Combinatorics and Reasoning: Representing, Justifying and Building Isomorphisms (Mathematics Education Library)

Combinatorics and Reasoning Mathematics Education Library VOLUME 47 Managing Editor A.J. Bishop, Monash University, M...

Author: Carolyn A. Maher | Arthur B. Powell | Elizabeth B. Uptegrove

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Combinatorics and Reasoning

Mathematics Education Library VOLUME 47 Managing Editor A.J. Bishop, Monash University, Melbourne, Australia

Editorial Board M.G. Bartolini Bussi, Modena, Italy J.P. Becker, Illinois, U.S.A. B. Kaur, Singapore C. Keitel, Berlin, Germany F. Leung, Hong Kong, China G. Leder, Melbourne, Australia D. Pimm, Edmonton, Canada K. Ruthven, Cambridge, United Kingdom A. Sfard, Haifa, Israel Y. Shimizu, Tennodai, Japan O. Skovsmose, Aalborg, Denmark

For further volumes: http://www.springer.com/series/6276

Carolyn A. Maher · Arthur B. Powell · Elizabeth B. Uptegrove (Editors)

Combinatorics and Reasoning Representing, Justifying and Building Isomorphisms

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Editors Dr. Carolyn A. Maher Rutgers University Graduate School of Education 10 Seminary Place New Brunswick, NJ 08901 USA [email protected]

Dr. Arthur B. Powell Rutgers University Department of Urban Education 110 Warren Street Newark, NJ 07102 USA [email protected]

Dr. Elizabeth B. Uptegrove Felician College Department of Mathematical Sciences 223 Montross Avenue Rutherford, NJ 07070 USA [email protected]

Series Editor: Alan Bishop Monash University Melbourne 3800 Australia [email protected]

ISBN 978-0-387-98131-4 e-ISBN 978-0-387-98132-1 DOI 10.1007/978-0-387-98132-1 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010926950 © Springer Science+Business Media, LLC 2010 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

This book is dedicated to the Kenilworth students who participated in the longitudinal study and from whom we continue to learn so much. We thank you for your continuing commitment, abundant trust, and generous sharing of how mathematical ideas and ways of reasoning are built.

Preface

Our research project on mathematical learning focuses on the accomplishments of a cohort group of learners from first grade though high school and beyond, concentrating on their work on a set of combinatorics tasks. We describe their impressive mathematical achievements over these years. We illustrate in detail the processes by which students learn to justify solutions to combinatorics problems that were challenging for their age and grade level. Based on transcribed video data and learners’ inscriptions, we provide a careful and detailed analysis of the process by which mathematical ideas are developed, discussed, modified, expanded, and justified. Our work underscores the power of attending to basic ideas in building arguments; it shows the importance of providing opportunities for the co-construction of knowledge by groups of learners; and it demonstrates the value of careful construction of appropriate tasks. Moreover, it documents how reasoning that takes the form of proof evolves with young children and it discusses the conditions for supporting student reasoning. We present in this book strong and compelling evidence that under appropriate conditions and with minimal intervention, learners can develop sophisticated ideas about proof and justification, generalization, isomorphism, and mathematical reasoning at an early age and can continue to refine and expand those ideas over time, developing increasingly sophisticated presentations and representations. We also describe an extension of this work with groups of undergraduate students, noting similarities and differences between the reasoning of the original cohort group of younger students and that of the college students. We include a detailed discussion of all the mathematical tasks, which can be used in classrooms from elementary school to the graduate college level.

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Acknowledgements

We are deeply grateful for the many colleagues who have made this book possible and would like to acknowledge their contributions. We thank Fred Rica, Principal of Harding Elementary School, Kenilworth New Jersey, whose vision made the study possible, and the faculty and administration at Kenilworth who gave their support and encouragement to students and researchers. We thank the many graduate students, video crew and videographers, especially Roger Conant, Ann Heisch, Lynda Smith, and Elena Steencken, as well as the dedicated researchers for their hard work and insights: Alice S. Alston, Robert B. Davis, John M. Francisco, Barbara H. Glass, Regina D. Kiczek, Judith H. Landis, Amy M. Martino, Ethel M. Muter, John J. O’Brien, Marcia O’Brien, Ralph Pantozzi, Manjit K. Sran, Maria Steffero, Lynn D. Tarlow, and Dina Yankelewitz. We are particularly grateful to Robert Speiser for his enthusiasm and support, as well as his invaluable help with task design. We thank the Sussex County Community College students who participated in the study. We thank the staff of the Robert B. Davis Institute for Learning, and in particular, Marjory F. Palius for research assistance and continued generous help, Robert Sigley for his overall knowledge and management of the collection as well as his invaluable technical help, Patricia Crossley for her organization of the data for the studies, and Manjit K. Sran and Dirck Uptegrove for their wonderful illustrations and artwork. We are grateful for support for the longitudinal study by: (1) the National Science Foundation with grants: MDR-9053597 (directed by R. B. Davis and C. A. Maher) and REC-9814846 (directed by C. A. Maher), and (2) from the New Jersey Department of Higher Education, the Johnson and Johnson Foundation, the Exxon Education Foundation, and the AT&T Foundation. Any opinions, findings, and conclusions or recommendations expressed in this book are those of the authors and do not necessarily reflect the views of the funding agencies.

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Contents

Part I

Introduction, Background, and Methodology

1 The Longitudinal Study . . . . . . . . . . . . . . . . . . . . . . . . Carolyn A. Maher

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2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Carolyn A. Maher and Elizabeth B. Uptegrove

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Part II

Foundations of Proof Building (1989–1996)

3 Representations as Tools for Building Arguments . . . . . . . . . . Carolyn A. Maher and Dina Yankelewitz

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4 Towers: Schemes, Strategies, and Arguments . . . . . . . . . . . . Carolyn A. Maher, Manjit K. Sran, and Dina Yankelewitz

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5 Building an Inductive Argument . . . . . . . . . . . . . . . . . . . Carolyn A. Maher, Manjit K. Sran, and Dina Yankelewitz

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6 Making Pizzas: Reasoning by Cases and by Recursion . . . . . . . Carolyn A. Maher, Manjit K. Sran, and Dina Yankelewitz

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7 Block Towers: From Concrete Objects to Conceptual Imagination Robert Speiser

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Part III Making Connections, Extending, and Generalizing (1997–2000) 8 Responding to Ankur’s Challenge: Co-construction of Argument Leading to Proof . . . . . . . . . . . . . . . . . . . . . . Carolyn A. Maher and Ethel M. Muter 9 Block Towers: Co-construction of Proof . . . . . . . . . . . . . . . Lynn D. Tarlow and Elizabeth B. Uptegrove 10

Representations and Connections . . . . . . . . . . . . . . . . . . . Ethel M. Muter and Elizabeth B. Uptegrove

89 97 105

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Contents

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Pizzas, Towers, and Binomials . . . . . . . . . . . . . . . . . . . . . Lynn D. Tarlow

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Representations and Standard Notation . . . . . . . . . . . . . . . Elizabeth B. Uptegrove

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So Let’s Prove It! . . . . . . . . . . . . . . . . . . . . . . . . . . . . Arthur B. Powell

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Part IV

Extending the Study, Conclusions, and Implications

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“Doing Mathematics” from the Learners’ Perspectives . . . . . . . John M. Francisco

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Adults Reasoning Combinatorially . . . . . . . . . . . . . . . . . . Barbara Glass

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Comparing the Problem Solving of College Students with Longitudinal Study Students . . . . . . . . . . . . . . . . . . . . . Barbara Glass

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Closing Observations . . . . . . . . . . . . . . . . . . . . . . . . . . Arthur B. Powell

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Appendix A Combinatorics Problems . . . . . . . . . . . . . . . . . . .

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Appendix B Counting and Combinatorics Dissertations from the Longitudinal Study . . . . . . . . . . . . . . . . . . . . . .

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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Introduction

Carolyn A. Maher, Arthur B. Powell, and Elizabeth B. Uptegrove

The theoretical foundation for the program of research on which this book is based comes from recognition that individual learning takes place within a community. The members of that community have access to and are influenced by the ideas of others. Individual learners are interconnected with other members of the community; engagement with others opens up possibilities for sharing and comparing representations of ideas and for revising existing schemes and building new ones. In the activity of problem solving, learners bring forth, communicate, and compare ideas. They explore whether the ways that others represent ideas correspond with their own representations, thereby extending their personal repertoires of tools for dealing with new ideas. In this way further learning takes place and understanding deepens (Davis & Maher, 1997; Maher, Martino, & Alston, 1993; Maher & Davis, 1990). The data for this book come from a long-term program of research detailing the collective building of mathematical ideas, which we call the longitudinal study. In this book, we explore student work in one of the mathematics strands of the longitudinal study: counting and combinatorics. It investigates how students’ reasoning evolved from elementary and high school years to college. The reasoning of learners is documented by their actions – that is, what they do, say, build, and write – as they work on strands of tasks. In studying how participants make sense of the complexity of problems, we trace the representations they share, the heuristics they invent and apply, and the modifications they make in building arguments and in offering justifications for solutions. The authors of the constituent 17 chapters relate how an ordinary group of school children manifest over a 12-year period an extraordinary array of mathematical ideas that they discursively build and how – with time – their ideas modify and mature as they reason and justify their ideas. The book reports episodes from a long-term study of how mathematical ideas and ways of reasoning are built by students over time. The study has produced over 4,500 h of video, over several sites, involving far more

C.A. Maher (B) Graduate School of Education, Rutgers University, New Brunswick, NJ, USA e-mail: [email protected]

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data than can be presented here. However, we have selected narratives that feature the voices of several children, as interpreted by a variety of researchers, to weave together a bigger story about how students can educate us about the multifaceted nature of mathematical development. In an important sense, the really big story is still being written as our work in preserving and further analyzing those 4,500 h of video through the Video Mosaic Collaborative1 continues to reveal new narratives. We invite readers to view the videos at http://www.video-mosaic.org/. Along with the narratives offered in the book chapters that follow, these videos enable readers to trace in detail the development of counting/combinatorics ideas and ways of reasoning of learners over more than a decade. Thus, while only some of the children’s voices appear in this book, we are indebted to all of them for sharing their developing mathematical ideas over time and in divergent contexts, which we continue to study and consider how these children’s extraordinary mathematical reasoning may inspire the fields of mathematics education, teacher education, and the learning sciences. To structure a story that emerges from the chapters, the editors have divided this book into four parts. The two chapters of the first part, respectively, provide historical background of the research study from which the details of the later chapters emerge and describe the design of the study. The first chapter describes the study and the purpose of the research, how the study began, and the conditions under which the research was conducted. It also briefly describes the mathematical ideas and ways of reasoning that emerged from the study. (The details are presented in later chapters.) The second chapter presents the method of the study, its design, including selection of participants, data collection, and analysis, as well as the strand of tasks on which participants were invited to work. The chapter also discusses the importance of the task design for helping learners to develop ways of reasoning. The second part of the book contains five chapters. These chapters chronicle the work of the study’s participants over a 7-year period from grades 2–8, tracing the development of their mathematical ideas, heuristics, and forms of reasoning. In particular, the reader will learn how the participating children represented their ideas; developed schemes and strategies; reasoned in specific ways; built inductive arguments; reasoned by cases and by recursion; and connected numbers in Pascal’s Triangle to results of previous problems. The authors of Chapter 3 discuss how young children use representations to express their mathematical ideas while building a solution to a particular counting problem (the shirts and jeans 1

The Video Mosaic Collaborative is a research and development project sponsored by the National Science Foundation (award DRL-0822204) directed by C.A. Maher, G. Agnew, C.E. Hmelo-Silver, and M.F. Palius that is leveraging the Rutgers Community Repository to preserve the unique video collection amassed by The Robert B. Davis Institute for Learning at Rutgers University through two decades of research with over four millions dollars of grant funding from the NSF (awards MDR-9053597, REC-9814846, REC-0309062 and DRL-0723475). In addition to preserving the video collection, new tools are being developed for conducting design research and an empirical study that use the videos in the context of teacher education. The editors gratefully acknowledge this considerable support from the National Science Foundation and wish to clarify that all views expressed in this book are those of the authors are not necessarily those of the NSF.

Introduction

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problem, described fully in Appendix A, along with all combinatorics problems discussed herein). They show how children structure their representations in response to requests to justify their problem solution and build convincing arguments to early counting problems. The authors of Chapter 4 and 5 discuss students’ work on different versions of the towers problems (which involve determining how many towers can be built of various heights when selecting from cubes of various numbers of colors). They show the emergence of different forms of reasoning (cases, contradiction, recursion, and induction) and how, motivated by the need to find the sample space for a basic probability exploration, students revisit the inductive argument for building towers. Chapter 6 discusses how participants collaboratively build representations that help them use reasoning by cases and by recursion to develop justifications for their solutions to classes of pizza problems. (Pizza problems involve determining how many pizzas it is possible to make when selecting from various numbers of toppings and under various other constraints.) Completing this part of the book, Chapter 7 presents the results of an interview with 13-yearold Stephanie, who discusses the relationship between the towers problems and the binomial expansion, including how the towers answers can be found in Pascal’s Triangle. The six chapters of the book’s third part closely examine the mathematical work of the research participants during their high school years. It shows how the students built important connections using sophisticated mathematical reasoning. In these chapters, the story revolves around the students’ proof making, use of representations, acquisition of standard notation, and forging of conceptual connections among isomorphic problems. Specifically, Chapter 8 shows that as they revisit their representations and arguments, students refine representations and clarify arguments. In Chapter 9, students working in groups on towers problems are seen to find and generalize formulas, using methods including controlling for variables, justification by cases, and induction. Chapter 10 shows how a tenth-grade student’s binary notation helped his group form connections among the pizza and towers problems, the binomial expansion, and Pascal’s Triangle. Chapter 11 details how representations are a source for making connections in solutions to pizza and tower problems, resulting in the students mapping the structure of the solution of these problems to Pascal’s Triangle and how their increasingly sophisticated use of representations led to further development of mathematical reasoning and justification. Chapter 12 discusses how students moved from personal to standard notations in order to express in general form their understanding of solutions to the pizza and towers problems and to extend their understanding in creating an isomorphism from the numerical results in those problems to Pascal’s Triangle. The chapter also shows how the students’ understanding of extensions of the pizza and tower problems led to their understanding of the addition rule for Pascal’s Triangle. The final chapter of Part III, Chapter 13, reveals how as high school seniors, days before graduation, the students used their understanding of relationships between the pizza and tower problems and Pascal’s Triangle to solve a third isomorphic problem – the Taxicab Problem. (This problem involves finding the number of routes from the starting point – the taxicab stand – to various points on

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a rectangular grid.) They recognized the isomorphism, used it to make conjectures about the new problem, saw the need to prove their conjectures, and provided a convincing argument. This chapter concludes by examining some of the extraordinary mathematical accomplishments of the cohort group of students. The last part of the book, consisting of four chapters, takes stock and looks forward. Chapter 14 examines the epistemological growth of the students, viewed from their own perspectives. Students’ reflections on their learning over the years challenges common views about student engagement in learning, and gives insight into how students view their own sense making in doing mathematics. Chapter 15 examines a different student population – college undergraduates – and their work with the set of combinatorics problems. The chapter shows that when adult college students are asked to justify ideas and make convincing arguments, an understanding of mathematical reasoning, proof, and generalization can emerge. In Chapter 16, Glass compares the strategies developed by children and older learners for solving the combinatorics problems and discusses the implications for adult learning. In closing, Chapter 17 presents the epistemological and methodological contributions of the book. We argue that students must be actively and purposely engaged in their learning so as to take ownership of and be proud of their accomplishments. Mathematics educators and teachers need to create opportunities for students to engage in ways similar to those described in this book. We have shown that in a program of carefully selected tasks, with minimal intervention by educators who pay careful attention to students’ arguments and justifications, students can perform mathematically at high levels. In addition to developing mathematical competency, students who participated in the study gained confidence and a sense of empowerment and were successful in their career choices. They learned to trust their own mathematical ability and they did not rely on outside authority for validation. This confidence, sense of empowerment and propensity to reason carefully has been carried over outside their mathematical work; these students found that the knowledge and ways of working that they gained through their participation in the longitudinal study continues to help them in many other areas of study and employment.

Contributors

John M. Francisco Secondary Mathematics Education, Department of Teacher Education & Curriculum Studies, University of Massachusetts Amherst, Amherst, MA 01003, USA, [email protected] Barbara Glass Sussex County Community College, Newton, New Jersey, USA, [email protected] Carolyn A. Maher Graduate School of Education, Rutgers University, New Brunswick, NJ, USA, [email protected] Ethel M. Muter 5280 Antioch Ridge Drive, Haymarket, VA 20169, USA, [email protected] Arthur B. Powell Department of Urban Education, Rutgers University, Newark, NJ, USA, [email protected] Robert Speiser 799 E 3800 N, Provo, UT 84604, USA, [email protected] Manjit K. Sran Mathematics Department, Monroe Township High school, 1629 Perrineville Road, Monroe Township, NJ 08831, USA, [email protected]; College of Business and Management, DeVry University, 630 U.S. Highway One, North Brunswick, NJ 08902, USA, [email protected] Lynn D. Tarlow Department of Secondary Education, The City College of the City University of New York, New York, NY 10031, USA, [email protected] Elizabeth B. Uptegrove Department of Mathematical Sciences, Felician College, Rutherford, NJ, USA, [email protected] Dina Yankelewitz Department of General Studies, The Richard Stockton College of New Jersey, Pomona, NJ, USA, [email protected]

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Part I

Introduction, Background, and Methodology

Chapter 1

The Longitudinal Study Carolyn A. Maher

1.1 Theoretical View Where do new ideas come from? Our view is that building new ideas is a process; new ideas come from old ideas that are revisited, reviewed, extended, and connected (Davis, 1984; Maher & Davis, 1995). Building new ideas also involves the retrieval and modification of representations of existing ideas. The representations that a learner builds for a mathematical idea or procedure can take different forms – physical objects or actions on objects, words, and symbols, for example. As the learner’s experience increases, old representations become elaborated, extended, and linked to new ones (Maher, 2008; Davis & Maher, 1997). The problem tasks that are posed to learners are critical to their learning (Francisco & Maher, 2005); they should be well defined, open-ended, and open to extension and generalization. The connections that the learner makes when analyzing and developing solutions to these problems provide further opportunity for growth in knowledge. Students are encouraged to revisit earlier problems because requirements to justify and generalize solutions can help students to see underlying mathematical structure. It is a widely accepted view that when learners understand the fundamental structure of a subject, the gap between “elementary” and “advanced” knowledge is reduced (Bruner, 1960). There is increasing evidence that learners, under certain conditions, can build meaningful, mathematical relationships and understand the structure of mathematical problems at an early age. For example, a study of Norwegian children indicated that even as young as Grade 3, learners are able to unearth the underlying structure of the mathematics of problem tasks (Torkildsen, 2006). A central component of the learning process is encouraging students to communicate their ideas. Sfard (2001) suggests that students learn to think mathematically


C.A. Maher et al. (eds.), Combinatorics and Reasoning, Mathematics Education Library 47, DOI 10.1007/978-0-387-98132-1_1, C Springer Science+Business Media, LLC 2010

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by participating in discourse about ideas – arguing, asking questions, and anticipating feedback. We have emphasized that justifying ideas in problem solving is an essential component of mathematical reasoning (Maher, 2002, 2005, 2008; Maher & Martino, 1996a; Martino & Maher, 1999). Learners, in communicating their ideas, share personal mental images – representations. When students make their representations public, they have an opportunity to talk further about them, compare them, and later revisit them. Similarities and differences in ideas naturally emerge. When learners try to convince others that their answers are correct, they can reorganize and reformulate their representations so as to make convincing arguments. In summary, students learn mathematics by engaging in the process of building their own personal representations, communicating them as ideas, and then providing support for those ideas by reorganizing and restructuring representations. Our view is that this process is a necessary prerequisite both for developing the idea of mathematical proof and for making suitable connections between problems of equivalent structure by building isomorphisms. In this book, we discuss how a group of students developed new and increasing sophisticated mathematical ideas by revisiting, reviewing, extending, and connecting old ideas that they had begun developing in first grade. They developed and modified representations that became increasingly elaborated and extended. They participated in serious mathematical discourse. And ultimately they built a strong and durable understanding of the solutions to a set of mathematical tasks. Our longitudinal work is important because it reveals the processes that these learners used to build structural understanding of solutions to mathematical tasks.

1.2 Background of the Study The longitudinal study began in 1987 in Kenilworth, New Jersey. This was during a time when behaviorism mainly governed mathematics instruction. It was a time before the reform movement in the United States emphasizing conceptual understanding had made its entry. The K-8 Harding Elementary School in the working-class community of Kenilworth, New Jersey, was typical of others at that time. Half-hour sessions were devoted to mathematics, and mathematics instruction was mainly rote. The rule was drill and practice for carrying out memorized procedures. For the most part, even the brightest students from the school did not excel when they moved on to high school mathematics classes, only doing average work. Most members of the community and most teachers had rather low expectations for student advancement. But Fred Rica, principal of the Harding Elementary School, had higher expectations for the students in his school. Formerly an elementary grade classroom teacher in Kenilworth, Fred Rica knew his staff and students well. Like other concerned educators, he knew when the system was not serving its student population. He turned to Rutgers University for help with instruction, first in language and literacy and then in mathematics. It was shortly after this professional development work that Fred Rica and Carolyn Maher created a partnership between the Kenilworth Public

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Schools and Rutgers University. It should be noted that the Rutgers–Kenilworth partnership, with its focus on students building meaning of mathematical ideas and working collaboratively with each other, began long before the National Council of Teachers of Mathematics published its reform standards. Initially, the project began as a teacher development intervention in mathematics. The Rutgers University team of researchers and graduate students worked for 3 years to help teachers build an understanding of the mathematics they were expected to teach and to learn to be attentive to the developing understanding of their students. (See Davis & Maher, 1993; O’Brien, 1994, for a detailed study of the teacher development project.) The project could not have survived the early years without the full support and active participation of the Kenilworth school administration. In particular, principal Rica actively participated in the teacher-training sessions, encouraged teachers to become involved, and made sure that students who were involved in the study were available to the researchers. Original financial support for the partnership came from the Kenilworth school district and through volunteer efforts of the Rutgers team. The Kenilworth school district continued to fund the study for several years as a component of its mathematics teacher development mission. The Rutgers research group received outside funding for the research from two National Science Foundation grants. The first grant awarded to Principal Investigators Robert B. Davis and Carolyn A. Maher was when the students were in Grade 4; the second grant awarded to Principal Investigator Carolyn A. Maher was when students were in high school.

1.2.1 Teacher Development Component It is not surprising that the teachers at the Harding Elementary School were not prepared to teach mathematics with understanding. What is surprising was the expectation of principal Fred Rica that the teachers were capable, with some professional development and classroom support, of understanding the mathematics they were expected to teach. In fact, this view was remarkable for its time. The teacher development team was made up of mathematics education doctoral students who had considerable experience in schools; its first members were Alice S. Alston and Judith H. Landis. The team worked closely with Fred Rica and his teachers to establish a program of activities that involved not only videotaped teacher workshops and classroom sessions, but also study of those workshops and sessions. The Rutgers team worked directly with students and with their teachers, first observing classroom sessions and later collaborating with the teachers in the design and implementation of lessons. Alice Alston also worked in the classrooms alongside the teachers. Principal Rica obtained school funding to support teachers’ summer work to revise the existing curriculum. Two years of summer professional development assisted by John O’Brien and Alice Alston resulted in a movement from a “drill and kill” approach to one in which students’ building of mathematical understanding

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was central. Curriculum revisions included the use of more engaging and thoughtful lessons for the students and the introduction of manipulatives that allowed students to build models of their solutions. Some Kenilworth teachers who participated in the teacher development programs also became involved in classroom action research. As teachers were introduced to new resources and tools, they developed new units and piloted them during the school year. Through course work opportunities at Rutgers, some teachers studied the mathematical learning of their own students (Landis & Maher, 1989; Landis, 1990; Maher 1988; O’Brien, 1994).

1.2.2 Intervention Design The Rutgers team was interested in what mathematical concepts students could learn with minimal intervention from teachers. Classrooms were organized so that children might work together and collaborate on problem tasks. Children were encouraged to use each other as resources in their investigations, to construct models of solutions with available tools, and to revisit tasks and discuss their strategies and solutions. An important observation during the first 3 years was that students produced arguments that took on a variety of forms of reasoning to support their solutions to the problems. By Grade 4, it became increasingly clear to researchers that students’ reasoning, in a natural way, took the form of proof. Children began their investigations by searching for patterns, organizing solutions, searching for completeness, deriving strategies for keeping track and checking, and then reorganizing justifications into arguments that were proof-like in structure. Using each other as resources, children freely shared ideas, questioned each other, argued about the reasonableness of ideas, and became comfortable in sharing and communicating with each other. What encouraged both the school staff and the university collaborators was the enthusiastic feedback from students. The children enjoyed talking about their ideas; they engaged with each other with energy and enthusiasm, becoming increasingly more comfortable making their ideas public. Their way of working underscored a demand for sense making, which then evolved as a cultural norm. This book explores student work for one of the mathematics strands of the longitudinal study: counting and combinatorics. It investigates how students’ reasoning evolved over the course of the longitudinal study that continued from elementary and high school years to college.

1.3 Longitudinal Study: Grades 1–3 In order to study the effectiveness of the intervention, the Rutgers team decided to follow a class of students throughout their elementary grades as they worked on mathematical investigations that were not part of the school curriculum. The study began with a class of 18 first-grade students from the Harding School. These children, randomly assigned to one of three first grades, became the initial focus

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group; they were together for Grades 1–3 as part of the school design. Throughout the study, students engaged in strands of thoughtful mathematics activities designed by the researchers. Although the mathematical investigations were not part of the curriculum, the concepts that were introduced would later become part of the regular school mathematics curriculum.

1.4 Longitudinal Study, Grades 4–8 After Grade 3, the students were distributed among different classrooms, according to school policy. However, the principal worked with Rutgers researchers to facilitate maintenance of a focus group of 12 students for research purposes. When families moved and new families entered the district, the composition of the focus group changed, but an attempt was made to maintain a group of comparable background and interest. Although some students stayed with the study from the start (and are still in touch today), some students moved from the district and new students joined. During middle school, the school arranged for the cohort group to continue working with researchers during school hours, 4–6 times a year in two 90-min sessions and one 45-min session each time.

1.5 Longitudinal Study: High School Years In 1996 the high school in Kenilworth was closed, as the school district became part of a regional system. The community joined forces to protest the merger and succeeded after 1 year. Hence, the first year of high school (ninth grade) proved disruptive for the students, although some math problem-solving sessions were conducted with small groups of students during that year in local homes, usually on Saturdays. After Kenilworth de-regionalized and the students returned to Kenilworth for the remaining 3 years of high school, groups of students resumed participation in the longitudinal study in informal, after-school sessions that were held during the year, usually on Friday. While students no longer met with researchers during regular class hours, 14 students (some from the original group of first graders and others who had joined the study at various times during middle school and high school) made time in their schedules to meet after school about 4–6 times a year for problem-solving sessions that lasted 1–2 h or longer. This group included ten students who had been with the study since Grade 1, two students who had joined the study in Grade 6, and two who joined in high school (Grade 11).

1.6 Longitudinal Study: Beyond High School All students in the focus group applied to Rutgers University, and all were accepted – a remarkable achievement for the district. However, not all students attended Rutgers; they attended a variety of universities, public and private; besides

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Rutgers, these included Cornell University, Kean University, St. John’s University, and the University of Pennsylvania. Majors included accounting, American studies, animal science, computer science, criminal justice, economics, engineering, English, and mathematics. All are now either employed or in graduate school. Some of the students have continued to meet occasionally with researchers during and after college. They do not generally work on problems (although sometimes old problems are revisited), but they talk about how being in the study has affected them, their academic careers, and their future plans. In the next chapter, we detail how the study was conducted and we discuss selected problems that formed the cornerstone of the student investigations over the years.

Chapter 2

Methodology Carolyn A. Maher and Elizabeth B. Uptegrove

2.1 Introduction In this chapter, we discuss how data were collected and analyzed, and we briefly describe some results, which will be more fully explored in later chapters. We summarize student work on fundamental problems and note how this work led to exceptional growth in the students’ mathematical understanding. Researchers (professors at the Rutgers University Graduate School of Education and their students) conducted all problem-solving sessions with the students; the sessions were always videotaped with one or more cameras. Researchers observed, described, and coded the videotape data, and they kept written and electronic files of the emerging theoretic, analytic, and interpretative ideas about the students’ mathematical behaviors. Researchers paid careful attention to children’s use of inscriptions, the connections they made between and among codes, and their emerging and extended ideas and ways of reasoning. Critical events in children’s reasoning were flagged and transcribed and transcripts were coded according to the research questions. The connected series of events that formed a trace led to the emergence of a narrative (Maher & Martino, 1996a; Powell, Francisco, & Maher, 2003). The videotapes, researcher notes, and student notes did not capture every interaction or every case of student learning. Some students sat silently during discussions; but they had quietly absorbed a problem or quietly developed a solution that came to light some time later in a different situation. Therefore, although we can make inferences about what is observed, we cannot assume that a student who is quiet does not understand. By videotaping children as they worked together on mathematical tasks over long periods of time, we were able to trace the origin and development of their mathematical ideas. We observed what children said to one another and showed to one another. We used videotapes and transcripts to study the meanings that children gave



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to mathematical situations and to note the different representations they made public. A detailed analysis of data made it possible to trace the origin and evolution of children’s arguments. Our data indicate how children expressed their ideas through spoken and written language, through the physical models they built, through the drawings and diagrams they made, and through the mathematical notations they invented.

2.2 Theoretical Perspectives Guiding our work is the view that children come to mathematical investigations with theories they can modify and refine. We observe them do so in settings that combine personal exploration and suitable social interaction. The theories we consider can include criteria to decide (1) what, at some given moment, needs to be investigated, (2) how to conduct such an investigation, (3) what key features need to be explored in detail, (4) when useful progress has been made, and, given such progress, (5) if further investigation might be needed. We have found that theories of this kind often empower striking and effective ways for children to work conceptually with mathematical ideas, often using concrete objects as specific anchors for their thinking.

2.3 Selected Problems Mathematics arose from the need to count, measure, and calculate, but the discipline evolved to include abstraction, logical reasoning, and the search for and analysis of patterns. Good mathematical problems are therefore those which give rise to the need for abstraction, systematization, and pattern recognition. A focus of the study was therefore to select problems that would give rise to these needs. Another focus of the longitudinal study was on doing problems that were not part of the regular curriculum, because it was important for the students to come to the problems fresh, without pre-taught algorithms. A major strand of the longitudinal study therefore consisted of problems in combinatorics, because in working on these problems, students can find the need to organize their work systematically, look for patterns, and generalize their findings; also, counting problems were at the time outside the regular elementary school curriculum and therefore unfamiliar to students. In addition, these problems lend themselves to the use of multiple personal representations that can be shared. Freudenthal (1991) cites the study of combinatorics as “a most important matter for reinvention” (p. 53), specifically because combinatorics can be learned through paradigmatic examples and because problems in combinatorics give rise to the need for convincing proof, including mathematical induction. Another purpose of the longitudinal study was to provide an environment in which certain socio-mathematical norms could be established to elicit in children

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sense making, argumentation, and justification in mathematics. As Yackel and Cobb (1996) and Cobb, Wood, Yackel, and McNeal (1992) suggest, an appropriate social context must be created to encourage students to try to convince others of the truth of the mathematical ideas that they build. The longitudinal study was structured to investigate and track the nature of the schemes that students developed and the methods that the students used to build and retrieve representations to solve mathematical tasks. In addition, the study attempted to trace how students shared ideas and how these ideas were adapted and assimilated by other students. Appendix A describes all the combinatorics problems that students worked on over the years. We summarize here some example problems, along with brief accounts of strategies and representations used by students and forms of reasoning that developed.

2.3.1 Shirts and Jeans Students worked on the shirts and jeans problem at the end of second grade and again at the beginning of third grade (1989 and 1990): Stephen has a white shirt, a blue shirt, and a yellow shirt. He has a pair of blue jeans and a pair of white jeans. How many different outfits can he make?

During second grade, most students drew pictures of outfits; some drew lines between shirts and jeans, and others made lists of outfits. Notational choices influenced the way they reasoned about the data. For example, Stephanie used “bluewhite” to stand for the white shirt/blue jeans outfit, and also for the blue shirt/white jeans outfit. Contextual issues also played a role in the problem solving. For example, Dana discarded the white jeans/yellow shirt outfit on grounds that the resulting outfit did not match and was thus not fashionable. That different students got different answers was not problematic for the children; in second grade, students seemed comfortable with the notion that answers varied between three and seven outfits. They willingly shared their interpretations and strategies and talked to each other about their findings. In third grade, when the children were again presented with this problem, they did not remember how they had solved the problem earlier, nor did they remember their earlier answers. Of particular interest is that evidence of further elaboration of earlier strategies emerged. Students used and built on strategies of their second-grade partners. For example, Stephanie indicated different outfits by drawing lines between drawings of shirts and jeans, as Dana had done in second grade. By third grade, techniques for checking and for keeping track, such as controlling for variables, were complete. Earlier ideas and strategies were refined to produce complete, elegant solutions rather quickly. Students built on their heuristics to solve more complex extensions of the problem to include belts and hats as parts of outfits. What was especially significant for the researchers was the evidence of how students built on earlier ideas and, without intervention or approval from researchers,

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continued their problem solving, driven by earlier heuristics and sense making to produce correct solutions that they could justify.

2.3.2 Towers Early in third grade (1990), students were given the four-tall tower problem for the first time: Your group has two colors of Unifix cubes. Work together and make as many different towers four cubes tall as is possible when selecting from two colors. See if you and your partner can plan a good way to find all the towers four cubes tall.

The definition of a tower is an ordered sequence of Unifix cubes, snapped together. Each cube can also be called a block. Each tower has a bottom and a top. The height of a tower is the number of its cubes. We say two towers are the same if their colors match, block by block, from top to bottom. Unifix cubes are interlocking cubes that come in various colors (typically blue, red, yellow, white, and green). In fourth grade (1991), students worked on the five-tall tower problem. Then in fifth grade (1992), they revisited the four-tall version. In 10th and 11th grades, they were asked to provide a justification for the n-tall tower problem. Students discussed variations and generalizations of the solution and they used their organization of towers by cases and knowledge of the binomial expansion to build an understanding of how Pascal’s triangle grows. Their work on the towers problems also illustrates how their representations changed over the years. At first, they used Unifix cubes to build towers. Eventually, they turned to drawings and codes, for example, using letters R and Y to mean red and yellow cubes. In some cases, a more general code emerged; some students would use X and O or 0 and 1 to indicate any two colors. More details on these emerging strategies are given in Sections 2.2 and 2.3.

2.3.3 Pizzas In order to introduce a variation of the tower problem and to investigate how students reasoned with an isomorphic problem, the researchers introduced the set of pizza problems. When the students were first given the problem in fifth grade, they interpreted the task as allowing different toppings on each half of the pizza, an alternative that they knew that was available in some pizza restaurants. In response to their interest in counting the varieties allowing toppings on half a pizza, the researcher asked them to solve it with only two toppings available. This pizza with halves problem is as follows: Kenilworth Pizza has asked up to help them design a form to keep track of certain pizza sales. Their standard plain pizza contains cheese. On this cheese pizza, one or two toppings could be added to either half of the plain pie or the whole pie. How many choices do customers have if they could choose from two different toppings (sausage and pepperoni) that

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could be placed on either the whole cheese pizza or half a cheese pizza? List all possibilities. Show your plan for determining the choices. Convince us that you have accounted for all possibilities and there could be no more.

The strategy that the students developed for the solution established the heuristic that was applied later when there were five toppings available, again, allowing some or no toppings on half the pizza. The final problem, the five-topping pizza problem, proved a trivial special case for the pizza with halves problem that they first solved successfully: The local pizza shop offers a plain cheese pizza. On this cheese pizza, you can place up to five different toppings. How many pizzas is it possible to make?

Pizza with Halves was the first of several variations of the pizza problem that the students worked on over the years. It illustrates a basic philosophy of the study – we did not start students off with easier problems and then progress to the more difficult ones. Instead, students began with the more difficult versions of the problems, which required them to tackle several challenges at once – organization (making sure no pizzas were repeated and none were omitted), notation (how to distinguish between pepperoni and peppers, for example), and forming a valid argument – how to convince the researchers (and themselves) that they had the right answer. Looking at students’ answers to the pizza problems over the years, we see growth in organization and in representations. At first, students drew fairly accurate renditions of pizzas; they drew circles to indicate pizzas, and inside those circles were wavy lines to indicate sausages and smaller circles to indicate pepperoni, for example. When they had to answer a question involving half pizzas, they drew lines down the middle of their pizza circles to show both halves, and they listed all the pizzas using full words (for example, “whole plain, half sausage half plain”). Eventually, they turned to codes, starting with single letters or combinations of letters (to distinguish between peppers and pepperoni, for example) and then moving to more abstract symbols such as 0s and 1s. These representations and organizational strategies are discussed more fully in Chapter 6. In 11th grade, some students investigated Pascal’s triangle and Pascal’s identity (the addition rule for Pascal’s triangle). Using the metaphor of the pizza problem, they explained how the triangle grows by explaining how the number of possible pizzas grows as new toppings become available. In an extraordinary session lasting over 2 h one evening in 1999, students generated a slightly nonstandard but mathematically correct equation for Pascal’s identity using standard combinatorial notation:

N X

+

N X+1

=

N+1 X+1

A detailed description of the students’ work on Pascal’s identity is given in Chapter 12.

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2.3.4 Taxicab The group that had generated Pascal’s identity was introduced to the taxicab problem in 12th grade (2000): All trips originate at the taxi stand, in the upper-left corner of a grid. The problem is to find the shortest route to three specific points on the grid and to determine the number of shortest routes to each point.

Their work on this problem is discussed in greater detail in Chapter 13. It is interesting to note that by this time, the students, without prompting, solved the general problem, in addition to answering the specific questions. For any point on the grid, they showed why the general answer was correct, and they demonstrated the connection to isomorphic problems (towers and pizzas) and to the binomial expansion. What is also interesting from this session is that the students took on the roles of eliciting justifications from each other. Their pursuit of explanations that made sense and that connected to earlier tasks was quite remarkable.

2.4 Concluding Remarks The purpose of the longitudinal study was not to teach the students particular topics in combinatorics or other areas of mathematics. Instead, the aim was to establish a culture where the correctness of an answer came from the sensemaking of the students, rather than from the authority of the researcher. We asked students questions about what was convincing, what made sense, and how they developed their answers. In justifying their answers, students usually exceeded our expectations. We were impressed by the seriousness with which students approached the problems and the collegiality of their work, as well as by the forms of reasoning they developed. In the early years of the study, children began to use inductive reasoning, to organize work by cases, and to think about justification through contradiction. By middle school, these forms of reasoning were more sharply defined, and other forms of reasoning emerged, such as controlling for variables. In high school, students began the process of building isomorphisms, using their own notation as well as standard notation to describe how some problems were related to each other and ultimately to Pascal’s triangle. In the following chapters, we provide details on the specific problems, the specific strategies and representations used by the students, and the specific results they generated. In the next chapter, we discuss the students’ earliest work on combinatorics problems, the second- and third-grade work on shirts and jeans.

