SCIENCE EDUCATION ISSUES AND DEVELOPMENTS
SCIENCE EDUCATION ISSUES AND DEVELOPMENTS
CALVIN L. PETROSELLI EDITOR
Nov...

Author:
Calvin L. Petroselli

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!

SCIENCE EDUCATION ISSUES AND DEVELOPMENTS

SCIENCE EDUCATION ISSUES AND DEVELOPMENTS

CALVIN L. PETROSELLI EDITOR

Nova Science Publishers, Inc. New York

Copyright © 2008 by Nova Science Publishers, Inc.

All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Science education issues and developments / Calvin L. Petroselli, editor. p. cm. Includes index. ISBN-13: 978-1-60692-604-8 1. Science--Study and teaching. 2. Education. I. Petroselli, Calvin L. Q181.S37342 2008 507.1--dc22 2007031841

Published by Nova Science Publishers, Inc.

New York

CONTENTS Preface

vii

Expert Commentary Commentary

Why do We Lose Physics Students? Ronald Newburgh

1

Research and Review Articles Chapter 1

A Numerical Landscape Robert Adjiage and François Pluvinage

Chapter 2

Learning in and from Science Laboratories: Enhancing Students' Meta-Cognition and Argumentation Skills Avi Hofstein, Mira Kipnis and Per Kind

59

The Crisis in Science Education and the Need to Enculturate All Learners in Science Stuart Rowlands

95

Chapter 3

Chapter 4

Understanding Student Affect in Learning Mathematics Marja-Liisa Malmivuori

Chapter 5

The Challenge of Using the Multimodal Aspects of Informal Sources of Science Learning in the Context of Formal Education Krystallia Halkia and Menis Theodoridis

Chapter 6

Chapter 7

Understanding Scientific Evidence and the Data Collection Process: Explorations of Why, Who, When, What, and How Heisawn Jeong and Nancy B. Songer Constructivist-Informed Classroom Teaching: The Importance and Potential of Motivation Research David H. Palmer

5

125

151

179

201

vi Chapter 8

Chapter 9

Chapter 10

Chapter 11

Index

Contents Oral Communication Competencies in the Science Classroom and the Scientific Workplace F. Elizabeth Gray, Lisa Emerson and Bruce MacKay

223

Supporting Future Teachers Learning to Teach Through an Integrated Model of Mentoring Pi-Jen Lin

239

Strategies to Address Issues and Challenges Faced by Instructors in General Education Introductory Astronomy Courses for Non-Science Majors Michael C. LoPresto Getting it to Work: A Case of Success in Sustaining Science Professional Development Betty J. Young, Sally Beauman and Barbara Fitzsimmons

257

271 283

PREFACE This book presents significant new analyses in the field of science education. This is hardly another field in education which is more important for a country's future than science education. Yet more and more students elect to concentrate on other fields to the exclusion of science for a variety of reasons: 1. The perception of degree of difficulty, 2. The actual degree of difficulty, 3. The lack of perceived prestige and earnings associated with the field. 4. The dearth of good and easy to use texts. 5. The lack of society in comprehending the significance of science and creating attractive incentives for those who enter the field. Chapter 1 - This study concerns the field of mathematics education. Today, for almost any technology, attaining the most advanced level relies on using digital systems. Therefore, the authors focus on number acquisition and use, emphasize major discussions about related topics, and introduce our personal contribution. The authors consider three areas: numbers in society, at school, and in the field of education. Numbers in society: Modern society needs two kinds of number users: the first has to deal with numbers, the second has to work with numbers. According to the framework for PISA assessment, dealing with numbers concerns every citizen facing situations in which the use of a quantitative… reasoning… would help clarify, formulate or solve a problem. Working with numbers is what a professional (marketing specialist, engineer, physician, artist…) does when he has to cope with numerical theoretical frameworks, or when he collaborates with a mathematician. This brings two types of questioning: what does competence in mathematics mean? What level of achievement is desirable in number-learning to meet either numerical need? Numbers at school: The authors examine what is proposed for teaching such a broad subject to 7-to-18-year-olds. The authors first observe and question the educational system, the designer of curricula, scenarios for teaching, training programmes, and national assessments. Secondly, the authors question the notions of problem solving and modelling as mere responses to mathematics-teaching issues. The authors then focus on what really happens in a standard classroom, particularly how teachers apply recommendations and directives, and how the generalization of assessments affects their practice. Numbers in the field of education: Three aspects are considered: epistemological, cognitive, and didactical. The authors distinguish four related levels which the authors have named: numeracy (competence linked to whole numbers), rationacy (competence linked to ratios and rational numbers), algebracy (competence linked to algebra), and functionacy (competence linked to calculus). The cognitive aspect evokes the essential issue of semiotic

viii

Calvin L. Petroselli

registers for representing and processing numerical objects, considers the discipline-ofexpression aspect of mathematics, and other issues taken into account by numerous researchers, such as process and object…. The didactical is devoted to exposing our conceptual framework for teaching numbers and understanding their learning. An experiment in ratio teaching is described and analysed. Chapter 2 - Laboratory activities have long had a distinctive and central role in the science curriculum and science educators have suggested that many benefits accrue from engaging students in science laboratory activities. More specifically, it has been suggested that, when properly developed, there is a potential to enhance students’ conceptual and procedural understanding, their practical and intellectual skills and their understanding of the nature of science. Research findings, however, have proven that “properly developed” laboratory work is less frequent than hoped for and that meaningful learning in laboratories is demanding and complex. The 21st century has offered new frames for dealing with the potential and challenges of laboratory based science teaching. This is an era of reform in which both the content and pedagogy of science learning and teaching are being scrutinized, and new standards intended to shape meaningful science education have emerged. The National Science Education Standards (National Research Council, 1996) and other science education literature (e.g. Lunetta, Hofstein and Clough, 2007) emphasize the importance of rethinking the role and practice of school laboratory work in light of these reforms. The new frames, however, also relates to the development in the understanding of human cognition and learning that has happened during the last 20 years. In the following chapter attention will be given to research on learning in and from the science laboratory. More specifically, the presentation will focus on the science laboratory as a unique learning environment for the following teaching and learning aspects: • •

Argumentation and the justification of assertions Development of metacognitive skill

It is suggested, that these are important aspects with a natural place in the science laboratory. They have, however, been neglected both regarding development of practical experiences provided to the student as well as in research on the effectiveness of practical work that is conducted in the context of science learning. A new approach is needed in which these two aspects are coordinated and seen in accordance with the general practice of teaching and learning in school science. Chapter 3 - There is a crisis in science education. Over the past two decades many organisations such as the American National Science Foundation, the Australian Audit of Science, Engineering and Technology and the UK’s Royal Society and the Confederation of British Industry, have reported a serious decline in students enrolling in science subjects and the failure of the science curriculum to inspire learners and to meet national needs. However, quite apart from instrumental reasons such as a national interest for having more scientists, science education is important for cultural reasons. Science permeates every aspect of modern life and arguably full citizenship in a technological society necessitates the understanding of science. Based on how the world is, science promotes critical thinking, a concern for evidence and an objectivity that is independent of personal opinion or the dictate of kings - yet few individuals have an elementary understanding of science. The failure of science education is

Preface

ix

reflected in science’s lack of popularity evident in the rise of mysticism, the rise in consensus of intelligent design, the postmodernist attack and the closing of many UK university science departments. There have been calls to remedy the situation, such as school visits by scientists and engineers, or overhauling the science curriculum by teaching the science deemed relevant to the everyday experience of children, consistent with the constructivist idea that there is a “children’s science”. However, there is no guarantee that exposure to the working lives of scientists will promote an interest in science and, moreover, science is not based on making sense of experience. To generate the interest and motivation of young learners requires an engagement with the nature of science (NOS) involving meta-discourse with the history and philosophy of the discipline. Contrary to the current wisdom of science educationalists, NOS has more to do with the rule-governed abstract possible world of the thought-experiment than hypothesis testing with a clipboard of data. Even the most concrete thinkers may be capable of thinking in the abstract and mechanics, because of its history and logical character (as opposed to the “soft sciences” such as ecology), provides the perfect opportunity to do this. This article consists of three parts: 1. Public perception of science and scientific literacy and understanding. 2. Why NOS is essential to science education. 3. Why “children’s science” and conceptual change, the largest domain in science education research, has failed to promote scientific understanding. Chapter 4 - Student affect has been one area of interest in mathematics education for decades. This applies in particular to rather large surveys of students in The United States since 1970’s. In general, education studies on affect have much focused on affective factors in the contexts related to mathematics achievements, learning of mathematics or solving mathematical problems. This is understandable since mathematics and mathematical problem solving carry many kinds of cognitive and sociocultural features that are not easily attached to the other school subjects. For example, the abstractness of mathematics and the differences in the symbol systems used in mathematical language set high demands on cognitive processes and also detach mathematics from the context and experience of everyday life. Furthermore, general views of mathematics as a difficult and demanding subject have caused it to be highly regarded and have been generally used to measure academic abilities. Mathematics tend to have a ritual value in societies that then cause powerful experiences with and important differences in mathematical performance. After showing passionate interest in human cognition and cognitive processes, education research paradigms have recently created new opportunities for and even laid emphasis on studies of student affect. Constructivism, together with applied socio-cognitive, cultural and contextual views of learning and education, has enriched our knowledge of affect in mathematics education research, as well. This theoretical chapter first discusses some conceptual features of affective factors traditionally applied in education research and especially in mathematics education studies. This short overview will then be followed by consideration of some of the most significant and often used affective variables in mathematics education research. More recently presented views of affect with cognition in learning will be considered as an introduction to the here suggested theoretical framework for understanding student affect in learning mathematics. Especially, perspectives on the coexistence of affect and cognition, on self-related cognitions and self-regulation are applied in constructing this suggested theoretical framework. It represents a dynamic, humanistic and socio-cognitive, viewpoint on the functioning and development of students´ powerful affect in their learning processes.

x

Calvin L. Petroselli

Chapter 5 - In this work, an attempt has been made to study the plethora and the diversity of informal sources of science learning and the ways formal education may benefit by making use of these sources in its everyday school practice. Informal sources of science knowledge have many forms: they use several means of presenting scientific information, take place in several environments and use several ways to compose their “text”. Each one of them has its own communication codes and uses multiple ways (modes) to present its “meaning”. The material coming from them is chaotic, because it is diverse in terms of the means used, the purposes and the targets stated, the audience addressed, etc. To study them it is helpful to categorize them. Thus a three dimensional model has been developed. Each dimension describes one system of taxonomy: The first one refers to the environment and the conditions under which science learning takes place; the second refers to the way a science “text” is made up and the codes used; while the third one refers to the kind of mode used in the science “text”. Furthermore, the different learning environments in which informal science learning takes place have been studied. Three different learning environments have been distinguished: the organized out-of-schools visits to institutions and organizations (science museums, science centers, zoos, botanic gardens etc.), the students’/teachers’ personal navigation in several sources outside school and the use of informal sources of science learning by the teachers within their everyday classroom practice. The study reveals their particular characteristics, as well as their power and limitations. It also suggests ways of using them effectively in the context of science classroom. Chapter 6 - What is scientific evidence? How should scientific data be collected? These questions comprise essential components of scientific reasoning that are not well understood by students. This chapter explores conceptual challenges students face in inquiry-rich classrooms with respect to the notion of scientific evidence and the related data collection process. As students seek out evidence to support their inquiry, they are likely to ask and need to answer questions such as these: Why collect data? Who collects data? When should data be collected? What counts as scientific evidence? and How should scientific data be collected and analyzed? After examining conceptual issues involved in answering these questions, this chapter proposes that understanding what it means to collect scientific data and what scientific evidence is requires a complex understanding that involves conceptual, procedural, and epistemological knowledge. Chapter 7 - A constructivist paradigm has dominated science education research in recent years. According to this view, students use their existing preconceptions to interpret new experiences, and in doing so, these preconceptions may become modified or revised. In this way, science learning proceeds as children actively reconstruct their ideas as they become presented with new information. However, the implications of constructivism for classroom teaching are still open to question. This position paper refers to the science education literature to argue that strategies to arouse and maintain student motivation should be a crucial component of constructivist-informed classroom teaching. This is because constructivism is universally accepted to be an active process – students must make an effort to reconstruct their ideas, so it follows that if they are not motivated to make that effort then no learning will occur. However, extant models of constructivist classroom teaching make little if any mention of student motivation. In these models, the focus has typically been on strategies to elicit students’ prior conceptions and to guide and monitor their progress towards more scientific conceptions, but the motivational impetus for this process has received little attention. Perhaps one reason for this is that there are relatively few studies of student motivation in the

Preface

xi

science education literature. Another possible reason is the lack of a unified theory of motivation, which means that there is no clear consensus on how best to motivate students in the classroom. In view of this situation, there is a need for studies which can clarify motivational strategies in science classrooms. “Situational interest” is one motivation construct which appears to offer considerable potential, yet it has been largely ignored by science education researchers. Situational interest occurs when a particular situation generates interest in the majority of students in the class – a spectacular science demonstration might arouse transient situational interest even in students who are not normally interested in science. The potential of this construct lies in the fact that studies outside of science have shown that when situational interest is aroused on a number of occasions it can result in longterm personal interest and motivation in the topic. It is thus a potentially powerful construct for science education, and is one which should be further explored. Chapter 8 - This chapter investigates the importance of oral communication training in undergraduate scientific education. The authors examine the status of oral communication training in New Zealand universities and the debate concerning employer attitudes to this issue. The specific relevance of these issues to science education is explored through analysis of a case study and a qualitative and quantitative study of the attitudes of students and employers in science-related industries. Cronin, Grice and Palmerton (2000), Dannels (2001), and Morello (2000) argue that to significantly develop the rhetorical flexibility necessary to communicate competently, oral communication skills training needs to be discipline-specific and firmly contextualized in the genres, expectations, and conventions of the particular field. Responding to this call, a number of recent studies have examined the role of oral communication skill development in specific fields as diverse as design education, archaeology education, and engineering. This chapter moves the discussion of discipline-specific oral communication instruction to undergraduate science education. The recent inclusion of an oral communication component within a compulsory science communication class at Massey University, New Zealand remains a contentious issue. Possibly seeing oral communication training as a low priority in terms of student skills, knowledge, or preparedness for a future scientific career, both students and faculty have resisted the inclusion of oral communication into course curricula and assessment. The researchers designed a study to clarify whether oral communication skills were important to employers in science-related industries, what science employers meant by oral communication skills, and which skills they prioritized. At the same time, the team surveyed science students to better understand their attitudes to training in oral communication. Study findings strongly support the importance of oral communication skills in sciencebased employment in New Zealand. Science employers indicate that they require and value highly a wide variety of oral communication skills. The study also reveals that while science employers and university science students agree that oral communication skills will be important in scientific careers, the majority of employers find the desired level of these skills in new graduates only sometimes or occasionally. The retention of oral skills teaching and assessments, as currently exemplified by the Communication in the Sciences course at Massey University, is clearly indicated; study findings also make a strong case for an extended focus on oral competencies in undergraduate science education. Chapter 9 - The purpose of this article is to introduce an integrated model of mentoring for supporting future teachers learning to teach under the impact of teacher education reform

xii

Calvin L. Petroselli

of Taiwan, particularly, in the internship. This article begins with the introduction of teacher education reform and is followed by the description of the impact of teacher education on quality control. Then, it includes a brief description of six integrated reach projects investigated by teacher educators. One of the integrated research projects that was designed to improve mentors’ competence of mentoring for supporting future teachers learning to teach is reported in detailed and an integrated model of mentoring is developed. Finally, the views of mentors and the future teachers are described briefly and the issues of mentoring are addressed. Chapter 10 - The challenges currently faced by instructors of introductory general education college astronomy courses are numerous. Before effective instruction can even begin, student misconceptions must be addressed. This alone is a daunting task since astronomy is a field in which there are many misconceptions. If dispelling misconceptions is achieved, then effective methods of instruction must be identified and used. Since current research shows that most students learn very little from lectures, other approaches need to be employed. This then means that resources must be either located or created before implementation can occur. The recent movement to stress understanding of concepts rather than memorization and the regurgitation of facts requires that students be engaged and prompted to think critically, or scientifically. This is a challenge in itself, since, as useful as it may be, many non-science students are not used to thinking in this manner. In fact, many students come to class not even aware what science actually is, not a body of facts and figures, but rather a process of investigation. Mathematical illiteracy is not only rampant in our society, but in many cases condoned. Because of this, many non-science majors are math-phobic. They cringe at the site of an equation or graph, even if it is only used to explain a concept and they are not even required to actually use it. Many students are members of Carl Sagan’s “Demon Haunted World” mistaking not only astrology, but television shows, tabloid articles and internet sites about the “paranormal” for science. Many have learned all they know about science from movies and television. Also, some have deeply engrained religious beliefs that prevent them from approaching scientific ideas with an open mind. These challenges are not insurmountable. What follows are the details of various strategies that have been developed and employed to address these issues and challenges with the goal of improving instruction and the entire experience of introductory astronomy for both the students and the instructors. Chapter 11 - This article presents a case of a successful partnership between a university and nine school districts. Science educators, science and engineering faculty from the University joined forces with local school districts to attract funding and implement a high quality K-8 science curriculum supported by new materials and on-going professional development. There are five broad themes to the strategies that contributed to the success of the lasting the partnership: taking the load off central office administrators so that a high quality science curriculum with supportive PD “just happens” with another office managing the details, high quality communication among all partners, management/oversight/control, formative assessment of the quality of professional development implementation with redesign, and documenting results (e.g., parent interest, state-level school site visits, teachers’

Preface

xiii

sense of preparedness to teach science, student achievement outcomes, and continued support by the University administration and faculty).

In: Science Education Issues and Developments Editor: C. L. Petroselli, pp. 1-4

ISBN: 978-1-60021-950-4 © 2008 Nova Science Publishers, Inc.

Expert Commentary

WHY DO WE LOSE PHYSICS STUDENTS? Ronald Newburgh1 Harvard Extension School, Cambridge, MA 02138, USA

ABSTRACT Why is it that we have such great difficulty in retaining students in physics? The answer lies, I believe, in the way most of us, as physics teachers, think. There is a wide spectrum in thought processes. Of the two major types one is linear (i.e. sequential) and the other lateral (i.e. seeking horizontal connections). Those who developed physics – from Galileo to Newton to Einstein to Heisenberg - were almost exclusively linear thinkers. The paradigm for linear thought is Euclidean thinking. Many physicists chose physics for their career as a result of their exposure to geometry. A consequence of this is that textbooks are usually written in a Euclidean format. Thus many beginning students look on physics as an exercise in Euclidean logic, with the attendant certainty that it implies. The sense of discovery is lost. Many students, male and female, do not recognize that the Euclidean format, though efficient, is not a valid description of how we do physics. Their way of approaching problems is different but just as valid. Too many physics teachers refuse to recognize the limitations of this approach, thereby causing would-be students who do not think in a Euclidean fashion to leave. Only when physics teachers are willing to make the effort to understand and even encourage other ways of thinking, will all students look on physics as a welcoming discipline.

The loss of physics students is an alarming trend. I believe that most physicists would agree that physics is an essential component for the development of an educated person. Certainly this belief was accepted in the eighteenth century when the study of natural philosophy was considered a sine qua non if a person were to be considered educated. Is this belief any less essential today? Moreover, the question of great importance to the nation. The renewal of our intellectual capital is a necessity for the maintenance of our standard of living. Yet the number of students who major in physics, as well as in other hard sciences and engineering, is continually 1

e-mail: [email protected]

2

Ronald Newburgh

declining. This is especially true if we consider the number of native-born students. Until recently we have been able to cover the deficit with foreign-born students who remained in the country after graduation. However, for various reasons, many are now returning to their homelands. If we accept the need for more people trained in physics, we must try to answer the question posed in the title. The answer lies, I believe, in the way most physicists think, or perhaps more accurately, the way they appear to think. The term linear thinking is often used to describe scientific thinking. Frequently it is a synonym for logical thinking. This term is quite facile, and I feel that we should go beyond it. In this paper I wish to examine the influence of Euclid, not on the actual doing of physics but rather on the presentation of completed research and the writing of our textbooks. I suggest that Euclid may be the proximate cause of the flight from physics. Though I have no statistics, many physicists with whom I have spoken have said that their introduction to geometry was the reason for their going into physics. Einstein in his Autobiographical Notes [1] wrote the following. “At the age of 13 I experienced a second wonder of a totally different nature [The first was the gift of a compass when he was 5.] : in a little book dealing with Euclidean plane geometry, which came into my hands at the beginning of a school year. Here were assertions, as for example the intersection of the three altitudes of a triangle in one point, which – though by no means evident – could nevertheless be proved with such certainty that any doubt appeared to be out of the question. This lucidity and certainty made an indescribable impression on me. That the axiom had to be accepted unproved did not disturb me. In any case it was quite sufficient for me if I could peg proofs upon propositions the validity of which did not seem to me to be dubious. ... If thus it appeared that it was possible to get certain knowledge of the objects of experience by means of pure thinking, this “wonder” rested upon an error. Nevertheless, for anyone who experiences it for the first time, it is marvelous enough that man is capable at all to reach such a degree of certainty and purity in pure thinking as the Greeks showed us for the first time in geometry.”

