so, but with varying degrees of awareness and perhaps reinterpreted in a globalized world and culture of the web 2. We want to conclude by re-proposing the table as a possible navigator for this millennium: Chinese Classification (in the order)

Music, discipline related to hearing and also to sophisticated processes that regulate vibrations. It is the practice of the inner qì. It concerns also with embodiment referred to the musical rhythms. Today we know that it is related to natural language and numbers learning. Wei-ch’i (Relationship between partial modelling and holistic view with a strategic intention.). Writing (holistic representation, algebraic structure , variable, parameter). Painting. (It is considered a quite direct and subjective discipline, accessible to all)

Western classification, by Marziano Capella, philosopher of late Latin age (IV-V century A.D.) Trivio (artes sermocinales): Grammar, Rhetoric and Dialectics Quadrivio (artes reales): Arithmetic, Geometry, Astronomy and Music

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