Part II

Foundations of Proof Building (1989–1996)

Chapter 3

Representations as Tools for Building Arguments Carolyn A. Maher and Dina Yankelewitz

Date and Grade: Tasks: Participants: Researchers:

May 30, 1990 and October 11, 1990; Grades 2 and 3 Shirts and Jeans Dana, Jaime, Michael, and Stephanie Carolyn A. Maher and Amy M. Martino

3.1 Introduction In this chapter, we discuss the children’s earliest work on combinatorics problems, the second- and third-grade efforts on the shirts and jeans problem: Stephen has a white shirt, a blue shirt, and a yellow shirt. He has a pair of blue jeans and a pair of white jeans. How many different outfits can he make? Convince us that you have them all.

In examining their problem solving, we focus on the children’s early use of personal representations. When introducing mathematics to children, it is important to invite them to use their personal representations to express their ideas and ways of reasoning (NCTM, 2000). These representations are the basic elements that children draw upon to express their ideas as they begin to engage in more abstract and logical reasoning. Children’s representations and how they are connected and related are foundational building blocks toward more sophisticated processes that lead to the creation of new mathematical ideas. When children are provided with opportunities to “reinvent” mathematics, they are in a better position later on to recognize their own need for abstraction, generalization, and logical reasoning (see Chapter 13).



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3.2 Representation as a Tool for Problem Solving Davis (1984) has written extensively about the role of representation in mathematical thinking. According to Davis, representations are mental models that allow for the association between the properties of a mathematical idea and the idea itself. These ideas are not stored in the mind in words or pictures, and so when we explore what these intangible representations are, we are only approximating their true nature. Davis stated that doing mathematics involves a series of steps, similar to those of a computer executing a program, through which the student must cycle one or more times. First, in an attempt to make sense of the problem, the student builds a representation for the input data. Second, the student searches his memory for knowledge that will assist in solving the problem. Finally, the student maps the data representation with the knowledge representation. When the mapping seems accurate enough to tackle the problem at hand, the student uses techniques associated with the knowledge representation to solve the problem. Students use representations that they build to make sense of and attribute meaning to the mathematics that they are doing. They use mathematical tools, which, according to Davis and Maher (1997) include mathematical notation, spoken and written language, physical models, drawings, and diagrams. According to Maher and Martino (1996a), students who are encouraged to build and use multiple representations as they work on problems become sense-makers and active members of the mathematical community. The use of different tools to build and express ideas allows students to make connections between different representations and understandings and to better understand the mathematics that they are learning. In addition, when students build and express multiple external representations, this allows observers (such as teachers, researchers, and fellow classmates) to better understand the students’ ideas. Using representations to make sense of problems and using representations to communicate ideas are therefore the building blocks of effective argumentation.

3.3 Early Counting Task Strand – Shirts and Jeans As noted in the previous chapter, the researchers in the longitudinal study aimed to provide the students with mathematical problems for which they had no algorithms and which would afford them opportunities to find patterns, be systematic, and generalize findings. Combinatorics problems were well suited to these goals. In the sections that follow, we will consider the specific mathematical ideas, fundamental both to combinatorics in particular and to mathematics in general, that are elicited by the tasks that were used in the longitudinal study. The shirts and jeans task (above) introduces the fundamental counting principle, a key idea in combinatorics.

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Fig. 3.1 A diagram and an organized list for displaying the shirts and jeans solution

In solving this problem, students may abstract the mathematics underlying the real-world situation; they may come to realize that the number of combinations of shirts and jeans is equivalent to the product of the number of shirts and the number of jeans. This type of multiplication is generally the most difficult for children to model, comprehend, and apply (Cathcart, Pothier, Vance, & Bezuk, 2006). This problem also introduces the need for notation or symbols to represent the real-world items described in the problem. So this problem creates a need for a bridge between the real-world situation presented and the mathematical ideas that will provide a solution. The problem also provides students with a chance to realize that an organization of the facts (by means of a diagram or an organized list) can help them to find a solution; this need for structure is fundamental to mathematics. The problem also requires students to think about how to justify their solution to others and convince others that they have found all the combinations. It has the potential to give rise to the need for direct or indirect arguments (Fig. 3.1). There are six combinations. In the figure, the letter “B” indicates a blue item, “W” indicates a white item, and “Y” indicates a yellow item. The shirt colors are listed on the left and the jean colors are on the right. The blue shirt can be combined with either the white jeans or the blue jeans to form an outfit; so two (and only two) outfits can be made using the blue shirt. The same is true for the white shirt as well as for the yellow shirt. Therefore, there are 2 × 3 or 6 possible combinations. All possible combinations are accounted for, since any other attempted combinations will be duplicates of the ones listed above. For example, there cannot be a third outfit formed using the blue shirt, because only white and blue jeans are available. The same is true for the other color shirts.

3.3.1 Second-Grade Problem Solving The students worked on the “shirts and jeans” problem in the second and third grades. The strategies of three students, Dana, Stephanie, and Michael (see Fig. 3.2), are analyzed and discussed in Martino (1992), Maher and Martino (1992a), and

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Fig. 3.2 Dana, Stephanie, and Michael, grade 2

Maher and Martino (1996). In addition, selections of the video were included in the Private Universe Project in Mathematics (Harvard-Smithsonian Center for Astrophysics, 2000), and the mathematical thinking and representations of these students are discussed there. In the second grade, all three students drew pictures of shirts and jeans to represent the items in the problem, and they used the pictures in their attempts to find different combinations of the shirts and jeans. Stephanie drew three shirts and labeled them “w” for white, “y” for yellow, and “b” for blue. She drew two pairs of jeans, similarly labeled “b” and “w.” She then began to make a list, writing the letter symbolizing the shirt directly above the letter that represented the jeans that together comprised an outfit. She then numbered the combinations that she found. However, when recording the fifth combination, she erased the “w” that she initially wrote to make a combination of a white shirt and blue jeans, and, in its place, wrote a “y” to show the combination of a yellow shirt and blue jeans (see Fig. 3.3). She then told the researcher that she had found five combinations and she was convinced that she had found them all.

Fig. 3.3 Stephanie’s grade 2 written work

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Fig. 3.4 Dana’s grade 2 written work

Dana, early on, expressed verbally her understanding of the structure of the solution when she indicated that each of the three shirts could be combined with each pair of jeans. She said, “He can make all three of these shirts with that outfit” (Martino, 1992, p. 47). It can be concluded from this statement and the subsequent problem-solving steps that she took that she had built a scheme that closely matched the problem solution. As Martino (1992) notes, “From her explanation it can be inferred that Dana possessed a key strategy for exhausting all possible combinations” (p. 48). Dana used a strategy of connecting her representations of shirts and jeans that she had drawn with lines as is shown in Fig. 3.4. An interesting decision that Dana made was not to draw a line between the yellow shirt and white jeans because a yellow shirt and white jeans do not “go together.” She then used Stephanie’s strategy of listing and numbering the combinations, so Dana also arrived at a solution of five combinations. However, Stephanie’s solution lacked the combination of a yellow shirt and blue jeans, while Dana’s was missing the combination of a yellow shirt and white jeans (the combination that did not match, according to Dana). We can conclude that she was aware of all possible outfits but her sense of fashion resulted in her rejecting the yellow shirt and white jeans. Michael’s strategy differed significantly from that of his classmates. He drew diagrams of the different color shirts and jeans, but said that he had arrived at three combinations: a white shirt with white jeans, a blue shirt with blue jeans, and a yellow shirt with yellow jeans (see Fig. 3.5). Although Stephanie and Dana pointed out that the shirts and jeans did not have to be the same color, Michael did not make any changes to his own solution.

3.3.2 Third-Grade Problem Solving In the third grade, the students were again given the shirts and jeans task. Stephanie and Dana again worked together, and they immediately began to draw diagrams to represent each item in the problem. Stephanie then suggested that they draw lines to

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Fig. 3.5 Michael’s grade 2 written work

show each combination, and they arrived at a total of six combinations. When questioned about why they drew lines to show the combinations, Stephanie explained that that was to ensure that they do not make any duplicate combinations. Figure 3.6 shows Stephanie’s drawing with numbered lines to keep track of the outfits. Dana, in her grade 3 drawing, again used a tree representation to form all shirts and jeans outfits as indicated in Fig. 3.7.

Fig. 3.6 Stephanie’s grade 3 written work

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Fig. 3.7 Dana’s grade 3 written work

Fig. 3.8 Michael’s grade 3 written work

Michael worked on this task with another student, Jaime. This time, Michael used the strategy of connecting lines between the shirts and jeans to represent the possible outfits, but, unlike Stephanie and Dana, he drew lines between the words in the problem, rather than between drawings of the shirts and jeans as indicated in Fig. 3.8. For example, he drew a line between the word “white” and the word “blue,” signifying an outfit of a white shirt and blue jeans. Using this strategy, Michael also arrived at a solution of six combinations. He used a strategy similar to that used by Stephanie in the second grade: he listed the combinations by writing the letter representing the color shirt above the letter representing the color jeans.

3.4 Cognitive Implications and Differences Observed In the second grade, none of the three students arrived at the correct number of outfits, although the way they solved the problem gave evidence that they were

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building schemes that could account for some or all of the outfits. All three students drew pictures in order to model the shirts and jeans, and all three used notation (first letter abbreviations) to indicate the colors of the shirts and jeans. Their representations of the problem included letters and line diagrams. It is important to note that Dana showed evidence of building the scheme for controlling for variables in the second grade. If not for her sense of style, Dana would have arrived at the correct answer of six outfits in the second grade. In the third grade, all three students arrived at a correct solution and none recalled that their correct solution was different from the solution they found in grade 2. This time, Stephanie offered a justification, explaining that the lines that they drew between the shirts and the jeans ensured that they accounted for all possibilities and also that they had not counted any combination more than once. Stephanie also used a system of counting that enabled her to keep track of her outfits. The children’s representations show that they are aware of the components of the problem and how they are put together.

3.5 Discussion This task prepares the groundwork for future tasks in combinatorics. It invites students to bring forth personal representations and it offers opportunities for sense making, so that students can begin to discuss how they arrived at their solutions and how they know their solutions are correct. Also, different ways of reasoning can be explored while students can learn how to formulate organizational schemes that can help them solve other problems in mathematics. In addition, the real-world setting of the problem shows the direct connection between the mathematical ideas and the world that students know. It also gives students a chance to think about the influence that real-world considerations have on mathematics, as can be seen from Dana’s sense of fashion and her insistence that one combination of shirts and jeans cannot constitute an outfit. Thinking about real-world considerations (which are important and which can be ignored) is a necessary step on the journey to mathematical understanding and abstraction. It is interesting to note the variety of strategies and approaches that were used in this problem. For example, in second grade, the three children used three different strategies to solve the problem. Although none of the strategies produced a correct answer, it should be emphasized that arriving at the correct answer was not the goal. In fact, when asked as third graders what answer they gave in grade 2, these students all responded that they found six outfits. For the researchers, the goal of the sessions was not for the children to give the correct answer; we were confident that they would eventually succeed. Our primary goal was to engage the children in thoughtful problem solving that could trigger growth in schema. We wanted solutions to be meaningfully constructed by the students. As the following chapters will reveal, a pattern of working with the students in the longitudinal study was to revisit problems and solutions in cycles so that earlier ideas could be built upon and new representations could be revealed. As we

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studied the progress made by the students over the years, we gained insight into their developing ideas and ways of reasoning. We observed the heuristics they used and the schemes that were later retrieved and modified. Over the months, there is evidence of durable learning. Also, there is evidence that children learned from each other as revealed in elements of one another’s strategies reappearing in their second attempt at solving the problems. More importantly, perhaps, is that each student incorporated strategies in unique ways. For example, Michael’s use of lines and notation to show combinations (see Fig. 3.8) differed significantly from the way that Stephanie and Dana used lines and notation. This suggests the importance of encouraging the use of students’ personal representations in building solutions. These episodes also demonstrate some benefits of group work. The contribution of each student to the cumulative body of knowledge enables students to arrive at their own solutions while simultaneously benefiting from the knowledge of others. As the data indicate, the children built durable schemes to solve the mathematical tasks that they were given. They used personal representations, as was seen in Dana’s and Stephanie’s second-grade work, to make meaning of the problem situation and to make an organized, systematic attempt to solve the problem. In addition, the data show that the representations and arguments that were originally built by individual students were used to effectively communicate to others the schemes upon which they had been built. This can be inferred from the students’ work in the third grade, when these schemes and representations were assimilated into the solution strategies and representations of classmates. In this chapter, we have seen that, at an early age and at the beginning of their investigations, students worked to make sense of the mathematics; they readily communicated their ideas and built on others’ ideas in forming their solutions. In Chapter 4 , we follow two of these students – Dana and Stephanie – along with a third student, Milin, as they work on more problems designed to help them explore fundamental ideas in algebra and combinatorics.

Chapter 4

Towers: Schemes, Strategies, and Arguments Carolyn A. Maher, Manjit K. Sran, and Dina Yankelewitz


1990–1992; Grades 3 and 4 Towers Dana, Jeff, Michelle I., Milin, and Stephanie Carolyn A. Maher and Amy M. Martino

4.1 Introduction In the previous chapter, we examined the representations, strategies, and problem-solving schemes used by four second- and third-grade students to build their solution to the shirts and jean problem (which was to determine how many outfits could be formed from three different shirts and two different pairs of jeans and to provide a convincing argument of the solution). In their effort to make sense of the components of the problem and to monitor their work, the students developed various notations to represent the data and illustrated the use of certain strategies. In this chapter, we examine how those students and others in the longitudinal study build on those representations and strategies in their work on some towers problems. (A towers problem involves determining how many towers can be built of a given height from a specified number of colors of Unifix cubes, small plastic cubes that can be stacked together. Because Unifix cubes have a vertical orientation – they have a top and a bottom – so do towers. An n-tall tower is one that was built from n Unifix cubes. Appendix A provides an analysis of solutions to the towers problems.) In this chapter, we examine the representations and strategies such as looking for patterns, guess and check, and controlling for variables that were used by students as they worked on the towers task. We trace students’ use of heuristics and ways of reasoning that were exhibited in their earlier problem solving with shirts and jeans. Finally, we trace the growth in students’ mathematical reasoning as their arguments and solutions took on proof-like forms. C.A. Maher (B) Graduate School of Education, Rutgers University, New Brunswick, NJ, USA e-mail: [email protected]


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As students were introduced to new problems and worked to make sense of the problem tasks, we observed growth in their knowledge as evidenced by the models they built, the identification of new and more elaborate patterns, and the structure of the arguments they provided in support of their solutions (Maher, 2002). Older ideas were elaborated and expanded upon. Students’ active engagement in the problem solving gave opportunity to build new ideas and methods of argumentation. As they attempted to resolve issues that could not be solved with their existing schemes, new schemes were built to accommodate the conditions of the problems. The structure of the towers problem served as an assimilation paradigm (Davis, 1984) for students’ later work with problems of similar structure, providing the students with a conceptualization that we see used in later years to tackle more complex combinatorial problems.

4.2 Stephanie We discuss here Stephanie’s emerging strategies as she worked on the towers problem in the third and fourth grades.

4.2.1 Stephanie Grade 3, Class Session The third-grade students in the study were asked to find all possible combinations of four-tall towers that can be made when selecting from two colors (in this case, red and blue). The strategies of a number of these students are documented and discussed in Martino (1992). We present here a discussion of Stephanie’s strategies as she worked on the task in the third grade (see Fig. 4.1).

Fig. 4.1 Dana (left) and Stephanie (right), grade 3 tower exploration

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Towers: Schemes, Strategies, and Arguments

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Fig. 4.2 Stephanie’s four-tall opposites

RED

BLUE

Stephanie and Dana began by working independently. Stephanie built ten towers. She began by making a four-tall tower and then its “opposite,” that is, a new tower of the same height with the second color in the corresponding position. Her first five towers included two sets of “opposites,” such as the tower with four blue cubes and its opposite, the tower with four red cubes (see Fig. 4.2). Dana also initially built ten towers, including two pairs of opposites. Then the two girls decided to combine efforts, and Stephanie took each of Dana’s towers in turn and checked it against her own to see if it was a duplicate. STEPHANIE: DANA: STEPHANIE:

Everything we make, we have to check. Everything we make. . . Let’s make a deal. Everything we make, we have to check. All right. I’ll always make it and you’ll always check it. Okay, you make it and I’ll check it.

When a duplicate was found, it was dismantled and returned to the pile of cubes. After this process, Dana and Stephanie now had 14 tower combinations. Stephanie suggested that Dana build new towers while she checked each new tower against the existing ones to ensure that it was not a duplicate. They finally eliminated all duplicates, and after attempting to find more combinations but not succeeding, they concluded that there were only 16 combinations, since they had checked many times and could not find new towers. This activity was marked by a number of emerging strategies. First, Stephanie and Dana used trial and error to find as many towers as they could. In addition, both thought of finding a tower and its opposite in an attempt to generate as many towers as possible, but neither used this strategy extensively or consistently. Further, the two decided to compare results and eliminate duplicates, and ultimately used this strategy of elimination to find the remaining tower combinations. Stephanie and Dana’s attempt to prevent duplication of combinations as they worked on the towers task is reminiscent of their strategy for solving the shirts and jeans task in the second and third grades (see Chapter 3 for an in-depth discussion). As they worked on the solution to that task, they used lines to ensure that they counted each combination of clothing once and only once. They explained to the researcher:

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RESEARCHER: STEPHANIE:

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What are these lines that you drew? You drew lines between the shirts and the pants. So that we could make sure; so instead of we didn’t do that again and say, “Oh, that would be seven, eight, nine, 10.” We just drew lines so that we can count our lines and say, “Oh we can’t do that again, we can’t do that again.”

As they worked on the towers task, Stephanie and Dana again were careful to check each combination against the others to ensure that there were no duplicates. RESEARCHER:

STEPHANIE:

How could you be sure that you haven’t made any of them twice or that one of you got them all? Is there a way that you could be sure? Well, there is a way. You could take one, like say we could take this one, this red with the blue on the bottom and we could go, we could compare it to every one. And the ones that match – that don’t match, put back; and the ones that do match, eliminate.

Stephanie and Dana were then asked to predict how many three-tall towers they could build. Stephanie first predicted that there would be the same number – 16; and other groups predicted that there would be more three-tall towers than fourtall towers. Upon experimentation, they found that removing one cube from each of their four-tall towers resulted in duplicates, or “pairs,” and they concluded that there were only eight combinations of three-tall towers (see Fig. 4.3). During an interview the next day, Stephanie explained why there were fewer three-tall towers than four-tall towers. RESEARCHER: STEPHANIE:


What do you think you learned from what you did? Well, we learned that . . . you might think there’d be more because there are less blocks so there’s more combinations you can make. There’s less because once you take one block off, say you have red, red, red, red, and you have red, red, red, blue. Once you take red, one red away and one blue away, they’re the same. Oh . . . So then you don’t have more. You haveHave less.

Fig. 4.3 From four-tall to three-tall

RED

BLUE

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Stephanie and Dana began to make conjectures and inferences based on their previous knowledge; both went on to suggest that there would be more five-tall towers than four-tall towers.

4.2.2 Stephanie: Grade 4, Class Session In the fourth grade, on February 6, 1992, the students were asked to find all possible five-tall towers, selecting from two colors (in this case, yellow and red). Stephanie and Dana began to make towers along with their opposites (see Fig. 4.4); and they checked their work as they progressed to prevent duplication. DANA: STEPHANIE: DANA: STEPHANIE:

And then I got another idea. Well, tell me it so I can do the opposite. I’m going to do the red – this, thatShow me. Oh, okay, and I’ll do the red – and I’ll do it with the red at the top.

At one point, they realized that an individual tower could be turned upside down to create a new tower. Dana called this new tower “cousin” (see Fig. 4.5). They used this strategy to find more possible arrangements. After forming as many towers as they could using this strategy of trial and error, they arrived at 32 different towers, arranged in pairs with a tower and its opposite and a tower and its cousin. Dana also considered different ways of arranging specified sets of towers that she referred to as “families.” An example of Dana’s “family” is the elevator pattern consisting of exactly one red cube (see Fig. 4.6). In her discussion with the researcher, Dana justified that there could only be five towers in this family because “it only goes up to five blocks.” Her reasoning indicates an argument by contradiction of the

Fig. 4.4 Stephanie and Dana’s group work

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Fig. 4.5 Dana’s tower and cousin

RED

YELLOW

Fig. 4.6 Dana’s family of one red cube and four yellow cubes

RED

YELLOW

given condition that the towers should be five-tall. If another red were added, the result would be a six-tall tower. RESEARCHER: DANA: RESEARCHER: DANA:

Are there any other members of this family? No. Why not? Because it only goes up to five blocks.

Stephanie and Dana located other “families” of towers with exactly two red cubes (see Fig. 4.7). Stephanie explained to the researcher: With two [red cubes] together, you can make four. With one [yellow cube] in between, you can make three. With two [yellow cubes] in between, you can make two. With three [yellow cubes] in between, you can make one. But you can’t make four in between or five in between or . . . anything else because you can only use five blocks.

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Fig. 4.7 Families of five-tall towers with exactly two red cubes

RED

YELLOW

By the close of this whole class activity, Stephanie and Dana had begun to explore an exhaustive method of finding the combinations of towers that were five cubes tall. This marked Stephanie’s first use of a partial argument by cases as she worked on the towers task.

4.2.3 Stephanie: Grade 4, Interviews Stephanie’s work on problems involving towers continued throughout the fourth and fifth grades (see Maher & Martino, 1996a, 1996b, 1997) and again in grade 8 (see Chapter 7). Stephanie’s growth in understanding of the idea of a mathematical proof is further documented by Martino and Maher (1999) and Maher and Speiser (1997b). Data from these episodes are presented here with attention to the emergent strategies that Stephanie used while working on the tower tasks. In an interview following the class session described above, Stephanie, using red and blue cubes, extended her family organizations of opposites, cousins, and elevators, to include a new organization, the “staircase” pattern (see Fig. 4.8). She discovered that introducing additional patterns sometimes resulted in duplicate towers that needed to be eliminated by checking. She said, “Yeah, we kept – we kept finding different patterns, but we didn’t check it with the other patterns.”

Fig. 4.8 Stephanie’s use of different patterns resulting in duplicates

RED

BLUE

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The interviewer asked Stephanie if there was a way she could be sure of how many towers of a specific type could be made. STEPHANIE: RESEARCHER: STEPHANIE: RESEARCHER: STEPHANIE:

RESEARCHER: STEPHANIE: RESEARCHER:

STEPHANIE:

I guess . . . a very lucky guess. Is there anything else possible for towers with exactly one blue? No. Why are you convinced? Because if there are towers of five, you can only build that many [with one blue cube]. You can’t really be convinced for everything because there’s no absolute way . . . you can’t go and say “I’m right.” [referring to the set of towers with one blue cube that Stephanie had shown] Well, this is an absolute way. Yeah, this is one of the absolute ways. This absolute way is when you looked at only one blue and I wonder if you could find absolute ways for looking at maybe two blues, three blues, or four blues. You could. Yeah, it is possible to have a certain number and get it right.

With this exchange, Stephanie demonstrated that the elevator pattern provided a convincing argument for justifying the number of towers with exactly one (or four) of a color. She also seemed to consider that other organizations, such as exactly two of a color, could be convincing. She began to consider families of towers as belonging to cases that could be justified individually to create the mutually exclusive and exhaustive set of cases for building an argument for finding all five-tall towers. In the latter part of the session, Stephanie used letters O and B to represent two colors. She made a grid with rows and columns to represent different six-tall towers. Notice, in Figs. 4.9 and 4.10, that Stephanie kept the entries in two rows constant, the top two rows in Fig. 4.9 and the bottom two rows in Fig. 4.10. Notice, also, in both figures Stephanie applied her elevator pattern while holding both rows constant.

Fig. 4.9 Towers with top two rows constant

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Fig. 4.10 Towers with bottom two rows constant

BLACK

WHITE

Fig. 4.11 Stephanie’s global organization

In a subsequent interview, Stephanie shared with her classmates the strategy of controlling for variables, that is, keeping the color of a cube in a particular position constant. This method of controlling for variables was useful to her in keeping track of larger number of towers (Maher & Martino, 1996a). During an individual interview on March 6, 1992, Stephanie presented a complete argument by cases. She was able to produce a global organization for four-tall towers. In her justification, she focused on number of white cubes yielding five categories of towers: towers with no white cubes, towers with exactly one white cube, towers with exactly two white cubes, towers with exactly three white cubes, and towers with exactly four white cubes (see Fig. 4.11).

4.3 Milin 4.3.1 Milin: Grade 4, Class Session During the February 6 class session, Milin worked with Michael on the five-tall towers task. Milin’s work has been referred to in earlier publications (Alston & Maher, 2003; Maher & Martino, 1996a) and was analyzed in greater detail by Sran (2010). Together with Michael, Milin began by using the strategy of building a tower using trial and error and then making an opposite for each tower. Michael and Milin

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Fig. 4.12 Milin’s cousin and opposite ways of making pairs

RED

YELLOW

Fig. 4.13 Milin’s cases of red cubes separated by one, two, and three yellow cubes

RED

YELLOW

Fig. 4.14 Class discussion and sharing of solutions

always paired their towers. Milin sometimes used the “opposite” strategy to make a pair and other times he utilized the “cousin” pairing, by inverting a tower. At the conclusion of the group work, the boys found all 32 towers. The strategies used by the two students on February 6, 1992, included: trial and error, building an opposite tower to complete a pair by switching the color of each cube, building an opposite tower to complete a pair by inverting the original tower, and monitoring work by checking for duplicates by comparing to previous towers (see Fig. 4.12). Milin noted that there were three possible combinations of towers in which the red cubes were separated by one yellow cube, two in which the red cubes were separated by two yellow cubes, and one in which the red cubes were separated by three yellow cubes (see Fig. 4.13). During the sharing session (see Fig. 4.14), the children were attentive as they listened to the findings and strategies used by their classmates.

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4.3.2 Milin: Grade 4, Interviews In an interview on February 7, the day after the class session, Milin made sets of towers using the elevator method for moving a cube of one color to each floor of the tower and by moving two cubes of one color the same way. He then found the remaining combinations by trial and error, and by grouping towers together with their opposites. Although Milin believed that he had found all combinations, he was only able to provide a convincing argument for his elevator patterns and his solid towers (see Fig. 4.15). Later during this interview, Milin began to consider simpler cases, and he said that there were four towers that could be built that were two cubes tall, and two that could be built that were one cube tall. Milin continued exploring simpler cases after this interview and brought the cubes home to further explore his idea. During the second interview 2 weeks later (on February 21), Milin reported that there were 16 four-tall towers. Later on in the interview, Milin showed towers that were one-, two-, and three-cubes tall, and he recorded the number of combinations that were possible for each (see Fig. 4.16). Fig. 4.15 Milin’s partial organization by cases and opposites

RED

Fig. 4.16 Milin’s one-, two-, and three-cube tall towers

YELLOW

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Fig. 4.17 Milin’s inductive reasoning with “families”

BLACK

LIGHT BLUE

Then, in a third interview (on March 6), he showed that larger towers could be built from smaller ones. For example, one can build four two-tall towers from the two one-tall towers (a blue cube or a black cube) by placing either a blue or a black cube on the blue cube and then placing either a blue or a black cube on the black cube. Milin showed that groups of larger towers could be included in the family of the smaller tower from which it was built (see Fig. 4.17). Later during the March 6 interview, Milin suggested that his rule for generating taller towers from shorter towers breaks down after five-tall towers. Toward the end of this interview, he retracted this claim, and he suggested that there were 64 possible combinations of six-tall towers. When asked if his pattern would hold for towers taller than five, he said it should, indicating, “We followed the pattern till five. Why can’t it follow the pattern to six?”

4.3.3 Small Group Interview: March 10, 1992 – Grade 4 Three weeks later, in a small group interview, arguments were presented by a group of four children for accounting for all possible towers, three-tall, selecting from two colors. This group sharing is referred to as “The Gang of Four” (see Fig. 4.18). It was conducted so that the children could share their strategies and arguments for

Fig. 4.18 Milin, Michelle, Jeff, and Stephanie (left to right)

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building towers of a variety of heights in earlier investigations. A simpler version of the problem was chosen deliberately for this session, as the evaluation was intended to identify the forms of reasoning and methods of justification that the children used to convince themselves and one another of the validity of their solutions (Maher & Martino, 1996b). The session began with the researcher asking the students how many sixtall towers could be built. Milin answered, “probably 64.” He was asked to explain why, and he described his inductive rule: multiply the previous answer by two. MILIN: RESEARCHER: MILIN: RESEARCHER: MILIN:

Well, because there was a pattern. What’s that? You just times them by two. Times what by two? The towers by two, because one is two, and then we figured out two is two, and then, I mean four, and then-

Milin used inductive reasoning to justify his solution, extending the problem beyond the three-tall case given to the group. He said that there were two one-tall towers, four two-tall towers, and eight three-tall towers. He was asked to re-explain how he progressed from four to eight towers. In this clip, he noted that a cube of each color could be added to the top of the shorter tower to build the taller tower. RESEARCHER: MILIN: RESEARCHER: MILIN: RESEARCHER: MILIN: RESEARCHER: MILIN: RESEARCHER: MILIN:

Why eight? That’s what Jeffrey asked about. I know. Go ahead. Let Milin persuade Jeff. If you do that, you just have to add – for each one of those you have to addEach one of what? These four? Yeah. You have to add one more color for each one. Which way are you adding it? Where are you putting that one more color, Milin? No, two more colors for each one. SeeSo this one with red on the bottom and blue on the top. You could put another blue or another red.

Later, Milin explained the logic behind the leap from two-tall towers to three-tall towers using inductive reasoning. He was able to demonstrate his doubling rule with each individual tower. In his own words, he noted that there were two possible cubes to be added to each three-tall tower to make a four-tall tower: “This was for three, so you could add two for each one of the three.” Milin explained this doubling rule by drawing two three-tall towers from a two-tall tower by adding a different color cube on top. He took his first two-tall tower with a blue cube on the bottom floor and a red cube on the top floor and generated two three-tall towers by first adding a red cube on the third floor and then adding a blue cube on the third floor (see Fig. 4.19). It is interesting to notice how Milin chose to draw the towers rather than use actual cubes as he had during his individual interview.

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Fig. 4.19 Milin’s representation showing his doubling rule

Fig. 4.20 Stephanie’s representation of an argument by cases

In this session, Stephanie presented an argument by cases to account for building all possible towers, three-tall. She represented the towers in a grid using letters B and R, for blue and red cubes as indicated in Fig. 4.20. Details of the session are described in Maher and Martino (1996b). She showed that there was only one way to form a tower without any blues. Then she showed that there were three combinations of two red cubes and one blue cube using the staircase pattern. She then used an argument by contradiction to show that this pattern could be used to show that there were only three possible combinations. Stephanie said, Well, there’s no, there’s no more of these because if you had to go down another one you’d have to have another block on the bottom. But then you have with three blues – well, not with three blues. I’ll go like this.

Stephanie used the staircase pattern to argue by contradiction that there could not be a fourth arrangement of two red cubes and one blue cube. What is of interest here is that Stephanie felt the need to prove that her argument by cases was complete and convincing, even though no one had challenged her answer. Stephanie continued her argument by cases by describing all the possible combinations of two blue cubes and one red cube. Stephanie’s organization was interesting in that she separated the case of two blue cubes into sub cases: two adjacent blue cubes and two non-adjacent blue cubes. When her classmates pointed out that these two cases could be grouped into one broader group, Stephanie insisted on continuing her explanation as she had originally presented it. The entire conversation follows, starting with Stephanie’s description of the “all red” tower. STEPHANIE: RESEARCHER: STEPHANIE: RESEARCHER:

All right, first you have without any blues, which is red, red, red. Okay, no blues. Then you have with one blue – Okay.

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MILIN: JEFF: STEPHANIE: RESEARCHER: STEPHANIE: RESEARCHER: STEPHANIE: RESEARCHER: STEPHANIE: MILIN: RESEARCHER: MICHELLE: STEPHANIE: RESEARCHER: STEPHANIE:


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Blue, red, red; or red, blue, red; or red, red, blue. All right. You could put blue, blue, red; you could put red, blue, and blue. You could put blue, red, and blue. You could put . . . Yeah, but that’s not what I am doing. I’m doing it so that they’re stuck together. Okay. There should be one – there could be one with one red and then you could break it up and there’s one with two reds and there’s one with three reds and then . . . Ah, but see – you did the same thing, but there’s the blue. See, there’s all reds and there’s three reds, two reds. There should be one with one red. And then you change it to blue. Well, that’s not how I do it. Let’s hear how Steph – we’ll hear that other way; that’s interesting. Okay, now, so what you’ve done so far is – One blue, two blue. Okay, no blues One blue, two blue. One blue, and two blues, but Milin just said you don’t have all two blues, and you said that – why is that? All right, so show me another two blues. With them stuck together, because that’s what I am doing. In that case, no. Okay, so now what are you doing, Stephanie? What if you just had two blues and they weren’t stuck together, you could – But that’s what I’m doing. I’m doing the blues stuck together. Okay. Then we have three blues, which you can only make one of. Then you want two blues stuck apart – not stuck apart; took apart. Separated? Yeah, separated. And you can go blue, red, blue right here.

Although Stephanie insisted on explaining her method of using two categories of towers with two blue cubes during this session, she later indicated (in a written assessment) that she understood the arguments of Milin and Michelle. At that time, Stephanie organized her cases as her classmates had suggested, producing a more elegant proof by cases (Maher & Martino, 1996a). Toward the end of the session, the students used Milin’s argument by induction as a stepping-stone to generalize the solution to the towers problem. Their progression of this understanding is documented by Maher and Martino (1997, 2000) and Maher (1998).

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Stephanie, during interviews preceding “The Gang of Four,” noticed a relationship between the height of towers and the total number produced and conjectured a “doubling rule.” During the “Gang of Four” session, she made reference to her doubling pattern and offered that there would be 1,024 ten-tall towers that could be built selecting from cubes of two colors. However, during that session, she chose to justify her solution to the three-tall tower problem with an argument by cases (Maher & Martino, 2000). In Chapter 5, while working on another problem, “Guess My Tower,” we see Stephanie learn why the doubling rule works as she investigates Milin’s inductive argument.

4.4 Summary of Strategies and Justifications Figure 4.21 outlines the strategies, representations, and forms of justification used by Stephanie and Milin during the five sessions on the towers problems. Both Stephanie and Milin began by using trial and error and justifying their solution empirically. They both progressed to more sophisticated strategies and forms of justification. Stephanie looked for patterns and controlled for variables to eventually formulate her justification using cases. Milin considered simpler cases and then recognized the recursive nature of the problem, arriving at his inductive justification. Both Milin and Stephanie arrived at a complete justification of their solution during The Gang of Four session. In addition, both students chose not to use the Unifix cubes to represent their towers but instead used notations in a grid (Stephanie) and drawings (Milin) to represent the different tower combinations. STEPHANIE Strategies Class Session Interview 1 Interview 2 Interview 3 Gang of Four

Tools

MILIN Justification

Trial and error; Unifix cubes Patterns, Opposites; cousins, partial cases Staircase Partial cases Drawings Pattern recognition and symbols Controlled for variables Staircase; Controlled for variables Staircase, Controlled for variables; Pattern recognition

Strategies

Drawings and symbols Drawings and symbols

Emergent cases

Trial and error; Opposites; Staircase Patterns; partial cases; simpler problem Considered simpler cases Inductive pattern recognition

Grid with symbols

Case argument

Inductive pattern recognition

Partial cases

Tools

Justification

Unifix cubes Partial cases

Drawing and Inductive argument symbols

Unifix cubes Partial Cases Unifix cubes Partial induction Unifix cubes Emergent induction

Fig. 4.21 Strategies, representations, and justifications used by Stephanie and Milin

4.5 Discussion The Gang of Four session evidenced particular structures and modes of reasoning by Milin and Stephanie in their justification of their solutions to the towers task. The students built and refined their representations over a period of time in which they

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had the chance to reflect upon the task, recognize emergent patterns, and choose schemes that best matched the representation that they had formed. Stephanie used symbols within a matrix to organize the towers by cases; Milin used drawings of towers to explain how they grew. The call for justification of the three-tall tower task enabled Stephanie and Milin to make public the schemes that they had built earlier. There are some similarities in Stephanie’s early use of representations for both the towers task and shirts and jeans tasks. In the second grade, Stephanie listed the outfit combinations by using the initials of each color and recording the combinations in a vertical format (see Fig. 3.2). When working on the towers task, she again used initials for the colors of cubes using a grid organization to show the different towers. Stephanie also used the heuristic of controlling for variables as she organized her tower combinations, a strategy that her partner Dana had used in the shirts and jeans task. Milin’s strategies of considering simpler cases and pattern recognition were powerful tools in his building of an inductive argument. As will be seen in Chapter 5, both students’ schemes proved durable as Stephanie and her classmates folded back to reflect on their earlier work to make sense of more complex combinatorial tasks in later grades. Importantly, these data show the advantage to revisiting tasks, group discussions about ideas, and sharing strategies. All of these components play a key role in the formulation and refinement of justifications. Stephanie and Milin, after having had multiple opportunities to think about and justify their ideas, presented a compelling argument to classmates during the group evaluation setting. As is evidenced in later years, unique aspects of the discussions that continued among this community of learners further triggered the development of more complex cognitive structures, triggered by the students’ need to produce justifications for combinatorial tasks of ever-increasing complexity. In Chapter 5, we follow Stephanie and other classmates as they continue to work on understanding Milin’s inductive argument for building towers as they retrieve earlier frames and cognitive structures revealed during the Gang of Four work.

Chapter 5

Building an Inductive Argument Carolyn A. Maher, Manjit K. Sran, and Dina Yankelewitz

Date and Grade: Task: Participants: Researchers:

February 26, 1993; Grade 5 Guess My Tower Bobby (Bobby is called Robert in later chapters), Matt, Michelle I., Michelle R., Milin, and Stephanie Carolyn Maher, Alice Alston, and Amy Martino

5.1 Introduction In the previous chapters, we followed the strategies, schemes, and arguments built by second-, third-, and fourth-grade students as they worked on combinatorial tasks. In this chapter, we trace how Stephanie and her classmates tried to make sense of the inductive method of generating towers. This strategy was originated by Milin, but it was eventually adopted by many other students. We attempt to identify the moments at which individual students gained ownership of the inductive argument and explained their new understanding to others.