Now I am no Einstein, but what he wrote is an exact description of my feelings on my first exposure to geometry, an experience that I can describe only as an epiphany. What has been the influence of Euclid on physicists? Think back on your own doctoral dissertation. In doing the research that led to the dissertation, the frustrations were enormous for nearly all of us – whether we were experimentalists or theorists. There were false starts, apparatus that did not work or broke, equations that did not describe the phenomena, and those that were insoluble. In conversations with advisors and fellow students we tossed out ideas, rejected some, argued about others and finally obtained definitive experimental results and developed a coherent description and resolution of the problem. One can hardly describe those years as an exercise in Euclidean logic. However, in writing the dissertation, we made the work conform to the Procrustean bed of Euclid. Very few dissertations describe or include the false steps. As a rule we present the research as starting from a carefully articulated thesis. It is designed to investigate and prove the effect of B on A and show how it relates to the principle C as enunciated by the eminent Dr. Pangloss. There is a sense of a monotonic progression – b follows a, c follows b, and therefore d is proven. The thesis is written with a strong sense of inevitability.

Why do We Lose Physics Students?

3

When the graduate student becomes a research scientist and teacher, both his scientific papers and textbooks have the same sense of inevitability. The format of our textbooks is Euclidean. Newton’s laws, Hamilton-Jacobi theory, and Maxwell’s equations are often presented as quasi-axioms in advanced texts. Elementary texts emulate them. After all, it is an efficient way of presenting complex material. It is just this approach that can confuse the beginning student. Seeing the material in this form and remembering Euclid, he will look on physics as a deductive discipline – exactly that which it isn’t. At the same time the instructor says that physics is an experimental science so that we discover our laws by induction. This contradiction, usually unrecognized, can create an intellectual malaise, especially for the questioning student. The logical certainty with which most textbooks are written is mirrored by the teacher in his presentation of the material. The problems accompanying the text do nothing to contradict this impression. Discussion of problem solving, whether in the text or by the teacher, usually becomes an application of sterile algorithms. Teachers talk of physics as an adventure in discovery, but most approach the subject in terms of meeting requirements such as the SAT’s. The laboratories become fixed exercises in which the student must confirm some principle already established. He knows the answer before he does the experiment. The result is that most students, in spite of all protestations to the contrary, look on physics as a matter of memorization. A few attempt to find the basic principles underlying the subject but then misuse them as the basis of deductive reasoning. It’s hardly surprising that many students decide that the subject is uninteresting and even illogical. No wonder they major in other fields. One teacher whose approach is a notable exception to this is Mazur [2] of Harvard. He lectures (or rather talks for 10 minutes), then poses a multiple choice question to the class. Using a computer interface, each students selects an answer. The results are recorded on a computer. Mazur then asks them to discuss their answers with the person beside them. This can take two or three minutes. They then vote again. Usually the second round leads to an overwhelming majority of correct answers. This method is one of real learning because the students are truly teaching themselves. Now I yield to no one in my admiration for Euclid. He has been an inspiration to many of us. We understand his genius but also see his limitations. Unfortunately there are many who do not follow his way of thinking. These, I submit, are the students we lose, both women and men,. By presenting alternate approaches to students, specifically uses of lateral thinking, false starts that must be corrected, and lessons that are discoveries not memorization, we can retain more students. We should remember that lateral thinking is essential to the formation of analogies, an activity that one cannot describe as Euclidean. Doing science without analogies seems to me an impossibility. At the same time I recognize that the introduction of alternate approaches must be a time consuming process. It is far less efficient than our current methods. It also requires teachers with greater flexibility, less rigidity, teachers who do not demand that the students parrot their analyses. Frost in his Mending Wall [3] writes of someone who “will not go beyond his father’s saying”. Mazur has gone beyond. If we are serious about the need for producing more scientists, so should we all. I feel that Euclid would agree.

4

Ronald Newburgh

A POSTSCRIPT I realize that I have not conducted a scientific survey of the number of us who were influenced, indeed changed by Euclid in our formative years. However, many conversations going back to the 1940’s lead me to believe that the number is great. I also recognize that I have not proposed a specific program (or programs) for improving physics instruction. Too often, it seems, we believe that there is a single solution to a problem, if we could but find it. Just as there are many types of students, there are many ways to teach. I ask only that as teachers we are more open to the views of students and less rigid in our own thinking.

REFERENCES [1] [2] [3]

Albert Einstein, “Autobiographical Notes’ in Albert Einstein, Philosopher-Scientist, edited by Paul Schilpp (Library of Living Philosophers, Evanston, IL, 1949), pp. 9,11. Eric Mazur, Peer Instruction A User’s Manual, (Prentice Hall, Upper Saddle River, NJ, 1997). Robert Frost, “Mending Wall” in The New Modern American and British Poetry, edited by Louis Untermeyer (Harcourt Brace, New York, NY, 1941).

In: Science Education Issues and Developments Editor: C. L. Petroselli, pp. 5-57

ISBN: 978-1-60021-950-4 © 2008 Nova Science Publishers, Inc.

Chapter 1

A NUMERICAL LANDSCAPE Robert Adjiage1 and François Pluvinage ABSTRACT This study concerns the field of mathematics education. Today, for almost any technology, attaining the most advanced level relies on using digital systems. Therefore, we focus on number acquisition and use, emphasize major discussions about related topics, and introduce our personal contribution. We consider three areas: numbers in society, at school, and in the field of education. Numbers in society Modern society needs two kinds of number users: the first has to deal with numbers, the second has to work with numbers. According to the framework for PISA assessment, dealing with numbers concerns every citizen facing situations in which the use of a quantitative… reasoning… would help clarify, formulate or solve a problem. Working with numbers is what a professional (marketing specialist, engineer, physician, artist…) does when he has to cope with numerical theoretical frameworks, or when he collaborates with a mathematician. This brings two types of questioning: what does competence in mathematics mean? What level of achievement is desirable in number-learning to meet either numerical need? Numbers at school We examine what is proposed for teaching such a broad subject to 7-to-18-year-olds. We first observe and question the educational system, the designer of curricula, scenarios for teaching, training programmes, and national assessments. Secondly, we question the notions of problem solving and modelling as mere responses to mathematics-teaching issues. We then focus on what really happens in a standard classroom, particularly how teachers apply recommendations and directives, and how the generalization of assessments affects their practice. Numbers in the field of education Three aspects are considered: epistemological, cognitive, and didactical. We distinguish four related levels which we have named: numeracy (competence linked to whole numbers), rationacy (competence linked to ratios and rational numbers), algebracy 1 IUFM d'Alsace, 141, avenue de Colmar, 67089 STRASBOURG Cedex ,Personal address: 2, rue des Roses, F67170 MITTELHAUSEN,Tel : 00 33 (0)3 88 51 43 27, Mobile : 00 33 (0)6 16 35 32 01, [email protected], [email protected]

6

Robert Adjiage and François Pluvinage (competence linked to algebra), and functionacy (competence linked to calculus). The cognitive aspect evokes the essential issue of semiotic registers for representing and processing numerical objects, considers the discipline-of-expression aspect of mathematics, and other issues taken into account by numerous researchers, such as process and object…. The didactical is devoted to exposing our conceptual framework for teaching numbers and understanding their learning. An experiment in ratio teaching is described and analysed.

INTRODUCTION Today, for almost any technology, attaining the most advanced level relies on using digital systems. Number acquisition has therefore become a central issue. As a consequence, we have decided to devote this chapter to numbers, learning them, and teaching them. We emphasize major discussions about related topics, and introduce our personal contribution. We will consider mathematics education from early childhood to grade 12. We mainly study the educational problems related to real numbers. Only a small subsection concerns complex numbers because they are mostly studied at university level. Mathematicians and advanced users of numerical domains face either the world of exact computation or the world of approximate computation. In the mathematical subject classification of the American Mathematical Society (2007, section 11), one finds a number theory section some of whose subsections (e.g. 11A Elementary number theory, 11D Diophantine equations, 11Y Computational number theory) are centred on exact computation. When performing exact computations, mathematicians consider certain subsets of real numbers (e.g. Ζ, Θ, Θ( 3 ), Ζ/2Ζ…) as separate entities. When performing approximate computations, educated people usually consider the field of real numbers as a whole. Nevertheless, although ancient Greek mathematicians already knew integers, rational numbers, and some irrational numbers like the square root of two, the construction of real numbers was only completed during the 19th century. This long period of maturation should suggest that mastering real numbers cannot be achieved quickly. Our main goal is to show that there are actually various stable levels of numerical acquisition. Mathematical problemsolving requires understanding the situation referred to, and then, adequate processing. Thus, for numerical problems, we can distinguish a level of access and a level of process. An individual facing a problem may: • • •

process the involved numerical objects correctly, when his own level allows him to both access and process the problem understand the statement but give an incorrect solution, when his own level allows him to access but not yet to process the problem not know how to proceed.

Modern society needs two kinds of number users. The first has to deal with numbers: “In real-world settings, citizens regularly face situations when shopping, travelling, cooking, dealing with their personal finances… in which the use of a quantitative… reasoning… would help clarify, formulate or solve a problem” (OECD, 2006, p.73). The second has to work with numbers. This is what a professional (marketing specialist, engineer, physician, artist…) does

A Numerical Landscape

7

when he has to cope with numerical theoretical frameworks, or when he collaborates with a mathematician. What level of achievement is desirable in number-learning to meet either numerical need? Nowadays, this level is mainly defined in terms of competence, or competencies, by educational systems whereas it was previously mostly related to contents. The emphasis put on mastering real-world problems, in order to found well-advised judgments, by many institutions like OECD’s PISA2 project or the European parliament, reinforces this tendency. Piaget (1967, p.65), referring to Kant’s epistemological analysis of mathematics and physics, states: “Knowledge, [and thus acquiring knowledge] entails the relationship between subject and object”. Does not focusing on competencies, generally understood as a potential of action, neglect this relationship and therefore diminish the learning process? Does it not lead to mainly considering the product of teaching, when it has worked, instead of the development of this relationship? Briefly, does it not emphasize the end and thus put the means in the background? We first try to specify what society expects from number users, whether occasional or professional. We then observe and question the way educational systems design curricula, scenarios for teaching, training programmes, and national assessments, in order to enable people to take charge of the societal needs. We focus on modelling and applying mathematics, which has become a predominant topic, presented by many educational systems as a major response to mathematics-teaching issues. How are these official instructions and recommendations taken into account by teachers? What really happens in the classroom will be the theme of the fourth section. We thereafter introduce our framework of number acquisition. We emphasize cognitive aspects related to the specificity of mathematical objects: “Unlike material objects, however, advanced mathematical constructs are totally inaccessible to our senses… Indeed, even when we draw a function, or write down a number… the sign on the paper is but one among many possible representations of some abstract entity, which by itself can be neither seen or touched… for the mathematician…. It is important merely to know the rules or laws by which they may be combined.” (Sfard, 1991, p. 3). The semiotic nature of mathematical objects allows immediate access to them, whether they are already known or to be constructed. object “… unlike external mediation of instruments, semiotic mediation may be internalised, i.e. it may be transformed into infra conscious processing and thus become transparent from a phenomenological viewpoint” (Duval, 1998, p. 175). We then question the notion of competence, and its relation to knowledge. We wonder what should be the extent of this notion, for it to allow designing teaching objectives and assessments for number learning. We consider the epistemological aspect of number learning, focussing on numbers as objects inserted into a coherent network with its ruptures and continuities (Brousseau, 1997, pp. 79-99). Following our questioning about coherence and competence in number acquisition, we define four fields of competencies that allow to clearly specify the level of competence of an individual intended to deal or work with mathematics.

2

Programme for International Student Assessment, which assesses the abilities of 15-year-old students from 41 countries (including 30 of the most developed) to apply learning to problems with a real-world context.

8

Robert Adjiage and François Pluvinage

NUMBERS IN SOCIETY Dealing with Numbers In order to define the social expectations for dealing with numbers, we can consult documents with rules or regulations that contain numbers. Understanding these rules is a social obligation, and legal pursuits may result from their lack of application, or from incorrect application. Rules requiring calculations are present in most fields of human activity, so that we shall consider a sample of official documents. Let us first assert that in all the documents that we have consulted, there are three levels in dealing with numbers. We shall describe these later in this paper. Therefore, the nature of the encountered phenomenon does not actually depend on the field of activity one chooses. At the moment we were writing this text, we read that World Water Day 2007 is celebrated each year on 22 March3 (by the way note the wrong use of the number 2007 in this sentence). As it appears anyway that water is something vital and possibly scarce, we suggest limiting our investigation to the domain of water: its consumption and its use. Texts of rules or regulations can refer to numbers in such a way that knowledge of decimal numbers and arithmetic operations, or numeracy, allows the reader to act as required. Such is the case in the following Mexican law, about consumers’ shares for water consumption. Each magnitude and arithmetic operation is completely described in a rhetorical way in this text (followed by its English translation).

Cámara de Diputados del H. Congreso de la Unión LEY DE CONTRIBUCIÓN DE MEJORAS POR OBRAS PÚBLICAS Y FEDERALES DE INFRAESTRUCTURA HIDRÁULICA Artículo 7 – II Nueva Ley D.O.F. 26/12/1990 Tratándose de acueductos o sistemas de suministro de agua en bloque realizados exclusivamente con inversión federal, el monto de la contribución obtenida en el artículo anterior se dividirá entre la capacidad de suministro del sistema, medida en metros cúbicos por segundo, y el cociente obtenido se multiplicará por el volumen asignado o concesionado por la Comisión Nacional del Agua a cada usuario del sistema, medido en metros cúbicos por segundo y el resultado será el monto de la contribución a cargo de cada contribuyente. Article 7-2 For aqueducts or water supply systems built with federal investment only, the amount of the contribution defined in the preceding article will be divided by the global capability of the system in cubic meters by second, and the obtained quotient will be multiplied by the volume in cubic meters per second attributed to each consumer, and the result will be the amount to be paid by each customer. 3

Extract from the UN-water webpage < http://www.unwater.org/wwd07/flashindex.html >: 'Coping with Water Scarcity' is the theme for World Water Day 2007, which is celebrated each year on 22 March.

A Numerical Landscape

9

The following general part of a US federal text introduces tables (we reproduce only one table, but there are several). Compared to the preceding document, this one supposes from its readers practice in consulting mathematical presentations (some other documents also present diagrams). But it deals similarly with numbers: Every arithmetic operation is well defined (see below). Moreover, the section of the document devoted to definitions includes magnitudes and units not necessarily known by everyone, e.g. the so-called degree-day. We conclude that the expected competence is only numeracy. Hereafter we shall see that in a more specialised part of the same document, the reader is supposed to deal with formulas.

CODE OF FEDERAL REGULATIONS [2006] 10cfr434-- Part 434_Energy Code For New Federal Commercial And MultiFamily High Rise Residential Buildings Degree-day, heating: a unit, based upon temperature difference and time, used in estimating heating energy consumption. For any one day, when the mean temperature is less than a reference temperature, typically 65ºF, there are as many degree-days as degrees Fahrenheit temperature difference between the mean temperature for the day and the reference temperature. Annual heating degree days (HDD) are the sum of the degree-days over a calendar year. § 434.518 Service water heating. 518.1 The service water loads for Prototype and Reference Buildings are defined in terms of Btu/h per person4 in Table 518.1.1, Service Hot Water Quantities. The service water heating loads from Table 518.1.1 are prescribed assumptions for multi-family high-rise residential buildings and default assumptions for all other buildings. The same service water-heating load assumptions shall be made in calculating Design Energy Consumption as were used in calculating the Energy Cost Budget.

Table 518.1.1.—Service Hot Water Quantities Building type Assembly Office Retail Warehouse School Hotel/Motel Restaurant Health Multi-family High Rise Residential (2)

Btu/person-hour (1) 215 175 135 225 215 1110 390 135 1700

(1) This value is the number to be multiplied by the percentage multipliers of the Building Profile Schedules in Table 513.2.b. See Table 513.2.a for occupancy levels. (2) Total hot water use per dwelling unit for each hour shall be 3,400 Btu/h times the multi-family high rise residential building SWH system multiplier from Table 513.2.b. 4

The British thermal unit (BTU or Btu) is a unit of energy used in North America (1 Btu ≈ 1.055 Joule). See http://en.wikipedia.org/wiki/British_thermal_unit .

10

Robert Adjiage and François Pluvinage

We do not reproduce Table 513.2 in which we could find the multipliers. Observe a possible difficulty for interpreting certain units of magnitudes: Slash refers to division as usual in arithmetic, but dash indicates product in degree-days and Btu/person-hour When reading official texts, we are often surprised by the importance of the use of common language, even in cases in which the use of formulas would simplify heavy verbal formulations. Some legal texts however introduce mathematical formulas, which shorten expression. See for example a French text taken from J.O. (Official Journal of French Republic, 12-29-2002, n 303, p. 60 059 text 5) and followed by its English translation. Avis et communications Ministère de l'écologie et du développement durable Avis relatif à des délibérations des agences de l'eau AGENCE DE L'EAU ARTOIS-PICARDIE Délibération n° 2002-A-063 du 4 octobre 2002, NOR: DEVE0210424V Article 9 Mesure indirecte des volumes prélevés 1. Calcul du prélèvement en fonction de l'énergie électrique consommée Le volume prélevé est obtenu par application de la formule suivante : P = 250 × W/Z avec: P : volume prélevé en mètres cubes durant la période soumise à redevance ; W : énergie électrique consommée mesurée au compteur, exprimée en kWh ; Z : hauteur théorique minimale d'élévation en mètres.

MINISTRY OF ECOLOGY AND DURABLE DEVELOPMENT (FRANCE), ANNOUNCEMENT OF DECISIONS TAKEN BY THE WATER AGENCY OF ARTOIS-PICARDIE (10/4/2002) Article 9 Indirect measurements of volumes of pumped water 1. Water volume computed with measurement of consumed electrical energy The volume P in cubic meters is obtained using the following formula: P = 250 × W/Z, where W is the measured electrical energy in kWh and Z the theoretical minimal height of elevation in meters.

We find a formula in this text, but we can assert that using this formula does not require algebraic processing: substituting numbers for letters and then perform arithmetic calculation is sufficient to find the result. Therefore, the reader is only expected to be at numeracy level. But a question arises: Under what conditions does dealing with a formula entail the use of an effective algebraic language? Nevertheless, many texts, generally including at least ratios and proportions, require a more advanced knowledge from their readers. The following text belongs to this category. Drinking Water Standards

Priority Rulemakings

Arsenic The Safe Drinking Water Act requires EPA to revise the existing 50 parts per billion (ppb) standard for arsenic in drinking water. EPA is implementing a 10 ppb standard for arsenic. Ground Water Rule EPA proposed a rule which specified the appropriate use of disinfection in ground water and addressed other components of ground water systems to assure public health protection. Lead and Copper EPA estimates that approximately 20 percent of human exposure to lead is attributable to lead in drinking water.