5.2 Early Ideas In third and fourth grades, the children continued to build powerful strategies and schemes as they worked on the tower problems (see Chapter 4). To support their solutions, students followed two different approaches. Stephanie and others made extensive use of argument by cases. Milin, over a series of task-based interviews, built an inductive argument and he was able to use an inductive argument to show how to generate the number of combinations of towers of any height. Stephanie also



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observed this doubling rule (that the number of towers of height n was double the number of towers of height n – 1). Stephanie conjectured a doubling rule from her successful case-building justifications of towers of different heights.

5.2.1 Stephanie’s Individual Interview: May 15, 1992 On May 15, 1992, Stephanie participated in another interview. When asked whether she had thought about the problem since the small-group session, she showed the interviewer a sheet of paper on which she recorded the “doubling method” of finding towers of a specific height. The interviewer introduced the idea of using a tree diagram to show a recursive pattern that could be used to generate towers and to organize the tower combinations, similar to Milin’s inductive scheme. The researcher showed Stephanie the first two levels of the tree diagram and then asked Stephanie to extend the tree to include three-tall and four-tall towers. Stephanie responded by producing a partial extension of the tree organization as indicated in Fig. 5.1.

5.2.2 Written Assessments for Stephanie and Milin: June 15, 1992 At the end of the fourth grade on June 15, 1992, the children participated in a written assessment (see Fig. 5.2). The children worked in pairs on this assessment to provide a convincing justification of the towers task. Stephanie and Milin were partners and provided individual written work of their solution. (See Fig. 5.3 for Stephanie’s written work and Fig. 5.4 for the work of Milin.) In her letter, Stephanie gave an elegant argument by cases to show that she found all the towers, and then used a doubling pattern to predict taller towers, offering a general method. She said, “All you have to [do] is find the no. [number] for the problem before and mulity [sic] by 2.” Notice that the representations included ideas from both Stephanie and Milin, with the generation of numbers of towers as the height increases. Milin made a grid of towers three-tall using letters “B” and “G” to represent the two colors. Then he paired the two sets of “opposites” and two sets of “cousins” (see Fig. 5.4). He also demonstrated an understanding of the doubling rule in his written work. Notice, too,

Fig. 5.1 Stephanie’s partial extension of the tree diagram

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Fig. 5.2 Assessment June 15, 1992

Fig. 5.3 Stephanie’s June 15, 1992, written assessment

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Fig. 5.4 Milin’s June 15, 1992, written assessment

on page 4 of both children’s work that they were considering the building of threetall towers selecting from three colors. Stephanie showed some of those towers; Milin wrote in the lower corner 3 × 1 = 3 and beneath it, he wrote 3 × 3 =.

5.2.3 Written Assessments for Stephanie and Milin: October 25, 1992 At the beginning of the fifth grade, individual assessments were given on October 25. The children worked alone and produced individual written work (Fig. 5.5). In a follow-up individual written assessment at the beginning of fifth grade, Stephanie again used an argument by cases, and she checked her solution by using the “doubling” method (see Fig. 5.6). It’s interesting that again Stephanie relied on

Fig. 5.5 Assessment October 25, 1992

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Fig. 5.6 Stephanie’s October 25 written assessment

her case argument to justify her solution. (Refer to Maher & Martino, 1996a, for further details.) Milin, in his October 25 letter, again explained his doubling rule and drew all eight three-tall towers in a grid. He then also included the one- and two-tall towers (see Fig. 5.7).

Fig. 5.7 Milin’s written response to October 25, 1992, assessment

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5.3 Investigating Inductive Reasoning Further activities with towers led several students to make advances in understanding of inductive reasoning previously introduced by Milin to some classmates. We discuss below the traveling of ideas within a small community of students, initiated first, by Milin; then, from Milin to Michelle I.; then from Michelle I. to Matt; then from Matt to Stephanie, Bobby, and Michelle R.; and then from Stephanie to the entire group. These ideas were triggered by Milin’s inductive argument.

5.3.1 Stephanie and Matt’s Beginning Exploration Later in the fifth grade year (on February 26, 1993), the children worked on a problem called “Guess My Tower,” in which they were asked to decide what kind of tower was most likely to be selected at random from a box containing first all possible three-tall towers and then all possible four-tall towers (see Appendix A). The problem called for a revisiting of tower building, enabling the researchers to monitor the durability of students’ strategies and arguments. Since solving the Guess My Tower problem required the building of a sample space for all possible events, it required that the students revisit the question of finding the total number of four-tall towers. We have seen that earlier, Stephanie had already built and justified by cases all possible four-tall towers and used the doubling method for determining the total number of towers. She was exposed to Milin’s inductive argument during previous sessions on March 10 and June 15. Stephanie and Matt worked together on this problem first, using paper and pencil and then by building actual towers using Unifix cubes. They found all eight towers (see Fig. 5.8). The researcher asked Stephanie and Matt to predict how many four-tall towers they would find. Stephanie remembered the pattern that she had noticed the previous year. She said, “Oh, I remember the way that you could make sure how many. It was

Fig. 5.8 Stephanie and Matt’s towers

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whatever number you got from the last one, you multiply by two, and then you get the number of how many there will be for the next one.” Stephanie predicted, using the doubling rule, 16 four-tall towers, but they were only able, using a trial and error strategy, to find 12 towers. Stephanie insisted that there should be 16 based upon her confidence in the doubling rule. She shared with Matt the doubling pattern. STEPHANIE:

MATT: STEPHANIE:

Well, a couple of us figured out a theory because we used to see a pattern forming. If you multiply the last problem by two, you get the answer for the next problem. But you have to get all the answers. See, this didn’t work out because we don’t have all the answers here. I thought we did. No. I mean all the answers, all the answers we can get . . . I don’t know what happened! Because I am positive it works. I have my papers at home that say it works. I know that you had to multiply it [the total number of towers of a given height] by something. Maybe it wasn’t two because I know it worked. Maybe it was adding two.

Stephanie and Matt continued to attempt to find more four-tall towers, but they could not find more than 12. Stephanie continued to assert that the total would have to be 16. STEPHANIE: MATT: STEPHANIE:

I don’t know how it worked. I know it worked. I just don’t know how to prove it because I’m stumped. Steph! Maybe it didn’t work! Oh no. No. Because I’m pretty sure it would . . . I think we goofed because I’m still sticking with my two thing. I’m convinced that I goofed, that I messed up because I know that . . .

Stephanie’s memory of the doubling pattern and the fact that it had been used to verify her solution to the towers problem in the past was sufficient for her to remain convinced of its validity even when faced with a discrepancy in their own results.

5.3.2 Milin’s Explanation and Michelle’s Aha! As Stephanie and Matt worked to find more towers, Milin attempted to convince his partner Michelle I. that the inductive method of generating towers was a valid way of accounting for the total number of towers of a given height. Recall that both Milin and Michelle I. participated in the Gang of Four session almost a year earlier on March 10, 1992, in which Stephanie gave her argument by cases and Milin introduced his inductive argument. (Refer to Chapter 4 for details.) Michelle I. was introduced to Milin’s argument the previous year in the group discussion and gave evidence of some understanding of the doubling pattern introduced in the session. In fact, building from Stephanie’s organization of towers by cases, Michelle placed “2” above each representation of a tower entry from each of the three-tall towers, to

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Fig. 5.9 Milin explaining inductive reasoning to Michelle I.

Fig. 5.10 Michelle I. continues Milin’s argument to four-tall towers

indicate that two new four-tall towers can be generated from each. In this session, however, Michelle told Milin that she did not understand his explanation of how he generated new towers using an inductive argument. Milin then explained his reasoning for a second time (see Fig. 5.9). Once Milin finished explaining his reasoning for building three-tall towers by adding a red cube or a yellow cube to the tower from the previous stage, Michelle I. interrupted Milin’s explanation, extending his reasoning to towers four cubes tall. She explained the method to the researcher, commenting, “This is a lot simpler, from the last time we explained it.” Michelle’s explanation demonstrated that she had come to her own understanding of Milin’s method and was able to extrapolate the number of towers four-tall using her solution to three-tall towers (Fig. 5.10). Michelle also shared her understanding of the inductive method with Stephanie and Matt (see Fig. 5.11).

5.3.3 Matt’s Explanation and Stephanie’s Aha! Matt seemed immediately to grasp Michelle’s explanation of Milin’s inductive generation of towers. As Michelle I. continued to generate taller towers, Matt joined in the explanation, commenting “and so it’s a family tree” (see Fig. 5.12). Matt then proposed that Milin’s method might be connected with the doubling pattern that

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Fig. 5.11 Michelle I. shares her understanding with Stephanie and Matt

Fig. 5.12 Michelle I. and Matt’s family tree

Stephanie had been using. The researcher suggested that Matt be given an opportunity to explain to the group how Milin and Michelle I. had generated towers. Matt eagerly complied, showing that one could find the total number of towers of any height (see Fig. 5.13), by using a tree diagram to build towers in an organized fashion, explaining.

Fig. 5.13 Matt explaining to Bobby and Michelle R. the “family tree”

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Stephanie and Matt moved on to talk to Bobby and Michelle R., who showed them that they, also, had found 16 towers. Stephanie commented, “When we multiplied it out we got 16. . .. But we weren’t able to find that many. We were only able to find . . . like 12.” After Matt, joining Michelle I., had demonstrated his understanding of the inductive pattern, the researcher asked Stephanie to explain Matt’s reasoning to Michelle R. who still was not convinced about the inductive method. RESEARCHER: STEPHANIE:

RESEARCHER:

I want to know how you are going to get to two-high. Okay. Once there’s no more, there’s absolutely, positively no more, you can’t build any more with one. So you go to the next number. And the next number is two. Okay? So you have four of two. That’s a big jump for me, Stephanie. You’re jumping too fast from four to two. I don’t know how they came. I don’t know how they grew.

Stephanie, who did not provide an explanation of how the towers were growing from one- to two-tall, was interrupted by Matt who nudged her aside, saying, “move over” and began to explain how the towers were growing. As Matt explained to Michelle R. and Bobby, an attentive Stephanie looked on. MATT:

RESEARCHER: MATT:

RESEARCHER: MATT:

All right. Now, from here, you add an opposite, an opposite, an opposite or the same color on. So then you add the yellow and the red on to the last one. So you have . . . What do you think of that, Michelle and Bobby? . . . Do you understand what he is saying? . . . So you have a red on the bottom . . .. You add a red or a yellow on top. You have the same yellow on the bottom, but you add a red or a yellow on top. Then you have . . . Is that all you can do? That’s it. And then, well, for this you have red, yellow, yellow, red, like that.

As Matt finishes explaining the generation of three-tall towers from the two-tall towers, Stephanie joins Matt and explains how two four-tall towers can be generated from each three-tall tower. A confident and elated Stephanie (see Fig. 5.14) offers to explain their method to the entire class declaring Yes! I knew it! I knew it! I knew it! . . . I told him all along, I was right . . .

Stephanie demonstrated her understanding using the cubes to build four-tall towers and, counting the towers, commented STEPHANIE: RESEARCHER:

STEPHANIE:

I understand. I’m just very happy that my rule worked. Your rule worked. But what . . . you know what I think is really valuable . . . for people to understand is to know why that rule worked. Well, I know what it is now. I, I figured it out! But I’m just happy that it worked.

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Fig. 5.14 I knew it!


Because you see how you can forget a rule, but if you know why it worked . . . Yeah, yeah.

5.3.4 Stephanie’s Sharing Milin’s Family Tree Finally, during a whole class discussion, Stephanie confidently explained the reasoning behind her doubling pattern to her classmates as shown in Fig. 5.15. STEPHANIE:

I have one red, okay? And I have a yellow and from each of these you can make two because all you have to do is you add on . . . you can add on a red to a red and a yellow to a red . . . and for the yellow you can add on a red to the yellow and a yellow to the yellow, okay?

Fig. 5.15 Stephanie explains to the group

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MICHELLE I.: STEPHANIE:

ALL: STEPHANIE:

So you don’t have to look for duplicates. Then each one of these has two, like, okay? If this is like a family tree . . . the mother, the parents . . . the mother, the parents . . . and then six kids, okay? Well, no. Actually eight kids . . . then they have eight kids and each one of them has two kids. And this one, you can add one red, one yellow, one, yellow, one red, one red, one yellow . . . . And on, and on, and on, and on. ’Cause each one of them is different . . . you keep adding on. And then here you can add the exact same pattern.

5.4 Discussion

Argument

Other Participants

Primary Participant

Interaction Type

Date

Event

Throughout the session, the students had many opportunities to reconstruct earlier ideas and share them with others. They had occasion to revisit earlier ideas and they were encouraged to explain and re-explain their arguments. Communication of ideas, encouraged by the researcher, was the students’ responsibility. Clearly the students took ownership and learned together and from each other. Figure 5.16

Michelle I., Stephanie, Jeff

Inductive

Milin, Stephanie, Jeff Milin, Michelle I., Jeff

Extension of Stephanie’s cases to doubling

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Michelle I. Milin Milin, Matt, Stephanie Stephanie, Bobby, Michelle R. Michelle R. and Bobby

Argument by cases, noting of the doubling pattern Doubling pattern; organization by cases Doubling rule Cases; doubling; 3 colors Induction, doubling, Organization by cases, verification using the doubling pattern Explains inductive pattern Illustrates inductive pattern Presents inductive pattern Presents inductive pattern Leads presentation of inductive pattern

Fig. 5.16 The building of an inductive argument (PT: partner talk; SG: small-group discussion; RT: researcher talk; WC: whole class discussion; I: interview; E: evaluation; WA: written assessment)

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shows a trace by which the members of the group built an understanding of the inductive argument for the growth of the towers. Gaining ownership of a mathematical idea involves a process by which the learner takes responsibility for knowing. Faced with a conflict in one’s understanding, a learner can work with others to express what may be clear or unclear in their understanding so that an obstacle can be resolved. Faced with a conflict, students can be motivated to find resolution. Value added is that there is personal gain and confidence in one’s ability as a problem solver and achieve understanding. Children, given the opportunity to share ideas, can contribute to each other’s growth in understanding. In this chapter we have shown how sharing and discussion helped students attain ownership of mathematical ideas. In the next chapter, we show how students made use of the strategies developed here in solving new problems in counting.

Chapter 6

Making Pizzas: Reasoning by Cases and by Recursion Carolyn A. Maher, Manjit K. Sran, and Dina Yankelewitz

Date and Grade: Tasks: Participants: Researcher:

March and April 1993; Grade 5 Pizzas with halves; Pizzas Amy-Lynn, Ankur, Bobby (Robert), Brian, Jeff, Matt, Mike, Michelle I., Michelle R., Milin, Romina, and Stephanie. Carolyn Maher

6.1 Introduction In previous chapters, we followed the students in Kenilworth as they worked on the shirts and jeans task and the towers problems in the second through fifth grades. In this chapter, we discuss five sessions during which these students worked to make sense of the pizza problems. These problems presented new challenges and required the students to adapt the representations and solution strategies that they had previously formed to meet the needs of these tasks. (Portions of the data analyzed here are described in Bellisio (1999), Muter (1999), and Tarlow (2004).) As we trace the students’ problem-solving attempts, we will identify the forms of justification and reasoning that were used by the students, the methods of notation that they used, and the heuristics and strategies that were developed as the students worked to resolve the complexity of the problems. During the first four sessions, the students worked on the pizza with halves task: A local pizza shop has asked us to help them design a form to keep track of certain pizza sales. Their standard “plain” pizza contains cheese. On this cheese pizza, one or two toppings could be added to either half of the plain pizza or the whole pie. How many choices do customers have if they could choose from two different toppings (sausage and pepperoni)



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During the fifth session, the students were presented with three related tasks; they used the strategies that had been developed for the more complex pizza with halves task to tackle these variations.

6.2 First Session: Initial Pizza Explorations During the first session, the students worked in two groups as they made their first pizza explorations. We will trace the discussion of the two groups separately, though they worked simultaneously on the task.

6.2.1 Group 1 Five students (Jeff, Matt, Michelle I., Milin, and Stephanie) worked together. They started by drawing pictures to represent the pizza combinations. They settled their differences in notation as they began to talk about what their drawings represented. STEPHANIE: MATT: MICHELLE I.: MATT: JEFF: MICHELLE I.: STEPHANIE: MATT: MICHELLE I.: MATT: JEFF: STEPHANIE: MATT: MICHELLE I.:

P equals cheese. S is sausage. Why don’t you just put P for plain? P is plain. P is pepperoni. Plain is cheese. Hold on! Nobody is explaining this to me. What is PE? Pepperoni. Or just regular P would be pepperoni. Okay, P is pepperoni. One pie, all cheese. [Jeff draws a circle and writes C in it.] What is half a pie? HP? No, we’re just drawing.

Just as Stephanie and Dana had discussed their method of notation with each other as they worked on the shirts and jeans task in the second grade (see Chapter 3), the fifth-grade students were careful to make sure that their notations matched before they began to discuss their findings. This group of students used notation as well as drawings to identify their pizza combinations. Matt’s suggestion to use a notation of HP to signify half a pie signaled the need to clarify what would be symbolized using letters and what would be drawn.

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In addition to discussing their method of notation, they discussed which pizza combinations were distinct and which were identical and could only be listed once. MICHELLE I.: JEFF: MICHELLE I.: JEFF: STEPHANIE:

Then we can switch the side so that the sausageNo, it’s not going to matter which side. It’s going to be the same pie no matter what side sausage is on No, it’s a different side, though. It’s the same pie but it’s a different side. Nobody’s going to care if it’s this or if it’s like this. [He points to two circles, one with c/s and the other with s/c.] Nobody’s going to call up and say I want the pepperoni on the left side of the pizza.

Stephanie’s real-world contextualization of the problem settled the argument, and all the students agreed that the two combinations (sausage/cheese and cheese/sausage) were not different in practice. The students worked to find and justify their pizza combinations. After some discussion, four of the five agreed that there were ten possible pizzas, while Milin was still unsure that they had found all combinations. The group of students discussed their ideas with each other and looked at their own work. Matt found that he had named one pizza S and another one CS but both represented the same pizza, since all pizzas contained cheese (see Fig. 6.1). Matt and Stephanie then discussed how they should organize their list of pizzas. Matt made a chart with C, P, and S as headings, listed all the possible pizza combinations for each column, and then eliminated the duplicate combinations as indicated in Fig. 6.2. Stephanie then showed him a different way of organizing pizzas: You just show how many different combinations you can make with like two little things, three little things, four little things . . . So you put the one toppings here, all the ones with one topping on top. And then the ones with two toppings you put those in the second row, and the ones with three toppings, you put them in the third row.

Fig. 6.1 Matt’s first picture of half pizzas

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Fig. 6.2 Matt’s organized list of pizza combinations

Stephanie’s method of organization is of note here. When Stephanie worked on the towers problem (see Chapters 4 and 5), she organized the towers by cases. She first considered towers containing no cubes of a color, then one of that color, then two of that color, three of that color, and so on. Now when she began to organize her pizza combinations, she did the same, first considering pizzas with only one topping, then those with two toppings, and finally those with three toppings. At the close of this session, the five students were convinced of their solution; they had begun to systematize their combinations and discuss their differences in organization. We will now contrast the work of the second group of students and identify key differences in their strategies and representations.

6.2.2 Group 2 Amy-Lynn, Ankur, Bobby (Robert), Brian, Mike, Michelle R., and Romina were in the second group. As the students began their investigation, Ankur, assisted by Brian, took a leading role. He verbally listed six possibilities; when asked to show that he had all the possibilities, he directed the others to write down the choices as he dictated them. Five of the others worked with Ankur and also contributed by calling out possible combinations. Each student wrote the list of pizzas that was compiled by the group (see Fig. 6.3). Ankur’s list, like the lists of the other members of the group, used words to describe the pizza combinations. However, his drawings at the end of the list showed

Fig. 6.3 Ankur’s written work on March 1, 1993

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his move toward using letters to symbolize toppings. Of note as well is his division of the pizzas into halves as well as fourths, which may have been a factor that resulted in the students’ confusion about the total number of combinations. Mike worked alone. From the start, he used drawings of circles to represent pizzas, with small circles to represent pepperoni and wavy lines to represent sausage (see Fig. 6.4). He did not use any letters to symbolize his toppings. This method may have contributed to his confusion and duplication of pizzas. After a few minutes, the students decided to discuss the possibility of having more than one topping on half of a pizza. They then started to make lists together, as the students listed pizzas aloud and wrote them down, and Mike drew another set of pizza pictures, as Amy-Lynn labeled them (see Fig. 6.5). After this phase, some students had a list of 11 pizzas, while Brian conjectured that there were 12. Amy-Lynn found that there was a duplicate in her list of 11 pizzas and pointed out to Mike that he also had one repeated combination. ROMINA: ANKUR: AMY-LYNN: BRIAN:

It’s the same thing if you turn it around. No it’s not because of, half of it is mixed and the other half of it is pepperoni. We have that. We have half plain, half pepperoni sausage. Mixed.

Fig. 6.4 Mike’s initial written work (on two pages) on March 1, 1993

Fig. 6.5 Mike and Amy-Lynn’s version of the group’s work (two pages)

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That’s mixed. That isn’t mixed though [pointing to the ninth pizza].

In earlier years, as students worked on the shirts and jeans task and the towers problem, they were careful to eliminate duplicate combinations. As the students worked on the pizza with halves task, they did the same, and like the other group they had lively discussions about what was considered a duplicate. Their differences in their definition of mixed pizzas, however, were not resolved until the next session. The students carefully color-coded the pizza representations, but the first session was over before they had a chance to analyze or organize their list.

6.3 Second Session: Further Pizza Explorations The groups remained static between the first two sessions, and the students returned the next day and continued working on the problem. However, the students from the two groups had discussed the problem between the sessions and shared their solutions with each other.

6.3.1 Group 1 Michelle I. began the discussion by commenting that the other group had found more than ten combinations, and that this group’s solution from the previous session was therefore probably incomplete. Stephanie insisted that their original solution was correct. She said, “I mean, maybe they’re wrong. Did you think of that?” This group of students began to make charts to organize their pizza combinations. Matt was first to complete his chart, which showed the ten combinations categorized according to the number of distinct pizza toppings on each pizza as indicated in Fig. 6.6. Michelle’s chart, as indicated in Fig. 6.7, showed a total of 13 combinations. Milin told Michelle I. that she had duplicates in her list. MILIN: MICHELLE I.: MILIN:

She’s wrong. C P . . . Look, C. Yeah, cheese, pepperoni, sausage. He has it, but this is cheese pepperoni. That means cheese pepperoni versus cheese, that doesn’t count.

Fig. 6.6 Matt’s list of pizza combinations

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Fig. 6.7 Michelle I.’s list of pizza combinations

MICHELLE I.: JEFF: MATT: MILIN: MATT:

Yes it does because cheese and pepperoni are mixed together. On the other side is just cheese. But that would be the same as that, though. Think about it. Watch. It has to be halves. It already has cheese. We’re only working with halves, not quarters.

Stephanie then asked the group why they wrote a C for cheese at some times and not at others. Jeff insisted that cheese was not a topping. JEFF: STEPHANIE: JEFF: STEPHANIE: JEFF: STEPHANIE: MICHELLE I.: JEFF:

Cheese is another way of saying plain. You get a plain, cheese pizza. [Jeff writes C = plain.] Then why do we put cheese here and here and here? If we put plain that would get confused with that. Why couldn’t we jut put like that? What does that mean? Half of the pizza. One side sausage and one side plain. You could do that. It doesn’t matter, but we decided to put cheese there.

Now that the students had finally settled their differences about notation for a second time, they concluded that there were ten combinations. However, they first discussed whether a pizza with sausage and pepperoni mixed together was the same as or different from the pizza that had sausage on one half and pepperoni on the other. The researcher asked the students to explain their solution and justify it. She reviewed their drawings and lists and asked Matt and Milin why and how their lists differed from one another’s. They reported that Matt had organized the pizzas according to the number of toppings, while Milin had three categories: whole pizzas, half pizzas, and mixed pizzas.

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6.3.2 Group 2 As part of the research protocol, researchers had photocopied the students’ work from the previous day, kept the originals, and returned the copies to the students. When the students returned the next day, they were disappointed that the photocopies did not show their color-coding. They were joined by Researcher 1, who asked them to think about organizing their work before taking the trouble to draw elaborate diagrams. They decided to check whether all students had the same list of pizzas. As pizzas were checked off, Mike made a list of the accepted pizzas. The students realized that some of their pizza combinations were identical, such as the “half pepperoni half sausage plain and half pepperoni-plain half sausage.” A question arose as to whether a half pepperoni/half sausage and pepperoni pizza was allowed, and the group decided that it was. They again came up with a list of ten pizzas. As they tried to justify this answer, Amy-Lynn made a list of the pizza combinations. The students were convinced that there were only four whole pizza combinations, but were not sure that they had found all half pizza combinations. Ankur suggested that Amy-Lynn’s list be revised to show the different categories of pizzas. Michelle wrote this new list, first writing the four whole pizza combinations, and Ankur pointed out that there were only three half pizzas that did not contain any mixture of two toppings. They then listed the pizzas that contained topping mixtures on one of the two halves. Michelle labeled her cases “whole” and “halves.” The researcher then asked them to point out the difference between the first three half pizzas and the remaining four. RESEARCHER: BRIAN: BOBBY:

Okay, what was special about these first three halves? They were different from the others. They weren’t mixed. They were in the halves. If you want to use plain on one side, there’s only two possible ones, plain on one side and pepperoni has to be on the other side or sausage.

Brian then justified Bobby’s claim using an argument by contradiction: Because plain is considered like a topping so with plain, only two other toppings, because that is all they give you . . . so you can’t use plain again.

They were not yet prepared to state that they could justify the entire solution, although Brian argued that one case, the “whole” pizzas case, was fully accounted for. After further discussion, they concluded that all the toppings were used in all possible combinations for each case, and so they reported that they felt they could justify their solution.

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6.4 Third Session: Getting the Right Answer At the third session, the researcher discussed with the students the different kinds of arguments that they had used. When they separated the pizzas into categories of whole, half, and mixed, they were using an argument by cases. In previous investigations with towers, they had also used an inductive argument. They argued by contradiction when they had stated that no more “whole” pizzas were possible because all toppings were used. Researcher 1 also discussed the importance of using a notation that clearly represented their ideas and communicated the meaning of the ideas clearly to others. The session ended with the following discussion: ANKUR: RESEARCHER:

JEFF: RESEARCHER: CHORUS: RESEARCHER: CHORUS: RESEARCHER: JEFF: RESEARCHER:

Did we get that problem right? What do you think? Is this a question? Ankur? Did you get the problem right? How many of you believe that you got the problem right? Yeah, can you tell us the answer to it? What is the answer to it? Ten. Ten. Have you proved it? Yes! Then why are you asking me? ‘Cause maybe, you’re the oneIt’s up to you. You shouldn’t have to ask me.

6.5 Fourth Session: Giving the Solution At this session, the children were asked to write a letter describing their work on the pizza with halves problem to Drs. Davis and Alston. Their letters give the solution, but do not provide a justification for the answer. As an example, Mike’s letter is shown in Fig. 6.8. Amy-Lynn’s written explanation is of note in the method she described. She wrote, “We moved the toppings around to make new ones. When we made one, we made the opposite.” She then listed the three cases that her group had delineated (whole, half, mixed) and listed the combinations that could be formed for each case. Although Amy-Lynn did not explain what she meant by her terminology of “opposites,” this word is reminiscent of the students’ explorations with towers in the third, fourth, and fifth grades. (See Chapters 4 and 5.) During those activities, the students worked on finding tower combinations, and they consistently searched for towers and their opposites, which were the towers with colors reversed. AmyLynn’s reference to this strategy indicates that the students used similar heuristics as they tackled two problems that, on the surface, seem to need very different problem-solving tactics.

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Fig. 6.8 Mike’s letter on March 5, 1993

6.6 Fifth Session: Additional Justifications This session, lasting 2 h and 30 min, provided an opportunity for students to work on four different pizza problems and to expand their repertoire of justifications. They continued the pizza with halves problem, and they also worked on variations of the four-topping pizza problem, including thin and thick crust and the four-topping pizza with halves problem.

6.6.1 Problem 1: Pizzas with Halves At the beginning of this session, the students spent about 40 min discussing the three “pizzas with halves” cases. The “whole pizzas” case proved to be problematic, because this case needed to be defined more clearly. Brian and Ankur provided a precise definition: ANKUR: BRIAN: ANKUR:

. . . ‘cause if you take one slice, they’re all gonna be the same with the mix. If you take one slice and you take another slice and you compare it, it would have sausage and pepperoni on both. They would both have the same.

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Fig. 6.9 Final decision on cases for the pizzas with halves problem

This definition clearly marked the difference between “whole” pizzas and other pizzas. The students continued by placing the pepperoni/sausage “whole” pizza in a subcategory of its own and also placing the three “mixed” pizzas as a subcategory under the “half pizzas” category. Their final set of cases is given by the table in Fig. 6.9.

6.6.2 Problem 2: The Four-Topping Pizza Problem The students then considered the Four-Topping Pizza Problem: Kenilworth Pizza has asked up to help design a form to keep track of certain pizza choices. They offer a cheese pizza with tomato sauce. A customer can then select from the following toppings: peppers, sausage, mushrooms, and pepperoni. How many choices for pizza does a customer have? List all the possible choices. Find a way to convince each other that you have accounted for all possibilities.

It took about 15 min for the students to find 16 pizzas by randomly generating combinations of toppings. Ankur suggested an organizational strategy: ANKUR:

RESEARCHER: ANKUR: RESEARCHER: ANKUR:

Okay. You start with the first one, that’s P. And you mix it with the second one. That’s P slash S. And then you start with the first one again, skip the second one and go to the next one. That’s M; P slash M. Then you start with P again and mix it with the fourth one, PE. And then you start with the S since that’s the . . . ‘cause you can’t use plain. We start with S and mix it with M. Where’s that? S M. Then we start with S and PE, right here. And we start with M and PE. S and P is right here, the first one. [He points to P/S.] Okay. So why is it you can’t go M with P? Because you already have it. P M. [He points to P/M.]

This marked the first use of a recursive strategy on pizza problems. When questioned by the researcher about their solution, Brian explained that they were sure

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that they had found all possibilities because “we have an order.” The students were very confident about their solution. As Brian and Ankur told the members of another group: ANKUR: BRIAN: ANKUR: BRIAN:

Sixteen. And we can prove it. Sixteen. And you guys can’t prove it. And we can prove it.

6.6.3 Problem 3: Another Pizza Problem Following the four-topping pizza problem, the students considered a problem called Another Pizza Problem: Kenilworth Pizza was so pleased with your help on the first problem that they have asked us to continue our work. Remember that they offer a cheese pizza with tomato sauce. A customer can then select from the following toppings: peppers, sausage, mushrooms, and pepperoni. Kenilworth Pizza now wants to offer a choice of crusts: regular (thin) or Sicilian (thick). How many choices does a customer have? List all the possible choices. Find a way to convince each other that you have accounted for all possible choices.

Within less than a minute, without writing anything, students were proclaiming the answer of 32 pizza combinations. Mike provided the explanation: Well, since there’s sixteen to make with those toppings, you put a Sicilian crust on it. That’s sixteen. Plus then you put a regular on it, and that’s 32. Sixteen and sixteen.

Thus Mike provided a succinct explanation of the students’ reasoning by recursion.

6.6.4 Problem 4: The Final Pizza Problem The session concluded with the Final Pizza Problem, which included components of all the previous problems: At customer request, Kenilworth Pizza has agreed to fill orders with different choices for each half of a pizza. Remember that they offer a cheese pizza with tomato sauce. A customer can then select from the following toppings: peppers, sausage, mushrooms, and pepperoni. There is a choice of crusts: regular (thin) or Sicilian (thick). How many different choices for pizza does a customer have? List all the possible choices. Find a way to convince each other that you have found all possible choices.

The entire group worked on this problem for about 50 min. Ankur, Brian, Jeff, and Romina jumped from answer to answer without seriously considering reasons why the numbers would be an answer to the question. When Matt proposed a method based on Ankur’s previous method, they began to focus on finding a justified solution. Matt began with the 16 pizzas that were the answer to the first pizza problem. Then he made half pizzas, using cheese on one side with each of the remaining 15

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Fig. 6.10 Brian, Romina, Jeff, and Ankur’s written work on April 2, 1993

Fig. 6.11 Matt’s written work on April 2, 1993

combinations on the other half of the pizza. The next pizza had peppers on one half and each of the remaining 14 combinations on the other half. Matt continued this recursive procedure to arrive at an answer of the sum of 1–16, times 2 (to account for the two different choices of crust). Refer to Fig. 6.10 for Matt’s work. Although his procedure was correct, Matt’s answer was incorrect, due to an arithmetic mistake. Matt was unable to convince the other students to consider his procedure which is represented in Fig. 6.11. After some of the students left, a subgroup consisting of Ankur, Brian, Matt, Milin, Mike, and Stephanie continued to work on the problem. Romina proposed a new categorization procedure: First you’re gonna have your wholes and then you’re gonna have half with one topping on one side and two toppings on the other side. Then you’re gonna have one topping, three toppings. Four top- I mean, one topping with four toppings. And then you’re gonna go to

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C.A. Maher et al. [sighs] here. I’ll write it down. You’re gonna have a whole first. And then you’re gonna have all your wholes, which equals up to, I think, five. And then you’re gonna have half of it with one topping, half of it with two. Then you’re gonna keep on doing that until you come to four. Then you’re gonna go two; one, two; oh, two three.

After some discussion, the students were ready to listen to Matt’s argument. This time they agreed that his method would count all the possible combinations and not include any duplicates, and they accepted the recursive method as a valid strategy of justifying the solution to this complex task. The children, pleased with their work and solutions, expressed jubilation at the end of the session.

6.7 Discussion During these five sessions, the students used two kinds of justifications: proof by cases and recursive arguments. These forms of reasoning had been seen in the schemes displayed in students’ earlier work on the towers problem. Now we see how the students retrieve, build upon, and extend earlier schemes to reason about pizza problems, despite differences in surface features. As the students folded back, activated, and drew on previously built cognitive structures, they thought about new ways of applying these strategies and they worked on settling differences in understanding and notation. They were animated and engaged in these interactions. In addition, we see students adopting some strategies of others as they worked on new problems; we see the students sharing ideas with their group members and attempting to convince them of the validity of their methods. In this way, ideas traveled across the learning community and were applied and adapted to new challenges. The episode shows the continual development of cognitive structures that were built earlier. In Chapter 7, we examine in detail the work of one of the members of this group, Stephanie. We observe how, 3 years later, she expanded on the explanations discussed here and in Chapter 5 and explored connections between these problems and Pascal’s triangle, showing further elaboration of her cognitive structures.

Chapter 7

Block Towers: From Concrete Objects to Conceptual Imagination Robert Speiser

Date and Grade: Tasks: Participant: Researchers:

March 13 and 27, April 17, 1996; Grade 8 Binomials and towers Stephanie Carolyn A. Maher and Robert Speiser

7.1 Introduction In previous chapters, we looked at the development of various forms of reasoning in students working in a classroom in small group settings. In this chapter, we focus on an individual student – we examine Stephanie’s development of combinatorial reasoning. In previous chapters, we saw how Stephanie, working with others and on her own, made sense of the towers and pizza problems. In this chapter we see how Stephanie extended that work. In her examination of patterns and symbolic representations of the coefficients in the binomial expansion, using ideas from earlier explorations with towers in grades 3–5, she examined several fundamental recursive processes, including the addition rule in Pascal’s Triangle. This chapter centers on how children can build fundamental mathematical understanding, over time, through extended task-based explorations. They create models, invent notation, and justify, reorganize, and extend previous ideas and understandings to address new challenges. By the time of the interviews that we report here, we had been observing Stephanie for 8 years. Her work in combinatorics began in grade 2, with the shirts and jeans problem (refer to Chapter 3). Even at this early stage, she would validate or reject her own ideas and the ideas of others, based on whether they made sense to her or not. Stephanie would monitor and often refer to ideas and conclusions of other group members and would often integrate the ideas

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of partners into her work and discourse. This constant, extended process of evaluation and revision helped her to keep track of data and to reconsider, strengthen, and extend her explanations (Davis, Maher, & Martino, 1992; Maher & Martino, 1991, 1992a, 1992b). In grade 3, Stephanie was introduced to investigations with block towers (see Chapter 4) that enabled her to build visual patterns of her ideas, such as the local organization within specific cases, based on ideas like “together,” “separated,” “how much separated.” She recorded tower arrangements first by drawing pictures of towers and placing a single letter on each cube to represent its color, and later by inventing a notation of letters to represent the colored cubes. Stephanie’s working knowledge about towers, gained over long periods of time through very concrete explorations, led as, we shall see, to powerful and personally meaningful new ways to work with mathematical ideas.

7.2 Theoretical Perspectives We believe that children come to mathematical investigations with theories that can be built upon, modified, and refined. In turn, children’s theories and their ways of working with these theories help us, as researchers, to constitute our own conceptions of children’s emerging work and thought, and so affect the way we build the discourse, day by day, that we will share with them. In the task-based interviews that we report, we, too, will seek to build a theory. Our emphasis on building theory informs directly how we structure research interviews. Initially, one interviewer will engage the child in a specific exploration, seeking to estimate the working theory that might guide the child’s thinking. Later, in the same interview or in a subsequent follow-up interview, key ideas noted so far will be pursued primarily by the child, who initiates, and then increasingly directs, the discourse. In such interviews, we frequently begin with very concrete discussions, followed by what might be called a “teaching phase” intended to investigate deeper connections. In such interviews, children will sometimes make powerful connections early and so break the flow we might naively have imagined. We have come to view such “unique outcomes” as potential opportunities to gain important insights from the children that we study. Therefore, when a child’s connection appears to break the flow, the interviewer, on principle, will invite more detailed explanation. In Mindstorms (1980), Seymour Papert reflects on how he built his personal mathematical understanding – an understanding that inspired his later work – based on his personal experience, as a young child, playing with gears. In a similar way, some of Stephanie’s key mathematical understandings can be traced to her activities, in the early grades, when she used block towers to investigate conceptually important counting problems. The specific arguments that Stephanie investigates below were first developed and explained in Speiser’s paper (1997), where block towers underpin a concrete microworld for productive exploration. These arguments, shaped specifically within

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the given microworld, were triggered by the early “Gang of Four” investigations (Maher & Martino, 1996a, 1996b, chap. 4), which first describe the reasoning and argument that enabled Stephanie and three other children, at age 9, to discover the idea of mathematical proof, as they built and then debated strategies for counting block towers. Building from this work with towers (and inspired by the young Papert) we seek precise, particular descriptions (1) of how Stephanie actually does strong mathematics based on towers, and (2) of what, specifically, might constitute its strength.

7.3 Setting Stephanie participated in the longitudinal study starting in first grade. Stephanie and her classmates were challenged in their mathematics classrooms to build solutions to problems and construct models of their solutions. This setting, which for Stephanie continued to grade 7, encouraged differences in thinking that were discussed and negotiated. In fall 1995, Stephanie moved to another community and transferred to a parochial girl’s school. Her mathematics program for grade 8 was a conventional algebra course. Stephanie continued to participate in the longitudinal study through a series of individual task-based interviews. A subset of these interviews provides the data for this chapter.

7.4 Guiding Questions The following questions guide our analysis in order to consider, systematically, the ways in which Stephanie’s past experience is drawn upon: (1) How does Stephanie work with towers in building images and understandings for higher mathematical ideas? (2) What is the role of past experience in building new ideas? (3) How are her ideas modified, extended, and refined over time? Data come from two of eight individual task-based interviews of Stephanie. The interviews were videotaped with two cameras, positioned to capture in detail what was said, written and built and to include less tangible data such as tone of voice, speech tempo, and where people look while they converse and work. Transcripts and analyses of the interviews were made and verified by a team that included several graduate students in addition to both authors. Stephanie’s written work prepared for the interview, and several observers’ notes, provide further data. The teaching experiment was conducted over a 6-month interval (November 8, 1995, to May 1, 1996). Each interview, approximately one and one-half hours in length, would typically begin with inquiries about the mathematics that Stephanie currently studied in eighth-grade algebra, both to open opportunities to talk about that mathematics and to explore her thinking about fundamental mathematical ideas.

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7.5 Results To introduce each data segment, we provide a brief discussion of the mathematics Stephanie was invited to explore. On this basis, we can more clearly understand each segment as a momentary snapshot of Stephanie’s emerging understanding. The correspondence between binomials and towers. During the March 13, 1996, interview, Stephanie, unprompted, made a connection to towers, by examining her symbolic representation of the expansions of (a + b)2 and (a + b)3 . STEPHANIE: RESEARCHER: STEPHANIE:

RESEARCHER: STEPHANIE: RESEARCHER: STEPHANIE:

So there’s a cubed [a3 ]. That’s 1. And there’s three a squared b [3a2 b] and there’s three a b squared [3ab2 ] and there’s b cubed [b3 ]. [Interviewer writes 1 3 3 1 under 1 2 1 as Stephanie speaks.] Isn’t that the same thing? What do you mean? As the towers. Why? It just is.

Stephanie asserts (in her own way) that each three-high tower gives a noncommutative monomial of degree 3 in a and b, and she has indicated that these non-commutative monomials, indexed by the corresponding towers, collect to give the coefficients for the commutative monomials that appear in her expansions of (a + b)2 and (a + b)3 . Our interpretation, therefore, is that Stephanie visualizes towers (referring to mental models – she does not have plastic cubes at this point) to help her organize the terms that she collects. We believe that Stephanie reasons about polynomials based on her mental images of towers. Working at home before the interview, Stephanie had written out the first six powers of the binomial a + b, and brought her written calculations to the interview. The interviewer covered Stephanie’s paper, guessed the coefficients for the sixth-power expansion, and wrote down the terms in full. Her coefficients were the same as Stephanie’s, although one monomial was slightly different. Several minutes further in the conversation, Stephanie gives further evidence, that she proceeds by visualizing towers and then reasoning based on her mental images. RESEARCHER: STEPHANIE: RESEARCHER: STEPHANIE: RESEARCHER: STEPHANIE: RESEARCHER: STEPHANIE: RESEARCHER: STEPHANIE:

So you have two factors of a. Right? Um hm. You have one of those. One thing with two factors of a. One thing with two a’s in it. Um hm. I don’t want to think of a’s. I want to think of red. Okay [laughing]. Can you switch that a minute? Yeah. So now I have one thing with two reds. What thing can I be thinking of with two reds? That’s a tower that’s two high.

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Okay. And here I’m talking about two things. Um hm. One is. Red and . . . one is . . . . . . one is yellow. Is that possible in two high? Yeah. Having one red and one yellow? There are two of them? Yeah. Which two? ‘Cause the one is the red could be on the top or the bottom, with the yellow the same thing. What about b squared? Um. Two yellow.

In a March 27, 1996, interview, Stephanie is invited to explain to a second interviewer (unfamiliar with her recent work) what had happened in the March 13 interview described above. Here Stephanie begins with towers, then reviews the binomial coefficient notation C(n, r), working through a sequence of examples with increasing n. Stephanie remarks that “r is a variable,” which she understands can range from 0 to n. This observation shifts the level of abstraction upward from specific towers (as above) to patterns of formal symbols as in Pascal’s Triangle. Hence, at this point, n, the height, and r, the number of red blocks for given n, will both vary. This richer context triggers, with encouragement from Interviewer 1, a confident, detailed, and carefully presented recapitulation by Stephanie of the recursive construction of the towers of height n from the towers of height n – 1, as it had been introduced by classmate Milin in grade 4 and revisited in grade 5 (see Chapters 4 and 5). During a previous interview, on March 13, Stephanie also referred to Pascal’s Triangle, in particular to its addition rule, to make similar predictions, but she had done so in a conceptually quite different domain: to predict, in effect, the numbers of n-tall towers in each given case (of r red blocks, say, for given r) for new values of the height. Stephanie’s choice to center, in the present interview, directly on binomials strongly suggests that Stephanie now grasps the isomorphism between Pascal’s Triangle, which she had built, at first, to summarize her towers cases, and the array of coefficients for her polynomial expansions of the powers (a + b)n , for variable n. On this basis, further interviews were planned, with towers available to serve as concrete anchors to establish formal facts about the C(n, r), viewed either formally as binomial coefficients or as counts of combinations, or, more concretely, as the numbers of specific kinds of towers. Fermat’s recursion. One goal for the March 27 exploration was to offer Stephanie the tools she’d need to construct a formula, originally due to Fermat (Weil, 1984), that expresses the relationship between two successive binomial coefficients. In symbols, here is Fermat’s formula:

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C(n, r + 1) =

n−r r+1

· C(n, r)

(7.1)

This equation, applied repeatedly beginning with the simple case r = 0, leads directly to the standard formula for C(n, r). To make sense of this formula, it seems especially helpful in this setting to interpret Equation (7.1) in terms of towers. For concreteness, take red and yellow for the colors of the blocks available. On the right side, C(n, r) counts the towers of height n that have exactly r red blocks, hence n – r yellow blocks. Call these the original towers. On the left side, C(n, r + 1) counts the towers of height n with exactly r +1 red blocks. Call these the new towers. In concrete terms, Equation (7.1) tells us that the number of new towers can be found by multiplying the number of old towers by the number of yellow blocks in each and then dividing by the number of red blocks in a new tower. In the data below, the interviewers will fix n, the height of a tower, and then vary r, beginning either with r = 0 or r = 1, for which C(n, r) is either known to Stephanie or easily determined by inspection. For each r, the interviewers will then invite Stephanie to construct new towers from a given set of original towers and explore with her what she has found. The construction process Stephanie explores will work for any height n and any r < n. For concreteness, we explain this process when n = 4 and r = 1. In this case, we have four original towers, each with a red block in one of four available positions. From each given original tower, we can build new towers by replacing one of its three yellow blocks with a red block. For each of the four original towers, we can therefore build three new ones. Working in this way (we’ll say by day) we obtain four groups of three new towers. The 12 towers constructed in this way clearly include each possible new tower. For example, consider Fig. 7.1 as an example: Working by day, begin with one original tower four blocks high with one red block (shaded). We obtain three new towers, each with two red blocks, by replacing in turn each of the three yellow blocks in the original tower with a red block. The total we have just obtained instantiates the product (n – r)·C(n, r) on the right side of Equation (7.1), but – this is a key point – the new towers we just built are not distinct. In fact, each tower appears exactly twice among the 12, as the denominator r + 1 predicts. To understand how duplicates emerge, consider a particular new tower. This tower has exactly two red blocks. Each of these two red blocks can be replaced (working, we shall say, by night) with a yellow block, producing one of two original towers. This construction, which reverses what we did by day, shows

Day

Fig. 7.1 Working by day: replace each yellow block (unshaded) by a red block (shaded)

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that each new tower will appear exactly twice among the 12 we had constructed. In particular, there will be exactly six towers of height 4 that have exactly two red blocks. Because towers correspond to combinations, we have used the known result C(4, 1) = 4 to show that we have C(4, 2) = 6. For example, consider Fig. 7.2. By night, begin with a new tower four blocks high (n = 4) with two red blocks. We obtain two original towers, each with one red block, by replacing one of the two red blocks in the given new tower with a yellow block. Stephanie began to explore the construction shown in Fig. 7.1 during the March 27 interview, first with three-tall towers. Her blocks were blue and green. Continuing to four-tall towers, she next built the four towers with exactly one green block and guessed initially (but incorrectly, perhaps based on her experience with three-tall towers) that from each such original four-tall tower she could obtain two new ones. RESEARCHER: STEPHANIE: RESEARCHER: STEPHANIE: RESEARCHER: STEPHANIE:

RESEARCHER: INTERVIEWER 2: RESEARCHER: STEPHANIE:

I wonder why you get two of them. I don’t know. Maybe cause it’s bigger. What would that have to do with it? I don’t cause you have more room to build on. Tell me, can you explain to me? Oh, well, maybe it’s because like you already have one [green block] that’s taking up space, so you only have three places to move it. I gotcha, okay. Okay. So what would you predict if you were building towers five high? You’d have four.

Here we see Stephanie revise and then explain her observations, starting from a set of four-tall towers that she had physically built. On this basis, she extends her observations to a set of five-tall towers she has just imagined. So far, she knows that duplicates appear in the construction she discusses, but has not yet explored in detail how or why they do. Revisiting the same construction in the next interview session (April 17, 1996), Stephanie considers duplicates directly. This time (as in the earlier examples) her colors will be red and yellow, and the variable r will count the red blocks in a tower. After reviewing, for four-tall towers, the construction of new towers from original towers, with r = 0, 1, 2, and 3 in succession, the researcher invites Stephanie to

Fig. 7.2 By night, replace each of the red blocks (shaded) by a yellow block (unshaded)

Night

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predict how many duplicates she would expect for towers of height 5 and helps her build the towers that she needed as they proceed. They begin with a tower with five yellow blocks (the case r = 0). STEPHANIE: RESEARCHER: STEPHANIE: RESEARCHER: STEPHANIE: RESEARCHER: STEPHANIE: RESEARCHER: STEPHANIE: RESEARCHER: STEPHANIE: RESEARCHER: STEPHANIE:

RESEARCHER: STEPHANIE: RESEARCHER: STEPHANIE:

The first one’s one. There’s one of those times five. Why five? ‘Cause there’s five positions. Okay. Divided by one, ‘cause they come in groups of one. Um, hm. Five. Okay. So that’s five things taken one at a time. Yes. The second one Why don’t you write that down? Five things taken, equals five things taken one at a time. [Stephanie writes.] Okay. For the second one, um, there’s four spaces. But there’s – out of five – so its five times four and they’ll come up in groups of – I don’t know, um, that’s what we don’t know though. All right. So. Let’s – can we make these five? Just, here. Well, maybe they might come in groups of two? One. Let’s think about at least one of these. They might come in groups of two, I guess.

Here, just as they begin to build the five five-tall towers with one red block, Stephanie repeats, it seems, the mistaken guess that she had made earlier for fourtall towers. At this point, the interviewer arranges the five towers they have built in front of Stephanie and offers Stephanie the tower with its one red block in the top position. RESEARCHER: STEPHANIE: RESEARCHER: STEPHANIE:

RESEARCHER: STEPHANIE: RESEARCHER: STEPHANIE: RESEARCHER: STEPHANIE: RESEARCHER: STEPHANIE:

Okay. So what you’re saying here – move some of this aside – um, okay. Let’s think of that one. Okay. There are five. [Builds a tower with a second red block just below the first.] You have one like that. [She builds another tower with a second red block two spaces below the first.] One like that. Well, can you predict before you do it? Yeah, there’s going to be four from each. Four from each. Yeah. Okay. So – and what’s the each? How many make up each? How, wh, what do you mean? You’re saying, it’s four from this. Yeah, four from

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What doesOne. -each mean in this case? Oh! Like there’s going to be four from this one. Four from that one. Four from that one. Four from that one. Four from that one. Okay. So how many eaches? There’s five. Five eaches. Okay. Yeah. All right. So that, you say, five times four. Yes. I have that. I just don’t know what theRight. -bottom part – it So – and by the groups, you mean. The groupings you mean. Groups like – one after we’ve put them all out. Like how many groups, they’re going to come inI don’t know. I’m duplicates? I’m wondering. When you say you divide by Oh! ‘Cause that’s the number of duplicates – that there are.

Again, working by day, with towers on the table, Stephanie corrects her guess, but then – a new step – tries to go further. She has just built 20 towers in five sets of four. Now she proposes to restructure her set of 20 towers into a groups of duplicates, or at least to find the number of such groups. In effect, she has proposed the key step by herself: to find how many duplicates each new tower has. RESEARCHER: STEPHANIE: RESEARCHER: STEPHANIE:

But how do you know beforehand? Do you think there’s a way? [Building towers.] Oops. So if this, um, is going to be a pattern to this – the five times four – what do you think you would divide by? Five times four – what do you think I’d – um – maybe two.

Working backward from the known entry, 10, in Pascal’s Triangle, Stephanie confirms that she indeed will need to divide 20 by 2. The explorations then continue with ten towers of height 5 with two red blocks. Working by day, Stephanie predicts that 30 new towers can be built beginning with her ten originals. This time each new tower will have three duplicates. STEPHANIE: RESEARCHER: STEPHANIE:

Ten. So it would be ten times three and you divide by three. [Writes as she speaks.] And it worked? Yeah. And the next one, there is two spaces to put it and you have ten. So there’s ten times two, and you divide by two? [Continues writing.] And the last one – there’s one space to put it – it’s five times one divided by five equals one.

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In this exploration, in effect, Stephanie has explained how the corresponding row (1, 5, 10, 10, 5, 1) of Pascal’s Triangle emerges numerically from the pattern of Equation (7.1), which she has not yet seen. At each step, she connects the product (n – r)·C(n, r) directly to the operation of replacing one yellow block with a red block. We note, however, that Stephanie has not yet identified the denominator, r + 1, with the number of red blocks in a new tower. Instead, she seems to follow a numerical pattern that she has observed empirically. At this point, Stephanie does not yet seem able to explain why the number of duplicates that she observes must necessarily be r + 1. Nonetheless, we remain astonished, after 12 years, by the depth and strength of the connections Stephanie has made, based on her familiarity with towers. The addition rule in Pascal’s Triangle. By March 1996, as noted above, Stephanie already knew the additive pattern that relates successive rows of Pascal’s Triangle. In symbols, this addition rule can be expressed as follows: C(n − 1, r) = C(n, r − 1) − C(n, r)

(7.2)

According to this formula, each row of Pascal’s Triangle can be computed from the row before it, by adding each pair of successive entries in the row above. To connect this formula to combinations, and in this way make sense of it, we will read each term as a count of towers. Specifically, the first term on the right counts the towers of height n that have exactly r – 1 red blocks, while the second term counts towers of the same height, but with one additional red block. So interpreted, the right side of Equation (7.2) at least suggests that every tower of height n + 1 that has exactly r red blocks can be constructed from suitable shorter towers of height n, either by placing a red block on top of a tower of height n with r – 1 red blocks or by placing a yellow block on top of a tower of height n with exactly r red blocks. A special case is shown in Fig. 7.3: In the top row, we begin with two sets of towers of height 3 (n = 3): one tower with no red block (r = 0) and three towers with one red block (r = 1, shaded). To accomplish the recursion, attach a red block (R) on

R

Fig. 7.3 A specific example of the addition rule for Pascal’s Triangle

Y

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top of the single tower in the first set, and a yellow block (Y) on top of each tower in the second set, to produce four towers of height 4 with one red block. Again we work both day and night. By day, attaching blocks as shown in Fig. 7.3, it is not difficult to see that the resulting new towers of height n + 1 must be distinct. Then (by night), if we remove the top block of each possible tower of height n + 1 that has exactly r red blocks, it’s clear that all such towers have been counted on the right side of Equation (7.2). In the data segment soon to follow (later in the April 17, 1997, session) Interviewer 1, drawing several rows of Pascal’s Triangle, writes down the numbers 1 and 3 that correspond to the towers shown in the top row of Fig. 7.3. She is just about to write the number 4 below them, and then draw two diagonal lines, to associate the numbers 1 and 3 to the 4. RESEARCHER: STEPHANIE: RESEARCHER: STEPHANIE: RESEARCHER: STEPHANIE:

RESEARCHER: STEPHANIE: RESEARCHER: STEPHANIE: RESEARCHER: STEPHANIE: RESEARCHER:

STEPHANIE: RESEARCHER: STEPHANIE:

Okay. Um. Let’s explore, um – which one should we explore? [Draws lines as above.] Let’s do this one. Um, hm. Do you know what this one means? If you had to build this one, what would that tower look like? That one? What would that one look like? What would those two look like? [A pause, while Stephanie builds towers.] I think that one would be like this. [Stephanie has built the tower of height 3 with all yellow blocks, and she indicates the one that Interviewer 1 has drawn] and that one. Three high, no red. Like this. [She has just built the first two towers of height three with one red, as in Fig. 7.3.] Okay. Three high, exactly one red. Yes. Okay. Oh! Wait! [Builds the remaining tower.] Okay. Makes you dizzy after a while, doesn’t it? ‘Cause I think I see exactly one also. Even when you make it, I just believe you’re gonna do it. Okay. Now. When we’re doing this [points to the 1, the 3, and the 4 that she has written]. Um, hm. What’s different about these and this tower here [taps the number 4 in Pascal’s Triangle] that I call four? There. Well – it’s four high.

For a few lines, Interviewer 1 and Stephanie review the towers of height 3 that Stephanie has built and has physically in front of her. They easily agree that the number 4 that Interviewer 1 has written beneath the entries 1 and 3 should count the towers 4 blocks high that have exactly one red block. These towers have not yet been built, and they will not be built in the conversation that will follow.

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R. Speiser

I want to know from here Uh, hm. what you do to these [the towers of height 3]Well. -to get me, to get meWell, I’d build them higher. Well, don’t do it yet. Just think about it for a minute. Remember what they’re going to look like. Yeah. There’s going to be exactly one red. This would go here [she moves the all-yellow tower of height 3] and there would be red. No. No. We start with these [points to the number 4 again]. I don’t want you to touch these [indicates the towers of height 3]. I want you to tell me what you’re gonna do to these so that when you’re all doneUm, hm. -you end up with exactly one red. But you’ve got to make them all four tall.

This point is delicate. Stephanie knows (empirical experience!) the four-high towers with exactly one red block, and she can easily imagine them. But does she understand, working by day, how those towers can be built from the four towers of height 3 she has in front of her? STEPHANIE: RESEARCHER: STEPHANIE: RESEARCHER: STEPHANIE:

RESEARCHER: STEPHANIE: RESEARCHER: STEPHANIE: RESEARCHER: STEPHANIE: RESEARCHER: STEPHANIE: RESEARCHER: STEPHANIE:

I’m going to put a yellow here [points to the first tower on the right in the top line of Fig. 7.3], Okay. I’m going to put a yellow here [points to the next tower], Right. I’m going to put a yellow there [points to the third tower in the same group] and I’m gonna put a red there [points to the all-yellow tower]. Okay. So how many ways – how many do you end up with? Four. Four. So from the one three tall with no reds Um, hm. And the three three-tall with one red, right? Yes. You end up four four-tall with one red. Yes. Isn’t that neat? Yeah.

In this and later interviews, Stephanie first masters this way of working in continued conversations with Interviewer 1, but she then goes on, in later sessions with her peers, to teach the line of reasoning she begins exploring here to others.

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Fig. 7.4 Stephanie’s tower exploration, grade 8

A conceptual reflection. In these data, we see Stephanie refining and revising new ideas that she has built from raw materials she draws from prior experience with towers and combinations. This prior experience includes a variety of proofs (first by cases, later by induction), expressed concretely with block towers and more formally through language and notation that she and peers have personally developed and refined throughout their long collaboration. This process of revision and refinement, which we emphasize throughout, might be most clearly visible across the data we see here as a progressive movement from sets of towers that Stephanie built physically (see Fig. 7.4) toward sets of towers that Stephanie comes to imagine. These imagined towers (such as the final set of four above) are not simply visualized as static images from prior tasks; indeed, the new towers have been constructed based on new conceptual ideas that Stephanie has begun to build, in real time, as the interviews proceeded.

7.6 Discussion In an earlier paper, based on just a fraction of the data we considered here, we used the metaphor of text to state the following conclusions (Maher & Speiser, 1997b, p. 131): Images, patterns, and relationships have become mathematical objects that Stephanie sees and works with mentally to build abstractions. Our conversations with her elicited both spoken and written texts. These texts, together with our interpretations, anchor an analytic narrative of the development of certain mathematical ideas. Such texts (which we propose to view as work in progress) extend through time and serve as records of particular events upon which later texts can comment. Further, they can serve as raw material from which new texts can be composed. We revise our texts, and so does Stephanie, as our experiments proceed through detailed interactions with each other. Hence, as Stephanie’s developing judgment enters the discussion, her presentations offer raw materials that help to focus and direct the researchers’ later task designs and explorations. Our agenda for the interviews, seen as an emerging text, continues to be rewritten, reconsidered, and revised, often in direct response to goals that Stephanie pursues.

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After the interviews we have considered here, events that neither Stephanie, nor her peers, nor the researchers could foresee in 1996, would offer opportunities for everyone involved to deepen, reconsider, refine, and extend their previous perspectives and conclusions. In the discussion we have just presented, perhaps most striking is the heightened prominence we see of personal, conceptual imagination to address new problems and, in the process, to give form to new and powerful ideas. In the next chapters, we follow other students from the longitudinal study who also build on previous explorations to make sense of Pascal’s Triangle and rules for its generation.

Part III

Making Connections, Extending, and Generalizing (1997–2000)

Chapter 8

Responding to Ankur’s Challenge: Co-construction of Argument Leading to Proof Carolyn A. Maher and Ethel M. Muter


January 9, 1998; Grade 10 Towers Ankur, Brian, Jeff, Mike, and Romina Carolyn Maher

8.1 Introduction In previous chapters, we saw elementary students work in classrooms on counting problems presented by researchers. In this chapter, we observe a group of five high school students working under different circumstances. When the students in the longitudinal study entered high school, they no longer worked on problems in class. Instead, the students who remained in the study worked on problems in afterschool sessions scheduled by the researchers and for which the students rearranged their after-school schedules in order to attend. In addition, this session is unique because the students worked on a problem proposed by a fellow student. In this session, Ankur and Mike were invited to propose and solve their own problem. Ankur proposed a new towers problem, which became known as Ankur’s Challenge: Find all possible towers that are four cubes tall, selecting from cubes available in three different colors, so that the resulting towers contain at least one of each color.

Mike and Ankur’s approach was to start with the total number of four-tall towers built from three colors and then subtract the number that did not fulfill the stated criteria. They started by writing combinations of towers, using the numbers 1, 2, and 3 to represent colors red, blue, and yellow, respectively, and using a 0 to represent the duplicated color. At first, they omitted some towers in their count, but later in the session they discovered the missing set.



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As Mike and Ankur proposed a variety of numerical answers (72, 54, 45, and 36) to the problem, Jeff, Brian, and Romina joined the discussion. Jeff suggested, based on a preliminary listing, that the answer to Ankur’s challenge was 36. (His list had 37 towers, but he thought that it included a duplicate.) Although they worked separately, the two groups periodically exchanged comments and suggestions. The first firm finding on which all agreed was that the total number of four-tall towers that can be built when selecting from three colors is 34 (81) towers.

8.2 Romina’s Presentation of Proof At this point the dynamics of the working groups changed. Jeff joined with Mike and Ankur, who were working on developing a justification based on cases. Brian sat quietly, sketching out ideas on a sheet of paper. Romina worked on her own, at times thinking out loud, as illustrated here: You know, it might be 36. ‘Cause I’m working with sixes now. And okay, you put them, like you pair ‘em up. ‘Cause you’re only gonna have . . .

At this point, Romina put up her hands and indicated that she needed to collect her thoughts. She continued: Let me think first, organize my thoughts a little. We’re gonna have them together. Together like over here [indicating her list]. These are together. These are together. These are together. Like two of the same color together. And then, in like a pattern, like, we’ll put them somewhere and then we’ll switch them around, so I’m up to 24 now and I’m going to put them the same way here and here. So then that’s 30. And I put the same ones here and here. Here, here. I didn’t put them, and then there’s your 36.

Figure 8.1 gives Romina’s first list of possibilities. Romina explained that two cubes of the same color had to be in each tower and that she was using that fact to create a pattern. Using X, O, and 1 to represent the three colors, Romina listed the six towers that could be created using a single pattern (that in which the repeated color occupied the first two positions in the tower), and then she moved on to the next possible arrangement. This was the beginning of a justification based on cases. She explained that by using this method she had found 24 towers so far and that there were two additional groupings that she had yet to complete. Counting all of the towers in her list would mean that there were 36 possible towers as the answer to Ankur’s Challenge. At this point, Romina began to generalize her solution. Instead of listing all of the combinations, she listed only the cubes that were duplicated in each tower. As Romina tried to get the attention of Jeff, Mike, and Ankur in order to explain her solution to them, Brian interrupted and pointed out that she had duplicated one of the rows in her list. Undaunted, Romina reached for a clean piece of paper and began to redraw her table. Brian watched carefully as she worked, offering helpful suggestions as she created this version of her solution. She drew boxes showing the duplicated cubes in five of the six possible positions. With Brian’s

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Fig. 8.1 Romina’s first attempt to write out the towers of Ankur’s Challenge

assistance, Romina reviewed what she had already written and included the sixth row (Fig. 8.2). She was then ready to present her thoughts to Jeff, who had moved over to see what she and Brian were doing. She counted her rows to verify that she had the required six and proceeded to explain what she had done to Jeff. She said: You know we’re gonna have two of the same color. Right? Two of the same color, which stands for putting these and these, right? And you’re gonna have them in, and then the rest you fill up, right? And then you’re gonna have the . . . And there’s only two other ones you could have. So this, you have this one, you have to multiply it by two. Well, one, two,

Fig. 8.2 Romina’s generalization

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This attempt to convey the idea that each row represented a four-tall tower that had to contain two cubes of one color with one of second color and one of the third color was not convincing to Jeff. Romina did not explain the reason for multiplying by two. Romina and Brian decided that Romina would write her ideas more neatly, and she produced a third version of her diagram. As Ankur and Mike began talking to the researcher about their solution, Jeff and Romina interrupted with the information that they had found a solution of 36, and they could prove that it was correct. They asked Mike to pay attention to their explanation and, although he agreed, it became clear that he was still thinking about his own ideas for finding the complement to Ankur’s Challenge. Romina’s presentation improved as she related her ideas this second time. The discussion follows. ROMINA:

ANKUR: ROMINA: ANKUR: ROMINA: JEFF:

BRIAN:

So you have to organize them so that you don’t have any doubles. So either you can have them next to each other. You can have them separated by one. You have them on the ends, in the middle, two and fourth spot, and third and fourth spot. So that’s six. Okay. Now you, in the other spots, you can have an o and an x. Those are colors. Like these are three different colors – an o and an x and an x and an o. Right? Yes. So that’s six. Yes. So you have to multiply each of these six by two. But you couldn’t have like x x because that wouldn’t fit the requirement. So you multiply each one by two. So that would give you 12. Correct? ‘Cause that means you could have like this . . . Like the x is in the first, the o in the first spot.

This time, Romina explained the code and her organization, iterating all the possible positions for the duplicated color. She explained how she placed the remaining two colors in her diagram and said why she multiplied each row by two (to account for the fact that the two non-duplicated colors could switch positions). Jeff and Brian added clarification of the reason for multiplication by two. Ankur accepted Romina’s solution. Mike, however, remained fixed on his earlier strategy to justify that the remaining towers, those that formed the complement of Ankur’s Challenge, numbered exactly 45. He was not willing to accept the number 45 as the difference between 81, the total number of towers that they had agreed on at the outset, and the 36 combinations that Romina, Brian, and Jeff had found. Although Ankur and Jeff claimed that Romina’s proof should be sufficient to prove that 45 was the solution to the complement problem, Mike wanted an explicit proof:

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Fig. 8.3 Romina’s chalkboard version of her proof of Ankur’s Challenge

ANKUR: JEFF: MIKE:

The only way you could prove you were right is to prove the other side. We proved the other. That’s not enough for me. I want to prove the other.

At the end of the session, Mike asked Romina to restate her proof, admitting that he had not been paying attention the first time that she had presented it to the group. Romina graciously complied, further refining both her explanation of the proof and the diagram she utilized as she spoke while writing it on the chalkboard. Refer to Fig. 8.3. At the group’s next meeting, Romina brought a written copy of her proof of Ankur’s Challenge. Notice her further refinement in showing 36 possibilities in her written work shown in Fig. 8.4.

Fig. 8.4 Romina’s second version of her proof of Ankur’s Challenge

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In Romina’s written explanation, she indicated a realization that there were three different colors for selecting the blocks and four positions on a tower to place a block. She indicated, also, that there would be a double of one of the colors and a single of the two remaining colors. She now used a different notation – yellow (Y) and blue (B) for the single blocks and red (R) for the duplicate block. Romina wrote that placing the red cubes in all possible positions would produce six towers. She pointed out that there were only two possibilities for the remaining colors, thereby producing 6 × 2 or 12 towers with double red blocks. She then concluded that by considering the remaining 3 colors, 3 × 12 or 36 towers would yield all possible combinations, thereby producing an elegant justification for the solution to Ankur’s Challenge.

8.3 Discussion In their work on Ankur’s Challenge, Romina, Brian, Jeff, Ankur, and Mike demonstrated how they worked as problem solvers. Ankur posed an interesting problem, which he and Mike partnered to solve. They started by applying previous knowledge (how to find the number of towers when selecting from two colors) to a new situation (three colors are now available). They successfully found the total number and proceeded to use a subtraction strategy. They listed exceptions by case; when the notation they originally chose proved inadequate, they introduced a new notation. Later, when provided with an answer derived from a different approach, Mike continued to work on Ankur and his initial strategy. Although he accepted the direct approach explained by Romina from the other group, he was unsatisfied that Ankur’s and his approach, seemingly reasonable, did not work. Romina and her group used a direct approach. They brought together ideas and notations from the past as they constructed a solution for Ankur’s Challenge. They used a variation of the binary coding scheme that Mike had introduced the previous month. They profited from the strategy that Ankur had presented in the fifth grade of fixing one of the variables and then considering the possibilities that satisfied that case. As Romina explained her strategy to Brian, she formulated different ways to express her thoughts; Brian assisted by pointing out additional cases. As Romina tried to communicate her ideas so that Jeff would follow, she revised her representations. Mike, ready to hear about the solution of the other group, asked Romina to repeat her explanation. In response, she presented her work at the blackboard with further refinement of their representations. Her final written summary provided another opportunity for detail, refinement, and generalization. In summary, we can say that Romina and her group profited by using their personal representations, communicating them as ideas, and then providing support for those ideas by reorganizing and restructuring representations. Further, in each iteration of the argument, Romina made refinements and clarified her reasoning. This suggests advantages for students

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when afforded more than one opportunity to explain and write about their ideas. Each explanation has the potential to contribute to a deeper understanding and for multiple ways to represent ideas. Romina, 1 week later, shared a written solution to the problem, indicating an interest in refining her explanation and demonstrating her motivation for further thought and reflection. In the following chapters, we return to the pizza and towers problems and see how groups of students continue to refine their representations, clarify their reasoning, and extend and generalize their understanding of mathematical ideas by revisiting old ideas, communicating their findings, and listening to the findings of others.

Chapter 9

Block Towers: Co-construction of Proof Lynn D. Tarlow and Elizabeth B. Uptegrove


November 13, 1998; Grade 11 Towers Ali, Angela, Magda, Michelle, Robert, and Sherly Carolyn Maher, Alice Alston, Susan Pirie

9.1 Introduction In previous chapters, we observed elementary school students working to make sense of the towers problems by building representations, formulating conjectures, and defending their solutions in discussions with classmates and researchers. In this chapter, we observe a cohort of high school juniors as they engage in explorations and constructions in the towers problem. During this session, the students found and generalized formulas for solutions to the original towers problem (building towers when selecting from two colors of Unifix cubes) and extensions (with more than two colors of cubes), using methods including controlling for variables, justification by cases, and inductive reasoning.

9.2 Building Towers In the 2-h session, students worked in pairs on tower problems. They came up with a general rule for the number of possible towers of height n when selecting from x colors (xn ) and an explanation of that result based on an inductive argument based on generating all possible towers of a given height. Their arguments contained reasoning by cases, induction, and reasoning by contradiction. In addition, Robert produced an equation for the number of towers having exactly two cubes of one color (when selecting from two colors), for a tower of any height. E.B. Uptegrove (B) Department of Mathematical Sciences, Felician College, Rutherford, NJ, USA e-mail: [email protected]


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9.2.1 Angela and Magda Neither Angela nor Magda had previous experience with the towers problem, as they had both joined the longitudinal study in sixth grade. In this, their first experience with towers, they found all 16 towers, four-tall, selecting from two colors. Interestingly, they used strategies similar to those developed by the fourth and fifth graders that participated earlier in the study (see Chapter 4). The girls organized their work by cases: (1) one blue, (2) two blues, (3) three blues, and (4) four of the same color. The two single-color cases consisted of one tower each; the oneblue-cube and three-blue-cube cases exhibited a local organization; they built those towers by moving the single cube of one color into each of the four possible positions. When asked how they knew that they had all the towers with one blue cube, they described their organization: MAGDA: ANGELA:

The blue is in each position each time. Yeah, each possible position because there’s only four spots.

Initially, they had no support for accounting for the towers in the two-blue-cube case; they explained that they were unable to find any more. However, after they found four of the towers for this case, and they were asked how they knew they had them all, Angela alluded to a preliminary organization using controlling for variables strategy, that is, holding the top and bottom cubes constant. She said: Well, I mean, I don’t know how to explain it, there’s just like no other possibilities for it. I mean, there’s only four places, you have them, like you know, yellow on top, blue on the bottom, and the blue on top, yellow on bottom, then blue on top and bottom, and yellow on top and bottom.

As she was saying this, Angela found the two towers for this case that were missed. There are two towers with yellow on the top and blue on the bottom and two towers with blue on the top and yellow on the bottom; they had originally found only one of each of those pairs. When asked to determine the number of three-tall towers, Angela and Magda moved from building towers to drawing them, again with an organization by cases. The eight towers that they found were organized in three cases: (1) one blue cube, (2) two blue cubes, and (3) all one color. Each case was locally organized, as shown in Fig. 9.1. After thinking about their findings, they developed a general rule; according to what they called “Angela’s Law of Towers,” the number of n-tall towers when you have x colors to choose from is xn . Thus Angela and Magda not only provided a solution to the specific four-tall towers problem posed, but they also posed a generalization from towers with two colors to towers with x colors.

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Fig. 9.1 Angela and Magda’s list of three-tall towers

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9.2.2 Sherly and Ali Sherly and Ali were new to the study, participating for the first time in grade 11. Their tower building strategy was also similar to that of the third- and fourth-grade students who, when first encountering the towers problem, used the strategy of grouping towers in pairs of “opposites.” (When the towers are placed side by side, the cubes in corresponding positions are opposite colors.) They found all 16 four-tall towers using this strategy. When they were asked to predict the number of three-tall and then five-tall towers, they made a prediction based on patterns. They conjectured that since the number of four-tall towers is 16 (4 times 4), there would be 9 three-tall towers (3 times 3) and 25 five-tall towers (5 times 5). After realizing that there are only 8 three-tall towers, they revised their conjecture for five-tall towers to 24, which follows a pattern of multiples of 8. During the whole group discussion (mentioned below), they were exposed to the conjecture of the other groups (that there would be 32 five-tall towers), but they were not convinced. As part of their work on five-tall towers, Sherly and Ali were asked to investigate the case of five-tall towers with exactly two blue cubes. They organized this group of towers by cases: (1) two blue cubes together; (2) two blue cubes separated by one yellow cube; (3) two blue cubes separated by two yellow cubes; and (4) two blue cubes separated by three yellow cubes; using this organization they found all ten towers that satisfied this condition. Refer to Fig. 9.2 for a diagram of the “two blues separated by one yellow” case. During the discussion of this work, Sherly provided a proof by contradiction: When a researcher asked if the two blue cubes could be separated by four yellow cubes, Sherly noted that this could not happen, “because there would be six [cubes in the tower] then.” We see that even though Ali and Sherly had not previously been exposed to the way younger students in the longitudinal study worked, they had by the end of the session adopted some of the methods that had been developed earlier by their classmates. Fig. 9.2 Sherly and Ali’s organization for “two blues separated by one yellow”

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9.2.3 Michelle and Robert Michelle and Robert were both in the longitudinal study from the first grade. Although Robert and Michelle sat along side each other, they used different organizational strategies. Michelle built towers randomly; she said it was without “any set plan,” and she rearranged the towers into pairs that she called “twos.” (These

100 Fig. 9.3 Robert’s three cases of four-tall towers with exactly two blue cubes

L.D. Tarlow and E.B. Uptegrove Case 1: blue on top

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were pairs of opposites like those built by Sherly and Ali, described above.) Robert immediately built and organized the towers by cases, focusing on the blue cube; his five cases were zero, one, two, three, and four blue cubes. When asked to show that he had all the possible four-tall towers with exactly two blue cubes, Robert provided a justification based on controlling for variables. He showed how he held the upper blue cube in a fixed position beginning at the top, while he moved the lower blue cube down one position each time he built a new tower. When the lower blue cube had been moved down to all of the possible positions, the upper blue cube was moved down one position. The process was repeated until it was not possible to move either blue cube down. With these three cases, Robert demonstrated that there were six towers in the two-blue case. Refer to Fig. 9.3 for a diagram of Robert’s organization. Robert and Michelle next worked on three-tall towers. Robert showed that the three-tall towers could be built by removing the top cube from each four-tall tower (giving two identical sets of 8 towers) and then removing duplicates. This reasoning foreshadows Robert’s use of inductive reasoning later in the session. Robert and Michelle went on to chart their results for the total number of towers two-tall through five-tall. See Fig. 9.4. Their entry for two-tall towers is incorrect (the actual number is 4), but the rest of the numbers are correct. They found a pattern that they believed applied to towers of height 3 and taller, and Robert used that pattern to predict that there would be 64 six-tall towers. Their rule for the number of towers was 2h , where h is the height of the tower. The researcher then asked them to consider the question of how many five-tall towers would have exactly two blue cubes. Robert and Michelle built those towers, using Robert’s strategy of controlling for variables by holding the top cube fixed and then moving it successively lower in the tower. Robert and Michelle did more than

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was asked, finding that for towers with heights two, three, four, five, and six, there are one, three, six, ten, and fifteen towers with exactly two blue cubes, respectively. Robert predicted, based upon the pattern “plus two, plus three, plus four, plus five” (see Fig. 9.5), that for seven-tall and eight-tall towers, there would be 21 and 28 towers with exactly two blue cubes. Robert then looked for an explicit formula for the number of towers with exactly two blue cubes for any height. With his explanation for why a formula would be useful, we observe Robert thinking like a mathematician: What if someone just, they just did this problem for the first time, and they just came up with like how many two yellow for fifteen, and they wanted to find out for fourteen without doing it? How would they do that? . . . We are not going to sit down and write out one plus two plus three plus four plus five . . .