A Numerical Landscape

11

The quoted text is only part of a larger document from U.S. Environmental Protection Agency . The first pages of this document are devoted to a general presentation of public health protection. The considered text concerns everybody, so that it is supposed to be understood by any reader. Below, we present a second part of this document. It gives details of measures and procedures, so that we could think that it is only directed at specialists. But these specialists, in turn, are in charge of delivering the information to a large range of citizens who are supposed to understand the involved concepts. Electronic Code of Federal Regulations (e-CFR) e-CFR Data is current as of March 16, 2007 http://www.gpoaccess.gov/cfr/index.html (c)Lead and copper action levels. (1) The lead action level is exceeded if the concentration of lead in more than 10 percent of tap water samples collected during any monitoring period conducted in accordance with §141.86 is greater than 0.015 mg/L (i.e., if the “90th percentile” lead level is greater than 0.015 mg/L). (2) The copper action level is exceeded if the concentration of copper in more than 10 percent of tap water samples collected during any monitoring period conducted in accordance with §141.86 is greater than 1.3 mg/L (i.e., if the “90th percentile” copper level is greater than 1.3 mg/L). (3) The 90th percentile lead and copper levels shall be computed as follows: (i) The results of all lead or copper samples taken during a monitoring period shall be placed in ascending order from the sample with the lowest concentration to the sample with the highest concentration. Each sampling result shall be assigned a number, ascending by single integers beginning with the number 1 for the sample with the lowest contaminant level. The number assigned to the sample with the highest contaminant level shall be equal to the total number of samples taken. (ii) The number of samples taken during the monitoring period shall be multiplied by 0.9. (iii) The contaminant concentration in the numbered sample yielded by the calculation in paragraph (c)(3)(ii) is the 90th percentile contaminant level. (iv) For water systems serving fewer than 100 people that collect 5 samples per monitoring period, the 90th percentile is computed by taking the average of the highest and second highest concentrations. This text refers to ratios without mentioning how to process them. A precise idea of ratio seems to be necessary for understanding at least the last statement of each presented part (Lead and Copper in Priority Rulemaking, 3-iv in e-CFR). This would be a second level of numerical knowledge required. On the other hand, although the text refers to statistical notions such as percentiles, we will not go so far as stating that the intended readers are supposed to master statistics as the required processes are explicitly defined. In the Energy Code section mentioned above, the reader is expected to know how to process formulas, which implies familiarity with algebraic symbolic language (knowledge of

12

Robert Adjiage and François Pluvinage

symbol Σ and sub-indices, distributive rule). This brings us to the third level of mathematical social expectations, perceivable in the following quoted example (p. 466 of the Energy Code, loc. cit.). The overall thermal transmittance of the building envelope shall be calculated in accordance with Equation 402.1.2: Uo=ΣUiAi/Ao=(U1A1 +U2A2 + . . . +UnAn)/Ao (402.1.2) Where: Uo = the area-weighted average thermal transmittance of the gross area of the building envelope; i.e., the exterior wall assembly including fenestration and doors, the roof and ceiling assembly, and the floor assembly, Btu/(h·ft2·ºF) Ao = the gross area of the building envelope, ft2 Ui = the thermal transmittance of each individual path of the building envelope, i.e., the opaque portion or the fenestration, Btu/(h·ft2·ºF) Ui = 1/Ri (where Ri is the total resistance to heat flow of an individual path through the building envelope) Ai = the area of each individual element of the building envelope, ft2

Working with Numbers Professional mathematicians obviously work with numbers, and over the last three decades employment of mathematicians has been increasing in many sectors of human activity: astronomy, meteorology, aeronautic traffic, bank, marketing, health, quality control, industrial design, musical acoustics, and so on. If we add traditional mathematics users like physicists, and mathematics teachers, the result is a large number of people who dedicate much of their time to working with numbers. The major aims are to predict, to control, to optimise, and to decide (e.g. how to reduce cost or waste in industrial processing). Modelling (e.g. traffic regulation by programming traffic lights, designing a ship hull) is a usual tool at the present time. To be sure, computation and computers have largely contributed to such a development, particularly in designing models and checking their conformity with the reality they simulate. A classical use of number knowledge has been to discover relationships between realworld quantities, and thus find formulas. Renowned examples of results published during the 20th century are Pareto’s principle in economy (20% of the population earns 80% of the income) and Zipf’s law in linguistics. This law is a consequence of an economic principle (principle of least effort): A writer, or a speaker, has a more or less easy access to words, depending on the frequency in which he uses them (the more frequent, the easier). Originally, Zipf's law stated that, in a corpus of natural language utterances, the frequency of any word is roughly inversely proportional to its rank in the frequency table. (Wikipedia, 2007a). Let us for instance consider the following statistics related to a collection of 423 short TIME magazine articles (total number of term occurrences: N = 245,412).

A Numerical Landscape

13

Table 1. Top 15 terms of short TIME magazine articles Rank r

Word

Frequency f

r*f

r*f/N

1

the

15861

15861

0.065

2

of

7239

14478

0.059

3

to

6331

18993

0.077

4

a

5878

23512

0.096

5

and

5614

28070

0.114

6

in

5294

31764

0.129

7

that

2507

17549

0.072

8

for

2228

17824

0.073

9

was

2149

19341

0.079

10

with

1839

18390

0.075

11

his

1815

19965

0.081

12

is

1810

21720

0.089

13

he

1700

22100

0.090

14

as

1581

22134

0.090

15

on

1551

23265

0.095

The product A = r*f/N in the last column tends to be equal to about 0.1. (http://linkage.rockefeller.edu/wli/zipf/cmpsci546_spring2002_notes.pdf, p. 3)

More recent kinds of work with numbers are: digitalisation, cryptography, and compression (zip). A typical example of digitalisation is that a coloured pixel on a screen is represented by a triple of numbers of the interval [0, 255]. In cryptography, a widespread method is NSA, related to public and private keys. It is based on the quasi-impossibility of finding the factors of a given product of two large prime numbers. Image compression produces JPEG files. The latter require the use of wavelets, a kind of mathematical function that allows decomposing a given function into different frequency components. All these domains are being developed on the base of constant contact between specialists and users. This supposes a sufficient general level of education. Thus, collaborative working becomes rather difficult between people whose competence differ by more than one level, for instance between somebody who is working at the level of functional analysis and somebody who does not yet master algebraic writing.

CURRICULA AND ASSESSMENT; THE CASE OF MODELLING After analysing the societal needs related to number learning, we investigate in this section curricular genesis and development, i.e. the process by which curricula are conceived and designed by institutions, as well as instructions for implementation. How does this

14

Robert Adjiage and François Pluvinage

process take account of both societal needs of dealing with and working with numbers? We will mainly consider an issue that has been in the foreground these last few years: mathematical modelling. Following the main line of this paper, we will focus on the numerical aspects of this topic.

PISA, Modelling, and Curricula We choose modelling because the way this domain of mathematics has emerged in just a few years in many curricula for teaching, teacher training, and the related research, as well as the emphasis put on it, seems to be characteristic of a modern process of reforming educational systems in the occidental world. What is intended by modelling? Let us refer to the Discussion Document (Blum, 2004, pp.152-153), prior to ICMI Study 2004 (International Commission on Mathematical Instruction), aimed at raising some important issues related to the theory and practice of teaching and learning mathematical modelling and applications. The authors of this Discussion Document (14 people from 12 countries), are the members of the International Programme Committee for this ICMI Study. “The starting point is normally a certain situation in the real world. Simplifying it, structuring it and making it more precise… leads to the formulation of a real model of the situation…. If appropriate, real data are collected in order to provide more information…This real model, still a part of the real world… is mathematised…” Mathematical results are then derived, using mathematical methods… and then “re-translated into the real world, that is, interpreted in relation to the original situation”. Checking whether the results are appropriate and reasonable now validates the model. “If need be… the whole process has to be repeated with a modified or a totally different model”. “…The obtained solution of the original real world problem is stated and communicated”.

Modelling is an important theme, taken into account these last few decades by numerous studies and international conferences in the field of mathematics education (see for instance: Niss, 1987; Blum et al.1989; Galbraith et al., 1990; Lesh et al., 2002). Nevertheless, “…genuine modelling activities are still rather rare in mathematics lessons” (Blum, 2004, p. 150). Now PISA has defined and tested mathematical literacy: “The emphasis in PISA is ‘on mathematical knowledge put into functional use in a multitude of different situations and contexts’. Therefore, real situations as well as interpreting, reflecting and validating mathematical results in “reality” are essential processes when solving literacy-oriented problems.” (Blum, 2004, p. p. 151-152). Major countries of the occidental world, including the USA, appeared to perform less well than expected at PISA 2000 and 2003. (For detailed results, see NCES, 2001, p. 43; OECD, 2004, p. 53 and pp. 89-95). As a consequence “an intense discussion has started, in several countries, about aims and design of mathematics instruction in schools, and especially about the role of mathematical modelling, applications of mathematics and relations to the real world.” (Blum, 2004, p. 151). One finds for instance, on the web site of the U.S. Department of Education (2004), a press release entitled: PISA results show need for high school reform, in which Secretary of

A Numerical Landscape

15

Education Rod Paige said: “The PISA results are a blinking warning light”. In releasing the U.S. findings, Robert Lerner, commissioner of NCES, said, “PISA provides important information about education in the United States and in other industrialised nations, giving us an external perspective on U.S. performance. We need PISA in particular because it offers such a different measure of achievement, one that poses complex problems that students might realistically encounter in their lives.” (Ibid, 2004). National Center for Education Statistics (2002-2006) provides a huge amount of documents addressing PISA: access to PISA items and results, international comparisons, official and scientific reports, aids for educators…. The European Parliament on its part defined (2006, p.1) Key Competencies for lifelong learning. Concerning mathematics, it states that: “An individual should have the skills to apply basic mathematical principles and processes in everyday contexts at home and work, and to follow and assess chains of arguments” (2006, p.18 pdf version), putting the emphasis on applying mathematics in ‘everyday contexts’. What kind of echo to these statements and recommendations can we find in official learning curricula? We will give two examples drawn from a wealth of international material. In 1996, the New York State Board of Regents adopted learning standards for all content (subject) areas. Since then, the New York State Education Department (NYSED) has issued a series of core curricula, which provide an additional level of specificity to these learning standards. The Core Curriculum Standard 3, from Pre-kindergarten to grade 12 (New York State Education Department, 2005), appears as carefully thought-out and precise. It defines a mathematical proficiency relying on conceptual understanding, procedural fluency, and problem solving. In introduction to this Core Curriculum (ibid, p. 1), it is stated: “Most problems that students will encounter in the real world are multi-step or process problems. Solution of these problems involves the integration of conceptual understanding and procedural knowledge. …Many textbook problems are not typical of those that students will meet in real life. Therefore, students need to be able to have a general understanding of how to analyse a problem and how to choose the most useful strategy for solving the problem.” We first note this formulation that considers the complexity (‘multi-step’) of problems that students meet in the real world as a model for learning. The word ‘model’ and derived words like ‘modelling’ appear 88 times in the full document, throughout all grades, to which one can add 21 occurrences of expressions as: ‘everyday’ situations or ‘experiences or ‘real world’ problems or situations. In France, the Minister of National Education issued a document that defines a common base of knowledge and competencies5 to be acquired at the end of mandatory schooling. This document explicitly refers (MEN, 2006, p. 2) to two sources: the already quoted Recommendation of the European Parliament (2006, p.1) related to Key Competencies for lifelong learning, and “international assessments, particularly PISA. It specifies (ibid, pp. 5-6) that: “mathematics provide tools for acting, choosing, and deciding in everyday life”, and that “mastering the main elements of mathematics essentially depends on problem solving, particularly when the involved problems rely on situations stemming from the real world.” New learning curricula have therefore been published (or are in the process of being published). These mention as one of the main aims the capacity to use mathematics in different domains (everyday life, other disciplines). 5 Socle commun de connaissances et de compétences

16

Robert Adjiage and François Pluvinage

Thus, across many countries in the world: “Researchers and practitioners in maths education and policymakers have reached agreement that mathematics education should enable students to apply mathematics in their everyday life (PISA) and contribute to the development of active citizens (Council of the European Union following Lisbon Report, 2001, p.4, p. 13).” (LEMA, 2006, project-background page). Of course, this requires new interest and competencies in teachers who generally do not successfully integrate applications of mathematics into their course design and daily classroom practice. Appropriate pre- and inservice teacher training has therefore been, or is being, conceived. For instance, a European project, COMENIUS-LEMA (Learning and Education in and through Modelling and Applications) is being developed. Partners of the project are institutions from different countries: France, Germany, Hungary, Poland, Spain, and United-Kingdom. Mathematics education researchers from these countries, including one the authors of the present paper, participate or collaborate in this project, which “proposes to support teachers with development of their pedagogic practice in mathematical modelling and applications by developing a teacher training course”. Target groups are: “in-service and pre-service teachers at primary and lower secondary level and teacher trainers”. (LEMA, 2006, project page). We are now in measure to specify the leading strand, in the mathematical modelling case, of the current evolution in educational systems. An international assessment, PISA, has provided policy makers with benchmarks that allow to reconsider their systems' performances, and to identify potential strategies to improve, according to PISA’s Standards, student achievement. Their recommendations have determined important inflections of curricula intended to help students to better handle the real world. Researchers, already interested in this topic, have designed or are designing training programs for aiding teachers to take account of curricula and apply the related educational instructions in their lessons. Now, how are PISA-assessed competencies determined? PISA has been influenced by the Danish KOM Project, initiated by the Ministry of Education in order to profoundly reform Danish mathematics education from school to university, and its director, Morgen Niss: “It should be noted that the thinking behind and before the Danish KOM-project has influenced the mathematics domain of OECD’s PISA project, partly because the author is a member of the mathematics expert group for the project”. (Niss, 2003, p. 12). According to Winsløw (2005, p. 141-142) the empirical and theoretical bases of Niss’ model are of two orders: Scandinavian tradition and Project Pedagogy. Concerning the former, mathematics education has been considered since the seventies, in a society more and more influenced by mathematical models, as a mean of making sense of the world in which students live and will work, and, as a consequence, developing active citizenship. In particular, this tradition refuses to consider mathematics as politically neutral; it challenges a didactic that contents itself with passing on scientific knowledge, and thus is mostly oriented towards competence acquisition. Concerning the latter, students acquire knowledge when they feel a need for it, that is, according to real-world demands, and this explains the great interest of the KOM project in modelling. May one therefore consider that PISA results are absolute indicators (as often presented in press releases), although they in fact reflect only a particular measure of student performance? Moreover, must curricula be determined by assessments, risking the behaviourist shift feared by Brousseau (2007) and Chevallard (2002)? Reforming educational

A Numerical Landscape

17

systems on only the PISA basis could only improve students’ performance at… PISA. Is this our unique goal? What sort of achievement do we expect for students and for what purpose?

Mathematical Coherence, Modelling, and Curricula We suggest another thought process. First of all, we must specify competencies for lifelong number learning, relying on a framework that takes account of personal development, and mathematical and didactical coherence. By personal development, we mean development that leads to either work or deal with numbers. By mathematical and didactical coherence we mean that our references for interpreting and structuring society and individual needs in number learning are mathematical objects, concepts, notions, domains (whole numbers, rational numbers, algebra, calculus, analysis) and the related studies in mathematics education. Assessments would be of course based on these competencies. As we can note, this process is opposite to the process described above: It stems from individuals and their needs in number learning and leads to assessments via curricula and competencies. May modelling remain a central reference in such a process of defining objectives for lifelong number learning? For debating this important question, we first go back to the ICMI Discussion Document (Blum, 2004), which is a very complete overall study on the subject. This document brings a real framework not only for appreciating and analysing the considered complexity and scope of modelling, but also for exhibiting any subject relevant to this theme. We especially focus on the chapter devoted to “Examples of important issues.” We particularly note that: •

• •

The interrelations between applications and modelling, and mathematics, from both a purely mathematical and a didactical viewpoint, are strongly taken into account. This goes in the direction of fostering mathematical coherence, which is one of the main lines we have indicated for lifelong learning. Shaping or restoring the image of mathematics is an important aim of applicationand-modelling promoters. Authenticity of the involved material is one the main considered issues.

And this leads us to examine the type of material, i.e. real-world situations, available and frequently used when implementing teaching sequences related to modelling. Let us refer to two authors that have conducted and reported many classroom-based studies on this subject: A. Peter-Koop, and K. Maass who participates in LEMA. In one of her papers (2004, p. 457), the former mentions that she has resorted to Fermi problems for the reported experience: “Enrico Fermi, who in 1938 won the Nobel Prize for physics, was known by his students for posing open problems that could only be solved by giving a reasonable estimate. Fermi problems such as ‘How many piano tuners are there in Chicago?’ share the characteristic that the initial response of the problem solver is that the problem could not possibly be solved without recourse to further reference material”.

18

Robert Adjiage and François Pluvinage

She gives a bit farther the following criteria as guidelines for the choice of problems posed in the research on work with 3rd and 4th-graders: • •

• •

The problems should present challenges and intrinsically motivate cooperation and interaction with peers. The wording of the problems should not contain numbers in order to avoid that the children immediately start calculating without first analysing the context of the given situation, and to challenge pupils to engage in estimation and rough calculation and/or the collection of relevant data. The problems should be based on a selection of real-world-related situations that include reference contexts for third and fourth graders. The problems should be open-beginning as well as open-ended real-world-related tasks that require decision-making with respect to the modelling process.

Four problems have therefore been posed to 3th- and 4th- graders: 1) How much paper does your school use in one month? (paper problem). 2) How many children are as heavy as a polar bear? (polar bear problem). 3) Your class is planning a trip to visit the Cologne Cathedral. Is it better to travel by bus or by train? (cathedral problem). 4) There is a 3 km tailback on the A1 motorway between Muenster and Bremen. How many vehicles are caught in this traffic jam? (traffic problem).

K. Maass (2005, p. 4), gives a list of problems she submitted to 7th- and 8th- graders. Among them, we retain the numerical items: 1) How many people can be found in a 25km-long traffic jam? 2) How can different charges of diverse mobile contracts be clearly arranged depending on customers’ habits? [….?…] 3) Is it possible to heat the water required in Stuttgart-Waldhausen with solar collectors on the roofs? 4) What is the connection between the height of fall and the subsequent height of rebound of a ball?

We lastly report a problem that has been experimented for LEMA: Giant’s foot. The following photo was taken in an English amusement park. How approximately tall is the entire figure, of which we can only see the foot? All these situations are much more than a “dressing up” of a “mere” mathematical problem, and the modelling process consists of much more than “undressing” the real-world problem. In this sense they bring great progress compared to numerous problems one finds in usual textbooks. The declared guidelines are respected. The quasi-absence of numbers in the statement is a main feature of these problems, and we have personally observed that this results in more considering relationships between the involved quantities rather than starting haphazard calculation. For instance, we have observed many 5th-graders, dealing with the “giant” problem, that have tried to find the number of men (or visible barriers, or boot-

A Numerical Landscape

19

soles…) that one could put into the whole estimated length of the giant. This attitude brings them close to the proportionality model, whereas the presence of numbers frequently leads students to combining numbers, often by adding them, without reflecting the relationships linking the underlying quantities.

6

Figure 1. Giant’s foot .

One observes that many situations, almost all of them actually, rely on two pillars: data collection, and then applying a multiplicative model: proportionality, calculation of averages… So that pupils are supposed to resort to an already studied (and somehow standard) model. Very few existing situations lead to developing a mathematical model. K. Maass’s fourth problem quoted above is of that kind, although it mobilises linear functions. We have suggested, but not already experienced in classrooms, situations as car braking distance, which depends on the square of the speed. This could be a new issue to experiment. Great care should also be brought to situations that give real possibilities of validating the retained data and the findings. For instance, the traffic-jam problem (K. Maass, number one) depends on many choices made by students: the ratio cars/trucks, the average length of each, the average number of persons in a car… This problem is thus open-ended, the results depending on these choices, themselves having to be coherent with the retained time of the day or period of the year. Whatever ones choice, it seems difficult to directly validate the employed ratio cars/trucks or average length of cars and trucks, this information being not easily available, e.g. on Internet. On the other hand, it could be possible to check, e.g. on specialised radio channels, the number of persons involved in a given traffic jam, and this may validate the retained data. Can modelling effectively contribute towards promoting views of mathematics that extend beyond transmissive techniques to its role as a tool for structuring other areas of knowledge? (Blum, 2004, p. 161). Modelling is also intended to promote interaction, cooperation, and communication: “The real-world problems used in the study should intrinsically present challenges and thus motivate peer interaction during the solution process as opposed to problems that can be solved quite easily by an individual student.” 6

The authors are grateful to Richard Philipps from: http://www.problempictures.co.uk/, who has allowed them to use this picture.