Robert’s table of results indicates that he developed a rule: multiply the height by half the height “minus point five.” This corresponds to the formula for the number of combinations when selecting two objects from a set of h objects:

h ⋅ ( h − 1) h! ⎛h = = =h 2 ⎝ 2 ( h − 2 ) ! 2!

(( h 2 ) − .5)

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There was an extended discussion of this formula; the researchers wanted to know where 0.5 came from and how Robert thought of it. He said, “I don’t know, it just seems to work.” Again thinking like a mathematician, Robert wondered aloud if the formula would work for towers build from three colors. (He conjectured that the 2’s in the formula would be replaced by 3’s.) Although Robert did not pursue this thought, this led to a discussion of the general towers formula. A researcher asked the students to illustrate how powers of 2 gave the numbers of towers for various heights. In the following discussion, we see how Robert, initially confused about the meaning of 21 , comes to an understanding with the help of Michelle and goes on to describe the use of the general rule.

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Show me what it would be for two to the first. What would the towers be? Uh, two blues. This and this. [Robert indicates one tower with two blue cubes, and one with two yellow cubes]. It’s just ones, so it would just be one, one. Yeah, total combinations likeTwo to the first would be, what would they be, show me. Right here. No. No, no. That isn’t two to the first; this is two to the first. [Michelle holds one single blue cube and one single yellow cube.] Ah. Okay, so two to the first would these guys. Two to the second, according to your theory would be how much? Oh, right here. Four. We have them right here. [Robert indicates the four two-tall towers that were already built.] There’s four. How did you get from here [one-tall towers] to here [two-tall towers]? We just built on top of them, I guess.

Robert followed up by demonstrating how each one-tall tower would generate two two-tall towers: one with a blue cube on top and another with a yellow cube on top, for a total of four two-tall towers. Refer to Fig. 9.6. Later, when the researcher asked Robert to show that his three-tall tower list was complete, he introduced an inductive argument based on the procedure he used to generate the two-tall towers: ROBERT:

RESEARCHER: ROBERT: RESEARCHER:

And then here [the set of four two-tall towers] you have all you could have on the bottom, and you are just adding to the top, I guess. I guess you just take this [a two-tall tower] and add a blue and a yellow to the top. All the way through, and then – you know what I am saying? Does that make sense? Yes, it does. Does it keeping going?

Fig. 9.6 Robert generates four two-tall towers from the two one-tall towers

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Yeah, I guess it would keep going on forever. That’s why that thing works. Because you are just adding an extra set of two.

9.2.4 Group Work Robert and Michelle were invited to discuss their inductive argument with Magda and Angela. They set up towers as shown in Fig. 9.6. A portion of their discussion follows. MAGDA: ROBERT:

ANGELA: ROBERT: ANGELA: ROBERT: MICHELLE: ANGELA:

You kind of add that one on top of that color? We just took this [one of the two-tall towers] and added a yellow and blue, and took this [another two-tall tower] and added a yellow and a blue and took that and added a yellow and blue like for all of them. [Robert indicates each two-tall tower in turn.] Oh. Do that for all of them and they grow another row. You just keep going. Add yellow and blue to each one. That’s it. It looks nice. That’s very lovely She said it branches. It branches. Yes it does.

The researcher asked if this process would work with three different colors. Angela responded that there would be three branches and more towers (“There would be three little thingies. . . . It would be a lot bigger”), and she added, “Would you like to see our theory? It is x to the n.” Robert responded, “Yeah, we have that same theory.” He directed Michelle to change their formula from 2h (the total number of towers of height h when there are two colors) to xh , where x is the number of color choices and h is the height of the tower. Four students (Robert, Michelle, Magda, and Angela) worked as a group to write up this result; Robert dictated, Michelle did the writing, and Magda and Angela observed. Robert said they had to carefully explain their notation so that any reader would understand what they had written and so that all the students would be able to explain it. Refer to Fig. 9.7 for their write-up.

Xh X = # of colors h = height

Fig. 9.7 The group’s general towers result

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9.3 Discussion In this chapter, we discussed a session in which students working in small groups and in a larger group gave specific and general answers, with justifications, to mathematical questions. They found formulas to answer specific towers questions; they generalized the formulas to handle general towers questions; they often gave convincing justifications for their answers, using methods including controlling for variables, justification by cases, and induction; and they explained most of the patterns they found by showing how the patterns made sense in the context of the problems. Although Robert sometimes took the lead in moving further and creating understandable explanations, all students (even those separated and new to the study) worked diligently on the problems, took care to explain their thinking, and made convincing arguments and justifications. In the next chapter, we observe another cohort of 10th-grade students working on the towers and pizza problems. They too found ways to make sense of the problems, generalize answers, and provide convincing justifications for those answers. In addition, their work led them to a deeper understanding of the binomial coefficients and Pascal’s Triangle.

Chapter 10

Representations and Connections Ethel M. Muter and Elizabeth B. Uptegrove

Dates and Grade: Tasks: Participants: Researchers:

December 1997 through March 1998; Grade 10 Towers and pizzas Ankur, Brian, Jeff, Mike, and Romina Carolyn Maher and Robert Speiser

10.1 Introduction In the previous chapter, we viewed a cohort of high school students from the longitudinal study as they explored the towers problems. In this chapter, we observe a different cohort of students also exploring the towers and pizza problems. In the five sessions discussed here, spanning December 1997 through March 1998, the five students in this cohort were reintroduced to the towers and pizza problems, which they last explored in elementary school as described earlier in Chapters 5 and 6. They found general solutions to those problems and a way to organize their solution lists to prove that all solutions were present. They recognized that those problems were related to each other, to the binomial coefficients, and to Pascal’s Triangle. They made use of their understanding of the structure of those problems to form preliminary ideas about the meaning of Pascal’s Identity. We show how their development and use of a sophisticated general representation scheme helped them make these connections and generalize their knowledge.

10.2 Session 1: A Common Notation On December 12, 1997, when they were in the tenth grade, Ankur, Brian, Jeff, Mike, and Romina began to meet with the researchers as a group for 1- to 2-h after school sessions that continued throughout high school. At their first meeting, E.B. Uptegrove (B) Department of Mathematical Sciences, Felician College, Rutherford, NJ, USA e-mail: [email protected]


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the researchers proposed the pizza problem to them as they snacked – on pizza. When first presented with the four-topping problem in the fourth grade, the students had enthusiastically tackled the problem by randomly generating combinations. As described in detail in Chapter 5, in approximately 15 min they found all 16 possibilities by using an alphabetical coding scheme to represent each pizza as it was generated. In tenth grade, working on the three-, four-, and five-topping pizza problem, Jeff and Romina utilized an alphabetical coding scheme similar to the one that they used in fourth grade (p for pepperoni, m for mushroom, etc.). Ankur and Brian used a numerical coding scheme (1 through n for the n different toppings). Mike, however, worked alone. He selected a unique binary number coding scheme to keep track of the combinations. The binary representation became a useful tool in many of their future discussions of combinatorial problems. As the first four students discussed the problem, they realized that they needed a common notation and adopted the alphabetical model. They kept the plain pizza separated, and so they found 7 possible pizzas for the three-topping case (plus plain) and 15 pizzas for the four-topping case (plus plain). When they found 30 five-topping pizzas (plus plain), they realized it did not fit the pattern. They hypothesized that a doubling rule might be involved and they decided to rethink their solution. At this point, Mike re-entered the discussion in order to introduce his binary coding scheme. He proposed that pizzas be represented by binary numbers; a four-topping pizza would be represented by a four-digit binary number, with a 1 in the kth digit representing the presence of the kth topping and a 0 representing the absence of the kth topping. For example, all one-topping pizzas are represented by all fourdigit binary numbers with exactly one 1: 0001, 0010, 0100, and 1000. This coding scheme was more easily generalizable than the letter code scheme: to add another topping, just add another binary digit. After listening to Mike’s explanation of the binary code, the group made a connection between the two representations (letter codes and binary notation: use the letters that stand for toppings as column headers for the list of binary digits). Refer to Fig. 10.1 for their table; O stands for onion, M for mushroom, P for pepperoni, and S for sausage. Mike’s understanding of the binary system and the way it could be used to describe the solution to the pizza problem gave him an insight into a generalization about the number of pizzas that can be created from n toppings; Mike hypothesized that the answer to the n-topping pizza problem is 2n . The group discussed the numerical solution for some time – there was some confusion about whether the coefficient might be n – 1 or n + 1, or whether the answer might be 2n – 1, possibly

Fig. 10.1 Students’ table linking topping codes and binary notation [annotation added]

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due to the uncertainty about how to count the plain pizza, but ultimately all agreed that the solution was 2n . As they were wrapping up the session, the researcher asked if the problem reminded them of other problems, and Brian mentioned towers: “Every thing we ever do always is like the tower problem.” In order to investigate the possible relationship between the two problems, the students worked on the three-tall tower problem and concluded that the answer was the same as for the three-topping pizza problem; there are eight three-tall towers, just as there are eight possible threetopping pizzas. Because they were focused on relating the pizza toppings to the cubes’ colors, they concluded that the problems were similar but not identical. Ankur noted, for example, that a red-yellow tower is different from a yellow-red tower, but a pepper–pepperoni pizza is the same as a pepperoni–pepper pizza.

10.3 Session 2: Towers and Pizzas One week later, the students returned and resumed their discussion of a possible relationship between pizza and towers problems. Although they were asked to consider only the two-color towers problems, they kept returning to the question of how to count the possible number of towers when there were cubes of three or more colors. Looking at this issue led them to the realization that when the height of the towers is the only variable under consideration, the towers problem is identical to the pizza problem. This time, they mapped the height of the tower to the number of pizza topping choices (contrary to the previous week, when they attempted to map number of colors to number of pizza topping choices). A portion of their discussion follows. JEFF: MIKE: JEFF: MIKE: JEFF: ANKUR: JEFF: ANKUR: JEFF:

ROMINA: ANKUR:

If the only variable we’re changing is height, it stays the same. It would be the same as the pizza. What would that be like changing on the pizza, though? You could change the height, the number of toppings. Changing the height would be like changing the number of toppings. Yes. Changing the color would be like, what? Say what you just said again. All right. When we change the height of the box, from like two to three, it’s like changing the topping on the pizza from a possible two toppings to three toppings. Okay. Okay.

Brian and Mike returned to the question of changing the number of colors available for building towers, and Mike proposed that when there are three colors to choose from, there are nine possible two-tall towers (3 times 3). After a 10-min discussion, the other four students agreed. In the course of the discussion, they attempted to clarify the meaning of the base and exponent in each problem. In doing

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this, they were able to answer a question that they had not answered in the previous session, that of the meaning of the 2 in the formula 2n . In the previous session, they had determined that n represented the number of toppings, and in this session, they determined that the 2 stands for the two toppings choices: on or off the pizza: BRIAN: MIKE:

Two has to stand for something. It stands for something; n was the number of toppings and 2 is what – you could either have 0 or 1. You either have a topping or not.

Later, Romina presented the group’s findings to the researchers: ROMINA: MIKE: JEFF: JEFF: ROMINA: JEFF:

Okay. For the pizza problem, the 2n [meaning 2n ], the two represents either topping or no topping. Right? There’s two different possibilities for each. That’s why there’s two. We didn’t know, I don’t think we explained that last time, why it was two. Topping or no topping, and that’s what the two is. Now the n, Romina. Is toppings. The number of toppings.

Mike used binary notation again in this session, this time using it to represent the two colors of the towers problem. He noted that instead of relating binary digits to the presence or absence of pizza toppings, he could relate them to colors of cubes: “Zero is blue and one is red.” Figure 10.2 shows Mike’s table of solutions for both the two-topping pizza problem and the two-tall towers problem. The column headers 1 and 2 represent the two pizza topping choices and the two levels of the tower. The 1 and 0 represent topping/no topping and blue cube/red cube. Mike explained that if the labels at the top of the chart stood for pizza toppings (m for mushrooms and p for pepperoni), the zeros and ones would represent the presence or absence of the topping. If the labels stood for positions in towers, the zeros and ones would represent the color of a cube; e.g., one would represent blue and zero would represent red. During the second half of this session, at the researcher’s request, the students explored geometric interpretations of the binomial expansion. Figure 10.3 represents their drawing of (a+b)2 . They went on to spend over half an hour working on drawings and three-dimensional models for (a+b)3 , although no model was entirely satisfactory to them. These investigations can be seen as preparation for their later work describing the isomorphic relationship among the towers and pizza problems and the binomial expansion.

Fig. 10.2 Mike’s listing of two-tall towers and two-topping pizzas

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Fig. 10.3 Geometric interpretation of (a + b)2

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In this session, the students gave the researchers clear and convincing explanations of the isomorphism between the pizza and towers problems and of the meaning of the components of the formula, and it appeared that they had a firm grasp of the underlying structure of the problems. But a few months later (in sessions discussed later in this chapter), we see them return to their focus on relating number of pizza toppings to number of colors (instead of to the height of the towers), once again deciding that the problems were similar but not identical. This illustrates how important it is to revisit problems and re-examine solutions, to solidify and expand students’ understanding.

10.4 Session 3: Towers and the Binomial Expansion When the students met in January 1998 after the holiday break, they returned to the topic of towers. The researchers gave them a problem from fourth grade: When you are choosing from red and yellow cubes, how many five-tall towers can you build containing exactly two red cubes? They immediately answered “ten,” and they were then challenged to provide an explanation. Mike and Ankur provided a justification in approximately 2 min, using Mike’s binary coding scheme, with 0 representing a yellow block and 1 representing a red block. As Ankur explained their solution, their organization improved; they begin to control for variables by holding the red cube fixed in the top position and then moving the second red block into successively lower positions until it reached the bottom position. This process was repeated, holding the red cube fixed in the second and then third and fourth positions. Figure 10.4 shows their original list followed by the re-organized list of all ten towers. This

Fig. 10.4 Mike and Ankur’s two lists of five-tall towers with exactly two red cubes

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illustrates the importance of revisiting and re-explaining answers; in the process of explaining their solution, they organized the list so as to make it clear that all possibilities were accounted for and none were missing. The other group (Jeff, Romina, and Brian) worked on their solution for approximately 25 min. Their solution depended on first finding all possible five-tall towers; they recalled from previous work that there are 32 such towers. They built a justification based on cases; their cases were (1) all red: 1 tower, (2) one red and four yellow: 5 towers, (3) two red and three yellow: 10 towers, (4) three red and two yellow: 10 towers, (5) four red and one yellow: 5 towers, and (6) all yellow: 1 tower. While Brian, Jeff, and Romina were working on their solution, and Ankur and Mike were done, Ankur proposed a problem that became known as Ankur’s Challenge. The group’s work on this problem was discussed in Chapter 8. At the end of this session, the researcher introduced some of the notations of combinatorics. She told the students that asking how many five-tall towers have exactly two red cubes is the same as asking how many combinations there are when selecting two of five objects. She showed four different ways to write this, as shown in Fig. 10.5. She concluded with a discussion of the binomial expansion and Pascal’s Triangle. Following up on the previous session’s investigation of the binomial expansion, she wrote the expansion of (a + b) to powers 0–3, drew Pascal’s Triangle, and then asked the students to think about the relationship. (Refer to Fig. 10.6 for the researcher’s drawings). In the following excerpt, the researcher hinted about the relationships that the students were in the process of discovering. RESEARCHER:

ANKUR: RESEARCHER:

The question is, what’s the relationship here? How could you model it? How could you show this relationship? And why does it work? That’s the question. So that’s sort of the direction. Are you interested in knowing that? I think you have the bits and pieces to put it together. Some of the pieces are really small. They’re bigger than you think. You’ve been working on this for a long time. ⎛5⎞ 5C2 C5,2 ⎝2⎠

Fig. 10.5 Notation for selecting two of five objects

Fig. 10.6 The binomial expansion and Pascal’s Triangle

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Is this what we did today though? You’ve been dealing with some of this today. So think about it. So are all of the things we learned for the past 8 years sort of combined into one thing? Imagine that.

Immediately following that discussion, the researcher asked the students to make concrete the numbers in Pascal’s Triangle, by thinking about them in a “very real way” (linking them to towers problems). RESEARCHER:

ANKUR: RESEARCHER: ANKUR: RESEARCHER: ANKUR: RESEARCHER: ANKUR: RESEARCHER: ANKUR: RESEARCHER:

When you first came in here today, you produced that number ten. [She refers to the first 10 in row 5 of Pascal’s Triangle – 1 5 10 10 5 1.] Right? Yes. And what problem were you solving? Two were red and three something else. Okay. So you can think of that ten in a very real way, if you want to, right? Yeah. Can you think of those other numbers in a real way? Does that help? The 1 is, in 1 4 6 4 1, the 1 represents all red. The other 1 represents all yellow I guess . . . All red and all yellow for what? Of four high. So this is four high. [The researcher points to row 4 of Pascal’s Triangle.] And these are all red. [The researcher points to the first 1 in that row.]

This marks the first time the towers problem was explicitly linked with Pascal’s Triangle, when row 4 of Pascal’s Triangle was connected to the four-tall towers problem. Before the session ended, the researcher asked the students to think about the meaning behind the addition rule for Pascal’s Triangle in the specific case of how the 6 in row 4 was generated from the two 3’s in row 3. Although the students did not offer an answer at this session, it is noted here as the first time they were asked to think about Pascal’s Identity. In this session we see three instances where the students were invited to think about how all the individual problems could be related, both to each other and to abstract mathematical entities, but without any explicit instruction about how to make a connection between the problems.

10.5 Session 4: Pizzas, Towers, and Pascal’s Triangle In this session, three students (Ankur, Jeff, and Romina), in a first meeting with visiting researcher Robert Speiser, explored the relationships among the pizza problem,

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the towers problem, and Pascal’s Triangle, and (for the first time) they discussed Pascal’s Identity in terms of operations on physical objects (adding cubes to towers). When Professor Speiser asked the students about their recent work, the students did not mention a relationship between the towers and pizza problems. Instead, as they had initially done back in December, they maintained that the problems were different. Romina, recalling Ankur’s earlier reasoning, said that red-blue cubes on a tower are different from blue-red cubes, but sausage–pepperoni is the same as pepperoni–sausage. Ankur added that a five-topping pizza problem is like a fivecolor towers problem. Jeff agreed, saying that a tower could have two of the same color but that a pizza could not have pepperoni–pepperoni. Although all three had participated in the earlier, correct, discussion of the relationship between the towers and pizza problems, they recalled now only their own original ideas. They said nothing about the height of the tower being connected to the number of toppings, or how on-the-pizza/not-on-the-pizza could be made to correspond to blue/red cubes via the binary representations 0 and 1. The researcher reminded the group about the combinatorics notation that had been introduced a month earlier and she reminded them how the notation was related to the five-tall towers problem. She went on to demonstrate the binomial expansion and to ask explicit questions: What are the relationships, if any, among (a+b)5 , the five-tall towers problem, the five-topping pizza problem, and the fifth row of Pascal’s Triangle? This question is significant in terms of the students’ later work, as it represents the first time the students were asked to think about a four-way link, among the binomial expansion, the two combinatorics problems, and Pascal’s Triangle. The students were able to make the connection, evoking and expanding Ankur’s explanation from the previous month of how the four-tall towers problem could be found in row 4 of Pascal’s Triangle. (This also anticipated their night session explanation of entries in Pascal’s Triangle in terms of pizzas.) In the following excerpt, the students linked the binomial expansion to the towers problem. RESEARCHER: ROMINA: ANKUR: RESEARCHER: ANKUR: RESEARCHER: ANKUR:

What are the a’s and the b’s here? Colors. . . . a and b is red and blue. . . . What do you mean by red and blue? a is red and b is blue. That’s [red-blue tower] a b. So b a would be a blue red. So how, if you have them in front of you, how would they look different? Red and blue, red’s on top, and blue’s on the bottom. Blue’s on top and red’s on the bottom.

In the following episode, Ankur explains how to find the answers to the five-tall towers problem in row 5 of Pascal’s Triangle, and then Jeff and Romina locate the pizza answers in row 6. (Row 6 of Pascal’s Triangle contains the numbers 1 6 15 20 15 6 1.) ANKUR: JEFF:

This [1] is no red. Yeah.

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ANKUR: JEFF: ANKUR: ROMINA: ANKUR: JEFF: ANKUR: JEFF: ROMINA: JEFF: ROMINA: ANKUR: JEFF: ROMINA: ANKUR: ROMINA: ANKUR: JEFF: ANKUR: JEFF: ROMINA: JEFF: ROMINA: JEFF: ROMINA: JEFF: ROMINA: ANKUR:

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So there’s one with no red. There’s six with one red. . . . There’s fifteen with two reds. Twenty with three reds. Six with five reds. And one with noAnd one with noNo. No. Six reds. One with six reds. . . . All right. Now. What does that have to do with pizza? Just relate the tower problem to the pizza problem. Well, we’re saying that this [1] is a pizza with just plain. Yeah. That’ll be the plain pizza. Plain. This [6] is with all your six toppings. That’s with one topping. You can’t exactly relate these numbers to the pizza problem. Well, we’ll try really quick. Yeah. You can. ‘Cause this [1] is plain, just plain pizza. And what will the other 1 represent? With everything on it. Okay. So this is plain. Okay. Six withWith one of each. Fifteen is withTwo toppings. Just two toppings out of your six. Twenty is with three toppings. Fifteen is with the four toppings. Six is with the five toppings. Five toppings. And the other one isAnd the one is with all of them. Like the supreme. Is that good? Cool. We’re on fire today.

Thus Ankur, Jeff, and Romina used the two combinatorics problems they knew in order to explain the numbers in Pascal’s Triangle. This was the first time they were observed connecting the pizza problem to Pascal’s Triangle. Although Ankur had initially been reluctant to attempt a definition of the relationship between Pascal’s Triangle and the pizza problem (“You can’t exactly relate these numbers to the pizza problem”), he still participated in the discussion and at the end expressed satisfaction with their work. (“We’re on fire today.”) We noted earlier that the students received no special concrete rewards for participation in the study. Ankur’s remarks illustrate our belief that the intellectual enjoyment involved with solving difficult problems was a factor in the students’ continuing involvement in the study. After explaining the link between specific numbers in Pascal’s Triangle and the pizza and towers problems, the students described an instance of the addition rule in terms of towers problems. They explained the instance of Pascal’s Identity shown in

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Fig. 10.7 One instance of Pascal’s Identity which students linked to towers problems

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Fig. 10.7. They described the two 10’s in row 5 and of the 20 in row 6 as counting classes of five-tall towers, and they described the process by which the 20 (counting a different class of six-tall towers) could be generated from the two 10’s. The transcript below gives portions of their discussion. RESEARCHER:

What are those tens counting? And what does the twenty count?

JEFF: ANKUR: ROMINA: ANKUR:

The tens showThe tens show two of one color. And three of another. One color and two of another color. . . . That’s why it’s ten and ten. But then, at the top of each one, you can put eitherYou could either put a red or like blue. The first ten in that row of five high has two reds and three blues? We’re counting reds? Yes. And the second ten hasThree reds and-three reds and two blues. Now coming down here, the twenty is supposed to count the ones that have three of each. Three red. Three reds and three blues. Right. So how do the two tens add to give the twenty? Because in these ten, where there’s three reds and two blues, you want to make it three reds and three blues. So you put a blue on top of each one.

JEFF: RESEARCHER: ANKUR: RESEARCHER: ANKUR: RESEARCHER: ANKUR: JEFF: RESEARCHER: ANKUR:

This is the first instance where a connection was made between Pascal’s Identity and a specific concrete combinatorics problem.

10.6 Session 5: Towers, Pizzas, and Pascal’s Triangle Ankur, Jeff, Mike, and Romina attended this session, 4 weeks after session 3. Much of the session was devoted to attempts by Ankur, Jeff, and Romina to explain to Mike specific instances of Pascal’s Identity in terms of towers and the binomial expansion. This session also provided another example of the importance of revisiting problems and re-explaining solutions. In the February session, described above, the students had successfully explained the addition rule shown in Fig. 10.7 in

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Fig. 10.8 Another instance of Pascal’s Identity discussed in terms of towers problems

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terms of towers: to create the 20 six-tall towers that have three red and three blue cubes, add a blue cube to the 10 five-tall towers with two blue cubes and add a red cube to the 10 five-tall towers with two red cubes. But this time, when asked to explain the similar case shown in Fig. 10.8, they did not recall the previous (correct) explanation. Instead, they tried to find a solution that would involve disassembling and re-assembling existing towers, an approach that surprised and confounded the researcher. They correctly mapped the numbers 1 and 3 into tower groups (1 = white-whitewhite and 3 = blue-white-white, white-blue-white, and white-white-blue, as shown in Fig. 10.9). But they tried to explain the 4 by breaking apart the tower representing 1 tower and distributing its cubes among the other three towers. After the researcher questioned this method, Mike gave a different explanation for how to represent the 4. JEFF:

RESEARCHER: JEFF: ANKUR: RESEARCHER: ANKUR: ROMINA: MIKE:

ANKUR:

We’ve got this [the white-white-white tower, representing 1]. And we’re saying how this goes together. [Jeff has assembled the three towers each with one blue cube to represent the 3. Refer to Fig. 10.9.] We’re saying- [Jeff starts to dismantle the white-white-white cube.] No. No. Don’t take that apart. BecauseWell, that’s why I made this. So I could. We made another one so we can take that one apart. . . . And show you. You mean, you mean, you mean you get the four by taking something apart? You’re not taking it apart. You’re not taking it apart; you’re just seeing how they go together. . . . You don’t really have to take it apart to show this, ‘cause look. Each one, the reason why they combine, each one of these four blocks [towers] is going to have something added to them to equal the same thing. Yeah.

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Fig. 10.9 Students map specific towers to 1 and 3 in row 3 of Pascal’s Triangle

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MIKE:

ANKUR: MIKE: ANKUR: MIKE:


These blocks [towers] are going to have, they’re going to have a white block added to them. [Mike indicates the three three-tall towers with one blue cube.] They’re going to have a b added to them. And this one’s [the white-white-white] going to have a, a blue added to it. An a added to it. And they’re going to equal the same thing. That’s why you’re going to have the four. [Refer to Fig. 10.10 for a diagram of Mike and Ankur’s suggestions.]

The other students accepted Mike’s explanation and apparently this time they comprehended the process, as evidenced by their later work in the night session (see Chapter 12) and subsequent interviews. In addition, Ankur reiterated a link noted in the February session, observing that the a’s and b’s in the binomial expansion could be connected to the blue and white cubes, respectively, in the towers. The same connection was explained a year later during the night session. Next, the researcher asked the students to relate the tower problems to binary notation and the pizza problem; she said, “If you had to make up a pizza problem to model this row [row 2 of Pascal’s Triangle], what’s the pizza problem?” Ankur reiterated the position that he had taken in two previous sessions, that a peppers and pepperoni pizza is the same as a pepperoni and peppers pizza; it appeared that he had not yet firmly established that there was a connection between the two choices of cube colors for each cube in a tower and the two choices for each topping – on or off the pizza. Although the group noted that the nth row of Pascal’s Triangle could be linked to the n-topping pizza problem, they did not propose an explanation about how to use the numbers in the nth row to enumerate pizzas. When the researcher asked for clarification and Ankur insisted that there was no relationship between colors and pizza toppings, Mike interrupted with his own explanation. It is

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Fig. 10.10 Mike and Ankur illustrate 1 + 3 = 4 by adding cubes to towers

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interesting to note the similarity between this episode and the earlier one in which the students discussed how to connect pizza problems to Pascal’s Triangle. In the earlier episode, Ankur initially denied a connection. Also, in that episode, when the connection was established, Ankur, with Jeff and Romina, quickly caught on and proceeded as enthusiastic participants in the exploration and explanation process. In this episode, also, we see a student express satisfaction with the group’s intellectual achievement. (Romina said, “Oh, wow!”) Figure 10.11 illustrates the four two-tall towers the group made as part of this process. RESEARCHER:

ANKUR: MIKE: ANKUR: ROMINA: MIKE: ANKUR: MIKE:

ANKUR: MIKE: JEFF: ANKUR: ROMINA: MIKE: ROMINA: MIKE: ROMINA: ANKUR:

Now wait. Now I’m lost again. What, what, what was this? . . . [The Researcher indicates the single white and blue cubes representing row 2 of Pascal’s Triangle.] The colors don’t, don’t look at the colors. No. No. No. Just look at this [Pascal’s Triangle]. . . . But the colors don’t specifically represent anything. Yeah. Yes. It does. No, it don’t. Topping. [Mike points to the blue cube.] Or no topping. [Mike points to the white cube]. Just say like that. And if you look at it like this, you know. So all of the whites are no topping? Yeah. [Mike takes the white-white-white tower.] Then this is a plain pizza with a choice. If you had a choice of three toppings. All right. Okay. Okay. This [the blue-white-blue tower] would be a pizzaOh. With the one. Ooh. -with two different toppings, without the other, third topping. That’s what I was asking. Okay.

Pizza with Second Topping

Plain Pizza

Fig. 10.11 The students’ link between two-tall towers and two-topping pizzas

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JEFF: ROMINA: ANKUR: ROMINA: JEFF: MIKE: ANKUR: JEFF: ANKUR: JEFF: ANKUR: ROMINA: RESEARCHER: ANKUR: ROMINA: RESEARCHER: ROMINA:


. . . Well, yeah. Well, if you’re just saying that this [the whitewhite-white tower] is the pizza with three no toppings, it’s plain. It’s just a plain pizza. All right. All right. So that’s [blue-blue tower] two toppings. Yeah. Yeah. All right. So. That’s [white-white tower] . . . a choice of two, but you want it plain. You have a choice of two toppings. Yeah, so this is, this [blue-blue tower] is choice of two using two. This [blue-white tower] is choice of two using one. Two using one. This [white-blue tower] is choice of two using the other one. That’s using the other one. And that’s [white-white tower] using nothing. Yeah. And that’s all the possibilities? Yes. Yeah. You like that? Oh, wow!

During this episode, the other three members of the group immediately accepted and built upon Mike’s brief remarks. All he had to say was “topping” and “no topping,” and all three of the others began immediately to form connections between specific individual towers and specific pizzas. This represented the fifth discussion of the pizza problem in 4 months, and at least three members of the group apparently began this discussion without a clear idea of the essential feature of the problem (topping versus no topping), as opposed to a surface feature (the fact that the toppings could be selected in any order). But it appears that this discussion helped them finally to make sense of the isomorphic relationship, because the pizza problem was the one that the group selected during the night session a year later, to explain Pascal’s Identity.

10.7 Discussion During these five tenth-grade problem-solving sessions, the students worked independently, sometimes spontaneously splitting themselves into subgroups, sometimes working individually, but always sharing their ideas with the other members of the group. By sharing, the students were able to incorporate others’ ideas into their own understanding of the justifications. An example is Mike’s introduction of the binary notation code. Mike watched Jeff and Romina work for a while, listened as the other four students exchanged ideas, and then began focusing on his own paper. He later presented his conception of how the binary system mapped onto the

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solution of the pizza problem. Mike recalled an episode from an eighth-grade class and applied his previously constructed knowledge to a totally new situation. His introduction of a coding system, the zeroes and ones of the binary system, to the justification being built by the group of five students was an important contribution. It became the students’ notation of choice for future problems. Over the course of these sessions, we observed the students investigating problems that had been explored in earlier years, retrieving earlier ideas and images as they built solutions and justifications. These ideas and images sometimes reappeared just as they were formed in the prior occurrences. In the third session of the sophomore year, we see Mike and Ankur’s swift production of a justification for the number of five-tall towers with exactly two red cubes. They reproduced a justification by cases that had originally been built in their fifth-grade classroom, but using Mike’s binary notation. In addition, they offered a second justification, utilizing a strategy that depended on controlling for variable that was first introduced by Ankur while solving a pizza problem in grade five. His ownership of this strategy allowed him to adapt it for use in the isomorphic block tower problem. For the same problem, Jeff retrieved a strategy used during that same fifth-grade session. Mike and Ankur had enthusiastically participated in the whole classroom discussion, which culminated in a proof by cases. Although Jeff was in the room, he was not focused on the classroom discussion; he was looking at patterns in the towers that he built. Jeff’s partners, Romina and Brian, also had more difficulty in providing a justification during the session in their sophomore year. In grade five, while Mike and Ankur were active participants in the classroom discussion and Jeff was quietly pursuing his own line of thinking, Brian and Romina were in another classroom. Although they worked on the five-tall towers problem, their class did not offer a convincing justification for the answer to that problem. The difficulties experienced by these students as they worked on the block tower problems as tenth graders might be explained by the absence of some earlier experiences. They constructed the images and representations to the block tower problem for the first time in this tenth-grade experience. In the attempt to think about the potential connection between the pizza and block tower problems, the students came to discuss many powerful mathematical concepts. While they were able at an early point to determine that the answer to the n-topping pizza problem is 2n , they came to this number by recognizing the pattern of {2, 4, 8, 16, 32, . . .}. They determined that n represented the number of toppings, but did not provide a satisfactory explanation for the base 2 until they began the discussing the possible relationship between pizzas and block towers; then that they came to see that the base 2 represented the presence or absence of a topping. This realization came, not from working on the pizza problem, but instead as a result of their search for an answer to the three-color four-tall tower problem. Thus we see that the opportunity to work on open-ended problems and follow paths determined by the interest of the moment can lead to greater understanding of other problems. In this case, the opportunity to investigate an isomorphic problem provided the students with the tools necessary to complete the formulation of the imperfectly developed earlier idea.

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In summary, the students investigated isomorphic problems in combinatorics and used them to explore how Pascal’s Triangle grows and to make sense of Pascal’s Identity. Between December 1997 and March 1998, they first found general solutions to the pizza and towers problems, using letter and number codes and binary notation to enumerate the pizzas and towers. Then they organized their lists of solutions, organizing the pizza problem solutions according to number of toppings and the towers problems solutions according to the number of cubes of one color. These lists not only provided a way to show that all cases were present, but they also provided the means to associate those cases with the numbers in Pascal’s Triangle. In discussions with the researcher and other researchers, the students described the isomorphic relationship between the pizza and towers problems. Their extensive repertoire of representations proved essential; in this process, they made use of words, written inscriptions, and concrete materials (as when Mike held up a blue cube and a white cube and said “topping” and “no topping”). The opportunity to revisit problems also proved crucial, as students often needed to have two, three, or more discussions on the same topic before critical ideas were firmly established. In the next chapter, we observe another cohort of students who also make sense of the relationships among towers, pizzas, and Pascal’s Triangle. They brought their own experience, their own representations, and their own ideas to the problem, but they too used personal representations, communicated findings, and made generalizations that showed their increased understanding of these problems in combinatorics.

Chapter 11

Pizzas, Towers, and Binomials Lynn D. Tarlow


March 1, 1999; Grade 11 Pizzas, Towers, and Pascal’s Triangle Amy-Lynn, Angela, Magda, Michelle, Robert, Shelly, Sherly, and Stephanie Carolyn Maher, Alice Alston, Regina Kiczek, Ralph Pantozzi

11.1 Introduction In the previous chapter, we observed a cohort of tenth-grade students as they investigated the connections among the pizza problems, the towers problems, the binomial coefficients, and Pascal’s Triangle, leading to their increased understanding of the meaning of the numbers in Pascal’s Triangle and how the triangle grows. In this chapter, we see how another cohort of students, composed of two subgroups with different backgrounds (from students new to the study to students who had been in the study from the start) worked together, sharing their ideas, comparing representations, and discussing relationships among problems. Through their collaborative work, they too came to discover generalized rules for the pizza and towers problems, see how both problems were related to Pascal’s Triangle, and explain the meaning of Pascal’s Identity through the use of two different metaphors: solving the pizza problems and solving the towers problems. The eight students at this session were organized into two groups of four. Each group worked independently. The four students at Table A (Robert, Stephanie, Shelly, and Amy-Lynn) had participated in the tower and pizza investigations in grades 3–5 through five. Only Robert had participated in the previous 11th-grade tower investigation discussed in Chapter 9. Of the students at Table B (Angela, Magda, Michelle, and Sherly), only Michelle had explored the pizza problems in

L.D. Tarlow (B) Department of Secondary Education, The City College of the City University of New York, New York, NY 10031, USA e-mail: [email protected] C.A. Maher et al. (eds.), Combinatorics and Reasoning, Mathematics Education Library 47, DOI 10.1007/978-0-387-98132-1_11, C Springer Science+Business Media, LLC 2010

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the early grades, but all had been present at the previous 11th-grade tower session. This time, the students were given the four- and five-topping pizza problems: A local pizza shop has asked us to help design a form to keep track of certain pizza choices. They offer a plain pizza that is cheese and tomato sauce. A customer can then select from the following toppings: pepper, sausage, mushrooms, and pepperoni. How many different choices for pizza does a customer have? List all the choices. Find a way to convince each other that you have accounted for all possible choices. Suppose a fifth topping, anchovies, were available. How many different choices for pizza does a customer now have? Why?

11.2 Table A: A Connection Between Pizzas and Towers As the students at Table A began to talk about listing pizzas, Shelly complained that she “just did this in school, combinatorics stuff,” but she was not able to remember a formula, although she thought it involved factorials. Her remarks showed that she remembered the form but not the meaning; she said: I don’t know if it’s factorial or combination. I don’t know if you would just do like five factorial plus four factorial plus three factorial plus two factorial plus one factorial. . . . I can’t remember. That was the last section we did. It’s so pathetic.

Shelly was correct in that the solution can be seen as a sum (but of five combinations, not of five factorials), although the solution is often seen as a power of 2: two choices (on or off) for each of four toppings give 16 possible pizzas: 4 4 4 4 4 + + + + = 24 = 16 0 1 2 3 4 Nevertheless, Amy-Lynn agreed that factorials were appropriate, and so Shelly took the sum of 1! to 5! and got an answer of 153. Although no one questioned this use of factorials, the group decided that the answer needed to be verified; and so they started to list the possible pizzas. Stephanie and Shelly discussed possible strategies: STEPHANIE: SHELLY: STEPHANIE:

Do we just want to, um, plot out the pizzas, like with shirts and pants or towers? Do you know what I’m talking about? Yeah. The tree diagram type thing. Yeah, kind of like that. Or is there an easier way to do it that I’m just not thinking of?