20

Robert Adjiage and François Pluvinage

(Peter-Koop, 2002, p.563). Succeeding in promoting these attitudes in the classroom does not only depend on the available didactical material. Teachers are often rushed by their schedule, whereas modelling activities demand giving students the time for acquiring the knowledge required by the real world situation, for collecting data, for testing hypotheses, for elaborating or identifying an appropriate mathematical model, for discussing all this matter between peers. So that teaching is also of great importance. We have observed many sequences, e.g. related to the giant situation, where productive peer interactions have been noted, leading to unexpected but correct procedures for calculating the giant’s height. But for lack of time, lack of training, the teacher did not take these ideas into account and favoured the procedures that he imagined. So that part of the expected benefit, in terms of the image of mathematics and interest of working groups, may be lost or invisible. In any case, the challenge of modelling has not been sufficient to change this teacher’s usual practise.For concluding this section, we would say that modelling still challenges research, training, and teaching. Important progress has already been made concerning the epistemology and the material related to modelling. Other issues, which we have mentioned above, remain to be explored. Modelling is coherent with the personal development of an individual and what society expects from him, whether he is intended to work or to deal with mathematics. However, let us beware of excessive enthusiasm. Let us consider other teaching entries, which take account of purely numerical obstacles and coherence, particularly those that help students to become aware of continuities and ruptures from numeracy to mastering calculus and analysis, going through ratios and algebra, because these themes structure all numerical development. Anyway, being capable of advanced modelling, particularly developing numerical models (and not only resorting to taught models), relies on mastering these fundamental areas and their interrelations.

STUDYING NUMBERS: HOW LONG AND UP TO WHICH DEGREE OF ACCURACY? In the previous section, we examined the way curricula are or might be designed. Here, we are interested in the general features and progression of curricula related to number acquisition, and the way they are actually implemented in the classroom. We take advantage of this presentation for sketching some continuities and ruptures through general numerical areas: whole numbers, ratio and rational numbers, algebra, calculus and analysis, which we consider essential for describing and understanding this learning. Numbers together with their operations constitute the first objects in learning mathematics, and pupils continually face the learning of numbers. At elementary school, teaching numbers is present each year, and has the most important place among the mathematical topics. After presentation of natural numbers and arithmetic operations, students learn about positive decimal numbers and fractions. Later, they study negative numbers, but from this stage, learning numbers does not follow a continuous path in curriculum development. That means that some properties, concerning for instance exponents, are examined without introduction of new concepts. And the theory of numbers, including theorems such as Fermat’s theorem (if p is a prime number, then for any integer a, the number a p – a is divisible by p), is actually taught only in advanced courses. Nowadays numeral systems, like binary system or hexadecimal, are only considered as a topic for

A Numerical Landscape

21

prospective mathematicians or specialists in computing. At middle school the students are led to deal with some irrational numbers, canonical examples of them being 2 and π. But only a scarce minority of college students in scientific careers masters real numbers. Nevertheless, in many countries, core curricula in mathematics education formulate their objectives in such a way that a reader may think that at the end of grade 7 or 10, (it depends on the educational system) students are supposed to master real numbers. Moreover, complementary official documents, such as those of Federal Resources for Educational Excellence in the USA, strengthen this misleading impression. For example, many textbooks, or online documents, present a chapter or a list of exercises devoted to real numbers. If we carefully examine the content, it often appears that the presented properties (operations and order) do not characterise real numbers, but also apply to rational numbers. The most difficult one, and also the most important, the closure property (if a nonempty set of real numbers has an upper bound, then it has a least upper bound), is dismissed. Summarizing our experience as well as observations made by various researchers, we can assert that general tendencies while teaching mathematics are the following. • •

• • • • • •

At primary school, strong priority is given to teaching numbers, for instance in comparison with geometry or other topics. For this reason, magnitudes do not constitute an effective component of learning whole numbers, although researchers, such as Galperin quoted by Arievitch7 (2003, p. 280), have stated opposite educational principles. Teachers carefully introduce the mathematical tools (scripts and algorithms) Nevertheless, they do not systematically train students in mental calculation, nor present them arithmetical techniques such as casting out nines They do not consider computational tools as a positive resource of numerical problems They do not consider certain links between the various presented ways of expressing numbers, e.g. from repeating decimals to fractions At the secondary level, algebra progressively takes the place of numbers, but does not explicitly help to better understand numbers Teachers give little consideration to numerical proofs.

We are not surprised, in these conditions, that educational goals are difficult to achieve for many students. National and international assessments allow us to better understand what is currently expected from learning, and to observe the distance between the expectations (intended curriculum) and the obtained results (implemented curriculum). Learning numbers seems to be obvious for people who master them, for the apparent easiness that operative rules present. But if we want to build a mathematical world, rather than an arbitrary one, each general rule has to be justified and demonstrated, which leads us to consider both the universe of concrete applications and the management of representations. Let us report a case that we observed in a kindergarten classroom: An experienced teacher 7

The underlying assumption in Galperin’s approach to learning and activity is that there are two different types of actions: (1) an ideal action that is performed in the presence of objects and (2) an ideal action that is carried out separately, without any presence of objects.

22

Robert Adjiage and François Pluvinage

organized a play, called “our boat is sinking”, with a group of 12 pupils. In a song mentioning a boat, suddenly come the words “our boat is sinking, for evacuation you have to form groups represents a number that the teacher chooses. When the chosen number was of ”, where 3 or 4, the task was easy for the pupils, but later, a choice was 8. Then the natural question arises: How many pupils are missing for forming two groups of eight? The teacher recommended using an abacus (representation) without allowing the pupils to consider the concrete situation that they could directly build: e.g. a row of 8 facing the remaining pupils, then constituting as many pairs as possible and counting how many pupils in the row of 8 remained alone.

Figure 2.

All students would easily understand this situation, whereas the link between the problem and its numerical representation and processing using the abacus (2×8 – 12) was not obvious for such young children. In this elementary situation we observe a possible link between learning numbers and exploiting a concrete situation. The observed lesson was very attractive for the pupils, so that we can only point out here some “loss of learning”, like one says “loss of earnings” in a commercial relationship. But we have also observed most important sources of difficulty in other situations. For instance, in a course about speed and velocity observed at grade 6, activities were proposed in the textbook, such as comparing the constant velocity of two swimmers. One of them takes 1 min 30 sec for 100 m, and a second swims 250 m in 4 min. The textbook also proposes comparing different students’ approaches. But instead of this, the teacher’s conception, i.e. applying the general formula v= d and comparing ratios, prevailed:

t

a female student built a table for the first swimmer’s times, under the hypothesis of constant velocity. Her correct result, 3 min 45 sec for 250m, was considered wrong and erased by the teacher. The latter reminded that the problem statement gave for the second swimmer 250 m in 4 min and that the question was to compare swimmers’ velocity by comparing ratios.

Figure 3. “A mistake”.

It is however well known that most 6th-graders need for this comparison such intermediary steps before being able to understand and apply the formula. These steps mark stages in the acquisition of processing involving ratios and proportions, and thus in acquiring competence related to these notions.

A Numerical Landscape

23

What characterises the acquisition of competence is the length of learning and the need of a wide variety of activities. We have presented two examples, one referring to the acquisition of counting and calculating, the second to ratio and proportion. PISA2 assessment reveals that many 15-year-old people do not properly manage ratio problems, even if they have been taught for several years. At a little more advanced level, the teaching of algebra starts. But secondary school pupils encounter serious difficulties with algebraic processing. As an example, we give hereafter six typical multiple- choice questions extracted from a pre-test, and the table of answers given by a sample of Mexican students at the beginning of undergraduate engineering programs. Pocket calculators are not allowed. Q2. Calculate: 3 − 5

4 6

−1 ; -1; 12

38 ; Other 24 1 1 Q3. Calculate: − 2n − 2 2n 1 1 2n − 1 ; − ; ; Other 0; 2 n ( n − 1) 2n ( n − 1) 2 ; 24

Q4. Which of the following numbers is largest? 3.5; 2.46; 3.19; 0.546 -9.87 Q5. Calculate: 3 ( 5-8 ) + 3 ⎡⎣ 2 ( 4+9 ) -5 ( 2-6 ) ⎤⎦ 2

9; 30; -55; 229; Other Q8. The solution of −30 x + 4 ≤ 0 is:

2 2 ; ( −∞ , ∞ ) ; x ≤ ; Other 15 15 2 Q9. The solutions of 2 x − 3 x − 2 = 0 are: 2 2 −1 x1 = −2 and x2 = ; x1 = 2 and x2 = − ; x1 = 2 and x 2 = ; 3 3 2 2 x1 = −2 and x2 = − ; Other 3 x=

2 ; 15

x≥

Table 2. Empirical evidence - percentages of choice made by undergraduate students Choice Q2 % Q3 % Q4 % Q5 % Q8 % Q9 %

1 25 9 88 4 21 17

2 65 25 0 3 39 26

3 1 30 2 9 3 43

4 4 27 4 2 36 5

5 5 9 6 83 1 9

24

Robert Adjiage and François Pluvinage

In the table above, the percentage of right answers lies in bold in gray cells. There is a strong contrast between numerical questions (Q2, Q4 and Q5) and questions including variables (Q3, Q8 and Q9, marked by the symbol). For Q3, Q8 and Q9 only a minority chose the correct answer. We particularly observe in Q8 that students were able to recognize the fraction

2 as representing 4 (choices 1, 2 and 4), but then had difficulty in 15 30

algebraically managing the inequality: The three corresponding choices are frequent. This underlines the difference between processing ratios and fractions and processing algebraic expressions. For this reason, it seems necessary to distinguish a level of mathematical competence that consists in managing algebra. If we consider the curricula up to grade 12, we can find other topics related to numbers in courses of Algebra. For instance, in the 2005 core curriculum of the State of New York, complex numbers are introduced in Algebra 2 and Trigonometry (grade 12). But in this case, the label “Algebra” may be questioned. Historically, complex numbers in the form of square roots of negative numbers were considered for solving polynomial equations: third grade by Niccolo Tartaglia and Gerolamo Cardano, and fourth grade by Lodovico Ferrari (all these results obtained during the 16th century). But later attempts at formalising encountered paradoxes,

such

as

the

extension

(−1)(−1) = 1 = 1 on one hand, and to

of

the

−1 −1 =

property

(

−1

)

2

ab = a b

leading

to

= −1 on the other hand, hence 1

= -1! For this reason, in Euler’s time, mathematicians introduced the symbol i (for imaginary) instead of the controversial −1 . Both numbers, i and its opposite –i, have -1 as square (then are acceptable square roots of -1), so that the paradox could be explained. Nevertheless, the full extent of complex numbers arose only two centuries later, with the proof of the fundamental theorem of algebra, also known as the d’Alembert/Gauss theorem. This theorem states that every non-constant polynomial with complex coefficients has at least one root. In other words, the field of complex numbers is said to be algebraically closed. As a consequence, a polynomial of degree n has n roots, counting multiplicities. The first book in which this result was stated (for polynomials with real coefficients) was published in 1629. Its author Albert Girard gave as an example that the equation x 4 = 4 x − 3 has the four roots: 1, 1, −1 + i 2, − 1 − i 2 (in modern writing). But the first attempt at proving the theorem was only made by d’Alembert in 1746. The proof given by d’Alembert was incomplete. Only in 1806, J. R. Argand (1768-1822), a non-professional mathematician who geometrically interpreted i as a rotation of 90º in the plane, published a nearly satisfactory proof (Argand, 1874, pp. 90-91), and Gauss later produced another proof. Although the theorem is about algebra, all proofs used analysis, e.g. Liouville’s theorem: “In complex analysis, every bounded entire function is constant”. Indeed, if a polynomial P(z) does not have any root, it 1 is entire and bounded, therefore P(z) reduces to a constant. follows that the function P( z ) From these epistemological considerations, it appears that an important jump was necessary: from discovering algorithms for solving some equations, to first formulating, and then proving general results about polynomials. Therefore, we cannot consider that manipulating algebraic expressions should constitute the highest level of mathematical competence that teaching up to grade 12 has to implement. Stating general results about

A Numerical Landscape

25

functions, and solving problems, the solutions of which being functions, seem to be elements of a more advanced field of competence than simply solving numerical equations, or even parametrical equations.

COGNITIVE ASPECTS After examining institutional expectations as well as practices concerning the use and the acquisition of numbers, we will now introduce cognitive aspects of the general framework to which we refer. As already mentioned in the introduction, we focus on the semiotic features of mathematical objects, we regard numerical processes as if they were linguistic ones, and this leads us to propose a characterisation of algebra. We lastly consider the link between numbers and graphical representations.

Semiotic Registers Writing or representing numbers mobilizes various semiotic registers. Let us specify this assertion: “We do not have any perceptive or instrumental access to mathematical objects… as for any other object or phenomenon of the external world…”, “…the only way of gaining access to them is using signs, words, or symbols…” Duval (2000, p. 61). As seen above, realworld experiences are certainly a starting point for understanding the pertinence of numbers and then entering their universe. Duval’s assertion does not mean that we can do without referring to the real world for teaching numbers, and further large areas of mathematics. It only means that, at a certain point of learning, pure mathematical objects must be considered, if only because they can be dissociated from a particular context: for example fractions have to be detached from a measurement context, in which they are usually introduced, to apply to many contexts, e.g. mixture, enlargement… This is what an educated adult population can do, what 7th-graders, taken as a whole, are not yet able to do. This level of comprehension is certainly a condition for mathematical modelling. What Duval states is that symbols are the only way of accessing these pure mathematical entities. Now, there exist, and students are actually taught to face, many means of expression in mathematics. For instance, when expressing and processing rational numbers, we have at our disposal: fractions, of course, decimal numbers, two separate whole numbers (like in the expression “a player makes an average of 3 out of 4 basket attempts”), visual representations such as “pie charts” and number lines which are supposed to help students to better understand rational numbers… Duval (1995, pp. 15-85; 2000, p. 60-65) considers these diverse means of expression as separate and organised systems or “settings”, which he terms “semiotic registers”, necessary for mathematical activity and particularly teaching and learning, for at least four reasons (Duval, 1995, pp. 68-69; 2000, p. 62). • •

They are the only paths to mathematical objects, in the sense described above. They help students to distinguish a mathematical object from its representation: disposing of many representations allows considering what is invariable beyond these representations, thus outlining the underlying object.

26

Robert Adjiage and François Pluvinage •

They highlight all aspects of a mathematical object: each representation is partial, and we need many representations for accessing all the complexity of a mathematical object. For instance, a fractional representation such as 37

5

•

highlights the

multiplicative relationship between 37 and 5, whereas the decimal representation 7.4 of the same number highlights its location (between 7 and 8, closer to 7 than 8…). They give alternatives when processing: a fraction can be useful for interpreting a ratio, but decimal numbers may make the comparison of ratios easier.

Duval (1995, pp. 39-44; 2000, p. 63) distinguishes two kinds of processing. “Treatment”: 3 6 = , and “conversion”: transformation of an object within a given register, e.g. 5 10 transformation of the representation of a mathematical object into a representation of the 3 same object in another register, e.g. = 0.6 . He states that the latter, unlike the former, 5 entails a rupture in the means of representing and processing, and thus in thinking. Conversion between semiotic registers is a cognitive operation essential for objectifying mathematical entities. Treatment and conversion are necessary for expressing mathematical rules and properties that legitimate processing. Constraints are determinant in the distinction between simple illustration and a semiotic register. For instance, it is usual to represent an addition like 5 + 4 on the number line with an arrow joining 5 to the point located 4 units ahead. But we note but we note in upper Figure 4 that the number 4 does not explicitly appear (we have to count 4 units). Thus, we consider this representation as a relevant illustration for 5 + 4, but not for a treatment in a semiotic register. It is truly different when the same operation is made with an “additive slide rule”: this kind of continuous abacus presents two rulers side by side, the upper one sliding at the user’s demand, and a cursor. Nowadays, students may use a virtual tool: the model presented in Figure 4 was built with CABRI. In this case, the upper ruler has its origin at point S facing 5, and the point A of the cursor faces 4. The cursor shows the result 9 on the lower ruler. Note that the same configuration may also be read: 9 – 4 = 5. Thus, the “additive slide rule” allows converting a given configuration into two distinct arithmetic equalities: 5 + 4 = 9 or 9 - 4 = 5. Moreover we observe that, if S remains constant, moving the cursor allows the expression of any sum of the kind 5 + a. In other words, the configuration: S constant, cursor variable, is the translation of the sentence “Add 5” expressed in the natural language or verbal register. We can also process a sum like 5 + 4 + 3 without determining an intermediary result. Starting from the initial position (the two 0’s facing one another), we first slide the upper ruler 5 units forward and we put the cursor on 4, then we slide the upper ruler a second time in order to put its origin under the cursor, and finally we move the cursor to 3. The result appears on the lower ruler. All these considerations show that the additive slide rule constitutes a semiotic register with its own treatments, distinct from both verbal and arithmetic registers. Software like Excel or the worksheet in Open Office give an apparent opportunity to see the same number expressed in various settings. For a given cell of a worksheet, you may select various number formats: decimal number, fraction, percentage, etc.

A Numerical Landscape

27

Figure 4. representing 5 + 4 = 9 in two different ways.

In the fraction case, you must choose the denominator range (e.g. less than 10, 100…), making the involved representations and processing not exact, as is possible with software like Derive or Maple. If we calculate the sum: 11 + 23 with Derive or Maple in exact mode,

30 70 73 we obtain the correct result: . But suppose we enter each fraction in a cell of a worksheet, 105 e.g. cells A1 and A2, and then calculate the sum in cell A3. We choose, in the considered cells, fractions with denominator less than 100 (which is the largest option in Open Office) as number formats. We obtain 16/23 with Excel and 57/82 with Open Office. Such experiments or only their results could be interesting for 7-graders equipped with pocket calculators. Are the fractions 16/23 and 57/82 distant from or very close to one another? And in relation to the exact result 73/105…? Let us underline here that the distinctions we have introduced are important even when using software. Some software (with worksheets, as in our examples) has only one kind of “treatment” in Duval’s sense. Standard worksheets always operate with Binary Coded Decimal: even when

11 appears in a cell, the number taken into account by the computer is 30

0.32857142857…, and the software only converts when it displays the final presentation. Other software (as Derive or Maple) processes in distinct ways depending on the form of the given objects. We may say about the second that it uses distinct semiotic registers.

28

Robert Adjiage and François Pluvinage

Components of Numerical Acquisitions Our general theory of number acquisition and numerical structures is based on a relationship between three components, which are to be studied separately and then linked with one another: physical experience, mathematical properties, and semiotic representations. We assert that a lack in presentation of one of the three components, or of their links, induces difficulties of understanding for many students and leads to unstable learning. In “The four competences” section below, we present typical physical situations, which may provide students with the needed experience in the conditions of an adequate learning milieu. In the following, we would like to emphasize the link between mathematical properties and semiotic representations. Example: In an observed active 7th-grade class, students tried to solve the following problem: obtain 1000 with 8 “8’s”. It is important to distinguish the heuristic phase from the written presentation of the answer. We consider the latter as one step toward algebraic treatment, although no variable is required. We explain below that the reason for this is that, when writing the solution, one is lead to process mathematical “sentences”. We assert that, in the considered case, the processing is a tool for solving the problem, not only a way to communicate or explain ideas. Two correct written answers are: 1000 = 888 + 88 + 8 + 8 + 8 1000 = (8888 – 888)÷8 Many students gave in this situation “equalities” like: 888 + 88 = 976 + 8 + 8 + 8 = 1000. Explaining why this kind of writing does not respect mathematical rules, and thus is incorrect, is a true challenge for teachers. Indeed, the correct use of the equality symbol is not something purely formal, as many students believe, but it constitutes an important element of the mathematical construction. Other students used seven “8’s” instead of eight, e.g. the following response that we summarise by the equality: 1000 = (8 + 8)×8×8 – 8 – 8 – 8

(*)

The students did not furnish this equality, they performed non-ordered calculations during the heuristic phase, and this may be the reason why nobody, including the teacher, noticed that there are only seven “8’s”. When calculating, the focus is on the result 1000 and this leads to forget the rest. If there had been a complete final writing of (*), this “concluding sentence” would have encouraged reconsidering what precedes, and thus noticing the errors. Moreover, from the above inadequate writing, it is easy (for someone familiar with algebraic process) to obtain a correct answer by dividing and multiplying by 8: 1000 = ((8 + 8)×8 – (8 + 8 + 8)÷8)×8.

A Numerical Landscape

29

And this would be true processing of (*), which can be compared to processing a linguistic sentence. Using letters in calculations is not specific to algebra. One can use letters without being competent in algebra, e.g. substituting values for time and velocity in the formula d = vt. Conversely, one can do algebra without using letters. We assert that a fundamental step in mastering algebra is being able to process sentences. A sentence is the unit of meaning. Mathematics form sentences, like common language does. In mathematical writing, we can recognize nouns that are: numbers, variables, or more generally all kinds of mathematical objects; verbs: =, ≠,

SCIENCE EDUCATION ISSUES AND DEVELOPMENTS

CALVIN L. PETROSELLI EDITOR

Nova Science Publishers, Inc. New York

Copyright © 2008 by Nova Science Publishers, Inc.