Stephanie, Shelly, and Amy-Lynn then proceeded to use tree diagrams to represent possible pizzas, using letter codes for topping combinations. As an example, refer to Fig. 11.1 for a diagram of Shelly’s tree. All of the students except Robert started out by including plain as a topping to be combined with other toppings. After a discussion about real pizzas, they decided that this was unnecessary. As Shelly said, “A pepperoni is a plain with pepperoni.” They kept the plain pizza on their lists, though, and just crossed out the duplicates that resulted. They compared answers and, unprompted, decided to create a new

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Fig. 11.1 Shelly’s tree of all 16 pizzas when there are four toppings to choose from

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list according to number of toppings. Enumerating this way, the students found that with four toppings available, there were 16 possible pizzas: one plain, four with one topping, six with two toppings, four with three toppings, and one with four toppings (1 4 6 4 1). They recognized these numbers as following a pattern and belonging in Pascal’s Triangle, but they also realized that they needed more of an explanation of the answer. A portion of the discussion between Shelly and Stephanie follows. SHELLY: STEPHANIE: SHELLY: STEPHANIE: SHELLY: STEPHANIE: SHELLY: STEPHANIE: SHELLY: STEPHANIE: SHELLY:

One, four, wait a minute. One four six four one, so the next one will be one. This is theThe triangle. The triangle. Yeah. So the next one is one five ten ten five one. We’re done. [Shelly laughs.] But what does that mean? [All three girls laugh.] I don’t know. . . . But what it, like, what does one four six four one. That means nothing to me. It means nothing to me either, but it’s the pattern we saw. So we have a pattern, but how do we apply it to getting sixteen pizzas? That would be the problem.

Stephanie and Shelly followed up with a discussion of how the numbers in row 4 could be matched to the answers to the four-topping pizza problem. STEPHANIE: SHELLY: STEPHANIE: SHELLY: STEPHANIE:

So, well, okay, let’s figure this is saying that we have one plain pizza. Uh-huh. And then we have four pizzas with two toppings? With one. Because it’s the plain and then with the one topping. Okay. So we have four pizzas with one topping. And we have four pizzas with two toppings. Oh, no, we have six pizzas with two toppings, four pizzas with three toppings.

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And one pizza with four toppings. Okay.

Then they used the next row of the triangle to determine the number of possible pizzas with five available toppings, 32 (the numbers 1 5 10 10 5 1 representing the numbers of pizzas with zero through five toppings in order). After the group found the relationship between pizzas and Pascal’s Triangle, Researcher 1 asked the students to explain the addition rule for generating new rows of Pascal’s Triangle from existing rows (Pascal’s Identity) in terms of pizzas. The students had some trouble figuring out how that would work. Stephanie said, “I just can’t get past the fact that you can’t make a pizza out of other pizzas. I think maybe if it was applied to something else I could look at it differently.” Robert suggested that they try relating the towers problem to Pascal’s Triangle. Robert started making drawing in order to follow this line of thinking, but the others went back to thinking about pizzas. Stephanie and Shelly tried to use pizzas to try to explain one particular instance of the addition rule on Pascal’s Triangle: 1 + 3 = 4. See Fig. 11.2. Stephanie continued to maintain that using pizzas to explain Pascal’s Identity did not make sense, “because, like, one is no topping, so adding one to three doesn’t materialize another topping.” She understood that, in general the nth row of Pascal’s Triangle represented all the possible pizzas that could be made with n toppings available, but she could not see how one row generated the next one. She called over Researcher 4 and he suggested that she think of having made all eight three-topping pizzas and then finding that another topping had become available. Then Stephanie realized that each existing pizza could either acquire the new topping or stay the same. Therefore, the addition rule could be explained by thinking of each existing pizza either acquiring a new topping or staying the same. After Stephanie was satisfied that she understood how the addition rule worked, the researcher moved on to another topic with Robert. Robert had drawn rows 0–5 of Pascal’s Triangle, and to the right of each row he wrote the sum of the numbers and the sum expressed as a power of 2. (Refer to Fig. 11.3.) He gave the general rule: that the number of possible combinations for pizza toppings is given by 2 to the number of choices. Researcher 4 asked Robert if he had thought about what role the number 2 played. Robert replied that he remembered that with towers, the total number of tower combinations was “two to the something,” and the pizza situation was “the same thing.” Amy-Lynn was listening, and she concurred: “That was with a lot of the problems, they went by two and it had something to do with powers.”

1 1 1 1

Fig. 11.2 Two instances of Pascal’s Identity

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Fig. 11.3 Robert’s diagram of Pascal’s Triangle and powers of 2

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The question of the role of the number 2 was temporarily deferred, as Researcher 3 stopped by, and Stephanie demonstrated that she had made sense of the relationship between pizza problems and Pascal’s Identity by explaining two instances of the addition rule, first 1 + 3 = 4. (Refer to Fig. 11.2.) You already have three pizzas with one topping. And the plain pizza becomes the pizza with the new topping. Okay, so this becomes, instead of one plain pizza, this is one pizza with one topping. Cause this one’s getting like the pepperoni thrown into it. And that produces the four pizzas with one topping.

Then she explained how 3 + 3 = 6. (Refer to Fig. 11.2.) Now you already have three pizzas with two toppings. So these three pizzas with one topping get an extra topping added on. So these become three pizzas with two toppings. And then three pizzas with two toppings plus three pizzas with two toppings equal six pizzas.

Stephanie also noted that each pizza moved to two places in the row below; in one move the pizza remains the same and in the other move the pizza gets the new topping. For example, a pizza with peppers could be moved to the left and remain a pizza with peppers, or it could move to the right and become a pizza with peppers and mushrooms. Amy-Lynn then returned to the unanswered question about the role of the 2 in 2n , Robert’s expression for the number of possible pizzas when n toppings are available: “Maybe that’s where he got the two to the n; maybe that is where the two comes from.” Shelly and Stephanie agreed: SHELLY: STEPHANIE: RESEARCHER: STEPHANIE: RESEARCHER: STEPHANIE: RESEARCHER: ROBERT:

That makes sense. Yeah that’s where each of the twos come from. That was good, that was really good This one [the first 1 in row 3] only goes here [to the 3 in row 4]? Does it go here [to the 1 in row 4] too? Yeah, it drops down as a plain pizza. Because this one is the plain, I see. Okay, so your drop-down idea is that it stays the same. It stays the same once. And it changes once. Where I guess Amy got the two. Very interesting. Do you agree with this? Uh-huh.

Once the formula question was settled, the researcher asked the students to imagine what the numbers on Pascal’s Triangle might mean with respect to towers.

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Stephanie and Robert said that the numbers in row 3 (1 3 3 1) were for threetall towers, and Robert added that the height was the same as the number of available pizza toppings. Researcher 3 asked, “So, the ultimate question now is, what if you have n toppings?” Stephanie and Robert responded: “2 to the n.” The other students agreed; they wrote up their work as shown in Fig. 11.4. Stephanie then described how part of the row in Pascal’s Triangle that would represent n available toppings would look, “one, n and then whatever, and then, n, one” (1 n . . . n 1). As a group, Amy-Lynn, Shelly, and Stephanie explained what each of the numbers in the 1 3 3 1 row of Pascal’s Triangle would represent in terms of towers, using the colors blue and red. Researcher 1 asked: “So why does one plus three give you four? You have towers three tall. Now you have towers four tall.” Shelly responded, “‘Cause you’re just adding the extra block on.” When asked if they could visualize the three towers with one red block, Stephanie and Amy-Lynn responded together, “One with a red at the top, one with a red in the middle, and one with the red on the bottom.” The group was asked to make a picture of Unifix cubes to illustrate the addition 1 + 3 = 4 (the same process illustrated in Fig. 11.2 that they had explained earlier in terms of pizzas). Stephanie drew diagrams of towers, using b for blue Unifix cubes and R for red, as shown in Fig. 11.5. Stephanie and Shelly then explained: Each of the three-tall towers gets a cube added on top to become a four-tall tower. The tower with no red cubes gets a red cube (R), and the three towers that already have one red cube each get a blue cube (b). These towers represent the case of four four-tall towers with exactly one red cube in each tower. Shelly said that it was easier to explain the “two thing” with the towers because there are only two colors. Stephanie agreed and added that with all the pizza toppings, “it throws you off; you expect eight hundred pizzas.” Researcher 1 asked if there were another way to think about the pizzas. Robert supplied the explanation of the isomorphism: the number of available pizza toppings corresponds to the height

Fig. 11.4 Students’ justification for their generalization of 2n

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Fig. 11.5 Stephanie’s illustration using towers for 1+3=4

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R b b R b b b

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b b b R

of the tower, and the two colors would indicate whether or not a topping was on the pizza. Researcher 1 pointed to the towers that Stephanie had drawn and asked what the b in a tower would mean if you were thinking of pizzas. Stephanie answered that it would mean that you either had or did not have the topping. Researcher 1 then asked the group what a four-tall tower would represent in terms of pizzas. Robert, Stephanie, and Shelly explained together that each of the four cubes indicated whether or not you would choose each of the four toppings. Stephanie added that it would be, for example, mushrooms “going all the way across,” and each tower that had an R in that position would indicate that pizza did not have mushrooms on it. The other toppings would be represented in the same way.

11.3 Table B: Connection Between Pizzas and Pascal’s Identity Angela, Michelle, Magda, and Sherly started work on the four-topping pizza problem by making individual tree diagrams. This work was interrupted with a question about order: Angela asked whether a sausage and pepperoni pizza is the same as a pepperoni and sausage pizza. Although the consensus was that this did represent the same pizza, they decided to leave duplicates in the tree diagram and remove them later. After they prepared their tree diagrams, Magda noted that she originally had 24 combinations on her tree diagram for the four-topping pizza, but after crossing out duplicates, she was left with only one pizza that had exactly four toppings; this led the group to realize that they would have to cross out a lot of pizzas. Angela found 15 different pizzas that can be made when there are four topping choices. In order to confirm this answer, the group agreed that each member would work on a different case: Sherly would do the one-topping case, Angela would do the two-topping case, Michelle would do three toppings, and Magda would do four toppings. Magda had only one pizza; for the other three cases, Angela and Sherly made lists and Michelle used a tree diagram. They concluded that there were 15 total combinations; four with one topping, six with two toppings, four with three toppings, and one with four toppings, confirming Angela’s earlier answer. When Researcher 2 stopped by to ask about their solution, Michelle pointed out that they had not counted the plain pizza; therefore, they now had 16 pizzas.

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Next the group continued their work for the five-topping pizza question, with anchovies as the new topping. Again they distributed working on the different cases among the group, with Magda adding the five-topping case to her original fourtopping case, and the others keeping the same cases. Using this procedure, they found all 32 five-topping pizzas. Angela realized that the number of possible pizzas doubled with the addition of one topping, and so she conjectured that for the threetopping case, there would be eight possible pizzas. She confirmed this by finding those eight pizzas. Researcher 2 asked the students to explain their work. Sherly and Magda explained that they had found combinations by substituting the new topping, anchovies, for each of the other toppings in their previous combinations. In this way, they would not have to cross out answers. Seeing the doubling pattern with the pizzas, Magda recalled that there was also a doubling pattern in the towers problem, which they had not been able to explain before. Angela recalled the formula for towers from the previous session (“we came up with that whole like x to the n thing”). She did not observe that the problems were structurally similar, though, noting that order seemed to make a difference with towers (a tower with a red cube on top of a yellow cube is a different tower from one with a yellow cube on top of a red cube), whereas a pepperoni and mushroom pizza is the same pizza as a mushroom and pepperoni pizza. Researcher 2 asked the group to continue to explore finding pizzas with other topping choices. They gave the numbers for two choices (four) and one choice (two), for which Angela listed the possible pizzas. They predicted that with six choices, the number of possible pizzas would be 64. Then Magda had what was described as a breakthrough: She noticed that the numbers they were finding were also seen in Pascal’s Triangle. A discussion with Researcher 2, Sherly, and Angela ensued. Magda at first thought she was mistaken, but the two other students convinced her that she had seen the pattern of Pascal’s Triangle (see Fig. 11.6).

Fig. 11.6 Magda and Sherly discuss Pascal’s Triangle

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I am thinking. One, three, three, one. I don’t know, maybe it has something to do with this. Why don’t you put it on a new piece of paper cause you are about to give out I don’t know if that works though. Does it? Let’s see. No, it doesn’t work because we don’t have the four in here, or something. You know that, like Pascal’s. Pascal’s, yeah. Yeah But it doesn’t work, so scratch out my idea. No, wait just a minute. Explain what you see and what doesn’t work. Wait, wait. How does it work? But it doesn’t work for this one [row 5]. Six, four and four. Yes it does. What are you talking about? Yeah. Oh wow, it does. Maybe my idea works. Yeah, ‘cause then if you have zero toppings, there is only one [pizza]. Magda’s smart. Who would have figured?

The diagram of Pascal’s Triangle that Magda drew is shown in Fig. 11.7. The researcher then asked them to explain how the addition rule for Pascal’s Triangle (Pascal’s Identity) could be explained in terms of pizzas. After a lengthy discussion, Angela said that the 6 in row 4 (representing six pizzas with exactly two toppings) would be generated from the two 3’s in row 3 (representing three pizzas with one topping and three pizzas with two toppings): add the new toppings (peppers) to the three pizzas that had one topping. She said, “Just go, peppers/sausage, peppers/pepperoni, peppers/mushroom, right? There you go.” The three other pizzas that already had two toppings each (sausage/pepperoni, sausage/mushroom, and pepperoni/mushroom) would be unchanged and become part of the two-topping group at the next level.

Fig. 11.7 Magda illustrates how Pascal’s Triangle is related to the pizza problem

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At Magda’s suggestion, the group tested this rule by explaining other additions on Pascal’s Triangle using pizzas. When Magda was asked to explain the relationship between pizzas and Pascal’s Triangle, she responded: Okay, so we have the three 1-toppings ones, which was the sausage, pepperoni, and mushroom. So those are the three combinations for the 1-topping ones. Because we are moving to four toppings, we needed to add an extra topping. So we just added peppers because this stands for two topping pizzas, so we just added the peppers.

The group noted that the other three represented pizzas that already had two toppings, so those three pizzas did not change. In summary, the group found a pattern and a rule for the general pizza problem. Their organization helped them to see a relationship between the pizza problem and the numbers in Pascal’s Triangle. They were able to explain the meaning of Pascal’s Identity in terms of generating successive groups of pizzas, with more choices for toppings. Finally, they noticed that the pizza and towers problems had the same answers.

11.4 Discussion All of the students in this session solved the pizza problem and justified their solution using proof by cases. They connected their topping combinations to the numbers on Pascal’s Triangle and used answers to the pizza problem in order to explain the addition rule for Pascal’s Triangle. They also noted the doubling pattern as the number of available toppings increased. In addition, the students at Table A, who had participated in the longitudinal study and explored the tower and pizza problems in the earlier grades, explained their reasoning for the doubling rule using both pizzas and towers. Furthermore, they explained the addition rule for Pascal’s Triangle using towers as well as pizzas. Finally, the Table A group constructed a three-way isomorphism between the tower problem, the pizza problem, and the numbers on Pascal’s Triangle. We also note that although the students at Table A made an attempt to use a partially remembered formula, they did not accept the formulaic answer but went on to use their own methods to verify the answer; in the process, they abandoned that formula and instead generated their own correct formula. These 11th-grade students exhibited advanced reasoning skills as they thought about the problems, justified their solutions, and made connections. The students who had participated in the longitudinal study in the early grades demonstrated that they had benefited from exposure to thoughtful mathematical experiences over a long period of time. They retrieved representations that had been built years before and then modified those representations in order to build and extend their mathematical knowledge and make connections to other mathematical ideas. The students who came to the study later on also demonstrated the ability to think deeply about the problems and justify their solutions.

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The students were presented with challenging problems and given the responsibility to solve them. They did not work in isolation; rather, they were active participants in a learning environment where ideas were shared and discussed. In the course of making sense of their observations and of what their peers were saying and doing, they moved back and forth between their representations, which had become less concrete and more abstract and symbolic. They developed, modified, refined, and extended representations, which enabled them to solve increasingly more complex combinatorics problems. They built an understanding of fundamental mathematical ideas and used those ideas to justify solutions to problems. Using their personal notations, they extended their reasoning and made connections to other ideas in combinatorics. They were offered opportunities to work on rich, demanding problems, to think carefully about their ideas, and to discuss their ideas with their peers and with the researchers. Given the opportunity to think and reason together, the students constructed deep and powerful mathematical ideas. In this chapter and the preceding chapters, we observed students make sense of Pascal’s Triangle based on personal experience with combinatorics problems and making use of personal notation. In the next chapter, we see how one group of students made sense of standard mathematical notation by building on their personal notation and knowledge.

Chapter 12

Representations and Standard Notation Elizabeth B. Uptegrove


May 18, 1999; Grade 11 Towers, Pizzas, and Pascal’s Triangle Ankur, Brian, Jeff, Mike, and Romina Carolyn Maher and Regina Kiczek

12.1 Introduction In the preceding chapters in this section, we considered how students made sense of Pascal’s Triangle and isomorphic combinatorics problems using their own increasingly sophisticated and abstract representations. In this chapter, we see how one group built on those ideas in order to derive, explain, and record Pascal’s Identity (the addition rule for Pascal’s Triangle) using standard mathematical notation. This remarkable demonstration of how students can come to make sense of complex mathematical ideas was captured during the session that came to be referred to as the “Night Session,” since it took place on a weekday evening from 7:30 to 10:00 PM (Uptegrove, 2004). In Chapter 10, we described how, during their sophomore and junior years of high school, Ankur, Brian, Jeff, Mike, and Romina made use of the pizza and towers problems to develop complex mathematical notions: recognize isomorphisms, generalize findings, and represent ideas using their own personal representations, which had become increasingly sophisticated and symbolic over the years. In the session described in this chapter, they made use of standard combinatorial notation to communicate, clearly and concisely, the ideas about Pascal’s Triangle and Pascal’s Identity that they had previously developed. They derived Pascal’s Identity, wrote it in standard notation, and explained the meaning of the standard notation in terms of general versions of the pizza and towers problems.

E.B. Uptegrove (B) Department of Mathematical Sciences, Felician College, Rutherford, NJ, USA e-mail: [email protected]


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In the following sections we discuss the strategies used by the students to make sense of the standard notation. We note how the use of increasingly sophisticated notation accompanied the students’ building of general notions about the meaning of Pascal’s Identity. We show how their organizational strategies proved key in their making sense of the standard notation and of the relationships between the combinatorial problems, Pascal’s Triangle, and Pascal’s Identity. Further, we show that they found in the standard notation an essential tool for expressing their understanding of Pascal’s Identity in general form.

12.2 Summary of Earlier Student Work As discussed in Chapter 10, during sophomore- and junior-year problem-solving sessions, Ankur, Brian, Jeff, Mike, and Romina revisited and extended previous work on two familiar combinatorial problems. In the first session of their sophomore year of high school, they were reintroduced to the towers and pizza problems. In subsequent sessions, they found a way to organize their solution lists to prove that all solutions were present; they recognized that those problems were related to each other, to the binomial coefficients, and to Pascal’s Triangle; they found general solutions to those problems; and they used those problems to form preliminary ideas about the meaning of Pascal’s Identity. The key organizational decision – to organize pizzas by number of toppings and towers by number of cubes of a given color – not only helped the students show that

The coefficients of the binomial expansion: 1

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Fig. 12.1 Row 5 of Pascal’s Triangle

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they had all the solutions, but also helped them see the relationship between the two problems and Pascal’s Triangle. It enabled the students to realize, for example, that the fifth row of Pascal’s Triangle (1 5 10 10 5 1) contains not only the coefficients of the binomial expansion but also the solution to both the five-topping pizza problem and the five-tall towers problem. Refer to Fig. 12.1. When the students were introduced to the use of standard notation to describe the binomial coefficients, they were able to make use of the connection they had already formed between the binomial coefficients and the towers and pizza problems. They applied the standard notation to the pizza and towers problems; knowing how to generate the answers to the pizza and towers problems in their own notation enabled them to use the standard notation to describe the general pizza and towers answers and finally to express the general rule for building Pascal’s Triangle.

12.3 The Night Session In the night session Ankur, Brian, Jeff, Mike, and Romina returned to the investigations of Pascal’s Triangle that they had begun in their sophomore year. By the end of this session, they had written Pascal’s Identity in standard notation and provided a sound explanation of its meaning. They did this by looking at general forms of the towers and pizza problems, referring back to their previous explanations of specific instances of Pascal’s Identity in terms of towers and pizzas, and making use of the binary notation that had been introduced by Mike some 18 months earlier. At the beginning of the session, the first three students to arrive (Jeff, Mike, and Romina) talked about that day’s class work, which had been to find the coefficients of the expansion of (a+b) n . Jeff brought up what they called “choose” notation, the notation to denote combinations, using the nCr function on their calculators. In this episode, when Jeff was trying to explain how to find a particular coefficient of the expansion of (a+b)10 , Romina spontaneously introduced the towers problem with the words “ten high” and “two reds.” Jeff and Mike elaborated that this meant building towers ten cubes tall, selecting from two colors, and counting how many there are containing exactly two cubes of one of the colors (red). JEFF:

ROMINA: JEFF:

. . . If we were looking for a plus b to the tenth say, . . . it was 1 a to the tenth and then 10 a to the ninth b to the first, right? . . . [The next coefficient] was 45, but we were working on how to figure it out. We knew it was the choose thing, whatever that means. . . . What was it? Ten choose two? . . . Like, uh, was it N-C-R? [Jeff is referring to buttons on his calculator.] Two, is that how you do it? [Jeff writes 10 nCr 2.] Right? . . . And that equals 45, and that’s the answer. . . . We’re not really sure how all this works but it’s like . . .. If you have ten different. What is it? Ten different things . . . Ten high. Ten high. Ten high. How many.

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How many would have two reds, only two reds. How many would have two, two reds. One more time. If you have towers with ten high and two colors. How many different places can you put two reds in there? And like a would be one color and b would be the other color.

Their original explanation of “ten choose two” was so brief; it would have been difficult for anyone not familiar with their work to understand the references “ten high” and “how many would have two reds.” But the elaboration (although it still assumed knowledge of the towers problem) shows that they knew that the coefficients of the binomial expansion to the tenth power were related to the ten-tall towers problem. A few minutes later when the researcher asked the group to discuss the “choose” notation that Jeff had mentioned, Mike drew a few rows of Pascal’s Triangle on the board and explained that any row could be expressed in “choose” notation; for example, the row 1 3 3 1 could be called “3 choose 0” through “3 choose 3.” When the researcher asked the students to talk about the addition rule for Pascal’s Triangle in that notation, Romina suggested a new vehicle, the pizza problem, even though she had just used the towers problem in the previous explanation. Mike used the pizza problem to explain a particular case of addition: think of the nth row of Pascal’s Triangle as representing all the possible pizzas that can be made when there are n toppings to choose from, and think of generating new rows of Pascal’s Triangle as making new pizza toppings available. Then the pizzas represented by the first three in row 3 are the one-topping pizzas (when there are three toppings to choose from). You can either add the new topping to those three pizzas (making them two-topping pizzas) or let them remain one-topping pizzas. If you add the new topping, you now group them with the second three in that row (the pizzas that already have two toppings), resulting in six pizzas with two toppings. If you do not add the new topping, those three pizzas are added to the one pizza that had no toppings (and that had the new topping added to it), giving four pizzas with one topping. Figure 12.2 illustrates Mike’s explanation. A portion of their discussion is given below.

Fig. 12.2 Examples of Pascal’s Identity

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Let’s go to this one. This would be like three different places, I guess. [Mike indicates row 3, which is 1 3 3 1.] . . . That would be a plus b to the third. All right, let’s say you have like, here’s a number, all right? [Mike writes 000.] Zero means no toppings. One would be a topping. So first category is everything with no toppings. [Mike points to the first 1 in row 3.] And that’s your number for that one. [Mike points to 000.] That’s like, like binary numbers or something. Next would be- [Mike writes 001, 010, then 101.] There’s all the, the ones that have one topping. Right, you got to write that 0 at the end. You messed up. Last one should be a hundred, not a hundred and one. I knew that. [Mike changes 101 to 100.] There’s all the ones that have one topping. . . .. There’s your 3 choose 1 and there’s three different combinations you could put that. . . .. But, um, when you have a new – when you add another place, another topping. [Mike draws dashes to the right of the four numbers already there. Refer to Fig. 12.3.] That could be one or the other, one or the other, one or the other. So, it could be one or the other. It could be a zero or one, a zero or one, zero or one. [Mike writes 0 and 1 above each dash.] So all these threes would either move up a step onto the next category and have two toppings. [Mike points to the 6 in row 4.] Or they might stay behind and still only have one if they have the zero. [Mike points to the 4 in row 4.] So three get a topping, go to this one [Mike points to 6.] and three won’t, will stay. [Mike points to 4.] These three [Mike points to the first 3 in row 3.] with one topping won’t get one so, you know, you can put them in the same category as this one. That’s their four? Yeah. That’s four. . . .. And you know, the three that had two toppings won’t get any. [Mike draws a line from the second 3 in row 3 to the 6 in row 4. Refer to Fig. 12.2.] And you could put them in together with the ones that did get something. That’s why you would add.

After Mike explained the specific instances of 3+3=6 and 1+3=4 in terms of pizzas, the researcher (R1) rewrote row 3 of Pascal’s Triangle in standard combinatorial notation and asked the students to write other rows in that form and show an example of the addition rule. Figure 12.4 shows what they did. Their discussion follows.

000 01 00101 01001 100 01 Fig. 12.3 Binary listing of 3 choose 0 and 3 choose 1

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Fig. 12.4 Showing 3 + 3 = 6 in combinatorial notation

RESEARCHER: MIKE: RESEARCHER: JEFF: ANKUR: MIKE: JEFF: ANKUR:

Show me that 3 plus 3 is 6. Which ones would it be? . . . This one and that one. [Mike points from 3 choose 1 and 3 choose 2 to 4 choose 2.] . . . Okay, so you’re saying 3 choose 1 plus 3 choose 2 equals 4 choose 2. Right? Okay. So what’s 4 choose 2 plus 4 choose 3? . . . 4 choose 2 plus 4 choose 3? That would be, that would be 55 choose. . . . 5 choose 3. Why is he 5 choose 3? Because it’s always the one on the right. [Ankur means that the “choose” number of the sum is the same as the “choose” number of the rightmost addend.]

Mike observed that the bottom number indicated the number of toppings actually used, so that when a topping was added, the bottom number changed and when a topping was not added, the bottom number did not change. The researcher asked the group to continue by writing a general (nth) row of Pascal’s Triangle and to use that row to discuss the meaning of the addition rule. Figure 12.5 shows the two general rows that Jeff drew. In spite of the researcher’s suggestion to use lowercase n to indicate row number and r to indicate a number in the middle of the row (following standard usage), Jeff used uppercase letters N and X. Brian arrived after the group had been working for almost an hour; first Jeff explained to Brian how Fig. 12.5 related to that day’s work in their regular math class relating to Pascal’s Triangle: We’re explaining the general addition, the addition rule using chooses to fill out the triangle, and this here would be N choose X plus 1 and then N choose X plus 2 and so on to whatever N equals.

Then the group was asked to write the addition rule in general form. Figure 12.7 shows Jeff working at the board as they discussed the problem. This discussion follows.

Fig. 12.5 Rows N–1 and N of Pascal’s Triangle

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ N − 1⎟ . . . ⎜ N − 1⎟ . . . ⎜ N − 1⎟ ⎝0 ⎠ ⎝ N − 1⎠ ⎝X ⎠ ⎛ N ⎞ ⎛ N ⎞ ⎛ N ⎞ ⎛ N ⎞ ⎛ N ⎞ ⎛N ⎞ ⎛ N ⎞ ⎜ 0 ⎟ … ⎜ X − 2⎟ ⎜ X −1⎟ ⎜ X ⎟ ⎜ X+ 1 ⎟ ⎜ X +2 ⎟ … ⎜ N ⎟ ⎠⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠⎝

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Representations and Standard Notation ⎛ N⎞ ⎛ N ⎞ ⎛ N + 1⎞ + = ⎝ X ⎠ ⎝ X+ 1⎠ ⎝ X + 1⎠

139 ⎛ n ⎞ ⎛ n − 1 ⎞ ⎛ n − 1⎞ = + ⎠ ⎝ r ⎠ ⎝ r −1 ⎠ ⎝ r

Fig. 12.6 Pascal’s Identity in students’ notation and as shown in textbooks

RESEARCHER: ANKUR: MIKE: JEFF: MIKE: JEFF:

MIKE: JEFF: MIKE: ANKUR: MIKE: JEFF: RESEARCHER: JEFF: BRIAN: JEFF: BRIAN: JEFF: ROMINA: BRIAN: JEFF: BRIAN: JEFF:

BRIAN: JEFF:

Can you write it as an equation? Just like you wrote three plus three equals six. N plus, just that plus that. [Ankur points to the entries N choose X and N choose X + 1 in Fig. 12.5.] N choose X. N choose X plus N choose X plus one. [Jeff writes on the board as he speaks. Refer to Fig. 12.6.] Equals that. . . . Plus one equals that right there. [Jeff points to N+1 choose X+1.] . . . Then, well, that’s, that’s because this would be gaining an X and going into the X plus 1. [Jeff points to N choose X.] Yeah. And this would be losing an X. [Jeff points to N choose X+1.] No, no, not losing, not getting anything. Staying the same. And the top numbers have changed because you have more. Because you’re adding; you have more things [to choose from]. Say it so Brian can follow it because he wasn’t here for the earlier pizza discussion. What, what we’re doing is the next line of the triangle. Remember how today in class the other triangle was one, two. Yeah. Three, that whole row there. Well, that’s the increase in N and then the X plus one. . . . Say we’re doing pizzas. All right. If you add another topping onto it. You know how we get the triangle and how we go 1 2 1 and add those two together? Yeah. We were explaining why you add. All right, keep going. Because [Jeff points to N choose X+1.] . . . If it gets a topping, that’s why it goes up to the X plus 1. [When a new topping is available, the second (“choose”) number in the expression is increased by 1.] And in this one, it’s staying the same, right? [Jeff points to N+1 choose X.] And that’s why it’s going there. Make sense? Yes. It actually does. So, so that would be the general addition rule in this case.

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Fig. 12.7 Two versions of Pascal’s Identity

The students’ version of Pascal’s Identity is equivalent to a standard textbook version of this equation, with n equivalent to N+1 and r equivalent to X+1. (See Fig. 12.6) Following the production of the equation in combinatorics notation, the students were asked to convert that notation to factorial notation. They did so; their work is shown in Fig. 12.7. Earlier in this session, Mike had explained 3 + 3 = 6 (Fig. 12.3) by stating that the threes are from the three-toppings row of Pascal’s Triangle: the first three represents the three one-topping pizzas that become two-topping pizzas, and the second three represents the three two-topping pizzas that remain two-topping pizzas; the six represents the six two-topping pizzas that can be made when there is a fourth topping available. Now the students generalized this rule using the standard notation, which they called “choose” notation: N choose X gives the number of pizzas that have exactly X toppings when there are N toppings to select from and N choose X+1 gives the number of pizzas that have exactly X+1 toppings. Moving to the next row down in Pascal’s Triangle means that you increase the number of available toppings by one, and so N increases by one. Adding the new topping to the first addend (N choose X) and not adding the new topping to the second addend (N choose X+1) gives a group of pizzas with X+1 toppings when there are N+1 toppings to select from. The students had described instances of Pascal’s Identity in earlier sessions. For example, in their sophomore year, Ankur, Jeff, and Romina had provided a specific explanation using towers: In order to create a six-tall tower with exactly three red cubes, add a red cube to the five-tall towers that have exactly two red cubes and add a blue cube to the five-tall towers that have already have three red cubes. (This was discussed in more detail in Chapter 10.) But this night session explanation was the first explanation of Pascal’s Identity using standard notation to state a general result.

12.4 Durability of Understanding Three years after the night session, in individual interviews, Mike, Romina, and Ankur were asked to recall this work. In that time, all three took math classes in college, but none studied combinatorics. All three were able when prompted to write

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the formula, and all three were able to provide a cogent explanation of the addition rule. As an example, we discuss here the interview with Mike. Over the years in which he worked on the combinatorial problems, Mike progressed from drawings and codes through his personal (binary) representation system to the standard combinatorial notation. From the time he introduced his ideas about binary notation to his fellow students to his most recent interview over 5 years later, he demonstrated the ability to make sense of the problems and of the notation, both through the use of his chosen notation and through the use of the combinatorics tasks. Mike took the lead in devising representations, finding connections, and making sense of the tasks. He recognized structural similarities between problems; he moved between different representations with ease; and he extended, generalized, and reorganized his knowledge when he discussed it with others. In this interview, which took place when Mike was in his second year of college, the researcher (R1) showed Mike a diagram of Pascal’s Triangle and asked him to recall how the group used the pizza problem to think about Pascal’s Triangle. Mike spent a little time regenerating the meaning of specific entries in Pascal’s Triangle and then, in response to a question about a general rule, he reconstructed the formula that had been developed during the night session. A portion of their discussion follows. Mike began with the two-topping pizza problem. MIKE: RESEARCHER: MIKE:

Okay. If you had no toppings, that would be one pizza. Okay. So where is that on the triangle? Well, I’m going to just draw it. . . . And then we’ll find it. . . . If you’re using just one topping, you can make two possible pizzas with that. And then if you have all the toppings, that’s one. Right. And then automatically I see that relates to this row. [Mike points to row 2 of Pascal’s Triangle (1 2 1).] And I’m pretty sure it would go down, this is like a third topping and a fourth topping. [Mike points to rows 3 and 4 of Pascal’s Triangle.] Now I think the way I thought about it is, like, the row on the outside [leftmost entry in a given row] would be your plain pizza. And there’s only one way to make a plain pizza. And . . . the next one over would be how many pizzas you could make using only one topping, and then so on until you get to the last row [the rightmost entry in a given row] which is all your toppings. And, once again, you can only make one pizza out of that. . . .

The researcher then suggested that Mike work on a general rule: RESEARCHER:

MIKE: RESEARCHER: MIKE:

And at that session, I asked them to write an equation to show, for instance, how that might happen from one row to the next. So can you just do that, write. . . . Like a general equation? Well, that was what I was going for ultimately. . . . To give an amount for any spot in the row.

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Right. . . . All right, so I guess we’ll give, you know, the row a name. Call that r. And I guess the spot in the row, like, you know, zero topping, one topping. Call that, n sounds fine. [There is a pause; then Mike writes the left part of the equation shown in Fig. 12.7.] I’m just going to like work this out in my head and see if it actually works. [A few seconds later, Mike adds the right part of the equation.]

This equation, shown below, is equivalent to the textbook version and to the night session equation, although he used different variables. (Textbooks usually use “n choose r” instead of “r choose n,” and the sum is given on the right side.) We can see that Mike did not rely on symbol manipulation. He linked the numbers to a problem task that made sense to him, and then he expressed the relationships in that task in symbolic form. We conclude that Mike was reconstructing the substance rather than merely remembering the form. r r r−1 + = n n+1 n+1

12.5 Discussion Exploring previously unexamined complexities of the towers and pizzas problems was a mathematically challenging task of the sort recommended by Davis and Maher (1990) to foster students’ ability to engage in real mathematics – developing their own mathematical theories, for example. Conditions important for the development of new mathematical ideas were in place: these students had ample time for exploration of mathematical ideas and the opportunity to express their own ideas. The students’ existing representations were taxed by new questions about how to relate these problems to each other, to Pascal’s Triangle, and to the binomial coefficients and about how to represent a general instance of Pascal’s Identity. Hence, there was a need to reorganize existing knowledge and to use new tools for dealing with these new ideas. We have shown that these students did make use of a new tool – standard mathematical notation – for dealing with their ideas about Pascal’s Identity. When they first started working on the pizza and towers problems, Ankur, Brian, Jeff, Mike, and Romina built towers and drew pictures of pizzas. As early as middle school, they began instead to use symbolic notation. (For example, they used letter and number codes to stand for the objects they were investigating.) Besides continuing the use of codes during high school, the students also found increasing use for the standard notations of mathematical discourse. For example, the binary notation that they began to use in high school was more powerful than the letter codes because it was easily extended (adding a cube to the tower or a topping choice to

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a pizza corresponded to adding a binary digit) and it was applicable to both pizza and towers problems, thus making it easier for the students to identify the similar structures of the two problems. Using binary notation helped the students focus on the isomorphic structural aspects of the combinatorial problems (the duality of the choices) rather than the surface features (the different pizza toppings, for example). Binary notation was also an easily generalizable notation in three ways: (1) adding an extra digit corresponded to making a new pizza topping available and to increasing the tower’s height by one block; (2) adding a 1 corresponded to adding that newly available topping and to adding a block of the designated color to the tower; and (3) adding a 0 corresponded to not adding the new topping and adding a block of the other color. This idea that it was not necessary to know the current number (or names) of pizza toppings or the current height of the tower in order to describe what happened next was important in the students’ production of the general equation for Pascal’s Identity. During the course of discussions over 4 months of their sophomore year, these students first noted that the pizza and towers problems had the same answer in specific cases. Then they linked specific answers to the pizza and towers problems to specific entries in Pascal’s Triangle. Finally, they described the links among binomial coefficients, pizza toppings, and towers. (Blue block = a = topping on the pizza; white block = b = topping off the pizza.) During the night session, they built on their knowledge of these links in order to produce the general form of Pascal’s Identity. We claim that their ability to map corresponding mathematical structures among these three representations is a strong indication of their mathematical competency and it indicates more competency than, for example, simply being able to reproduce or use a memorized formula. The way these students organized their answers to the pizza and towers problems was a key organizational element that helped them to form connections among those problems and Pascal’s Triangle. They also made extensive use of their personal representations at the beginning of the process. But once those connections were formed, the students began to make general statements about Pascal’s Triangle and Pascal’s Identity, and they had less use for personal representations. Finally, although they were able to articulate general information about Pascal’s Triangle and Pascal’s Identity, they did not represent the generalizations symbolically until the night session. After they had made the association between Pascal’s Triangle and the combinatorial problems, the students demonstrated an ability to describe any selected entries in Pascal’s Triangle in terms of the combinatorial problems. For example, they described the numbers in row 6 as representing six-tall towers with zero through six red cubes, respectively. The fact that they could explain any instance suggested that they had an idea of the general rule; but without the standard notation, they could express their general ideas most easily by referring to specific examples. By the time of the night session, these students seemed to know general rules about generating Pascal’s Triangle, but they lacked the notation to express these rules in a concise way. They were at the point where they needed standard notation in order to proceed further.