All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Science education issues and developments / Calvin L. Petroselli, editor. p. cm. Includes index. ISBN-13: 978-1-60692-604-8 1. Science--Study and teaching. 2. Education. I. Petroselli, Calvin L. Q181.S37342 2008 507.1--dc22 2007031841

Published by Nova Science Publishers, Inc.

New York

CONTENTS Preface

vii

Expert Commentary Commentary

Why do We Lose Physics Students? Ronald Newburgh

1

Research and Review Articles Chapter 1

A Numerical Landscape Robert Adjiage and François Pluvinage

Chapter 2

Learning in and from Science Laboratories: Enhancing Students' Meta-Cognition and Argumentation Skills Avi Hofstein, Mira Kipnis and Per Kind

59

The Crisis in Science Education and the Need to Enculturate All Learners in Science Stuart Rowlands

95

Chapter 3

Chapter 4

Understanding Student Affect in Learning Mathematics Marja-Liisa Malmivuori

Chapter 5

The Challenge of Using the Multimodal Aspects of Informal Sources of Science Learning in the Context of Formal Education Krystallia Halkia and Menis Theodoridis

Chapter 6

Chapter 7

Understanding Scientific Evidence and the Data Collection Process: Explorations of Why, Who, When, What, and How Heisawn Jeong and Nancy B. Songer Constructivist-Informed Classroom Teaching: The Importance and Potential of Motivation Research David H. Palmer

5

125

151

179

201

vi Chapter 8

Chapter 9

Chapter 10

Chapter 11

Index

Contents Oral Communication Competencies in the Science Classroom and the Scientific Workplace F. Elizabeth Gray, Lisa Emerson and Bruce MacKay

223

Supporting Future Teachers Learning to Teach Through an Integrated Model of Mentoring Pi-Jen Lin

239

Strategies to Address Issues and Challenges Faced by Instructors in General Education Introductory Astronomy Courses for Non-Science Majors Michael C. LoPresto Getting it to Work: A Case of Success in Sustaining Science Professional Development Betty J. Young, Sally Beauman and Barbara Fitzsimmons

257

271 283

PREFACE This book presents significant new analyses in the field of science education. This is hardly another field in education which is more important for a country's future than science education. Yet more and more students elect to concentrate on other fields to the exclusion of science for a variety of reasons: 1. The perception of degree of difficulty, 2. The actual degree of difficulty, 3. The lack of perceived prestige and earnings associated with the field. 4. The dearth of good and easy to use texts. 5. The lack of society in comprehending the significance of science and creating attractive incentives for those who enter the field. Chapter 1 - This study concerns the field of mathematics education. Today, for almost any technology, attaining the most advanced level relies on using digital systems. Therefore, the authors focus on number acquisition and use, emphasize major discussions about related topics, and introduce our personal contribution. The authors consider three areas: numbers in society, at school, and in the field of education. Numbers in society: Modern society needs two kinds of number users: the first has to deal with numbers, the second has to work with numbers. According to the framework for PISA assessment, dealing with numbers concerns every citizen facing situations in which the use of a quantitative… reasoning… would help clarify, formulate or solve a problem. Working with numbers is what a professional (marketing specialist, engineer, physician, artist…) does when he has to cope with numerical theoretical frameworks, or when he collaborates with a mathematician. This brings two types of questioning: what does competence in mathematics mean? What level of achievement is desirable in number-learning to meet either numerical need? Numbers at school: The authors examine what is proposed for teaching such a broad subject to 7-to-18-year-olds. The authors first observe and question the educational system, the designer of curricula, scenarios for teaching, training programmes, and national assessments. Secondly, the authors question the notions of problem solving and modelling as mere responses to mathematics-teaching issues. The authors then focus on what really happens in a standard classroom, particularly how teachers apply recommendations and directives, and how the generalization of assessments affects their practice. Numbers in the field of education: Three aspects are considered: epistemological, cognitive, and didactical. The authors distinguish four related levels which the authors have named: numeracy (competence linked to whole numbers), rationacy (competence linked to ratios and rational numbers), algebracy (competence linked to algebra), and functionacy (competence linked to calculus). The cognitive aspect evokes the essential issue of semiotic

viii

Calvin L. Petroselli

registers for representing and processing numerical objects, considers the discipline-ofexpression aspect of mathematics, and other issues taken into account by numerous researchers, such as process and object…. The didactical is devoted to exposing our conceptual framework for teaching numbers and understanding their learning. An experiment in ratio teaching is described and analysed. Chapter 2 - Laboratory activities have long had a distinctive and central role in the science curriculum and science educators have suggested that many benefits accrue from engaging students in science laboratory activities. More specifically, it has been suggested that, when properly developed, there is a potential to enhance students’ conceptual and procedural understanding, their practical and intellectual skills and their understanding of the nature of science. Research findings, however, have proven that “properly developed” laboratory work is less frequent than hoped for and that meaningful learning in laboratories is demanding and complex. The 21st century has offered new frames for dealing with the potential and challenges of laboratory based science teaching. This is an era of reform in which both the content and pedagogy of science learning and teaching are being scrutinized, and new standards intended to shape meaningful science education have emerged. The National Science Education Standards (National Research Council, 1996) and other science education literature (e.g. Lunetta, Hofstein and Clough, 2007) emphasize the importance of rethinking the role and practice of school laboratory work in light of these reforms. The new frames, however, also relates to the development in the understanding of human cognition and learning that has happened during the last 20 years. In the following chapter attention will be given to research on learning in and from the science laboratory. More specifically, the presentation will focus on the science laboratory as a unique learning environment for the following teaching and learning aspects: • •

Argumentation and the justification of assertions Development of metacognitive skill

It is suggested, that these are important aspects with a natural place in the science laboratory. They have, however, been neglected both regarding development of practical experiences provided to the student as well as in research on the effectiveness of practical work that is conducted in the context of science learning. A new approach is needed in which these two aspects are coordinated and seen in accordance with the general practice of teaching and learning in school science. Chapter 3 - There is a crisis in science education. Over the past two decades many organisations such as the American National Science Foundation, the Australian Audit of Science, Engineering and Technology and the UK’s Royal Society and the Confederation of British Industry, have reported a serious decline in students enrolling in science subjects and the failure of the science curriculum to inspire learners and to meet national needs. However, quite apart from instrumental reasons such as a national interest for having more scientists, science education is important for cultural reasons. Science permeates every aspect of modern life and arguably full citizenship in a technological society necessitates the understanding of science. Based on how the world is, science promotes critical thinking, a concern for evidence and an objectivity that is independent of personal opinion or the dictate of kings - yet few individuals have an elementary understanding of science. The failure of science education is

Preface

ix

reflected in science’s lack of popularity evident in the rise of mysticism, the rise in consensus of intelligent design, the postmodernist attack and the closing of many UK university science departments. There have been calls to remedy the situation, such as school visits by scientists and engineers, or overhauling the science curriculum by teaching the science deemed relevant to the everyday experience of children, consistent with the constructivist idea that there is a “children’s science”. However, there is no guarantee that exposure to the working lives of scientists will promote an interest in science and, moreover, science is not based on making sense of experience. To generate the interest and motivation of young learners requires an engagement with the nature of science (NOS) involving meta-discourse with the history and philosophy of the discipline. Contrary to the current wisdom of science educationalists, NOS has more to do with the rule-governed abstract possible world of the thought-experiment than hypothesis testing with a clipboard of data. Even the most concrete thinkers may be capable of thinking in the abstract and mechanics, because of its history and logical character (as opposed to the “soft sciences” such as ecology), provides the perfect opportunity to do this. This article consists of three parts: 1. Public perception of science and scientific literacy and understanding. 2. Why NOS is essential to science education. 3. Why “children’s science” and conceptual change, the largest domain in science education research, has failed to promote scientific understanding. Chapter 4 - Student affect has been one area of interest in mathematics education for decades. This applies in particular to rather large surveys of students in The United States since 1970’s. In general, education studies on affect have much focused on affective factors in the contexts related to mathematics achievements, learning of mathematics or solving mathematical problems. This is understandable since mathematics and mathematical problem solving carry many kinds of cognitive and sociocultural features that are not easily attached to the other school subjects. For example, the abstractness of mathematics and the differences in the symbol systems used in mathematical language set high demands on cognitive processes and also detach mathematics from the context and experience of everyday life. Furthermore, general views of mathematics as a difficult and demanding subject have caused it to be highly regarded and have been generally used to measure academic abilities. Mathematics tend to have a ritual value in societies that then cause powerful experiences with and important differences in mathematical performance. After showing passionate interest in human cognition and cognitive processes, education research paradigms have recently created new opportunities for and even laid emphasis on studies of student affect. Constructivism, together with applied socio-cognitive, cultural and contextual views of learning and education, has enriched our knowledge of affect in mathematics education research, as well. This theoretical chapter first discusses some conceptual features of affective factors traditionally applied in education research and especially in mathematics education studies. This short overview will then be followed by consideration of some of the most significant and often used affective variables in mathematics education research. More recently presented views of affect with cognition in learning will be considered as an introduction to the here suggested theoretical framework for understanding student affect in learning mathematics. Especially, perspectives on the coexistence of affect and cognition, on self-related cognitions and self-regulation are applied in constructing this suggested theoretical framework. It represents a dynamic, humanistic and socio-cognitive, viewpoint on the functioning and development of students´ powerful affect in their learning processes.

x

Calvin L. Petroselli

Chapter 5 - In this work, an attempt has been made to study the plethora and the diversity of informal sources of science learning and the ways formal education may benefit by making use of these sources in its everyday school practice. Informal sources of science knowledge have many forms: they use several means of presenting scientific information, take place in several environments and use several ways to compose their “text”. Each one of them has its own communication codes and uses multiple ways (modes) to present its “meaning”. The material coming from them is chaotic, because it is diverse in terms of the means used, the purposes and the targets stated, the audience addressed, etc. To study them it is helpful to categorize them. Thus a three dimensional model has been developed. Each dimension describes one system of taxonomy: The first one refers to the environment and the conditions under which science learning takes place; the second refers to the way a science “text” is made up and the codes used; while the third one refers to the kind of mode used in the science “text”. Furthermore, the different learning environments in which informal science learning takes place have been studied. Three different learning environments have been distinguished: the organized out-of-schools visits to institutions and organizations (science museums, science centers, zoos, botanic gardens etc.), the students’/teachers’ personal navigation in several sources outside school and the use of informal sources of science learning by the teachers within their everyday classroom practice. The study reveals their particular characteristics, as well as their power and limitations. It also suggests ways of using them effectively in the context of science classroom. Chapter 6 - What is scientific evidence? How should scientific data be collected? These questions comprise essential components of scientific reasoning that are not well understood by students. This chapter explores conceptual challenges students face in inquiry-rich classrooms with respect to the notion of scientific evidence and the related data collection process. As students seek out evidence to support their inquiry, they are likely to ask and need to answer questions such as these: Why collect data? Who collects data? When should data be collected? What counts as scientific evidence? and How should scientific data be collected and analyzed? After examining conceptual issues involved in answering these questions, this chapter proposes that understanding what it means to collect scientific data and what scientific evidence is requires a complex understanding that involves conceptual, procedural, and epistemological knowledge. Chapter 7 - A constructivist paradigm has dominated science education research in recent years. According to this view, students use their existing preconceptions to interpret new experiences, and in doing so, these preconceptions may become modified or revised. In this way, science learning proceeds as children actively reconstruct their ideas as they become presented with new information. However, the implications of constructivism for classroom teaching are still open to question. This position paper refers to the science education literature to argue that strategies to arouse and maintain student motivation should be a crucial component of constructivist-informed classroom teaching. This is because constructivism is universally accepted to be an active process – students must make an effort to reconstruct their ideas, so it follows that if they are not motivated to make that effort then no learning will occur. However, extant models of constructivist classroom teaching make little if any mention of student motivation. In these models, the focus has typically been on strategies to elicit students’ prior conceptions and to guide and monitor their progress towards more scientific conceptions, but the motivational impetus for this process has received little attention. Perhaps one reason for this is that there are relatively few studies of student motivation in the

Preface

xi

science education literature. Another possible reason is the lack of a unified theory of motivation, which means that there is no clear consensus on how best to motivate students in the classroom. In view of this situation, there is a need for studies which can clarify motivational strategies in science classrooms. “Situational interest” is one motivation construct which appears to offer considerable potential, yet it has been largely ignored by science education researchers. Situational interest occurs when a particular situation generates interest in the majority of students in the class – a spectacular science demonstration might arouse transient situational interest even in students who are not normally interested in science. The potential of this construct lies in the fact that studies outside of science have shown that when situational interest is aroused on a number of occasions it can result in longterm personal interest and motivation in the topic. It is thus a potentially powerful construct for science education, and is one which should be further explored. Chapter 8 - This chapter investigates the importance of oral communication training in undergraduate scientific education. The authors examine the status of oral communication training in New Zealand universities and the debate concerning employer attitudes to this issue. The specific relevance of these issues to science education is explored through analysis of a case study and a qualitative and quantitative study of the attitudes of students and employers in science-related industries. Cronin, Grice and Palmerton (2000), Dannels (2001), and Morello (2000) argue that to significantly develop the rhetorical flexibility necessary to communicate competently, oral communication skills training needs to be discipline-specific and firmly contextualized in the genres, expectations, and conventions of the particular field. Responding to this call, a number of recent studies have examined the role of oral communication skill development in specific fields as diverse as design education, archaeology education, and engineering. This chapter moves the discussion of discipline-specific oral communication instruction to undergraduate science education. The recent inclusion of an oral communication component within a compulsory science communication class at Massey University, New Zealand remains a contentious issue. Possibly seeing oral communication training as a low priority in terms of student skills, knowledge, or preparedness for a future scientific career, both students and faculty have resisted the inclusion of oral communication into course curricula and assessment. The researchers designed a study to clarify whether oral communication skills were important to employers in science-related industries, what science employers meant by oral communication skills, and which skills they prioritized. At the same time, the team surveyed science students to better understand their attitudes to training in oral communication. Study findings strongly support the importance of oral communication skills in sciencebased employment in New Zealand. Science employers indicate that they require and value highly a wide variety of oral communication skills. The study also reveals that while science employers and university science students agree that oral communication skills will be important in scientific careers, the majority of employers find the desired level of these skills in new graduates only sometimes or occasionally. The retention of oral skills teaching and assessments, as currently exemplified by the Communication in the Sciences course at Massey University, is clearly indicated; study findings also make a strong case for an extended focus on oral competencies in undergraduate science education. Chapter 9 - The purpose of this article is to introduce an integrated model of mentoring for supporting future teachers learning to teach under the impact of teacher education reform

xii

Calvin L. Petroselli

of Taiwan, particularly, in the internship. This article begins with the introduction of teacher education reform and is followed by the description of the impact of teacher education on quality control. Then, it includes a brief description of six integrated reach projects investigated by teacher educators. One of the integrated research projects that was designed to improve mentors’ competence of mentoring for supporting future teachers learning to teach is reported in detailed and an integrated model of mentoring is developed. Finally, the views of mentors and the future teachers are described briefly and the issues of mentoring are addressed. Chapter 10 - The challenges currently faced by instructors of introductory general education college astronomy courses are numerous. Before effective instruction can even begin, student misconceptions must be addressed. This alone is a daunting task since astronomy is a field in which there are many misconceptions. If dispelling misconceptions is achieved, then effective methods of instruction must be identified and used. Since current research shows that most students learn very little from lectures, other approaches need to be employed. This then means that resources must be either located or created before implementation can occur. The recent movement to stress understanding of concepts rather than memorization and the regurgitation of facts requires that students be engaged and prompted to think critically, or scientifically. This is a challenge in itself, since, as useful as it may be, many non-science students are not used to thinking in this manner. In fact, many students come to class not even aware what science actually is, not a body of facts and figures, but rather a process of investigation. Mathematical illiteracy is not only rampant in our society, but in many cases condoned. Because of this, many non-science majors are math-phobic. They cringe at the site of an equation or graph, even if it is only used to explain a concept and they are not even required to actually use it. Many students are members of Carl Sagan’s “Demon Haunted World” mistaking not only astrology, but television shows, tabloid articles and internet sites about the “paranormal” for science. Many have learned all they know about science from movies and television. Also, some have deeply engrained religious beliefs that prevent them from approaching scientific ideas with an open mind. These challenges are not insurmountable. What follows are the details of various strategies that have been developed and employed to address these issues and challenges with the goal of improving instruction and the entire experience of introductory astronomy for both the students and the instructors. Chapter 11 - This article presents a case of a successful partnership between a university and nine school districts. Science educators, science and engineering faculty from the University joined forces with local school districts to attract funding and implement a high quality K-8 science curriculum supported by new materials and on-going professional development. There are five broad themes to the strategies that contributed to the success of the lasting the partnership: taking the load off central office administrators so that a high quality science curriculum with supportive PD “just happens” with another office managing the details, high quality communication among all partners, management/oversight/control, formative assessment of the quality of professional development implementation with redesign, and documenting results (e.g., parent interest, state-level school site visits, teachers’

Preface

xiii

sense of preparedness to teach science, student achievement outcomes, and continued support by the University administration and faculty).

In: Science Education Issues and Developments Editor: C. L. Petroselli, pp. 1-4

ISBN: 978-1-60021-950-4 © 2008 Nova Science Publishers, Inc.

Expert Commentary

WHY DO WE LOSE PHYSICS STUDENTS? Ronald Newburgh1 Harvard Extension School, Cambridge, MA 02138, USA

ABSTRACT Why is it that we have such great difficulty in retaining students in physics? The answer lies, I believe, in the way most of us, as physics teachers, think. There is a wide spectrum in thought processes. Of the two major types one is linear (i.e. sequential) and the other lateral (i.e. seeking horizontal connections). Those who developed physics – from Galileo to Newton to Einstein to Heisenberg - were almost exclusively linear thinkers. The paradigm for linear thought is Euclidean thinking. Many physicists chose physics for their career as a result of their exposure to geometry. A consequence of this is that textbooks are usually written in a Euclidean format. Thus many beginning students look on physics as an exercise in Euclidean logic, with the attendant certainty that it implies. The sense of discovery is lost. Many students, male and female, do not recognize that the Euclidean format, though efficient, is not a valid description of how we do physics. Their way of approaching problems is different but just as valid. Too many physics teachers refuse to recognize the limitations of this approach, thereby causing would-be students who do not think in a Euclidean fashion to leave. Only when physics teachers are willing to make the effort to understand and even encourage other ways of thinking, will all students look on physics as a welcoming discipline.

The loss of physics students is an alarming trend. I believe that most physicists would agree that physics is an essential component for the development of an educated person. Certainly this belief was accepted in the eighteenth century when the study of natural philosophy was considered a sine qua non if a person were to be considered educated. Is this belief any less essential today? Moreover, the question of great importance to the nation. The renewal of our intellectual capital is a necessity for the maintenance of our standard of living. Yet the number of students who major in physics, as well as in other hard sciences and engineering, is continually 1

e-mail: [email protected]

2

Ronald Newburgh

declining. This is especially true if we consider the number of native-born students. Until recently we have been able to cover the deficit with foreign-born students who remained in the country after graduation. However, for various reasons, many are now returning to their homelands. If we accept the need for more people trained in physics, we must try to answer the question posed in the title. The answer lies, I believe, in the way most physicists think, or perhaps more accurately, the way they appear to think. The term linear thinking is often used to describe scientific thinking. Frequently it is a synonym for logical thinking. This term is quite facile, and I feel that we should go beyond it. In this paper I wish to examine the influence of Euclid, not on the actual doing of physics but rather on the presentation of completed research and the writing of our textbooks. I suggest that Euclid may be the proximate cause of the flight from physics. Though I have no statistics, many physicists with whom I have spoken have said that their introduction to geometry was the reason for their going into physics. Einstein in his Autobiographical Notes [1] wrote the following. “At the age of 13 I experienced a second wonder of a totally different nature [The first was the gift of a compass when he was 5.] : in a little book dealing with Euclidean plane geometry, which came into my hands at the beginning of a school year. Here were assertions, as for example the intersection of the three altitudes of a triangle in one point, which – though by no means evident – could nevertheless be proved with such certainty that any doubt appeared to be out of the question. This lucidity and certainty made an indescribable impression on me. That the axiom had to be accepted unproved did not disturb me. In any case it was quite sufficient for me if I could peg proofs upon propositions the validity of which did not seem to me to be dubious. ... If thus it appeared that it was possible to get certain knowledge of the objects of experience by means of pure thinking, this “wonder” rested upon an error. Nevertheless, for anyone who experiences it for the first time, it is marvelous enough that man is capable at all to reach such a degree of certainty and purity in pure thinking as the Greeks showed us for the first time in geometry.”