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We suggest that these findings point to one way that teachers can follow the recommendation by the NCTM (2000) to “use sound professional judgment when deciding when and how to help students move toward conventional representation” (p. 284). Teachers should aim to help students to develop a powerful organization, one that lends itself to a mapping onto formal notation. In that way, the formal notation can be seen as the solution to a problem that arises during the students’ own investigations: the problem of how to express in a general way the findings that the students have developed on their own. In this chapter, we have seen how this group of students learned about the relationships among well-known combinatorics problems and Pascal’s Triangle. In the following chapter, we observe the same students working on new problems in combinatorics and using what they learned about the pizza and towers problems and their relation to Pascal’s Triangle in order to make sense of that unfamiliar problem – the Taxicab Problem.

Chapter 13

So Let’s Prove It! Arthur B. Powell


May 5, 2000, Grade 12 The Taxicab Problem Brian, Jeff, Mike, and Romina Carolyn A. Maher and Arthur B. Powell

13.1 Introduction In previous chapters, we observed students throughout middle school and high school working on and making sense of two isomorphic problems in combinatorics – the towers problems and the pizza problems. In this chapter, we see how students just finishing high-school work on another isomorphic problem, demonstrating the application of techniques and ways of thinking that they developed throughout their previous years in the study. We further address the challenge that Davis (1992a) proposes to mathematics education researchers to investigate the emergence among learners of what lies at the core of mathematics: mathematical ideas. Here, a cohort of four high-school seniors – Brian, Jeff, Mike, and Romina – elaborates mathematical ideas and reasoning through work on the Taxicab Problem. They display criteria and techniques for justifying claims and an awareness of the power of generalizing, particularly as an aid to respond to special cases.

13.1.1 The Task The problem-solving session was held in a classroom during the late afternoon, after school hours. During the session, which lasted about 1 h and 40 min, the four

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students collaborated on a culminating, performance-assessment task of the research strand on combinatorics – the Taxicab Problem: A taxi driver is given a specific territory of a town, shown below. All trips originate at the taxi stand. One very slow night, the driver is dispatched only three times; each time, she picks up passengers at one of the intersections indicated on the map. To pass the time, she considers all the possible routes she could have taken to each pick-up point and wonders if she could have chosen a shorter route. What is the shortest route from a taxi stand to each of three different destination points? How do you know it is the shortest? Is there more than one shortest route to each point? If not, why not? If so, how many? Justify your answer.

13.2 Justifying Claims It is a non-trivial cognitive task for students to recognize which statements or claims in their mathematical discourse require justification or proof. This is particularly true if the students deem the claim to be obvious or if the students are in the midst of group problem solving with intellectual peers. On May 5, 2000, in the late afternoon, after school, and just a few weeks shy of their high-school graduation, Brian, Jeff, Romina, and Mike are seated around three sides of a trapezoidal-shaped table, on top of which are four black felt-tip markers, sheets of blank paper, and a problem statement. The statement is of a problem in which one is to determine in a given rectangular grid the number of different shortest paths between pairs of specified colored, endpoints (black and blue, black and red, and black and green). A researcher asks the four seniors to read the Taxicab Problem and to see whether they understand it. Jeff asks aloud whether one has to stay on the grid lines and whether they represent streets. The researcher responds, “Exactly.” Each student has taken a marker. Among themselves, they observe that from the black endpoint

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or “taxi stand,” five and seven are respectively the number of blocks it takes to reach the blue and red endpoints or “pick-up points.” Moreover, some assert that different routes to each point have the same length as long as one doesn’t go beyond the particular pick-up point. Especially noteworthy from cognitive and pedagogical viewpoints, Brian says to his colleagues, “So, let’s prove it!” After a few quiet moments, a discussion ensues as to how they know that their claim is true.

13.2.1 Generalizations, Isomorphisms, and Transitivity After further individual and collective work and discussions, Brian, Jeff, Mike, and Romina decide that to determine the number of paths between three specialized pairs of endpoints they need to generalize the problem. This moment is a watershed event in their mathematical work on the Taxicab Problem. Through their various heuristic actions, the students generate data that they consider reliable. They reflect on numerical patterns in their array of data, observe that it resembles Pascal’s Triangle, and conjecture that Pascal’s arithmetic array underlies the mathematical structure of the problem. How do they justify this conjecture? They embark on building an isomorphism between the Towers Problem and the Taxicab Problem since from previous experience they know that Pascal’s Triangle underlies the mathematical structure of the Towers Problem. The students’ strategy can be interpreted as justifying their conjecture by transitivity: (a) the mathematical structure of Pascal’s Triangle is equivalent to that of the Towers Problem and (b) the mathematical structure of the Towers Problem is equivalent to that of the Taxicab Problem; implying that (c) the mathematical structure of Pascal’s Triangle is equivalent to that of the Taxicab Problem. The students knew that (a) is true and demonstrate (b) to justify and conclude (c).

13.2.2 Reasoning and Justifying In the following sections, we discuss students’ methods of reasoning and ways of justifying their statements. 13.2.2.1 Realizing the Need to Discursively Build a Justification Two and a half minutes after receiving the task, Romina begins the first student-tostudent interaction. It centers on a question about a relation that she notices about which Romina invites her colleagues to comment. ROMINA: BRIAN: ROMINA: BRIAN: MIKE:

Isn’t it like anyway you goPretty much, because lookAs long as you don’t go like past it. [Facing Brian’s direction.] The first one- No. Well what if you go to the last one-

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You can go all the way down and go over and go down three and go over two. [Tracing the routes above the problem sheet with a black marker in his right hand.] Isn’t it- Don’t they all come out to be the same amount of blocks? Five. Five? Five? I got seven. Uh, which one- Yeah, we were both looking at the red one. I’m looking at blue. [Mike is tapping his pen on the grid along intersection points.] Yeah. Oh, okay. All right. I mean pretty much. As long as you don’t go like past it you’re fine. So it’s the same thing. So, let’s prove it.

Romina’s interrogative, “isn’t it like any way you go, they [the lengths of routes] all come out the same . . . as long as you don’t go past it [the pick-up point]?” suggests that she is aware of a relation among efficient (“as long as you don’t go past it”) paths or routes between the taxi stand and the red pick-up point. She observes that as long as one does not go beyond the red pick-up point that the numbers of blocks traversed or lengths of routes to red equal each other. Specialized to the red pick-up point, she expresses three awarenesses about relations among objects: (a) an efficient route will be a shortest route, (b) there can exist more than one shortest route, and, her central observation, (c) efficient routes have the same length. These three ideas are important and fundamental for progressing toward a resolution of the problem task. At first, Brian disagrees (“The first one- No, ’cause-”) and then, examining routes to the blue pick-up point, attempts to understand Romina’s remark (“You can go all the way down and go over and go down three and go over two”). Afterward, Jeff and Romina try to understand Brian’s assertion, “five,” for the number of blocks traversed by shortest routes between the taxi stand and the red pick-up point. Ultimately, Brian sees that they are speaking about routes to the red point (“Yeah, we were both looking at the red one.”). While, they understand that he is referring to the blue pick-up point (“I’m looking at blue.”). Taking up Romina’s observation for the red pick-up point along with his own for the blue point, Brian suggests, “So, let’s prove it.” Brian’s proposal is not immediately entertained. However, after about 1.5 min, Jeff poses a question that places Brian’s proposal back onto the agenda, and the students discuss how they know that Romina’s unchallenged assertion is true. JEFF: MIKE: ROMINA: MIKE:

So why- why is it the same every time? You’re going left and right. Ours is a four by one, right? It’s the only way to go. It’s the only way you can go. Yeah, it’s a four by one, unless you go backwards a couple of times.

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You can’t go, wellI know that would be dumb. [inaudible] the shortest route only if you go forward. But the only- You can’t go diagonal so you have to go up and down. So if the thing is down this many and Over that many, it’s the same It’s the sameIt’s the same area No matter how you do it, no matter how you do it it’s- you have toyou can’t get around doing that. [Pointing and gesturing around his grid] All right. You can’t get around going four down and right one ’cause -. All right, yeah. All right. You can’t go over there. You can’t get around doing that. Yeah. What if I were to go like to the red when I go one, two, three, four[Pointing at her problem sheet.] But they’re not asking for that. Five, six, seven. Five, six, seven. It’s the same thing. Like how- how am I going to- like how would IIt’s the same thing. It’s the same. -devise an area for that? Like this- this area up here? [Motioning with her pen on her grid, indicating the area of the rectangular space whose vertices are taxi stand and the red pick-up point.] Like plus and [Inaudible]. Well, it’s not area. It’s not area. It’s just aIt’s the perimeter. It’s like each one being one. One, two, three, four, five, six, seven. [Pointing at Romina’s paper and counting the length of a route to the red destination point.] [Jeff scratches his head.] All right. There’s no way you can get around going- [gesturing with his hands] Going seven blocks. No, yeah, I understand. Across that many and down that many because you can’t go diagonally. Can’t- [gesturing with his hands over his problem sheet across to the left and then down] Yeah. Can’t get around it, so- [gesturing with his hands] I mean, that’s the most sensible way I think to say that. Right? And they want to know how many though.

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Justifying Romina’s observation, reiterated by Jeff, or, equivalently, entertaining Brian’s proposal becomes a shared project of the participants. When Jeff poses his question (“why is it the same every time?”) and the others understand his “it” to mean “the set of efficient routes to a pick-up point.” Mike’s immediate response, coming just 4 s after Jeff finishes uttering his question, is in contrast to the silence that Brian’s proposal received almost 2 min earlier. The ensuing discursive exchange hints that the issue of the why Romina’s observation was true in general remained a concern of the participants and that they are only now prepared to tackle it. Mike explains that to reach a pick-up point, the shortest distances will always require one to move a fixed number of units down (south) and a fixed number across (east) and observes that within the grid one cannot travel diagonally. Brian reminds the others that only going forward will produce a shortest route. Mike generalizes his awareness to all routes. Jeff signals that he is convinced, saying, “I mean, that’s the most sensible way I think to say that.” In the process of the group’s discourse, Jeff and Mike help Romina to see that area is not an operative idea in this task. In the above conversational exchange, the participants engage in socially emergent cognition (Powell, 2006), providing discursive evidence to several ideas: (1) movement within the given portion of the taxicab plane goes left or right and up or down; (2) diagonal movements are not permissible; (3) the taxi stand and each pick-up point together define a rectangle in which the pair of points are located at opposite ends of a diagonal, and the problem task involves moving along the perimeter but does not concern the extent of space that a rectangle occupies; (4) the number of units down plus the number of units across are objects related by addition to produce the length of a shortest path; (5) any route to the blue pick-up point will involve four blocks down and one block across; and (6) each horizontal and vertical line segment of the grid can be considered as one unit in length. By the end of the exchange, Jeff, who in the form of a question reintroduced Brian’s proposal that they justify the idea that the length of efficient routes from the taxi stand to a pick-up point are equivalent, expresses satisfaction with Brian and Mike’s argumentation (“I mean, that’s the most sensible way I think to say that.”), checks whether the others agree (“Right?”), and reminds his colleagues that they can now turn their attention of the crux of their task: “And they want to know how many though.” The discursive exchanges in the three episodes quoted above are critical. They present the major occasion in which the participants ferret out the nature of the problem space and build fundamental ideas essential for investigating the problem task. The participants establish what are the basic objects of taxicab geometry (points and line segments or routes); basic awareness of the Taxicab Problem (there can be more than one shortest route to an intersection point in the taxicab plane); and implicitly note a distinguishing feature between Euclidean and taxicab geometries (how distance is measured). This distinction emerges when Mike observes that in the context of the problem task, one cannot travel diagonally, he touches upon the fundamental distinction between the metric of Euclidean geometry and that of taxicab geometry. Moving forward with the ideas they have built that were illustrated in the three episodes, the students shift their focus to delve further into the problem task and generate considerably more data.

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13.2.2.2 Generalizing to Specialize The students take a decisive turn in their investigation: they generalize the problem. Instead of determining the number of shortest paths between each of the three specialized pairs of endpoints, the work to uncover a pattern among the numerical values that represent the number of shortest paths between the taxi stand and any point on the grid. They first examine points in close proximity to the taxi stand. This decision proves to be a watershed event in their mathematical work on the Taxicab Problem.

13.2.2.3 Building Isomorphisms to Justify The transcription of the problem-solving session contains 1,869 turns of speech. The portion of the transcript that relates to the students building an isomorphism to justify their solution transpires over many turns of speech, spanning from turn 159 to turn 1,320. Space does not permit us to present a full illustration of the development of the ideas and reasoning that comprise the students’ work toward justifying their solution. They have continual discursive interactions with the aim of building an isomorphism between a rule for generating the entries of Pascal’s Triangle and the number of shortest routes to points on the taxicab grid. Early in their work, they manifest embryonic thinking about an isomorphism. Romina wonders aloud, “can’t we do towers on this?” (This group’s previous work on the towers problem is discussed in Chapters 4, 5, 10, and 12.) Her public query catalyzes a negotiatory interlocution among Mike, Jeff, and her. Jeff, responding immediately to Romina, says, “that’s what I’m saying,” and invites her to think with him about the dyadic choice that one has at intersections of the taxicab grid. Furthermore, he wonders whether one can find the number of shortest routes to a pick-up point by adding up the different choices one encounters in route to the point. Romina proposes that since the length of a shortest route to the red pick-up point is 10, then “ten could be like the number of blocks we have in the tower.” Romina’s query concerning the application of towers to the present problem task prompts Mike’s engagement with the idea, as well. As if advising his colleagues and himself, he reacts in part by saying, “think of the possibilities of doing this and then doing that.” While uttering these words, he points at an intersection; from that intersection gestures first downward (“doing this”), returns to the point, and then motions rightward (“doing that”). Similar to Jeff’s words and gestures, Mike’s actions also acknowledge cognitively and corporally the dyadic-choice aspect of the problem task. Through their negotiatory interactions, Mike, Jeff, and Romina raised the prospect of as well as provided insights for building an isomorphism between the Taxicab and Towers Problems. The prospect and work of building such an isomorphism reemerges several more times in the participants’ interlocution and, each time, they further elaborate their insights and advance more isomorphic propositions. Eventually, the building of isomorphisms dominates their conversational exchanges. Approximately 35 min after Romina first broached the possibility of relating attributes of the Towers Problem to the problem at hand, the participants reengage with the idea. Romina speculates that between the two problems one can relate “like lines over” to “like the color” and

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then “the lines down” to the “number of blocks.” What is essential here is Romina’s apparent awareness that each of the two different directions of travel in the Taxicab Problem needs to be associated with different objects in the Towers Problem. Romina uses this insight later in the session. She transfers the data that she and her colleagues have generated from a transparency of a 1-cm grid to plain paper. Their data are equivalent to binomial coefficients. She identifies one unit of horizontal distance with one Unifix cube of color A and one unit of vertical distance with one Unifix cube of color B: Like doesn’t the two- there’s- that I mean, that’s one- that means it’s one of A color, one of B color [pointing to the 2 in Pascal’s Triangle]. Here’s oneit’s either one- either way you go. It’s one of across and one down [pointing to a number on the transparency grid and motions with her pen to go across and down]. And for three that means there’s two A color and one B color [pointing to a 3 in Pascal’s Triangle], so here it’s two across, one down or the other way [tracing across and down on the transparency grid] you can get three is two down [pointing to the grid].

Furthering the building of their isomorphism, Mike offers another propositional foundation. Pointing at their data on the transparency grid and referring to its diagonals as rows, he notes that each row of the data refers to the number of shortest routes to particular points of a particular length. For instance, pointing to the array – 1 4 6 4 1 – of their transparency, he observes that each number refers to an intersection point whose “shortest route is four.” Moreover, he remarks that one could name a diagonal by, for example, “six” since “everything [each intersection point] in the row [diagonal] has shortest route of six.” In terms of an isomorphism, Mike’s observation points to two different ideas (1) it relates diagonals of information in their data to rows of numbers in Pascal’s Triangle and (2) it notes that intersection points whose shortest routes have the same length can have different numbers of shortest routes. Later in responding to a researcher’s question, the participants develop a proposition that relates how they know that a particular intersection in the taxicab grid corresponds to a number in Pascal’s Triangle. They focus their attention on their inscriptions in Fig. 13.1, which shows empirical data of shortest routes between the taxi stand and nearby intersection points. In array A, the green numbers (lighter shade inside sqiares) show empirical data of shortest routes between the taxi stand and nearby intersection points. Jeff wrote the 1 s on the side in blue (darker shade) to augment the appearance of the numerical array as Pascal’s Triangle. From the participant perspective, to the left of Jeff’s numbers, Romina wrote in green (lighter shade) the numbers 1, 2, and 3 to indicate the row numbers of the triangular array. Array B shows their drawing of Pascal’s Triangle. The first five rows contain empirical data; the remaining two rows contain assumed data values based on the addition rule for Pascal’s Triangle. Mike and Romina discuss correspondences between the two inscriptions. Referring to a point on their grid that is five units east and two units south, Romina associates the length of its shortest route, which is seven, to a row of her Pascal’s Triangle by counting down seven rows and saying, “five of one thing and two of another thing.” Mike inquires about her meaning for “five and two.” Both Romina

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Fig. 13.1 Participants’ data arrays A and B

and Brian respond, “five across and two down.” She then associates the combinatorial numbers in the seventh row of her Pascal’s Triangle to the idea of “five of one thing and two of another thing,” specifying that, left to right from her perspective, the first 21 represents two of one color, while the second 21 “is five of one color,” presuming the same color. Using this special case, Romina hints at a general proposition for an isomorphism between the Taxicab and Towers Problems.

13.3 Conclusion The narrative of these four students working on the Taxicab Problem has three sections. The first concerns their recognition of the need to justify an observation that they made immediately after reading the problem statement. The observation maybe simple but their recognition of the significance of the observation and that it needed to be justified before progressing on with resolving the problem is rather sophisticated. This sophistication in their mathematical work speaks to the sociomathematical norms (Yackel & Cobb, 1996) that they have developed through their longitudinal experience working on open-ended problems. This sociomathematical norm is subtle and akin to the way mathematicians work. The second section of the narrative pertains to their decision to seek a general solution to the problem and that such a solution would be easier than trying to count the number of shortest routes between each of the three pair of given endpoints. This is an instance of what can be called generalizing to specialize. That is, finding a general solution of a problem situation in order to answer more specific questions of the problem. Often the general case is easier to solve than special cases. Finally, the third section of the narrative revolves around not only with the recognition that claims needs to be justified but also with a particular proof strategy that

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emerged in the students’ attempt to justify their resolution of a generalized form of the Taxicab Problem. Important sociomathematical norms (Yackel & Cobb, 1996) are evident in the students’ mathematical interactions in the first and third threads: claims need to be justified and a problem’s solution needs to be connected or linked to attributes of the problem. These norms emerge from the mathematical interactions of students who have a collective history of problem solving through occasional interactions over their school years with researchers from Rutgers University who increasingly over the years left the students to structure their own mathematical investigations in response to given tasks. In this chapter and the previous chapters of this section, we have given the researchers’ perspectives on the students’ mathematical work. In Chapter 14, we examine this work from the point of view of the students.

Part IV

Extending the Study, Conclusions, and Implications

Chapter 14

“Doing Mathematics” from the Learners’ Perspectives John M. Francisco

Date and Grade: Tasks: Participants: Researcher/Instructor:

1999–2000; high school Clinical interviews Ankur, Brian, Jeff, Mike, and Romina John Francisco

14.1 Introduction The previous chapters focused on aspects of the cognitive development of the students in the longitudinal study. The present chapter looks into the epistemological growth of the students. During the longitudinal study, individual clinical interviews were conducted with the students with the goal of capturing the mathematical beliefs that the students might have developed in connection with their experiences in the longitudinal study. This chapter reports on the analysis of five such interviews. The results provide insights into the students’ views about mathematics and about how it should be learned and taught. The findings challenge the widespread view that students below college hold naïve epistemological views; support studies that show that students who experience constructivist learning environments tend to develop sophisticated epistemological beliefs and highlight the important of past mathematical experiences in framing individuals’ mathematical beliefs. Research on students’ views about mathematics can be placed within the field of personal epistemological beliefs. This is a field traditionally concerned with describing individuals’ views about the nature of knowledge and knowing. A substantial amount of research has been conducted within the field since the pioneering work of Perry (1970) with Harvard college students. Even more research has been associated with this field since the epistemological construct was expanded to include individuals’ views about learning, teaching, and intelligence through the work of J.M. Francisco (B) Secondary Mathematics Education, Department of Teacher Education & Curriculum Studies, University of Massachusetts Amherst, Amherst, MA 01003, USA e-mail: [email protected]


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Schommer (2002) and Schommer and Walker (1995) and some continental scholars (De Corte, Op’t Eynde, & Verschaffel, 2002; Leder et al., 2002). Even though the expansion has not been free of controversy, it is recognized that this expansion has brought the research on the field closer to classrooms practice. Students’ epistemological beliefs have been examined in relation to a variety of constructs. Students’ beliefs have been studied in relation to the students’ home and school environments (Hammer & Elby, 2002) and their teachers’ epistemological beliefs (Hofer, 1994; Lyons, 1990; Pirie & Kieren, 1992; Roth & Roychoudhury, 1994). There has been also extensive research that has examined students’ beliefs within disciplines (Carey & Smith, 1993; Ceci, 1989; Lampert, 1990; Konold, Pollatsek, Well, Lohmeier, & Lipson, 1993) as well as across disciplines (Case, 1992; Sternberg, 1989). However, a comprehensive review of the field (Pintrich & Hofer, 1997, 2002) suggested that a number of challenges remain to be addressed. One particular challenge is the need for more research on the epistemological beliefs of students below college level. Except for a few cases (Schoenfeld, 1989; Pehkonnen, 2002), most research in the field has remained at the college level. The review notes that there have been few studies involving students below college and even fewer below high school. The review further points out that lack of such research has resulted in students below college being assigned naïve epistemological beliefs only because research findings show that entering college students tend to hold such views. Another challenge is the lack of studies that have examined the epistemological beliefs of students who have experienced a constructivist learning environments. The few existing studies (e.g., Hofer, 1994) have been exploratory in nature. The present study grew out of a natural interest on the part of the researchers to examine the epistemological growth that participating students in the longitudinal study might have experienced in connection with the particular conditions in which they were asked to do mathematics. The researchers were particularly interested in the students’ beliefs about (1) success and failure in mathematics, (2) knowing mathematics, (3) learning and teaching mathematics, and (4) how the practices that they assigned to doing or learning mathematics compared with those in other disciplines. However, the researchers also sought to make a contribution toward deepening the research community’s understanding of the epistemological beliefs of students below college, particularly at the high school level, and of students who experienced constructivist mathematical environments. The researchers viewed the longitudinal study as a “learning experiment,” rather than a “teaching experiment,” through which they tried to understand how students construct mathematical ideas while working on open-ended mathematical tasks in particular conditions. However, there were no preconceived ideas about what students were to learn or how they were supposed to learn. Students’ constructed mathematical ideas and reasoning were results, not preconceived goals, of the research. This was consistent with a constructivist approach to learning in the sense that participants had plenty of opportunities to construct and accordingly revise their ideas without any guidance from the researchers, but rather within their own community of learners. The students were interviewed about their experiences in the longitudinal study, and from these interviews, inferences are made about their mathematics beliefs.

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Their conversations provide insights on their mathematical beliefs and challenge the widespread view that students below college level hold naïve views in contrast to studies that show students who experience constructivist leaning environments tend to develop sophisticated epistemological beliefs. However, the results also highlight the importance of past mathematical experiences in the development of individuals’ mathematical beliefs. This study used a phenomenological approach to the students’ experiences in the longitudinal study. Researchers avoided imposing any interpretive framework on the students (Wilson, 1977) and sought to infer the students’ epistemological views among the meanings that the students assigned to their experiences in the longitudinal study (Creswell, 1998). Overall, the approach was similar to Perry’s (1970) idea of inferring individuals’ epistemological beliefs from their reflections on educational experiences. Data for the present study consisted of 1-h videotaped individual interviews with the five participating students about their experiences in the longitudinal study. The four males and one female – Ankur, Brian, Jeff, Mike, and Romina – agreed to be interviewed and videotaped. However, it was the students’ long experience in the longitudinal study, starting in first grade, that constituted the major criterion for selecting the students to take part in of the interviews. Their long-term participation in the longitudinal study satisfies the criterion sampling method (Miles & Huberman, 1994) recommended for phenomenological studies. The interviews used a semi-structured interview protocol. There was a clear goal (i.e., capturing the students’ views about mathematics as a discipline with particular practices and criteria of validity), but the interview proceeded by eliciting and building on students’ reflections on their experiences in the longitudinal study to ascertain the students’ epistemological views. Typically, the interviews started with the open question, “What are your memories of the longitudinal study?” Then the researchers tried to steer the interviews toward obtaining insights on the students’ views on mathematics.

14.2 Findings The search for answers to the research questions generated five major themes about personal success and failure in mathematics, mathematics as sense making, mathematics as a discovery activity, mathematics as an activity involving discourse, and the relationship between mathematics and other disciplines. These themes are described below, along with supporting statements from the students. (Emphasis was added to quotes.)

14.2.1 Personal Success/Failure in Mathematics All of the students described themselves as confident and good in mathematics. Ankur even said, “I’m well above average in mathematics,” and modesty prevented

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Mike from describing himself as being better than a “normal kid.” There were differences, however, on how the students construed mathematical success. Mike and Ankur emphasized personal interest and hard work as the ultimate sources of success. They argued that those who like mathematics can be successful because they are willing to work harder in mathematics than those who do not like it. The other students stressed the importance of previous mathematical experiences and training. In particular, they singled out particular aspects of experiences in the longitudinal study such as collaborative work and opportunities to come up with ideas, as opposed to merely receiving them from teachers or experts, as having contributed to their success and confidence in mathematics. Romina further suggested that confidence and success is built over time: In fourth grade, I didn’t know who you were. Now we’re comfortable with you. You’ve been our teachers for 10 years. That’s what you’ve been to us, so now it’s easier, and we know what’s expected of us, what we have to do. Before we would wait for you to give us a little start or a little push and point us in the direction. Now you hand us a problem and you just kind of leave, and we just do it ourselves. We just start experimenting and see what we can give you.

She also suggested that lack of success can be a function of how success is defined. She explained that, although she generally felt confident in her abilities, she might not feel confident in situations where she is asked to engage in rule-based mathematics, as in textbooks, as opposed to ways that are personally meaningful: They might throw out, “Oh, do you know this rule?” I’m like, “No, but if you sit me down, maybe I know it.” I know it in my own way, not in their way. Everything I explain is in my own words, not in anyone else’s words. It’s not from some mathematician from a 1,000 years ago, because I don’t know that. I didn’t know what the pyramid [Pascal’s Triangle] was called. I just know everything in my own way. Everything has Romina’s definition to it.

There were no suggestions that the students considered success as a quality or trait that people are born with. On the contrary, a closer analysis of the students’ reflections suggests that the students converge on recognizing the importance of past mathematical experiences in promoting mathematical confidence and ability either directly or indirectly through motivation. Romina’s last statement also suggests that standardized testing has the potential to portray otherwise bright students as mathematically weak only because the students do not do mathematics in the ways prescribed in textbooks or by experts.

14.2.2 Knowing Mathematics as Sense Making The students’ reflections on their experiences in the longitudinal study emphasized the importance of understanding as opposed to memorization of concepts. For example, Mike reported gaining increased conceptual understanding in the longitudinal study: It feels different now because I know a lot more than I did before. If I were to solve the same problems, it would be easier. I understand a lot better too the whole concept behind

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each problem. Like, all the problems that have been given to us, I feel like, somehow, one is related to each other. When you’re little, you can’t really understand that.

Romina emphasized the importance of building durable understanding. In particular, she explained that it involved the ability to recall as well as reconstruct previously learned mathematical concepts: Because everything I do I understand, because it’s more than just the numbers to me. If you understand something from the beginning, you’re going to always understand it. You can’t forget something like that. And like an equation, I don’t really know any equations. It’s like things, I don’t know any solid equations, but I could explain to you something and work from there. And you’re likely to forget an equation.

Jeff related understanding to the ability to “explain” what one knows to others: The name really doesn’t matter. That’s neither here nor there. I mean, just knowing how to do it, that’s the important part, that’s what we learned. And that’s being able to do it, being able to teach it to somebody else, to explain it, to use it for what you need to use it for. That’s what really matters, not being able to know the name of it, or how to draw it up, or anything like that.

Brian emphasized the importance of developing the ability to “look deeper than just the surface” and of always asking “why,” qualities which he asserts that he gained in the longitudinal study: When I look back at things, I’m happy I got involved in this program. Because, I know at times, I seem very frustrated with it. But if I think hard, I really have gained a lot of knowledge, and I learned how to look into things deeper than just surface things like, “Why is it like this?” Now, I start thinking like that. And it helps me compute things in my mind better. Like, I really don’t know how to put this, but it just helps me in doing things other than math. I think more “in-depth” and very seriously about things.

Ankur’s idea of understanding was not as explicit as that of others. However, when recalling different mathematics experiences in different schools, he was clear about why he liked the one in which the teacher did not “teach out of the textbook”: it promoted understanding. At Harding [Elementary School], my math experience was, I’d say it was good. The teacher would teach, I’d understand, I’d participate, and it was just, I enjoyed it. And then we went over to the Springfield [Regional High School], and I did not enjoy geometry class at all. It was one of the first times that we used the textbook. I don’t remember in Harding using a math textbook. And the teacher would just simply teach out of the book, and assign homework, straight problems, and it wasn’t anything that I enjoyed. Then after Springfield, we came here [to the local Kenilworth High School] and I had Mr. Pantozzi [his mathematics teacher who was also involved with the longitudinal study as a researcher and graduate student] for 2 years. And I enjoyed that. His teaching style was like none other, and it works.

In particular, if Romina’s statement above further suggests that knowing or understanding mathematics has a personal dimension, Jeff’s statement suggests that understanding has a social or interpersonal dimension to knowing mathematics.

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Fig. 14.1 Ankur Interview

14.2.3 Mathematics as a Discovery Activity Ankur’s statement in the previous section suggests that he favored a mathematical environment where students [not teachers] came up with their mathematical ideas or knowledge as opposed to merely receiving them from teachers or textbooks. The implicit idea of learning as a discovery activity was present in the reflections of all students, albeit articulated in slight different ways. Mike argued that discovery learning helped the majority of students understand mathematics and emphasized exploration of concepts over time and group work during the discovery process: Kids can learn new things if they discover them themselves, and not if somebody tells them, I think that is a better way of learning. Like Mr. Pantozzi, he gives us some information, but basically, he lets us discover the things that a normal teacher would just tell us. Like we were learning about e [base of natural logarithms], and he told me that when he was in school, the teacher told them, “e is this, 2.7, whatever.” The teacher told him what it is. In our class, all we did was just explore e. We took days at a time, and I have a good understanding of it. I guess, in a normal class, only selected kids might understand it. But in a class where everybody’s working together, everybody’s a part of the teaching, everybody or at least the majority of kids will understand it.

Jeff also made reference to working on tasks over time and to group work, but he stressed the importance of mathematical arguments or discussions during group work. He asserted that participating in discussions was a better way than listening to teachers for students to build durable mathematical understanding:

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Fig. 14.2 Jeff Interview

Well when the teacher just comes out and tells you the answer, you find you can study it, you can get it for that test, but a few weeks later, a few days later, it doesn’t matter anymore, you don’t need to know it, and you’re onto the next thing. And that’s wasting your time. Because you spend the whole year running through this, you learn, say, twenty different things, but by the end of the year, you’ve forgotten them all, and you have nothing. If you would, say, argue for a couple days or weeks or whatever on different topics, you cover ten things, but when you walk away, you still know those ten things at the end. And that’s why it’s important to do that, and not just get the answers.

Romina also singled out mathematical discussions during mathematical activities. In particular, she added that disagreements during the discussions were an important cognitive mechanism by which students built new knowledge and how she, personally, learned mathematics: Because if you’re, like, passive, and I’m like, “This is what I think it is,” and everyone is, “Okay, that’s what it is,” we all sit back and we all take that and we never go any further. But if I disagree with someone, they’ll have to explain it to me, and if they’re explaining it, they’re either going to find something right, or they’re going to find something more. So, if I don’t agree with it, they’re going to explain it to me, but if they find something wrong, maybe I can help, and then someone else may disagree with me. And that’s how we get through everything. We just disagree. I’ve always had to argue to get somewhere, because they never actually told me where we were heading with anything. So, through arguing, that’s the only reason I know math.

Brian put the emphasis on hands-on experiences during mathematical activities. He argued that hands-on activities motivated students to do mathematics and helped them build durable understanding of mathematics:

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Fig. 14.3 Brian Interview

If I could change courses, I would make everyone hands-on because kids get tired of sitting there. But when you’re up doing things, time flies, and you have fun and you learn, which you retain more.

Ankur favored problem-solving activities involving interesting and challenging tasks and collaborative work, as opposed to teaching by the textbook: Right now, in my current math class, the teacher doesn’t use a book. I’d say he comes up with problems, and most of the problems are interesting problems, and a lot of them are challenging, and all the students participate. We enjoy working in groups. We help one another, and that helps out a great deal.

14.2.4 Mathematics as an Activity Involving Discourse Some of the students’ statements in the previous section, particularly those by Jeff and Romina about mathematical arguments, suggest the idea of doing mathematics as a discursive activity. The statements assign mathematical arguing the cognitive role of fostering knowledge acquisition. Jeff’s statement about understanding as explaining ideas also suggests the idea of arguments as a way of proving or establishing the validity of mathematical claims. In the statement below, Jeff is even more explicit about the arguments as way of proving mathematical claims:

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We didn’t know if we were right or wrong. You only knew so much, but I would have my idea about how to get to a certain point and you might have the same idea about how to get to it. But getting there was the hardest part. That is what we were arguing about, the right way to get there, the right way to make sure we covered the basis, how to make sure, how to prove what we needed to accomplish.

Mike suggests a similar idea, but puts it in the context of probability. However, he does not claim that arguing as proving only takes place in probability. Rather, he suggests that because uncertainty is more common in probability than in any other area of mathematics, arguments are more likely to take place in probability: The reason we argued about math, because math is like, when we do about probability, probability is an iffy subject. Like, sometimes, I mean the math says it’s right, but do you believe it’s right, and sometimes that influences your decision. That’s probably why we argue. I remember the problem with the World Series Problem [see Appendix A]. We had two different answers. I still don’t know which one is correct.

The students had a response for those who might claim that in group work, some students might not be engaged and so might not learn. Above, Mike suggested that discovery learning with group work can help the majority of students learn mathematics. Ankur’s statement below suggests a similar idea and tries to illustrate how it happens: Usually, you think that only one person in the group is learning, but if the group fully participates and everyone is involved, everyone in the group learns. When the Rutgers group comes over here, we all learn. I don’t think there is a case when someone doesn’t understand. Because if one person doesn’t understand, they’ll say something, or even if they’re quiet, someone else will suggest something, will ask them if they understand, or say “Could you explain it back to me” And that’s how everyone learns.

14.2.5 Mathematics and Other Disciplines The students had different responses on whether the practices that they associated with learning or doing mathematics were specific to mathematics or applicable to other subjects or real life. Ankur suggested that he thought that teaching out of the textbook, as opposed to the discovery method, was more suited to other subjects such as history than to mathematics: In ninth grade, I was in a different school, and the teacher there taught me differently. She [the ninth grade math teacher] taught more like a history teacher. A history teacher would simply teach out of the book, just go right down through the years, and you’d learn like that. But the math teacher, I wouldn’t think, a math teacher should teach like that. A math teacher usually teaches differently. I don’t know how to explain it, but it just seems that way. This teacher taught straight out of the textbook, you wouldn’t learn anything more, just simply what the book stated.

Romina was the most categorical of all the students in her response. She claimed that arguing about ideas was a learning practice specific to mathematics and could not be implemented in other subjects such as English and History:

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Fig. 14.4 Romina Interview

Well, math is where the most arguing is. Like, you can’t do this in other classes. It’s not like, in English, you read. You don’t argue. It’s there. It’s written. And in history, you don’t do the same. In math, it’s like, well especially the way I’ve been taught, because I have never actually had a math teacher that’s said, “This is the equation, put in the numbers, and do it.” I’ve always had to argue to get somewhere.

Brian and Mike, however, had different responses. They suggested that learning mathematical practices were also relevant to other disciplines. For example, Brian suggested that his history teacher also used a problem-solving approach as opposed to just telling students what to do: Well, the closest thing to my math class would have to be my history class. My history teacher is an incredible teacher. He always, like for instance, we’re doing the Cuban Missile Crisis thing, he set the class up into countries, and we had to all deal with the problems, instead of just sitting there and telling us. Next to Mr. Pantozzi, he gets us involved just as much as he does.

Mike argued that his longitudinal study experiences were relevant to other subjects. “I think it’s relevant to a lot of other subjects, like science, history; I guess you could apply it to, basically, all subjects” and claimed that he used his experiences in the longitudinal study, which he called a “type of thinking” in real life and other subjects: I guess I use the type of thinking in, like other subjects in school; I don’t know how you can apply it to life. It’s not hard to recognize what style of thinking you’re thinking of. I can’t compare it with someone else’s because I don’t know what they’re thinking. So, I think, yeah, I probably do use it in life, and other subjects in school.

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Fig. 14.5 Mike Interview

A closer analysis of the statements suggests that differences among the students’ responses regarding the applicability of mathematical learning practices to other subjects reflect differences in interpretation of the question asked. Jeff, Romina, and Ankur seem to have answered the question of whether the mathematical learning practices were actually taking place in other subjects. Mike and Brian, however, seem to have understood the question as asking whether they believed that those practices were applicable to other subjects.