Now I am no Einstein, but what he wrote is an exact description of my feelings on my first exposure to geometry, an experience that I can describe only as an epiphany. What has been the influence of Euclid on physicists? Think back on your own doctoral dissertation. In doing the research that led to the dissertation, the frustrations were enormous for nearly all of us – whether we were experimentalists or theorists. There were false starts, apparatus that did not work or broke, equations that did not describe the phenomena, and those that were insoluble. In conversations with advisors and fellow students we tossed out ideas, rejected some, argued about others and finally obtained definitive experimental results and developed a coherent description and resolution of the problem. One can hardly describe those years as an exercise in Euclidean logic. However, in writing the dissertation, we made the work conform to the Procrustean bed of Euclid. Very few dissertations describe or include the false steps. As a rule we present the research as starting from a carefully articulated thesis. It is designed to investigate and prove the effect of B on A and show how it relates to the principle C as enunciated by the eminent Dr. Pangloss. There is a sense of a monotonic progression – b follows a, c follows b, and therefore d is proven. The thesis is written with a strong sense of inevitability.

Why do We Lose Physics Students?

3

When the graduate student becomes a research scientist and teacher, both his scientific papers and textbooks have the same sense of inevitability. The format of our textbooks is Euclidean. Newton’s laws, Hamilton-Jacobi theory, and Maxwell’s equations are often presented as quasi-axioms in advanced texts. Elementary texts emulate them. After all, it is an efficient way of presenting complex material. It is just this approach that can confuse the beginning student. Seeing the material in this form and remembering Euclid, he will look on physics as a deductive discipline – exactly that which it isn’t. At the same time the instructor says that physics is an experimental science so that we discover our laws by induction. This contradiction, usually unrecognized, can create an intellectual malaise, especially for the questioning student. The logical certainty with which most textbooks are written is mirrored by the teacher in his presentation of the material. The problems accompanying the text do nothing to contradict this impression. Discussion of problem solving, whether in the text or by the teacher, usually becomes an application of sterile algorithms. Teachers talk of physics as an adventure in discovery, but most approach the subject in terms of meeting requirements such as the SAT’s. The laboratories become fixed exercises in which the student must confirm some principle already established. He knows the answer before he does the experiment. The result is that most students, in spite of all protestations to the contrary, look on physics as a matter of memorization. A few attempt to find the basic principles underlying the subject but then misuse them as the basis of deductive reasoning. It’s hardly surprising that many students decide that the subject is uninteresting and even illogical. No wonder they major in other fields. One teacher whose approach is a notable exception to this is Mazur [2] of Harvard. He lectures (or rather talks for 10 minutes), then poses a multiple choice question to the class. Using a computer interface, each students selects an answer. The results are recorded on a computer. Mazur then asks them to discuss their answers with the person beside them. This can take two or three minutes. They then vote again. Usually the second round leads to an overwhelming majority of correct answers. This method is one of real learning because the students are truly teaching themselves. Now I yield to no one in my admiration for Euclid. He has been an inspiration to many of us. We understand his genius but also see his limitations. Unfortunately there are many who do not follow his way of thinking. These, I submit, are the students we lose, both women and men,. By presenting alternate approaches to students, specifically uses of lateral thinking, false starts that must be corrected, and lessons that are discoveries not memorization, we can retain more students. We should remember that lateral thinking is essential to the formation of analogies, an activity that one cannot describe as Euclidean. Doing science without analogies seems to me an impossibility. At the same time I recognize that the introduction of alternate approaches must be a time consuming process. It is far less efficient than our current methods. It also requires teachers with greater flexibility, less rigidity, teachers who do not demand that the students parrot their analyses. Frost in his Mending Wall [3] writes of someone who “will not go beyond his father’s saying”. Mazur has gone beyond. If we are serious about the need for producing more scientists, so should we all. I feel that Euclid would agree.

4

Ronald Newburgh

A POSTSCRIPT I realize that I have not conducted a scientific survey of the number of us who were influenced, indeed changed by Euclid in our formative years. However, many conversations going back to the 1940’s lead me to believe that the number is great. I also recognize that I have not proposed a specific program (or programs) for improving physics instruction. Too often, it seems, we believe that there is a single solution to a problem, if we could but find it. Just as there are many types of students, there are many ways to teach. I ask only that as teachers we are more open to the views of students and less rigid in our own thinking.

REFERENCES [1] [2] [3]

Albert Einstein, “Autobiographical Notes’ in Albert Einstein, Philosopher-Scientist, edited by Paul Schilpp (Library of Living Philosophers, Evanston, IL, 1949), pp. 9,11. Eric Mazur, Peer Instruction A User’s Manual, (Prentice Hall, Upper Saddle River, NJ, 1997). Robert Frost, “Mending Wall” in The New Modern American and British Poetry, edited by Louis Untermeyer (Harcourt Brace, New York, NY, 1941).

In: Science Education Issues and Developments Editor: C. L. Petroselli, pp. 5-57

ISBN: 978-1-60021-950-4 © 2008 Nova Science Publishers, Inc.

Chapter 1

A NUMERICAL LANDSCAPE Robert Adjiage1 and François Pluvinage ABSTRACT This study concerns the field of mathematics education. Today, for almost any technology, attaining the most advanced level relies on using digital systems. Therefore, we focus on number acquisition and use, emphasize major discussions about related topics, and introduce our personal contribution. We consider three areas: numbers in society, at school, and in the field of education. Numbers in society Modern society needs two kinds of number users: the first has to deal with numbers, the second has to work with numbers. According to the framework for PISA assessment, dealing with numbers concerns every citizen facing situations in which the use of a quantitative… reasoning… would help clarify, formulate or solve a problem. Working with numbers is what a professional (marketing specialist, engineer, physician, artist…) does when he has to cope with numerical theoretical frameworks, or when he collaborates with a mathematician. This brings two types of questioning: what does competence in mathematics mean? What level of achievement is desirable in number-learning to meet either numerical need? Numbers at school We examine what is proposed for teaching such a broad subject to 7-to-18-year-olds. We first observe and question the educational system, the designer of curricula, scenarios for teaching, training programmes, and national assessments. Secondly, we question the notions of problem solving and modelling as mere responses to mathematics-teaching issues. We then focus on what really happens in a standard classroom, particularly how teachers apply recommendations and directives, and how the generalization of assessments affects their practice. Numbers in the field of education Three aspects are considered: epistemological, cognitive, and didactical. We distinguish four related levels which we have named: numeracy (competence linked to whole numbers), rationacy (competence linked to ratios and rational numbers), algebracy 1 IUFM d'Alsace, 141, avenue de Colmar, 67089 STRASBOURG Cedex ,Personal address: 2, rue des Roses, F67170 MITTELHAUSEN,Tel : 00 33 (0)3 88 51 43 27, Mobile : 00 33 (0)6 16 35 32 01, [email protected], [email protected]

6

Robert Adjiage and François Pluvinage (competence linked to algebra), and functionacy (competence linked to calculus). The cognitive aspect evokes the essential issue of semiotic registers for representing and processing numerical objects, considers the discipline-of-expression aspect of mathematics, and other issues taken into account by numerous researchers, such as process and object…. The didactical is devoted to exposing our conceptual framework for teaching numbers and understanding their learning. An experiment in ratio teaching is described and analysed.

INTRODUCTION Today, for almost any technology, attaining the most advanced level relies on using digital systems. Number acquisition has therefore become a central issue. As a consequence, we have decided to devote this chapter to numbers, learning them, and teaching them. We emphasize major discussions about related topics, and introduce our personal contribution. We will consider mathematics education from early childhood to grade 12. We mainly study the educational problems related to real numbers. Only a small subsection concerns complex numbers because they are mostly studied at university level. Mathematicians and advanced users of numerical domains face either the world of exact computation or the world of approximate computation. In the mathematical subject classification of the American Mathematical Society (2007, section 11), one finds a number theory section some of whose subsections (e.g. 11A Elementary number theory, 11D Diophantine equations, 11Y Computational number theory) are centred on exact computation. When performing exact computations, mathematicians consider certain subsets of real numbers (e.g. Ζ, Θ, Θ( 3 ), Ζ/2Ζ…) as separate entities. When performing approximate computations, educated people usually consider the field of real numbers as a whole. Nevertheless, although ancient Greek mathematicians already knew integers, rational numbers, and some irrational numbers like the square root of two, the construction of real numbers was only completed during the 19th century. This long period of maturation should suggest that mastering real numbers cannot be achieved quickly. Our main goal is to show that there are actually various stable levels of numerical acquisition. Mathematical problemsolving requires understanding the situation referred to, and then, adequate processing. Thus, for numerical problems, we can distinguish a level of access and a level of process. An individual facing a problem may: • • •

process the involved numerical objects correctly, when his own level allows him to both access and process the problem understand the statement but give an incorrect solution, when his own level allows him to access but not yet to process the problem not know how to proceed.

Modern society needs two kinds of number users. The first has to deal with numbers: “In real-world settings, citizens regularly face situations when shopping, travelling, cooking, dealing with their personal finances… in which the use of a quantitative… reasoning… would help clarify, formulate or solve a problem” (OECD, 2006, p.73). The second has to work with numbers. This is what a professional (marketing specialist, engineer, physician, artist…) does

A Numerical Landscape

7

when he has to cope with numerical theoretical frameworks, or when he collaborates with a mathematician. What level of achievement is desirable in number-learning to meet either numerical need? Nowadays, this level is mainly defined in terms of competence, or competencies, by educational systems whereas it was previously mostly related to contents. The emphasis put on mastering real-world problems, in order to found well-advised judgments, by many institutions like OECD’s PISA2 project or the European parliament, reinforces this tendency. Piaget (1967, p.65), referring to Kant’s epistemological analysis of mathematics and physics, states: “Knowledge, [and thus acquiring knowledge] entails the relationship between subject and object”. Does not focusing on competencies, generally understood as a potential of action, neglect this relationship and therefore diminish the learning process? Does it not lead to mainly considering the product of teaching, when it has worked, instead of the development of this relationship? Briefly, does it not emphasize the end and thus put the means in the background? We first try to specify what society expects from number users, whether occasional or professional. We then observe and question the way educational systems design curricula, scenarios for teaching, training programmes, and national assessments, in order to enable people to take charge of the societal needs. We focus on modelling and applying mathematics, which has become a predominant topic, presented by many educational systems as a major response to mathematics-teaching issues. How are these official instructions and recommendations taken into account by teachers? What really happens in the classroom will be the theme of the fourth section. We thereafter introduce our framework of number acquisition. We emphasize cognitive aspects related to the specificity of mathematical objects: “Unlike material objects, however, advanced mathematical constructs are totally inaccessible to our senses… Indeed, even when we draw a function, or write down a number… the sign on the paper is but one among many possible representations of some abstract entity, which by itself can be neither seen or touched… for the mathematician…. It is important merely to know the rules or laws by which they may be combined.” (Sfard, 1991, p. 3). The semiotic nature of mathematical objects allows immediate access to them, whether they are already known or to be constructed. object “… unlike external mediation of instruments, semiotic mediation may be internalised, i.e. it may be transformed into infra conscious processing and thus become transparent from a phenomenological viewpoint” (Duval, 1998, p. 175). We then question the notion of competence, and its relation to knowledge. We wonder what should be the extent of this notion, for it to allow designing teaching objectives and assessments for number learning. We consider the epistemological aspect of number learning, focussing on numbers as objects inserted into a coherent network with its ruptures and continuities (Brousseau, 1997, pp. 79-99). Following our questioning about coherence and competence in number acquisition, we define four fields of competencies that allow to clearly specify the level of competence of an individual intended to deal or work with mathematics.

2

Programme for International Student Assessment, which assesses the abilities of 15-year-old students from 41 countries (including 30 of the most developed) to apply learning to problems with a real-world context.

8

Robert Adjiage and François Pluvinage

NUMBERS IN SOCIETY Dealing with Numbers In order to define the social expectations for dealing with numbers, we can consult documents with rules or regulations that contain numbers. Understanding these rules is a social obligation, and legal pursuits may result from their lack of application, or from incorrect application. Rules requiring calculations are present in most fields of human activity, so that we shall consider a sample of official documents. Let us first assert that in all the documents that we have consulted, there are three levels in dealing with numbers. We shall describe these later in this paper. Therefore, the nature of the encountered phenomenon does not actually depend on the field of activity one chooses. At the moment we were writing this text, we read that World Water Day 2007 is celebrated each year on 22 March3 (by the way note the wrong use of the number 2007 in this sentence). As it appears anyway that water is something vital and possibly scarce, we suggest limiting our investigation to the domain of water: its consumption and its use. Texts of rules or regulations can refer to numbers in such a way that knowledge of decimal numbers and arithmetic operations, or numeracy, allows the reader to act as required. Such is the case in the following Mexican law, about consumers’ shares for water consumption. Each magnitude and arithmetic operation is completely described in a rhetorical way in this text (followed by its English translation).

Cámara de Diputados del H. Congreso de la Unión LEY DE CONTRIBUCIÓN DE MEJORAS POR OBRAS PÚBLICAS Y FEDERALES DE INFRAESTRUCTURA HIDRÁULICA Artículo 7 – II Nueva Ley D.O.F. 26/12/1990 Tratándose de acueductos o sistemas de suministro de agua en bloque realizados exclusivamente con inversión federal, el monto de la contribución obtenida en el artículo anterior se dividirá entre la capacidad de suministro del sistema, medida en metros cúbicos por segundo, y el cociente obtenido se multiplicará por el volumen asignado o concesionado por la Comisión Nacional del Agua a cada usuario del sistema, medido en metros cúbicos por segundo y el resultado será el monto de la contribución a cargo de cada contribuyente. Article 7-2 For aqueducts or water supply systems built with federal investment only, the amount of the contribution defined in the preceding article will be divided by the global capability of the system in cubic meters by second, and the obtained quotient will be multiplied by the volume in cubic meters per second attributed to each consumer, and the result will be the amount to be paid by each customer. 3

Extract from the UN-water webpage < http://www.unwater.org/wwd07/flashindex.html >: 'Coping with Water Scarcity' is the theme for World Water Day 2007, which is celebrated each year on 22 March.

A Numerical Landscape

9

The following general part of a US federal text introduces tables (we reproduce only one table, but there are several). Compared to the preceding document, this one supposes from its readers practice in consulting mathematical presentations (some other documents also present diagrams). But it deals similarly with numbers: Every arithmetic operation is well defined (see below). Moreover, the section of the document devoted to definitions includes magnitudes and units not necessarily known by everyone, e.g. the so-called degree-day. We conclude that the expected competence is only numeracy. Hereafter we shall see that in a more specialised part of the same document, the reader is supposed to deal with formulas.

CODE OF FEDERAL REGULATIONS [2006] 10cfr434-- Part 434_Energy Code For New Federal Commercial And MultiFamily High Rise Residential Buildings Degree-day, heating: a unit, based upon temperature difference and time, used in estimating heating energy consumption. For any one day, when the mean temperature is less than a reference temperature, typically 65ºF, there are as many degree-days as degrees Fahrenheit temperature difference between the mean temperature for the day and the reference temperature. Annual heating degree days (HDD) are the sum of the degree-days over a calendar year. § 434.518 Service water heating. 518.1 The service water loads for Prototype and Reference Buildings are defined in terms of Btu/h per person4 in Table 518.1.1, Service Hot Water Quantities. The service water heating loads from Table 518.1.1 are prescribed assumptions for multi-family high-rise residential buildings and default assumptions for all other buildings. The same service water-heating load assumptions shall be made in calculating Design Energy Consumption as were used in calculating the Energy Cost Budget.

Table 518.1.1.—Service Hot Water Quantities Building type Assembly Office Retail Warehouse School Hotel/Motel Restaurant Health Multi-family High Rise Residential (2)

Btu/person-hour (1) 215 175 135 225 215 1110 390 135 1700

(1) This value is the number to be multiplied by the percentage multipliers of the Building Profile Schedules in Table 513.2.b. See Table 513.2.a for occupancy levels. (2) Total hot water use per dwelling unit for each hour shall be 3,400 Btu/h times the multi-family high rise residential building SWH system multiplier from Table 513.2.b. 4

The British thermal unit (BTU or Btu) is a unit of energy used in North America (1 Btu ≈ 1.055 Joule). See http://en.wikipedia.org/wiki/British_thermal_unit .

10

Robert Adjiage and François Pluvinage

We do not reproduce Table 513.2 in which we could find the multipliers. Observe a possible difficulty for interpreting certain units of magnitudes: Slash refers to division as usual in arithmetic, but dash indicates product in degree-days and Btu/person-hour When reading official texts, we are often surprised by the importance of the use of common language, even in cases in which the use of formulas would simplify heavy verbal formulations. Some legal texts however introduce mathematical formulas, which shorten expression. See for example a French text taken from J.O. (Official Journal of French Republic, 12-29-2002, n 303, p. 60 059 text 5) and followed by its English translation. Avis et communications Ministère de l'écologie et du développement durable Avis relatif à des délibérations des agences de l'eau AGENCE DE L'EAU ARTOIS-PICARDIE Délibération n° 2002-A-063 du 4 octobre 2002, NOR: DEVE0210424V Article 9 Mesure indirecte des volumes prélevés 1. Calcul du prélèvement en fonction de l'énergie électrique consommée Le volume prélevé est obtenu par application de la formule suivante : P = 250 × W/Z avec: P : volume prélevé en mètres cubes durant la période soumise à redevance ; W : énergie électrique consommée mesurée au compteur, exprimée en kWh ; Z : hauteur théorique minimale d'élévation en mètres.

MINISTRY OF ECOLOGY AND DURABLE DEVELOPMENT (FRANCE), ANNOUNCEMENT OF DECISIONS TAKEN BY THE WATER AGENCY OF ARTOIS-PICARDIE (10/4/2002) Article 9 Indirect measurements of volumes of pumped water 1. Water volume computed with measurement of consumed electrical energy The volume P in cubic meters is obtained using the following formula: P = 250 × W/Z, where W is the measured electrical energy in kWh and Z the theoretical minimal height of elevation in meters.

We find a formula in this text, but we can assert that using this formula does not require algebraic processing: substituting numbers for letters and then perform arithmetic calculation is sufficient to find the result. Therefore, the reader is only expected to be at numeracy level. But a question arises: Under what conditions does dealing with a formula entail the use of an effective algebraic language? Nevertheless, many texts, generally including at least ratios and proportions, require a more advanced knowledge from their readers. The following text belongs to this category. Drinking Water Standards

Priority Rulemakings

Arsenic The Safe Drinking Water Act requires EPA to revise the existing 50 parts per billion (ppb) standard for arsenic in drinking water. EPA is implementing a 10 ppb standard for arsenic. Ground Water Rule EPA proposed a rule which specified the appropriate use of disinfection in ground water and addressed other components of ground water systems to assure public health protection. Lead and Copper EPA estimates that approximately 20 percent of human exposure to lead is attributable to lead in drinking water.

A Numerical Landscape

11

The quoted text is only part of a larger document from U.S. Environmental Protection Agency . The first pages of this document are devoted to a general presentation of public health protection. The considered text concerns everybody, so that it is supposed to be understood by any reader. Below, we present a second part of this document. It gives details of measures and procedures, so that we could think that it is only directed at specialists. But these specialists, in turn, are in charge of delivering the information to a large range of citizens who are supposed to understand the involved concepts. Electronic Code of Federal Regulations (e-CFR) e-CFR Data is current as of March 16, 2007 http://www.gpoaccess.gov/cfr/index.html (c)Lead and copper action levels. (1) The lead action level is exceeded if the concentration of lead in more than 10 percent of tap water samples collected during any monitoring period conducted in accordance with §141.86 is greater than 0.015 mg/L (i.e., if the “90th percentile” lead level is greater than 0.015 mg/L). (2) The copper action level is exceeded if the concentration of copper in more than 10 percent of tap water samples collected during any monitoring period conducted in accordance with §141.86 is greater than 1.3 mg/L (i.e., if the “90th percentile” copper level is greater than 1.3 mg/L). (3) The 90th percentile lead and copper levels shall be computed as follows: (i) The results of all lead or copper samples taken during a monitoring period shall be placed in ascending order from the sample with the lowest concentration to the sample with the highest concentration. Each sampling result shall be assigned a number, ascending by single integers beginning with the number 1 for the sample with the lowest contaminant level. The number assigned to the sample with the highest contaminant level shall be equal to the total number of samples taken. (ii) The number of samples taken during the monitoring period shall be multiplied by 0.9. (iii) The contaminant concentration in the numbered sample yielded by the calculation in paragraph (c)(3)(ii) is the 90th percentile contaminant level. (iv) For water systems serving fewer than 100 people that collect 5 samples per monitoring period, the 90th percentile is computed by taking the average of the highest and second highest concentrations. This text refers to ratios without mentioning how to process them. A precise idea of ratio seems to be necessary for understanding at least the last statement of each presented part (Lead and Copper in Priority Rulemaking, 3-iv in e-CFR). This would be a second level of numerical knowledge required. On the other hand, although the text refers to statistical notions such as percentiles, we will not go so far as stating that the intended readers are supposed to master statistics as the required processes are explicitly defined. In the Energy Code section mentioned above, the reader is expected to know how to process formulas, which implies familiarity with algebraic symbolic language (knowledge of

12

Robert Adjiage and François Pluvinage

symbol Σ and sub-indices, distributive rule). This brings us to the third level of mathematical social expectations, perceivable in the following quoted example (p. 466 of the Energy Code, loc. cit.). The overall thermal transmittance of the building envelope shall be calculated in accordance with Equation 402.1.2: Uo=ΣUiAi/Ao=(U1A1 +U2A2 + . . . +UnAn)/Ao (402.1.2) Where: Uo = the area-weighted average thermal transmittance of the gross area of the building envelope; i.e., the exterior wall assembly including fenestration and doors, the roof and ceiling assembly, and the floor assembly, Btu/(h·ft2·ºF) Ao = the gross area of the building envelope, ft2 Ui = the thermal transmittance of each individual path of the building envelope, i.e., the opaque portion or the fenestration, Btu/(h·ft2·ºF) Ui = 1/Ri (where Ri is the total resistance to heat flow of an individual path through the building envelope) Ai = the area of each individual element of the building envelope, ft2

Working with Numbers Professional mathematicians obviously work with numbers, and over the last three decades employment of mathematicians has been increasing in many sectors of human activity: astronomy, meteorology, aeronautic traffic, bank, marketing, health, quality control, industrial design, musical acoustics, and so on. If we add traditional mathematics users like physicists, and mathematics teachers, the result is a large number of people who dedicate much of their time to working with numbers. The major aims are to predict, to control, to optimise, and to decide (e.g. how to reduce cost or waste in industrial processing). Modelling (e.g. traffic regulation by programming traffic lights, designing a ship hull) is a usual tool at the present time. To be sure, computation and computers have largely contributed to such a development, particularly in designing models and checking their conformity with the reality they simulate. A classical use of number knowledge has been to discover relationships between realworld quantities, and thus find formulas. Renowned examples of results published during the 20th century are Pareto’s principle in economy (20% of the population earns 80% of the income) and Zipf’s law in linguistics. This law is a consequence of an economic principle (principle of least effort): A writer, or a speaker, has a more or less easy access to words, depending on the frequency in which he uses them (the more frequent, the easier). Originally, Zipf's law stated that, in a corpus of natural language utterances, the frequency of any word is roughly inversely proportional to its rank in the frequency table. (Wikipedia, 2007a). Let us for instance consider the following statistics related to a collection of 423 short TIME magazine articles (total number of term occurrences: N = 245,412).

A Numerical Landscape

13

Table 1. Top 15 terms of short TIME magazine articles Rank r

Word

Frequency f

r*f

r*f/N

1

the

15861

15861

0.065

2

of

7239

14478

0.059

3

to

6331

18993

0.077

4

a

5878

23512

0.096

5

and

5614

28070

0.114

6

in

5294

31764

0.129

7

that

2507

17549

0.072

8

for

2228

17824

0.073

9

was

2149

19341

0.079

10

with

1839

18390

0.075

11

his

1815

19965

0.081

12

is

1810

21720

0.089

13

he

1700

22100

0.090

14

as

1581

22134

0.090

15

on

1551

23265

0.095

The product A = r*f/N in the last column tends to be equal to about 0.1. (http://linkage.rockefeller.edu/wli/zipf/cmpsci546_spring2002_notes.pdf, p. 3)

More recent kinds of work with numbers are: digitalisation, cryptography, and compression (zip). A typical example of digitalisation is that a coloured pixel on a screen is represented by a triple of numbers of the interval [0, 255]. In cryptography, a widespread method is NSA, related to public and private keys. It is based on the quasi-impossibility of finding the factors of a given product of two large prime numbers. Image compression produces JPEG files. The latter require the use of wavelets, a kind of mathematical function that allows decomposing a given function into different frequency components. All these domains are being developed on the base of constant contact between specialists and users. This supposes a sufficient general level of education. Thus, collaborative working becomes rather difficult between people whose competence differ by more than one level, for instance between somebody who is working at the level of functional analysis and somebody who does not yet master algebraic writing.

CURRICULA AND ASSESSMENT; THE CASE OF MODELLING After analysing the societal needs related to number learning, we investigate in this section curricular genesis and development, i.e. the process by which curricula are conceived and designed by institutions, as well as instructions for implementation. How does this

14

Robert Adjiage and François Pluvinage

process take account of both societal needs of dealing with and working with numbers? We will mainly consider an issue that has been in the foreground these last few years: mathematical modelling. Following the main line of this paper, we will focus on the numerical aspects of this topic.

PISA, Modelling, and Curricula We choose modelling because the way this domain of mathematics has emerged in just a few years in many curricula for teaching, teacher training, and the related research, as well as the emphasis put on it, seems to be characteristic of a modern process of reforming educational systems in the occidental world. What is intended by modelling? Let us refer to the Discussion Document (Blum, 2004, pp.152-153), prior to ICMI Study 2004 (International Commission on Mathematical Instruction), aimed at raising some important issues related to the theory and practice of teaching and learning mathematical modelling and applications. The authors of this Discussion Document (14 people from 12 countries), are the members of the International Programme Committee for this ICMI Study. “The starting point is normally a certain situation in the real world. Simplifying it, structuring it and making it more precise… leads to the formulation of a real model of the situation…. If appropriate, real data are collected in order to provide more information…This real model, still a part of the real world… is mathematised…” Mathematical results are then derived, using mathematical methods… and then “re-translated into the real world, that is, interpreted in relation to the original situation”. Checking whether the results are appropriate and reasonable now validates the model. “If need be… the whole process has to be repeated with a modified or a totally different model”. “…The obtained solution of the original real world problem is stated and communicated”.

Modelling is an important theme, taken into account these last few decades by numerous studies and international conferences in the field of mathematics education (see for instance: Niss, 1987; Blum et al.1989; Galbraith et al., 1990; Lesh et al., 2002). Nevertheless, “…genuine modelling activities are still rather rare in mathematics lessons” (Blum, 2004, p. 150). Now PISA has defined and tested mathematical literacy: “The emphasis in PISA is ‘on mathematical knowledge put into functional use in a multitude of different situations and contexts’. Therefore, real situations as well as interpreting, reflecting and validating mathematical results in “reality” are essential processes when solving literacy-oriented problems.” (Blum, 2004, p. p. 151-152). Major countries of the occidental world, including the USA, appeared to perform less well than expected at PISA 2000 and 2003. (For detailed results, see NCES, 2001, p. 43; OECD, 2004, p. 53 and pp. 89-95). As a consequence “an intense discussion has started, in several countries, about aims and design of mathematics instruction in schools, and especially about the role of mathematical modelling, applications of mathematics and relations to the real world.” (Blum, 2004, p. 151). One finds for instance, on the web site of the U.S. Department of Education (2004), a press release entitled: PISA results show need for high school reform, in which Secretary of

A Numerical Landscape

15

Education Rod Paige said: “The PISA results are a blinking warning light”. In releasing the U.S. findings, Robert Lerner, commissioner of NCES, said, “PISA provides important information about education in the United States and in other industrialised nations, giving us an external perspective on U.S. performance. We need PISA in particular because it offers such a different measure of achievement, one that poses complex problems that students might realistically encounter in their lives.” (Ibid, 2004). National Center for Education Statistics (2002-2006) provides a huge amount of documents addressing PISA: access to PISA items and results, international comparisons, official and scientific reports, aids for educators…. The European Parliament on its part defined (2006, p.1) Key Competencies for lifelong learning. Concerning mathematics, it states that: “An individual should have the skills to apply basic mathematical principles and processes in everyday contexts at home and work, and to follow and assess chains of arguments” (2006, p.18 pdf version), putting the emphasis on applying mathematics in ‘everyday contexts’. What kind of echo to these statements and recommendations can we find in official learning curricula? We will give two examples drawn from a wealth of international material. In 1996, the New York State Board of Regents adopted learning standards for all content (subject) areas. Since then, the New York State Education Department (NYSED) has issued a series of core curricula, which provide an additional level of specificity to these learning standards. The Core Curriculum Standard 3, from Pre-kindergarten to grade 12 (New York State Education Department, 2005), appears as carefully thought-out and precise. It defines a mathematical proficiency relying on conceptual understanding, procedural fluency, and problem solving. In introduction to this Core Curriculum (ibid, p. 1), it is stated: “Most problems that students will encounter in the real world are multi-step or process problems. Solution of these problems involves the integration of conceptual understanding and procedural knowledge. …Many textbook problems are not typical of those that students will meet in real life. Therefore, students need to be able to have a general understanding of how to analyse a problem and how to choose the most useful strategy for solving the problem.” We first note this formulation that considers the complexity (‘multi-step’) of problems that students meet in the real world as a model for learning. The word ‘model’ and derived words like ‘modelling’ appear 88 times in the full document, throughout all grades, to which one can add 21 occurrences of expressions as: ‘everyday’ situations or ‘experiences or ‘real world’ problems or situations. In France, the Minister of National Education issued a document that defines a common base of knowledge and competencies5 to be acquired at the end of mandatory schooling. This document explicitly refers (MEN, 2006, p. 2) to two sources: the already quoted Recommendation of the European Parliament (2006, p.1) related to Key Competencies for lifelong learning, and “international assessments, particularly PISA. It specifies (ibid, pp. 5-6) that: “mathematics provide tools for acting, choosing, and deciding in everyday life”, and that “mastering the main elements of mathematics essentially depends on problem solving, particularly when the involved problems rely on situations stemming from the real world.” New learning curricula have therefore been published (or are in the process of being published). These mention as one of the main aims the capacity to use mathematics in different domains (everyday life, other disciplines). 5 Socle commun de connaissances et de compétences

16

Robert Adjiage and François Pluvinage

Thus, across many countries in the world: “Researchers and practitioners in maths education and policymakers have reached agreement that mathematics education should enable students to apply mathematics in their everyday life (PISA) and contribute to the development of active citizens (Council of the European Union following Lisbon Report, 2001, p.4, p. 13).” (LEMA, 2006, project-background page). Of course, this requires new interest and competencies in teachers who generally do not successfully integrate applications of mathematics into their course design and daily classroom practice. Appropriate pre- and inservice teacher training has therefore been, or is being, conceived. For instance, a European project, COMENIUS-LEMA (Learning and Education in and through Modelling and Applications) is being developed. Partners of the project are institutions from different countries: France, Germany, Hungary, Poland, Spain, and United-Kingdom. Mathematics education researchers from these countries, including one the authors of the present paper, participate or collaborate in this project, which “proposes to support teachers with development of their pedagogic practice in mathematical modelling and applications by developing a teacher training course”. Target groups are: “in-service and pre-service teachers at primary and lower secondary level and teacher trainers”. (LEMA, 2006, project page). We are now in measure to specify the leading strand, in the mathematical modelling case, of the current evolution in educational systems. An international assessment, PISA, has provided policy makers with benchmarks that allow to reconsider their systems' performances, and to identify potential strategies to improve, according to PISA’s Standards, student achievement. Their recommendations have determined important inflections of curricula intended to help students to better handle the real world. Researchers, already interested in this topic, have designed or are designing training programs for aiding teachers to take account of curricula and apply the related educational instructions in their lessons. Now, how are PISA-assessed competencies determined? PISA has been influenced by the Danish KOM Project, initiated by the Ministry of Education in order to profoundly reform Danish mathematics education from school to university, and its director, Morgen Niss: “It should be noted that the thinking behind and before the Danish KOM-project has influenced the mathematics domain of OECD’s PISA project, partly because the author is a member of the mathematics expert group for the project”. (Niss, 2003, p. 12). According to Winsløw (2005, p. 141-142) the empirical and theoretical bases of Niss’ model are of two orders: Scandinavian tradition and Project Pedagogy. Concerning the former, mathematics education has been considered since the seventies, in a society more and more influenced by mathematical models, as a mean of making sense of the world in which students live and will work, and, as a consequence, developing active citizenship. In particular, this tradition refuses to consider mathematics as politically neutral; it challenges a didactic that contents itself with passing on scientific knowledge, and thus is mostly oriented towards competence acquisition. Concerning the latter, students acquire knowledge when they feel a need for it, that is, according to real-world demands, and this explains the great interest of the KOM project in modelling. May one therefore consider that PISA results are absolute indicators (as often presented in press releases), although they in fact reflect only a particular measure of student performance? Moreover, must curricula be determined by assessments, risking the behaviourist shift feared by Brousseau (2007) and Chevallard (2002)? Reforming educational

A Numerical Landscape

17

systems on only the PISA basis could only improve students’ performance at… PISA. Is this our unique goal? What sort of achievement do we expect for students and for what purpose?

Mathematical Coherence, Modelling, and Curricula We suggest another thought process. First of all, we must specify competencies for lifelong number learning, relying on a framework that takes account of personal development, and mathematical and didactical coherence. By personal development, we mean development that leads to either work or deal with numbers. By mathematical and didactical coherence we mean that our references for interpreting and structuring society and individual needs in number learning are mathematical objects, concepts, notions, domains (whole numbers, rational numbers, algebra, calculus, analysis) and the related studies in mathematics education. Assessments would be of course based on these competencies. As we can note, this process is opposite to the process described above: It stems from individuals and their needs in number learning and leads to assessments via curricula and competencies. May modelling remain a central reference in such a process of defining objectives for lifelong number learning? For debating this important question, we first go back to the ICMI Discussion Document (Blum, 2004), which is a very complete overall study on the subject. This document brings a real framework not only for appreciating and analysing the considered complexity and scope of modelling, but also for exhibiting any subject relevant to this theme. We especially focus on the chapter devoted to “Examples of important issues.” We particularly note that: •

• •

The interrelations between applications and modelling, and mathematics, from both a purely mathematical and a didactical viewpoint, are strongly taken into account. This goes in the direction of fostering mathematical coherence, which is one of the main lines we have indicated for lifelong learning. Shaping or restoring the image of mathematics is an important aim of applicationand-modelling promoters. Authenticity of the involved material is one the main considered issues.

And this leads us to examine the type of material, i.e. real-world situations, available and frequently used when implementing teaching sequences related to modelling. Let us refer to two authors that have conducted and reported many classroom-based studies on this subject: A. Peter-Koop, and K. Maass who participates in LEMA. In one of her papers (2004, p. 457), the former mentions that she has resorted to Fermi problems for the reported experience: “Enrico Fermi, who in 1938 won the Nobel Prize for physics, was known by his students for posing open problems that could only be solved by giving a reasonable estimate. Fermi problems such as ‘How many piano tuners are there in Chicago?’ share the characteristic that the initial response of the problem solver is that the problem could not possibly be solved without recourse to further reference material”.

18

Robert Adjiage and François Pluvinage

She gives a bit farther the following criteria as guidelines for the choice of problems posed in the research on work with 3rd and 4th-graders: • •

• •

The problems should present challenges and intrinsically motivate cooperation and interaction with peers. The wording of the problems should not contain numbers in order to avoid that the children immediately start calculating without first analysing the context of the given situation, and to challenge pupils to engage in estimation and rough calculation and/or the collection of relevant data. The problems should be based on a selection of real-world-related situations that include reference contexts for third and fourth graders. The problems should be open-beginning as well as open-ended real-world-related tasks that require decision-making with respect to the modelling process.

Four problems have therefore been posed to 3th- and 4th- graders: 1) How much paper does your school use in one month? (paper problem). 2) How many children are as heavy as a polar bear? (polar bear problem). 3) Your class is planning a trip to visit the Cologne Cathedral. Is it better to travel by bus or by train? (cathedral problem). 4) There is a 3 km tailback on the A1 motorway between Muenster and Bremen. How many vehicles are caught in this traffic jam? (traffic problem).

K. Maass (2005, p. 4), gives a list of problems she submitted to 7th- and 8th- graders. Among them, we retain the numerical items: 1) How many people can be found in a 25km-long traffic jam? 2) How can different charges of diverse mobile contracts be clearly arranged depending on customers’ habits? [….?…] 3) Is it possible to heat the water required in Stuttgart-Waldhausen with solar collectors on the roofs? 4) What is the connection between the height of fall and the subsequent height of rebound of a ball?

We lastly report a problem that has been experimented for LEMA: Giant’s foot. The following photo was taken in an English amusement park. How approximately tall is the entire figure, of which we can only see the foot? All these situations are much more than a “dressing up” of a “mere” mathematical problem, and the modelling process consists of much more than “undressing” the real-world problem. In this sense they bring great progress compared to numerous problems one finds in usual textbooks. The declared guidelines are respected. The quasi-absence of numbers in the statement is a main feature of these problems, and we have personally observed that this results in more considering relationships between the involved quantities rather than starting haphazard calculation. For instance, we have observed many 5th-graders, dealing with the “giant” problem, that have tried to find the number of men (or visible barriers, or boot-

A Numerical Landscape

19

soles…) that one could put into the whole estimated length of the giant. This attitude brings them close to the proportionality model, whereas the presence of numbers frequently leads students to combining numbers, often by adding them, without reflecting the relationships linking the underlying quantities.

6

Figure 1. Giant’s foot .

One observes that many situations, almost all of them actually, rely on two pillars: data collection, and then applying a multiplicative model: proportionality, calculation of averages… So that pupils are supposed to resort to an already studied (and somehow standard) model. Very few existing situations lead to developing a mathematical model. K. Maass’s fourth problem quoted above is of that kind, although it mobilises linear functions. We have suggested, but not already experienced in classrooms, situations as car braking distance, which depends on the square of the speed. This could be a new issue to experiment. Great care should also be brought to situations that give real possibilities of validating the retained data and the findings. For instance, the traffic-jam problem (K. Maass, number one) depends on many choices made by students: the ratio cars/trucks, the average length of each, the average number of persons in a car… This problem is thus open-ended, the results depending on these choices, themselves having to be coherent with the retained time of the day or period of the year. Whatever ones choice, it seems difficult to directly validate the employed ratio cars/trucks or average length of cars and trucks, this information being not easily available, e.g. on Internet. On the other hand, it could be possible to check, e.g. on specialised radio channels, the number of persons involved in a given traffic jam, and this may validate the retained data. Can modelling effectively contribute towards promoting views of mathematics that extend beyond transmissive techniques to its role as a tool for structuring other areas of knowledge? (Blum, 2004, p. 161). Modelling is also intended to promote interaction, cooperation, and communication: “The real-world problems used in the study should intrinsically present challenges and thus motivate peer interaction during the solution process as opposed to problems that can be solved quite easily by an individual student.” 6

The authors are grateful to Richard Philipps from: http://www.problempictures.co.uk/, who has allowed them to use this picture.

20

Robert Adjiage and François Pluvinage

(Peter-Koop, 2002, p.563). Succeeding in promoting these attitudes in the classroom does not only depend on the available didactical material. Teachers are often rushed by their schedule, whereas modelling activities demand giving students the time for acquiring the knowledge required by the real world situation, for collecting data, for testing hypotheses, for elaborating or identifying an appropriate mathematical model, for discussing all this matter between peers. So that teaching is also of great importance. We have observed many sequences, e.g. related to the giant situation, where productive peer interactions have been noted, leading to unexpected but correct procedures for calculating the giant’s height. But for lack of time, lack of training, the teacher did not take these ideas into account and favoured the procedures that he imagined. So that part of the expected benefit, in terms of the image of mathematics and interest of working groups, may be lost or invisible. In any case, the challenge of modelling has not been sufficient to change this teacher’s usual practise.For concluding this section, we would say that modelling still challenges research, training, and teaching. Important progress has already been made concerning the epistemology and the material related to modelling. Other issues, which we have mentioned above, remain to be explored. Modelling is coherent with the personal development of an individual and what society expects from him, whether he is intended to work or to deal with mathematics. However, let us beware of excessive enthusiasm. Let us consider other teaching entries, which take account of purely numerical obstacles and coherence, particularly those that help students to become aware of continuities and ruptures from numeracy to mastering calculus and analysis, going through ratios and algebra, because these themes structure all numerical development. Anyway, being capable of advanced modelling, particularly developing numerical models (and not only resorting to taught models), relies on mastering these fundamental areas and their interrelations.

STUDYING NUMBERS: HOW LONG AND UP TO WHICH DEGREE OF ACCURACY? In the previous section, we examined the way curricula are or might be designed. Here, we are interested in the general features and progression of curricula related to number acquisition, and the way they are actually implemented in the classroom. We take advantage of this presentation for sketching some continuities and ruptures through general numerical areas: whole numbers, ratio and rational numbers, algebra, calculus and analysis, which we consider essential for describing and understanding this learning. Numbers together with their operations constitute the first objects in learning mathematics, and pupils continually face the learning of numbers. At elementary school, teaching numbers is present each year, and has the most important place among the mathematical topics. After presentation of natural numbers and arithmetic operations, students learn about positive decimal numbers and fractions. Later, they study negative numbers, but from this stage, learning numbers does not follow a continuous path in curriculum development. That means that some properties, concerning for instance exponents, are examined without introduction of new concepts. And the theory of numbers, including theorems such as Fermat’s theorem (if p is a prime number, then for any integer a, the number a p – a is divisible by p), is actually taught only in advanced courses. Nowadays numeral systems, like binary system or hexadecimal, are only considered as a topic for

A Numerical Landscape

21

prospective mathematicians or specialists in computing. At middle school the students are led to deal with some irrational numbers, canonical examples of them being 2 and π. But only a scarce minority of college students in scientific careers masters real numbers. Nevertheless, in many countries, core curricula in mathematics education formulate their objectives in such a way that a reader may think that at the end of grade 7 or 10, (it depends on the educational system) students are supposed to master real numbers. Moreover, complementary official documents, such as those of Federal Resources for Educational Excellence in the USA, strengthen this misleading impression. For example, many textbooks, or online documents, present a chapter or a list of exercises devoted to real numbers. If we carefully examine the content, it often appears that the presented properties (operations and order) do not characterise real numbers, but also apply to rational numbers. The most difficult one, and also the most important, the closure property (if a nonempty set of real numbers has an upper bound, then it has a least upper bound), is dismissed. Summarizing our experience as well as observations made by various researchers, we can assert that general tendencies while teaching mathematics are the following. • •

• • • • • •

At primary school, strong priority is given to teaching numbers, for instance in comparison with geometry or other topics. For this reason, magnitudes do not constitute an effective component of learning whole numbers, although researchers, such as Galperin quoted by Arievitch7 (2003, p. 280), have stated opposite educational principles. Teachers carefully introduce the mathematical tools (scripts and algorithms) Nevertheless, they do not systematically train students in mental calculation, nor present them arithmetical techniques such as casting out nines They do not consider computational tools as a positive resource of numerical problems They do not consider certain links between the various presented ways of expressing numbers, e.g. from repeating decimals to fractions At the secondary level, algebra progressively takes the place of numbers, but does not explicitly help to better understand numbers Teachers give little consideration to numerical proofs.

We are not surprised, in these conditions, that educational goals are difficult to achieve for many students. National and international assessments allow us to better understand what is currently expected from learning, and to observe the distance between the expectations (intended curriculum) and the obtained results (implemented curriculum). Learning numbers seems to be obvious for people who master them, for the apparent easiness that operative rules present. But if we want to build a mathematical world, rather than an arbitrary one, each general rule has to be justified and demonstrated, which leads us to consider both the universe of concrete applications and the management of representations. Let us report a case that we observed in a kindergarten classroom: An experienced teacher 7

The underlying assumption in Galperin’s approach to learning and activity is that there are two different types of actions: (1) an ideal action that is performed in the presence of objects and (2) an ideal action that is carried out separately, without any presence of objects.

22

Robert Adjiage and François Pluvinage

organized a play, called “our boat is sinking”, with a group of 12 pupils. In a song mentioning a boat, suddenly come the words “our boat is sinking, for evacuation you have to form groups represents a number that the teacher chooses. When the chosen number was of ”, where 3 or 4, the task was easy for the pupils, but later, a choice was 8. Then the natural question arises: How many pupils are missing for forming two groups of eight? The teacher recommended using an abacus (representation) without allowing the pupils to consider the concrete situation that they could directly build: e.g. a row of 8 facing the remaining pupils, then constituting as many pairs as possible and counting how many pupils in the row of 8 remained alone.

Figure 2.

All students would easily understand this situation, whereas the link between the problem and its numerical representation and processing using the abacus (2×8 – 12) was not obvious for such young children. In this elementary situation we observe a possible link between learning numbers and exploiting a concrete situation. The observed lesson was very attractive for the pupils, so that we can only point out here some “loss of learning”, like one says “loss of earnings” in a commercial relationship. But we have also observed most important sources of difficulty in other situations. For instance, in a course about speed and velocity observed at grade 6, activities were proposed in the textbook, such as comparing the constant velocity of two swimmers. One of them takes 1 min 30 sec for 100 m, and a second swims 250 m in 4 min. The textbook also proposes comparing different students’ approaches. But instead of this, the teacher’s conception, i.e. applying the general formula v= d and comparing ratios, prevailed:

t

a female student built a table for the first swimmer’s times, under the hypothesis of constant velocity. Her correct result, 3 min 45 sec for 250m, was considered wrong and erased by the teacher. The latter reminded that the problem statement gave for the second swimmer 250 m in 4 min and that the question was to compare swimmers’ velocity by comparing ratios.

Figure 3. “A mistake”.

It is however well known that most 6th-graders need for this comparison such intermediary steps before being able to understand and apply the formula. These steps mark stages in the acquisition of processing involving ratios and proportions, and thus in acquiring competence related to these notions.

A Numerical Landscape

23

What characterises the acquisition of competence is the length of learning and the need of a wide variety of activities. We have presented two examples, one referring to the acquisition of counting and calculating, the second to ratio and proportion. PISA2 assessment reveals that many 15-year-old people do not properly manage ratio problems, even if they have been taught for several years. At a little more advanced level, the teaching of algebra starts. But secondary school pupils encounter serious difficulties with algebraic processing. As an example, we give hereafter six typical multiple- choice questions extracted from a pre-test, and the table of answers given by a sample of Mexican students at the beginning of undergraduate engineering programs. Pocket calculators are not allowed. Q2. Calculate: 3 − 5

4 6

−1 ; -1; 12

38 ; Other 24 1 1 Q3. Calculate: − 2n − 2 2n 1 1 2n − 1 ; − ; ; Other 0; 2 n ( n − 1) 2n ( n − 1) 2 ; 24

Q4. Which of the following numbers is largest? 3.5; 2.46; 3.19; 0.546 -9.87 Q5. Calculate: 3 ( 5-8 ) + 3 ⎡⎣ 2 ( 4+9 ) -5 ( 2-6 ) ⎤⎦ 2

9; 30; -55; 229; Other Q8. The solution of −30 x + 4 ≤ 0 is:

2 2 ; ( −∞ , ∞ ) ; x ≤ ; Other 15 15 2 Q9. The solutions of 2 x − 3 x − 2 = 0 are: 2 2 −1 x1 = −2 and x2 = ; x1 = 2 and x2 = − ; x1 = 2 and x 2 = ; 3 3 2 2 x1 = −2 and x2 = − ; Other 3 x=

2 ; 15

x≥

Table 2. Empirical evidence - percentages of choice made by undergraduate students Choice Q2 % Q3 % Q4 % Q5 % Q8 % Q9 %

1 25 9 88 4 21 17

2 65 25 0 3 39 26

3 1 30 2 9 3 43

4 4 27 4 2 36 5

5 5 9 6 83 1 9

24

Robert Adjiage and François Pluvinage

In the table above, the percentage of right answers lies in bold in gray cells. There is a strong contrast between numerical questions (Q2, Q4 and Q5) and questions including variables (Q3, Q8 and Q9, marked by the symbol). For Q3, Q8 and Q9 only a minority chose the correct answer. We particularly observe in Q8 that students were able to recognize the fraction

2 as representing 4 (choices 1, 2 and 4), but then had difficulty in 15 30

algebraically managing the inequality: The three corresponding choices are frequent. This underlines the difference between processing ratios and fractions and processing algebraic expressions. For this reason, it seems necessary to distinguish a level of mathematical competence that consists in managing algebra. If we consider the curricula up to grade 12, we can find other topics related to numbers in courses of Algebra. For instance, in the 2005 core curriculum of the State of New York, complex numbers are introduced in Algebra 2 and Trigonometry (grade 12). But in this case, the label “Algebra” may be questioned. Historically, complex numbers in the form of square roots of negative numbers were considered for solving polynomial equations: third grade by Niccolo Tartaglia and Gerolamo Cardano, and fourth grade by Lodovico Ferrari (all these results obtained during the 16th century). But later attempts at formalising encountered paradoxes,

such

as

the

extension

(−1)(−1) = 1 = 1 on one hand, and to

of

the

−1 −1 =

property

(

−1

)

2

ab = a b

leading

to

= −1 on the other hand, hence 1

= -1! For this reason, in Euler’s time, mathematicians introduced the symbol i (for imaginary) instead of the controversial −1 . Both numbers, i and its opposite –i, have -1 as square (then are acceptable square roots of -1), so that the paradox could be explained. Nevertheless, the full extent of complex numbers arose only two centuries later, with the proof of the fundamental theorem of algebra, also known as the d’Alembert/Gauss theorem. This theorem states that every non-constant polynomial with complex coefficients has at least one root. In other words, the field of complex numbers is said to be algebraically closed. As a consequence, a polynomial of degree n has n roots, counting multiplicities. The first book in which this result was stated (for polynomials with real coefficients) was published in 1629. Its author Albert Girard gave as an example that the equation x 4 = 4 x − 3 has the four roots: 1, 1, −1 + i 2, − 1 − i 2 (in modern writing). But the first attempt at proving the theorem was only made by d’Alembert in 1746. The proof given by d’Alembert was incomplete. Only in 1806, J. R. Argand (1768-1822), a non-professional mathematician who geometrically interpreted i as a rotation of 90º in the plane, published a nearly satisfactory proof (Argand, 1874, pp. 90-91), and Gauss later produced another proof. Although the theorem is about algebra, all proofs used analysis, e.g. Liouville’s theorem: “In complex analysis, every bounded entire function is constant”. Indeed, if a polynomial P(z) does not have any root, it 1 is entire and bounded, therefore P(z) reduces to a constant. follows that the function P( z ) From these epistemological considerations, it appears that an important jump was necessary: from discovering algorithms for solving some equations, to first formulating, and then proving general results about polynomials. Therefore, we cannot consider that manipulating algebraic expressions should constitute the highest level of mathematical competence that teaching up to grade 12 has to implement. Stating general results about

A Numerical Landscape

25

functions, and solving problems, the solutions of which being functions, seem to be elements of a more advanced field of competence than simply solving numerical equations, or even parametrical equations.

COGNITIVE ASPECTS After examining institutional expectations as well as practices concerning the use and the acquisition of numbers, we will now introduce cognitive aspects of the general framework to which we refer. As already mentioned in the introduction, we focus on the semiotic features of mathematical objects, we regard numerical processes as if they were linguistic ones, and this leads us to propose a characterisation of algebra. We lastly consider the link between numbers and graphical representations.

Semiotic Registers Writing or representing numbers mobilizes various semiotic registers. Let us specify this assertion: “We do not have any perceptive or instrumental access to mathematical objects… as for any other object or phenomenon of the external world…”, “…the only way of gaining access to them is using signs, words, or symbols…” Duval (2000, p. 61). As seen above, realworld experiences are certainly a starting point for understanding the pertinence of numbers and then entering their universe. Duval’s assertion does not mean that we can do without referring to the real world for teaching numbers, and further large areas of mathematics. It only means that, at a certain point of learning, pure mathematical objects must be considered, if only because they can be dissociated from a particular context: for example fractions have to be detached from a measurement context, in which they are usually introduced, to apply to many contexts, e.g. mixture, enlargement… This is what an educated adult population can do, what 7th-graders, taken as a whole, are not yet able to do. This level of comprehension is certainly a condition for mathematical modelling. What Duval states is that symbols are the only way of accessing these pure mathematical entities. Now, there exist, and students are actually taught to face, many means of expression in mathematics. For instance, when expressing and processing rational numbers, we have at our disposal: fractions, of course, decimal numbers, two separate whole numbers (like in the expression “a player makes an average of 3 out of 4 basket attempts”), visual representations such as “pie charts” and number lines which are supposed to help students to better understand rational numbers… Duval (1995, pp. 15-85; 2000, p. 60-65) considers these diverse means of expression as separate and organised systems or “settings”, which he terms “semiotic registers”, necessary for mathematical activity and particularly teaching and learning, for at least four reasons (Duval, 1995, pp. 68-69; 2000, p. 62). • •

They are the only paths to mathematical objects, in the sense described above. They help students to distinguish a mathematical object from its representation: disposing of many representations allows considering what is invariable beyond these representations, thus outlining the underlying object.

26

Robert Adjiage and François Pluvinage •

They highlight all aspects of a mathematical object: each representation is partial, and we need many representations for accessing all the complexity of a mathematical object. For instance, a fractional representation such as 37

5

•

highlights the

multiplicative relationship between 37 and 5, whereas the decimal representation 7.4 of the same number highlights its location (between 7 and 8, closer to 7 than 8…). They give alternatives when processing: a fraction can be useful for interpreting a ratio, but decimal numbers may make the comparison of ratios easier.

Duval (1995, pp. 39-44; 2000, p. 63) distinguishes two kinds of processing. “Treatment”: 3 6 = , and “conversion”: transformation of an object within a given register, e.g. 5 10 transformation of the representation of a mathematical object into a representation of the 3 same object in another register, e.g. = 0.6 . He states that the latter, unlike the former, 5 entails a rupture in the means of representing and processing, and thus in thinking. Conversion between semiotic registers is a cognitive operation essential for objectifying mathematical entities. Treatment and conversion are necessary for expressing mathematical rules and properties that legitimate processing. Constraints are determinant in the distinction between simple illustration and a semiotic register. For instance, it is usual to represent an addition like 5 + 4 on the number line with an arrow joining 5 to the point located 4 units ahead. But we note but we note in upper Figure 4 that the number 4 does not explicitly appear (we have to count 4 units). Thus, we consider this representation as a relevant illustration for 5 + 4, but not for a treatment in a semiotic register. It is truly different when the same operation is made with an “additive slide rule”: this kind of continuous abacus presents two rulers side by side, the upper one sliding at the user’s demand, and a cursor. Nowadays, students may use a virtual tool: the model presented in Figure 4 was built with CABRI. In this case, the upper ruler has its origin at point S facing 5, and the point A of the cursor faces 4. The cursor shows the result 9 on the lower ruler. Note that the same configuration may also be read: 9 – 4 = 5. Thus, the “additive slide rule” allows converting a given configuration into two distinct arithmetic equalities: 5 + 4 = 9 or 9 - 4 = 5. Moreover we observe that, if S remains constant, moving the cursor allows the expression of any sum of the kind 5 + a. In other words, the configuration: S constant, cursor variable, is the translation of the sentence “Add 5” expressed in the natural language or verbal register. We can also process a sum like 5 + 4 + 3 without determining an intermediary result. Starting from the initial position (the two 0’s facing one another), we first slide the upper ruler 5 units forward and we put the cursor on 4, then we slide the upper ruler a second time in order to put its origin under the cursor, and finally we move the cursor to 3. The result appears on the lower ruler. All these considerations show that the additive slide rule constitutes a semiotic register with its own treatments, distinct from both verbal and arithmetic registers. Software like Excel or the worksheet in Open Office give an apparent opportunity to see the same number expressed in various settings. For a given cell of a worksheet, you may select various number formats: decimal number, fraction, percentage, etc.

A Numerical Landscape

27

Figure 4. representing 5 + 4 = 9 in two different ways.

In the fraction case, you must choose the denominator range (e.g. less than 10, 100…), making the involved representations and processing not exact, as is possible with software like Derive or Maple. If we calculate the sum: 11 + 23 with Derive or Maple in exact mode,

30 70 73 we obtain the correct result: . But suppose we enter each fraction in a cell of a worksheet, 105 e.g. cells A1 and A2, and then calculate the sum in cell A3. We choose, in the considered cells, fractions with denominator less than 100 (which is the largest option in Open Office) as number formats. We obtain 16/23 with Excel and 57/82 with Open Office. Such experiments or only their results could be interesting for 7-graders equipped with pocket calculators. Are the fractions 16/23 and 57/82 distant from or very close to one another? And in relation to the exact result 73/105…? Let us underline here that the distinctions we have introduced are important even when using software. Some software (with worksheets, as in our examples) has only one kind of “treatment” in Duval’s sense. Standard worksheets always operate with Binary Coded Decimal: even when

11 appears in a cell, the number taken into account by the computer is 30

0.32857142857…, and the software only converts when it displays the final presentation. Other software (as Derive or Maple) processes in distinct ways depending on the form of the given objects. We may say about the second that it uses distinct semiotic registers.

28

Robert Adjiage and François Pluvinage

Components of Numerical Acquisitions Our general theory of number acquisition and numerical structures is based on a relationship between three components, which are to be studied separately and then linked with one another: physical experience, mathematical properties, and semiotic representations. We assert that a lack in presentation of one of the three components, or of their links, induces difficulties of understanding for many students and leads to unstable learning. In “The four competences” section below, we present typical physical situations, which may provide students with the needed experience in the conditions of an adequate learning milieu. In the following, we would like to emphasize the link between mathematical properties and semiotic representations. Example: In an observed active 7th-grade class, students tried to solve the following problem: obtain 1000 with 8 “8’s”. It is important to distinguish the heuristic phase from the written presentation of the answer. We consider the latter as one step toward algebraic treatment, although no variable is required. We explain below that the reason for this is that, when writing the solution, one is lead to process mathematical “sentences”. We assert that, in the considered case, the processing is a tool for solving the problem, not only a way to communicate or explain ideas. Two correct written answers are: 1000 = 888 + 88 + 8 + 8 + 8 1000 = (8888 – 888)÷8 Many students gave in this situation “equalities” like: 888 + 88 = 976 + 8 + 8 + 8 = 1000. Explaining why this kind of writing does not respect mathematical rules, and thus is incorrect, is a true challenge for teachers. Indeed, the correct use of the equality symbol is not something purely formal, as many students believe, but it constitutes an important element of the mathematical construction. Other students used seven “8’s” instead of eight, e.g. the following response that we summarise by the equality: 1000 = (8 + 8)×8×8 – 8 – 8 – 8

(*)

The students did not furnish this equality, they performed non-ordered calculations during the heuristic phase, and this may be the reason why nobody, including the teacher, noticed that there are only seven “8’s”. When calculating, the focus is on the result 1000 and this leads to forget the rest. If there had been a complete final writing of (*), this “concluding sentence” would have encouraged reconsidering what precedes, and thus noticing the errors. Moreover, from the above inadequate writing, it is easy (for someone familiar with algebraic process) to obtain a correct answer by dividing and multiplying by 8: 1000 = ((8 + 8)×8 – (8 + 8 + 8)÷8)×8.

A Numerical Landscape

29

And this would be true processing of (*), which can be compared to processing a linguistic sentence. Using letters in calculations is not specific to algebra. One can use letters without being competent in algebra, e.g. substituting values for time and velocity in the formula d = vt. Conversely, one can do algebra without using letters. We assert that a fundamental step in mastering algebra is being able to process sentences. A sentence is the unit of meaning. Mathematics form sentences, like common language does. In mathematical writing, we can recognize nouns that are: numbers, variables, or more generally all kinds of mathematical objects; verbs: =, ≠,

Our partners will collect data and use cookies for ad personalization and measurement. Learn how we and our ad partner Google, collect and use data. Agree & close