14.3 Conclusions The analysis of the interviews with the five students who participated in this study suggests that the students (1) are confident in their mathematical ability, (2) emphasize mathematical understanding over memorization, and view mathematics as (3) a discovery activity and (4) a discursive activity. The results also suggest an agreement that that (5) the practices were not being implemented in the regular schools, except in a few isolated cases (a history teacher and Mr. Pantozzi). The findings suggest a few insights. The students’ emphasis on the importance of learning as a discovery activity suggests that they view themselves as learners as active participants in the construction and justification of their mathematical knowledge, and not as mere receivers of knowledge and truth from experts or textbooks written by experts. Within the domain of personal epistemological beliefs, such a view is held by individuals holding sophisticated or powerful personal epistemological beliefs. As a result, the findings of the present study challenge the aforementioned widespread belief that students below college hold naïve epistemological beliefs based on research that show that to be the case among freshmen college students. Given that the conditions of the longitudinal study were consistent with a constructivist approach to learning,

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the results also support findings from exploratory studies suggesting that students who experience constructivist learning environments tend to develop more sophisticated epistemological views than students who experience teaching approaches based on showing and telling students what to do. The students’ views about mathematics are also consistent with the nontraditional approach to mathematical learning and teaching, advocated by the research community and promoted through publications such as the 2000 Principles and Standards for School Mathematics of standards of the National Council of Teachers of mathematics (NCTM). This is evident in the students’ emphasis on durable mathematical understanding as opposed to memorization of concepts or procedures, discovery learning, convincing or explanatory arguments, and collaborative work. The students articulate the merits of such practices in enhancing learning for the majority of students and point out that these practices also motivate them to learn. This suggests powerful beliefs within the particular field of personal epistemological beliefs and within the larger field of mathematics education. Another dimension of the depth of the students’ mathematical views is reflected in the nature of the students’ articulation of cognitive process involved in doing or learning mathematics. The students provide different characterizations of mathematical understanding concepts with qualifiers such as conceptual, operational, durable, personal, and interpersonal. Understanding is also defined not only as recall but also as the ability to reconstruct previously learned ideas. Arguing is associated with knowledge acquisition as well as justification or proof for mathematical claims. There are also rich descriptions of the conditions in which learning, particularly discovery learning, takes place: hands-on activities, explorations, problem solving, interesting and challenging tasks, collaborative work as arguing or discussing ideas, work on tasks over time, and so on. The students’ ability to articulate in detail different cognitive aspects involved in learning is another measure of depth of the students’ mathematical beliefs. In particular, Romina’s idea about knowing or understanding mathematics as personal is particularly insightful. A great deal has been written about the issue under the idea of personal representation (Francisco & Maher, 2005; Maher, 2005; diSessa & Sherin, 2000; Davis, 1992b; Davis & Maher, 1990). The idea has been to encourage teachers to attend to and promote the conception between formally defined mathematics and students’ personal conceptualizations to promote understanding. Finally, it is particularly interesting that the students’ mathematical views mirror the particular conditions within which they engaged in mathematical activities in the longitudinal study. Under the idea that the longitudinal study was more of a “learning experiment” rather than “teaching experiment,” researchers encouraged the students to work collaboratively with other students; justify their reasoning to classmates; be the arbiters of whether or not a solution was correct based on whether it made sense; work on the same tasks over an extended period of time; and revisit similar or same task and refine their ideas and mathematical reasoning. Such conditions are reflected in the students’ thoughts about their experiences in the longitudinal study. This suggests that the importance of construing mathematical

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beliefs within the particular experiences in which students engage in mathematics. In particular, this highlights the importance of teachers paying closer attention to the kind of beliefs that they might be promoting in their students through their conscious or unconscious practices or beliefs in mathematics classroom. This is an area which remains largely unexplored, as few studies have examined the relation between mathematical beliefs and particular settings, whether within or across cultural settings. In summary, we provide here an existence proof that students below college level are capable of building powerful mathematical beliefs and insights about the cognitive processes and conditions involved in doing mathematics. However, we also emphasize the importance of examining mathematical beliefs within particular learning conditions. In this and preceding chapters, we followed students in the longitudinal study through elementary school, middle school, and high school, working on problems in combinatorics. In the following chapter, we look at a group of college students working on the towers and pizza problems, and we see how their work compares to that of the younger students.

Chapter 15

Adults Reasoning Combinatorially Barbara Glass

Date and Grade: Tasks: Participants:

Researcher/Instructor:

1998–2000; College Freshman Pizzas and Towers Danielle, Donna, Errol, Jeff C., Linda, Lisa, Mary, Melinda, Mike C., Penny, Rob, Stephanie C., Samantha, Steve, Tim, Tracy, and Wesley. (We use the initial C for “college” for Jeff, Mike, and Stephanie to distinguish them from the elementary students of the same names discussed in other chapters.) Barbara Glass

15.1 Introduction In the preceding chapters of this book, we have provided considerable evidence showing elementary and secondary school students’ success in solving open-ended problems, over time, under conditions that encouraged critical thinking. In this chapter, we address the question as to whether similar results can be achieved by liberal-arts college students within a well-implemented curriculum that includes a strand of connected problems to be solved over the course of the semester. From a perspective of conceptualizing reasoning in terms of solving open-ended problems, it was of interest to learn whether students in a liberal-arts college mathematics course could be successful in providing arguments to support their reasoning and in making connections in a problem-solving-based curriculum. Students enrolled in college-level mathematics courses might be expected to have already developed effective reasoning skills. Unfortunately this is too often not the case. This may be explained, in part, by a history of mathematics instruction in settings that devalue thinking and focus on rote and procedural learning.

B. Glass (B) Sussex County Community College, Newton, New Jersey, USA e-mail: [email protected]


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Often, in traditional mathematics classrooms, the answer key or the teacher is the source of authority about the correctness of answers; unfortunately, quick, correct answers are often valued more than the thinking that leads to the answer. Too often, teachers ask students to explain their thinking only when answers are wrong, emphasizing the product rather than the process of problem solving. Sanchez and Sacristan (2003) offer data to support this from studying students’ written work. They report that students are not accustomed to expressing mathematical ideas, and they offer as an explanation that the emphasis in schools is mainly on producing correct solutions. One consequence is that students tend to develop the belief that all problems can be solved in a short amount of time. Students often stop trying to build a solution if they are unable to solve a problem immediately. For example, in a survey, high-school students were asked to respond to the question “What is a reasonable amount of time to work on a problem before you know it’s impossible?” Schoenfeld (1989) reported that the largest response was 20 min and the average time was 12 min. Further, students view school mathematics as a process of mastering formal procedures. These rules are often removed from real-life experience and application. As a result, students can feel that answers need not make sense. It is not surprising, then, that students accept and memorize what they are told without making any attempt to deal with meaning (Schoenfeld, 1987). Since many of the students in this study were previously taught mathematics in this fashion, it would not be entirely surprising if they were unable to apply knowledge from previous mathematics courses to novel situations. Moreover, since a student’s willingness to think about a problem is influenced by notions about what mathematics is and what should be expected of students, it is not surprising when students do not display the level of reasoning of which they are capable. In this chapter, we examine how a small group of community college students enrolled in a liberal-arts mathematics class solved open-ended non-routine problems in which they had to build and justify a solution. The tasks were the towers and pizza problems and extensions of these tasks. Our questions were (1) How do college students solve non-routine mathematical investigations? (2) How do college students’ representations and level of reasoning contrast with those of younger students from a longitudinal study engaged in the same investigations? (3) What connections, if any, do the college students make to analogous problems and to the rules learned in previous classes? (4) To what extent, if any, do the college students justify and generalize their results?

15.2 The Study The study was conducted in a mathematics class for liberal-arts majors called Mathematical Concepts. The curriculum includes algebra and problem solving. Liberal-arts students also took a second mathematics course called Contemporary Mathematics that introduces logic, counting methods including combinations and permutations, probability and statistics, geometry, and a cluster of applications

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called “consumer math.” The two liberal-arts mathematics courses can be taken in either order, so some students in Mathematical Concepts had already taken Contemporary Mathematics and others had not. Most of the students in the Mathematical Concepts class take the course to fulfill the mathematics requirement, although a few take the course as an elective. The mathematical background of the students in this study varied widely. Some had taken college preparatory mathematics in high school, while others took only general mathematics courses. Some had already taken other college-level mathematics courses, while for others this was their first college-level mathematics course. When asked on a questionnaire about their mathematics background, many students described themselves as being very poor mathematics students who disliked and feared mathematics, while others stated that they liked mathematics and had always done well in mathematics classes. There was also a wide range of ages, some students having recently completed high school, with others not having taken any mathematics for many years. The study took place in a relatively new community college of moderate size in an area of New Jersey that ranges from rural to suburban with very little racial or cultural diversity. In the fall semester of 2000 there were 929 full-time and 1,357 part-time students enrolled. As with other community colleges, some of the students attend because poor academic records prevent them from being accepted elsewhere. Others are excellent students who attend the college for a variety of reasons including lower costs and the convenience of being close to their homes and places of employment. Nine classes ranging in size from 6 to 25 students were studied between 1998 and 2000. Sections of the course met for 15 weeks for two 75-min classes each week or three 50-min classes each week. The students spent approximately half of the class time working on various non-routine problems in a small group setting. After they worked together on these problems the students were encouraged to present their solutions to the class. In addition, a weekly problem-solving homework assignment was given. As a part of the assignment, students were required to give a written explanation of their solution method and a justification of how they knew that their solution was correct. Students also submitted write-ups of the problems done in class. Two groups from each class were videotaped as they worked on the towers and pizza problems. In addition, task-based interviews with ten representative students were videotaped. Students were selected because they were willing to be videotaped while participating in problem-solving sessions and willing to participate in videotaped follow-up interviews.

15.3 Student Solutions The students worked on the towers problem during the 8th or 9th week of the semester. By this time, they had become accustomed to working on problems and to justifying their solutions. The students began by working on the four-tall towers

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problem. They then were asked to consider the five-tall towers problem. Some groups also worked with three-color towers problems. The students worked on the pizza problem during the 13th or 14th week of each semester, first on the four-topping pizza problem and then on the five-topping pizza problem. After they solved the basic problems, some groups were asked to consider the pizza with halves problem, in which a topping could be placed on either a whole pizza or a half pizza.

15.3.1 Towers Problems Most of the college students used patterns or some other form of local organization immediately, and some immediately imposed a global organization scheme. An organization by cases according to the number of cubes of one color was the method chosen by six students. One group, which started the problem by randomly generating towers using a build and check method, switched to this organization by cases at the suggestion of Jeff C. He said, Here, put the ones that have three yellows and a red all together. [Danielle rearranges the towers.] Okay. So now we do three yellows and a red at the bottom, ’cause you don’t have that. [Jeff C. builds YYYR and hands it to Danielle.] And the ones that have two and two, put those together. [Danielle rearranges the towers.] Now the ones that have three reds and the other.

The cases were no cubes of the selected color, then one, two, three, and four of the selected color. All six students who selected organization by cases determined that there were two solid-color towers (one all of one color and one all the other color). All six used a staircase pattern to show that they had found all towers with three cubes of one color and one cube of the other color; refer to Fig. 15.1 for an example staircase pattern.

Fig. 15.1 Dana’s family of one red cube and four yellow cubes

RED

YELLOW

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The case with two cubes of one color and two cubes of the other color was more problematic. The students used a variety of methods to demonstrate that they had found all towers in this group. Two groups, Melinda’s group and Donna’s group, stated that they had found all towers because they were unable to find any more. But as the students in these groups spoke to the instructor, they began to organize their towers and move toward a proof by cases. However, both groups still stated that their justification for the claim that they had all towers with two of each color was that they could not find any more. Three of the students, Lisa, Errol, and Wesley, tried to argue that the number of towers is sixteen because four times four is sixteen. The instructor responded that they needed a reason why the answer should be four times four. Lisa then produced a proof by cases, although she had difficulty justifying the case of two cubes of each color. During her interview 7 weeks later, Lisa found an organization that accounted for all of the towers with two cubes of each color. Wesley rearranged his towers, but he offered no explanation for why his arrangement produced all possible towers. About 7 weeks later, during the interview, Wesley produced a similar arrangement and used it to account for all possible combinations with a proof by cases where his cases were (1) towers with four cubes of the same color together, (2) three cubes of the same color together, (3) two cubes of the same color together, and (4) no cubes of the same color together. Errol’s partner, Mary, offered a proof by cases. However, Errol wanted a proof that his numerical argument worked. As he continued to think about the problem, he rearranged the towers in a way that he thought showed that four times four was the correct answer. This arrangement grouped towers with a red on top together and towers with a yellow on top together. When the instructor continued to question him as to why this showed that the answer should be four times four, Errol turned to simpler cases in an attempt to verify his numerical argument. He then noticed the doubling pattern and used that to develop an argument by induction, abandoning the four-times-four argument. Five of the college students did a proof by cases for the five-tall towers. Each of these proofs referred to opposites (pairs of towers with opposite colors in the same positions). They also all used a staircase pattern to account for the towers with one cube of one color and four cubes of the other color. They used a variety of methods to justify the cases with two cubes of one color and three cubes of the other color. After Rob and his group had organized their towers by cases, they noticed that the number of five-tall towers was double the number of four-tall towers. They extended the doubling pattern to predict how many towers they would get if the towers were three tall, two tall, and one tall. They then built the one-tall and two-tall towers in order to test their theory. While justifying their answer to the five-tall towers problem, they referred both to their doubling pattern and to a proof by cases (Glass, 2001). The instructor asked the students to think of a reason why the number of towers doubled. After a few minutes, Rob explained to Steve that the number doubled because you could add either a red cube or a yellow cube to the bottom of each tower. He explained as follows, building from a generic original tower he called X.

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Okay, let’s say the top of our tower is X, X. [Rob writes an X on his paper.] Then we’re putting one on the bottom. For every X we can have a Y [yellow] down here, or for every X we can have a red [R] down here. So for each block we have, there are now two more things it could be. So before we just had X. This is X. [Rob picks up the solid red tower of four as an example.] Now we have XR and XY derived from this. XY and XR. [Rob holds up RRRRY and RRRRR.]

Steve demonstrated that he understood Rob’s explanation by using Rob’s procedure to build two-tall towers by adding cubes to the bottom of one-tall towers. Wesley built his five-tall towers by adding a red cube to the top of each of his towers of four. He then built the opposites of these towers to find all towers with a yellow cube on top. He justified that he had found all five-tall towers with an inductive argument, but he was unable to extend this reasoning to predict how many six-tall towers there are. However, during an interview 7 weeks later, Wesley correctly extended the doubling pattern beyond the case that went from four-tall to five-tall and predicted that there would be 64 six-tall towers. Jeff C. applied the fundamental counting principle to predict that there would be 32 five-tall towers. After Jeff’s group had produced those 32 towers, he used an inductive argument to show that they had found all possible towers by pairing each of the five-tall towers with the corresponding four-tall tower that would generate it. Errol used the inductive argument that he had developed while working with four-tall towers to predict that there would be 32 five-tall towers. Even though he did not build the five-tall towers in class, he used an inductive method to produce a list of all five-tall towers on his written assignment. Figure 15.2 shows Errol’s method. There are sixteen possibilities. To justify my answer we will start with the possibilities if the towers were two cubes high (R-red, W-white) W-W W-R We have 4 possibilities.

R-W

R-R

Now, if we want to go to towers three blocks high, we simply take the 4 towers we have and add a white block to the top and do the same with the red block (8 towers) W-W-W R-W-W R-W-R W-W-R & R-R-W W-R-W R-R-R W-R-R Now, for 4-cube high towers, we do the same thing: add a white block to the top of all eight 3-cube high towers and add a red to each of the eight towers. (This would also work if you put them on the bottom instead). (16 towers) W-R-W-W W-R-W-R W-R-R-W W-R-R-R

W-W-W-W W-W-W-R W-W-R-W W-W-R-R

&

R-R-W-W W-R-W-R R-R-R-W R-R-R-R

Fig. 15.2 Errol’s written justification of four-tall towers

R-W-W-W W-W-W-R R-W-R-W R-W-R-R

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B

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Fig. 15.3 Penny’s written justification of her answer to the four-tall towers problem

Penny, who had been absent the day that the class worked on the towers problem, did the problem at home. She invented a tree diagram strategy to produce an inductive argument for the four-tall and five-tall towers. Her written work is shown below. Her diagram is shown in Fig. 15.3. The answer is 24 = 16 because my first cube will be either blue or brown (2 choices) my second cube will be either blue or brown matched to a blue or brown first cube (4 choices). For each of those combinations I can use either a blue or a brown cube, doubling my possibilities to 8, and for each of those eight combinations I can add a blue or a brown cube which finally doubles my answer giving me 16 possibilities. (I am using big ’B’ for brown and a little ‘b’ for blue.)

Tim used a binary coding system to justify his conclusion that he had found all possible combinations. This is the same method used by tenth-grade Mike from the Rutgers longitudinal study to justify his assertion that there were 32 different five-topping pizzas. Several groups also had time to work on the three-color towers problem. Mike C.’s group and Rob’s group worked on four-tall towers, while Jeff’s group worked on three-tall towers. Rob and his partners applied the inductive method that they had developed for towers with two colors to solve the problem quickly. Jeff C. used the fundamental counting principle to calculate the number of towers, but he did not use an inductive method to build the towers. Mike’s group divided the problem into two cases: (1) towers with at most two colors and (2) towers with all three colors. Each student in the group used two of the three colors to build all 16 possible fourtall towers that contained those two colors. After eliminating the three duplicate one-color towers that they had produced using this method, they had a total of 45 towers. Then they worked on the second case, towers that contained at least one cube of each color. They each chose one of the colors and built all 12 towers with two cubes of that color and one cube of each of the other colors. Their solution for the three-color four-tall towers problem was thus 81.

15.3.2 Pizza Problems All the college students used a justification by cases approach to the pizza problem. The students created their two-topping lists systematically; they held one topping

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fixed and paired it with each of the other toppings. Then they moved to the next topping on the list. Some students paired each additional topping only with toppings that were below it on the list, reasoning that they had already accounted for the other combinations. Other students considered all pairs and then eliminated the ones that they already had. For the three-topping pizzas, some of the students again systematically went down the list of toppings, while others failed to exhaust both major and minor items. For example, after some students combined pepperoni and green peppers with all other possibilities, they moved to the pepper and mushroom toppings instead of exhausting all possibilities that contained pepperoni. Two students used a chart that was similar to the chart that fourth-grader Brandon had created when he did the pizza problem (Maher & Martino, 1998). Instead of the ones and zeroes that Brandon had used, these students made a check to indicate that a topping was on a pizza and left a blank space to indicate toppings that were not on the pizza. Interestingly, this is the same method that Brandon’s partner Colin used. Stephanie C., who was simultaneously enrolled in a statistics class, hypothesized that she could calculate the number of pizzas using combinations formulas. She and her partner Tracy systematically created a list of pizzas and compared the numbers to those that Stephanie C. had conjectured using her formulas, confirming Stephanie’s prediction. Several other students who had previously studied combinations tried to calculate the number of pizzas using combinations formulas. However, they did not understand combinations well enough to apply them to the problem correctly. For example, Melinda stated that the problem could be solved either by combinations or by permutations, but she could not remember which to use. Also, students who tried to use formulas attempted to do a single calculation instead of doing separate calculations for each number of toppings. In short, most students were not successful at using combinations. The correct combinations formula for the number of pizzas having exactly r toppings when there are n toppings to select from is n Cr =

n! n = r r! (n − r)!

To find the total number of four-topping pizzas, therefore, it is necessary to sum C 4 0 through 4 C4 (the number of pizzas with exactly 0–4 toppings).

15.3.3 Connections Between Problems During the pizza problem session, several students noticed a relationship between the towers problem and the pizza problem. For example, Rob and Samantha explained as follows: ROB: INSTRUCTOR:

So we decided the toppings are the block positions. Okay. So you have pepperoni at the top.

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INSTRUCTOR: SAMANTHA: ROB: INSTRUCTOR: ROB: INSTRUCTOR: ROB: SAMANTHA:

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Right. Because it was convenient. So onion would be the second block, sausage would be the third block, and mushroom would be the bottom block. Okay. What would your colors be? Orange and yellow. Yeah. What would orange be? Orange means it’s on the pizza. And yellow? It’s off. Off the pizza.

Jeff C. was not able to explain the isomorphism during class, but he did explain it a week later during an interview. JEFF C.:

INSTRUCTOR: JEFF C.: INSTRUCTOR: JEFF C.:

INSTRUCTOR: JEFF C.:

INSTRUCTOR:

What you could do to relate that to the topping problem. Is that, you could say, you could designate each spot for a topping. This is the pepperoni. This is the green pepper spot, and this is the sausage spot. [Jeff C. writes toppings by the drawing of the first tower.] Okay. Or onion, or whatever you want to have it. Whatever. So with nothing on it, with no toppings, there’s only one possibility. [Jeff C. points to the first tower he drew.] Now for one topping, there’s three possibilities for a one topping pizza. There’s pepperoni. [Jeff C. marks the first block of the second tower.] There’s green pepper [Jeff C. marks the second block of the third tower], and there’s sausage. [Jeff C. marks the bottom block of the fourth tower.] Okay. Then for a two topping pizza [Jeff C. draws three more towers.], there’re three possibilities. [Jeff C. moves the actual towers with one white cube.] Pepperoni and green pepper [Jeff C. marks the first and second blocks on the first tower.], pepperoni and sausage [Jeff C. marks the first and third blocks.], and green pepper and sausage. [Jeff C. marks the second and third blocks.] Okay. All right. Which is where the blue, the blue blocks are. [Jeff C. points to the towers with one white cube.] So in other words, if you took, you couldn’t flip that one around [Jeff C. takes WBB and turns it upside down and puts it next to BBW.] because then you’d have two of the same combination. You’d have two pizzas with pepperoni and green pepper. Okay.

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So those are the three possibilities. The ones with two toppings. And then for three toppings, there’s only one possibility – pepperoni, green pepper, and sausage [Jeff C. draws another tower on paper and marks all the blocks.], or blue, blue, and blue. [Jeff C. indicates the solid blue tower.]

Rob’s group tried to relate the pizza toppings to colors in the towers problem; they conjectured that each topping corresponded to a cube of a specific color. This is the same explanation that the third-grader Meredith had originally given for the relationship between the two problems. Meredith later revised her explanation to note correctly that the blocks in the towers corresponded to the toppings and the colors in the towers corresponded to the presence or absence of each topping (Maher & Martino, 1999). Unlike Meredith, however, the students in Rob’s group did not take their exploration any further and so did not revise their explanation. Melinda, Stephanie C., Wesley, and Lisa mentioned the relationship between the two problems during their interviews, approximately 1 week after the class session on the pizza problem. At this time, each student displayed an understanding of the isomorphism between the two problems. During her interview, Lisa first built the four-tall towers and then she produced a pizza problem chart similar to the one that she had made in class. As she completed the chart, she discovered that the rows of the chart looked like the towers. As a consequence, she was able to match the towers with the rows in her chart. A portion of Lisa’s chart is shown in Fig. 15.4.

Pep.

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Fig. 15.4 Lisa’s chart for the pizza problem

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Lisa remarked that the entries in her chart were like the binary system that the class had studied earlier in the semester.

15.3.4 Connections with Pascal’s Triangle While working on the pizza problem, Mike’s group noticed the doubling pattern and verified that it continued to hold for smaller numbers of available toppings. As Mike C. thought about why the number of pizzas should double, he discovered that the numbers from the pizza problem matched the rows of Pascal’s triangle. He was unable to explain, however, why the addition rule of Pascal’s triangle applied to pizzas. After Rob and Steve’s group completed the four-topping pizza problem, Steve remarked that they should look for a pattern before moving on to the five-topping problem. Rob noticed that the pizzas from the four-topping problem formed a row of Pascal’s triangle. After figuring out how the addition rule of Pascal’s triangle worked with pizzas, Rob used an inductive method and Pascal’s triangle to create a list of pizzas for the five-topping pizza problem. Rob’s explanation of the addition rule for the Pizza Problem was similar to that used by Mike and the other students in the Night Session, discussed in Chapter 12. Rob wrote,

Zero toppings Zero row One topping 1st row Two toppings 2nd row Three toppings

0 1 P

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Fig. 15.5 Rob’s drawing of Pascal’s triangle

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The reason this [Pascal’s triangle] works is because every time we add another topping we are increasing the possibility of choice, without losing the old ones. In other words, all the two topping pies in the two topping total still apply when there are three total toppings. Also all those that had two toppings, by adding a third topping, are now three topping pizzas. In this way we can see absolutely, positively, without a doubt, that we have all possible combinations – each new row is built by adding the old columns with the new topping. This once again is the Pascal’s triangle principle of adding old combinations with new possibilities to find new combinations.

Rob’s drawing of Pascal’s triangle is shown in Fig. 15.5.

15.4 Discussion The college students solved problems and justified their solutions in many of the same ways as the Kenilworth students had done in elementary school and high school. However, the college students solved the problems more quickly than the third and fourth grade children and, unlike the elementary school children, the college students did not generally rely on random checking as a primary strategy. All college students were able to solve the four-tall towers problem and start the fivetall towers problem within a 50-min or a 75-min period. Most also finished the five-tall towers problem within the same period. Several students also had time to work on extensions of the problem. However, the college students also showed less inclination than the Kenilworth students to think about problems for extended periods of time. Many stopped thinking about the problem after they had arrived at an answer, even when the instructor asked them to spend some more time considering the problem in order to reveal their thinking. After Rob and his group had built their towers and organized them by cases, they noticed the doubling pattern and they developed a proof by induction. This is similar to what Milin had done in the fourth grade. (Refer to Chapter 5 for details.) Mike C. also noticed that the number of towers and pizzas was doubling, but Mike C. was unable to provide a reason for the doubling pattern. It is interesting to note that in fourth grade both Stephanie and Milin had noticed the doubling pattern in the towers problem. Stephanie’s progression from pattern recognition to development of an inductive argument took about 8 months while Milin’s understanding developed more quickly (Alston & Maher, 2003; Maher & Martino, 2000). Perhaps Mike C. would also have recognized the reason for the doubling pattern if, like Stephanie, he had been given an extended time frame in which to develop his ideas. Mike C., however, was limited to about 8 weeks to develop his ideas about the problem. The college students made very few connections with previous mathematical knowledge, and most of these connections were trivial connections. Several students recognized that the problems were related to permutations or combinations, but most did not understand these concepts well enough to correctly apply them to the solutions of novel problems. This would suggest that learning about mathematics in an atmosphere in which students are told what to do does not enable

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them to develop genuine understanding. In contrast, the students in this study did demonstrate a high level of reasoning as they thought about the problems, justified their solutions, and made connections between problems and the mathematics they learned within the course. Some of the conditions of the Rutgers University longitudinal study, such as extended classroom sessions and revisiting the same problem several times within an extended time frame, could not be replicated because of the time constraints of a college course. However, many of the conditions that enabled the elementary and secondary students to become thoughtful problem solvers were duplicated. Both groups were given rich mathematical tasks and were encouraged to explain their reasoning and methods of solution and to justify their solutions to the problems. Both groups were engaged in thoughtful mathematics. They found patterns, developed methods of justifications, and provided justifications that their patterns were reasonable. It cannot be disputed that the students in the Rutgers longitudinal study benefited from exposure to rich mathematical experiences over an extended period of time. It is also significant that the students in this study, who had previously experienced a variety of generally traditional mathematics instruction, demonstrated that it is not too late to introduce rich mathematical experiences in a collegiate level mathematics class. The level of reasoning that these students demonstrated provides evidence that it is possible to experience thoughtful mathematics within a traditional 15-week college semester. In this chapter, we have shown how college students worked on the towers and pizza problems. In the following chapter, we will follow up on the work on these college students and other college students, comparing the work of these college undergraduates to the work of the longitudinal study high-school students (see Chapter 8) on the extension of the towers problem called Ankur’s Challenge (finding the number of four-tall towers, built when choosing from three cube colors, having at least one cube of each color).

Chapter 16

Comparing the Problem Solving of College Students with Longitudinal Study Students Barbara Glass

16.1 Introduction In this chapter we consider a variety of solutions to Ankur’s Challenge from students ranging from high school to graduate level study of mathematics. The problem was created by Ankur and first posed to tenth-grade classmates in an after-school session of the Rutgers longitudinal study (see Chapter 8). Find all possible towers that are four cubes tall, selecting from cubes available in three different colors, so that the resulting towers contain at least one of each color.

Since 1998, when Ankur first presented this problem to four high school classmates, we have given his problem to several cohorts of students enrolled in liberal arts mathematics classes and in graduate mathematics-education courses. Students presented their written work and gave further verbal explanations and clarifications of their solutions. Researcher notes provided the data for the oral explanations. We classified their forms of reasoning into four categories: (1) justification by cases, (2) inductive argument, (3) elimination argument, and (4) analytic method (use of formulas). We present here some representative solutions from the high school (H), undergraduate (U), and graduate (G) students, listed according to the arguments that were used.

16.2 Justifications by Cases We list below nine ways that students used justification by cases; one from high school students, five from undergraduates, and two from graduate students. All those who used justification by cases began with the observation that one color would

B. Glass (B) Sussex County Community College, Newton, New Jersey, USA e-mail: [email protected]


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appear twice and each of the other colors would appear once. Many then proceeded by selecting one color and listing all possibilities for that color in some organized fashion and then arguing from symmetry that the other cases would be the same. Details are given below for these justifications by cases.

16.2.1 Romina, Jeff, and Brian’s Solution (H) Rather than using particular colors, these high school students used the general codes 1, 0, and X to indicate the three colors, and they focused on the placement of the duplicate color, using 1 to indicate the color that was duplicated and X and O to indicate the other two colors. As shown in Fig. 16.1, they found six possible ways to place the two 1’s in a four-tall tower. They observed that each tower pictured in Fig. 16.1 represents two towers, one with the X and O as shown on the top of the diagram and the other with the X and O in the bottom position, giving 12 towers. Finally, arguing from symmetry, they noted that X and O could also be the duplicate color, and so there are 12 towers for the duplicate X case and 12 for the duplicate O case. They concluded that there are therefore 36 towers that fulfill Ankur’s condition. Fig. 16.1 Romina, Jeff, and Brian’s six prototype towers

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16.2.2 Joanne and Donna’s Solution (U) Joanne and Donna used red, blue, and green cubes. They found six ways to place two blocks of the same color in a tower containing four blocks, counting from the top: positions 1 and 4, 1 and 3, 1 and 2, 2 and 4, 2 and 3, and 3 and 4. Each

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Fig. 16.2 Joanne and Donna’s listing of towers

of those six possibilities gives six towers because there are three possibilities for the same color blocks and two possibilities for the remaining color. Six towers for each of six possibilities gives 36 towers. Refer to Fig. 16.2 for their enumeration of towers.

16.2.3 Rob and Jessica’s Solution (U) Rob and Jessica used yellow, red, and blue cubes. For their first case (which we designate as Case I), they chose yellow as the duplicated color. For Subcase A, they fixed the blue cube in the top position and then they moved the red cube into the second, third, and fourth positions. This gives a total of three towers for this subcase. For Subcase B, they fixed the red cube in the top position and moved the blue cube into the second, third, and fourth positions, giving another group of three towers. For Subcase C, they placed one yellow cube in the top position of the tower and the second yellow cube in the second, third, and fourth positions in turn. They noted that each position of the second yellow cube produces two towers, because the red and blue cubes can be reversed. Therefore, there are six towers for Subcase C. The total number of towers for Case I (yellow as the duplicated color) is therefore 12. They repeated this process for the case of two red cubes and two blue cubes, giving a total of 36 towers. Refer to Fig. 16.3 for Rob and Jessica’s list of 12 towers in Case I.

Fig. 16.3 Rob and Jessica: 12 towers with two yellow cubes

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16.2.4 Marie’s Solution (U) Marie used blue, red, and yellow cubes. She started by assuming that the blue cube was the color that appears twice. For the three cases, under this assumption, Marie controlled for variables. For the first case (Case 1) under this assumption, she fixed the position of the first blue cube on the top of the tower and then she moved the second blue cube to the second, third, and fourth positions. Reasoning from symmetry, Marie noted that there are two towers for each of those positions because the red and yellow cubes can be reversed. So for Case I, she found six towers. For Case 2, she fixed the first blue cube in the second position and moved the second blue cube into the two remaining possible positions. As noted above, reasoning from symmetry, Marie knew that there are two towers for each position; therefore, there are four towers for Case 2. For Case 3, Marie placed the two blue cubes in the third and fourth position; that gives two towers for this case. Refer to Fig. 16.4 for a diagram of the 12 towers that Marie found under the assumption that the duplicated color is blue. Again reasoning from symmetry Marie noted that this process could be repeated for red and yellow as the cube appearing twice, and so there are 36 towers.

Fig. 16.4 Marie’s three cases for blue

16.2.5 Bob’s Solution by Cases (U) Bob used red, blue, and yellow cubes. He wrote, “There has to be one color that appears twice, while the other two colors appear once. If the blue cube appears twice, keep the two blue cubes together and move to all possible positions. There are two towers for each position because the other two colors can be reversed. Next separate the two blue cubes by one and move into all possible positions. Again each position will give two towers. Finally place the two blue cubes in the first and fourth position, separated by two cubes, to give two more towers. This process can be repeated for each of the other colors.” Figure 16.5 shows Bob’s three cases for blue.

16.2.6 April’s Solution (U) April used blue, purple, and white (B, P, and W) cubes. She started by considering the case where blue is the top cube in the tower. April wrote, “Start with blue on the top. If there is also a blue in the second position, the third and fourth position

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Fig. 16.5 Bob’s three cases for blue

Fig. 16.6 April’s three cases: Blue on top, white on top, and purple on top

must be PW or WP in order to have all three colors in the tower. If the second cube is purple, the other two cubes must have at least one white cube. They can be WW, BW, WB, WP, or PW. If the second cube is white, the other two cubes must have at least one purple cube. They can be PP, PB, BP, PW, or WP. This gives 12 combinations with blue on top. There are also 12 combinations with white on top and 12 combinations with purple on top for a total of 36 towers.” Figure 16.6 is a diagram of April’s f three cases of blue on top, white on top, and purple on top.

16.2.7 Bernadette’s Solution (U) Bernadette used blue, purple, and white cubes. Her enumeration by cases was similar to April’s, except she worked from the bottom up; her three cases were: blue cube on the bottom of the tower, white cube on the bottom, and purple cube on the bottom. Figure 16.7 shows Bernadette’s three cases.

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Fig. 16.7 Bernadette’s three cases: blue, white, and purple on the bottom

Bernadette made an exhaustive list of all possibilities for towers with a blue cube on the bottom and the repeated the procedure for the other two colors. She wrote of her attempts to organize her work by finding a pattern: After a lot of trouble in class trying to figure this problem out, and with your help, I finally came to the conclusion that there are 12 possibilities with each color block on the bottom, with 36 possibilities in all. I know my answer is correct because of the pattern I found. For example, the blue block was the concrete tower for 12 towers. The blue block was positioned in every possible way, including one blue block at least always on the bottom. Then, the purple and white blocks would alternate positions. I showed this on my work. I did the same with the white and purple blocks, replacing the blue, getting only 36 possibilities of towers.

16.2.8 Tim’s Solution (G) Tim used the colors red, yellow, and green (R, Y, and G); he described three cases of towers with the required conditions and reasoned from symmetry that all three cases would have the same number of towers. He built the required towers by starting with four-tall towers built from two colors with exactly two cubes of each color. His diagram, shown in Fig. 16.8, illustrates the organization (holding the top cube fixed and moving the second cube of that color into all possible other positions) that shows that there are exactly six such towers. Once he had those six towers, he

Fig. 16.8 Tim’s diagram for the four-tall towers with two red and two green

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exchanged each of the red cubes for a yellow cube in order to fulfill the conditions of the problem; since this can be done in two ways (first exchange one red cube and then exchange the other red cube), this gave him 12 towers with two green cubes. Reasoning from symmetry, he concluded that there are 12 towers for each of the other two cases. He described his reasoning as follows: “Given three colors Red Yellow and Green, towers 4 tall containing at least one cube of every color will yield towers with 1R, 1Y, 2G; 1 R, 2Y, 1G; and 2R, 1Y, 1G. All these cases will be equal in number. Consider 2R and 2G. There are six towers that are four tall with 2R and 2G. Now exchange a Y for one of the two R’s in each tower. There are two ways to do this for each tower. Therefore there are 2 × 6 = 12 towers of [each of the cases] 1R, 1Y, 2G; 1R, 2Y, 1G; and 2R, 1Y, 1G, for a total of 36.”

16.2.9 Traci’s Solution (G) Traci used colors A, B, and C. First she found all towers that fulfilled the condition with color A on the bottom (Case 1). For her first subcase (AB), she placed a cube of color A on the bottom and a cube of color B next. Within this subcase, Traci first placed C in the third position; the top position can thus be any of the colors, giving three towers ABCA, ABCB, and ABCC. Continuing the AB subcase, Traci kept AB in place and determined that there were two more towers that started with AB and that did not have C in the third place; these are ABAC and ABBC. Thus for the AB subcase, she had a total of five towers. The second subcase was AC. Within this case, Traci first assigned B to the third position. As with subcase AB, the top position can be any one of the three colors, giving three towers ACBA, ACBB, and ACBC. To finish subcase BC, Traci left the first positions as AC and found two additional towers, ACAB and ACCB. Thus for subcase AC, there are again five towers. For the last subcase (AA), Traci assigned AA to the bottom two cubes; the two towers that can be made with AA are AABC and AACB. These three subcases cover all possibilities for the case in which color A is on the bottom; they give a total of 12 towers with A. Arguing from similarity, Traci found that there are also 12 towers with B on the bottom and 12 with C on the bottom, for a total of 36 towers. Traci’s written work is shown in Fig. 16.9.

16.3 Inductive Arguments Four students (three undergraduates and one graduate student) used inductive arguments, either in whole or as part of their justifications of their solutions.

16.3.1 Errol’s Solution (U) Errol’s solution combined proof by cases with an induction argument. He used red, blue, and white cubes, with a tree diagram and a horizontal code (for example, RBWR for red–blue–white–red). He fixed the first cube as red and worked with three

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Fig. 16.9 Traci’s justification by cases

subcases (second cube white, second cube blue, and second cube red). If the second cube is white, then Errol said that the third and fourth cubes would be one of the four combinations blue–white, white–blue, blue–red, or red–blue; Errol missed the fifth combination (blue–blue). Similarly, Errol noted that if the second cube is blue, then the third and fourth cubes would be one of the four combinations white–red, red–white, white–blue, or red–blue; here he missed the white–white combination. For his third subcase, Errol noted that if the first two cubes are red, then the third and fourth cubes would have the other two colors (white–blue or blue–white), for a total of two combinations. He thus found 10 of the 12 possible towers, and so when he multiplied this case by three (“since there are three colors involved, we can find out how many possibilities there are with one specific color on top, and multiply the answer by 3”), his answer of 30 was off by 6. Figure 16.10 shows Errol’s written work.

16.3.2 Christina’s Solution (U) Using colors A, B, and C, Christina built up the four-tall towers by starting with all nine possible two-tall towers that can be built when selecting from three colors. Using a strategy of controlling variables, she built these towers by adding A, B, and C to each of the three one-tall towers. (Fig. 16.11 shows her set of two-tall towers.) Christina then placed a cube of color A on the top of each two-tall tower, giving nine three-tall towers, as shown in Fig. 16.12. Each of the nine three-tall towers produced three four-tall towers when she added a fourth cube of each color to the bottom, for a total of 27 towers. After Christina crossed out the 15 towers that did not fulfill Ankur’s condition (see Fig. 16.13), she was left with 12 towers. She

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Fig. 16.10 Errol’s written justification

Fig. 16.11 Christina’s set of two-tall towers

Fig. 16.12 Christina’s set of three-tall towers with an A on top

Fig. 16.13 Christina’s set of 27 four-tall towers, with 12 meeting Ankur’s condition

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followed the same procedure for colors B and C and produced 24 more towers for a total of 36.

16.3.3 Bob’s Inductive Solution (U) Working with colors red, blue, and yellow, Bob started by making the six three-tall towers that have all three colors, as shown in Fig. 16.14. He demonstrated that all such towers were accounted for by controlling for variables: he held the top color fixed and moved the other two cubes in both possible positions. Fig. 16.14 Bob’s starting set of towers

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Then Bob built four-tall towers by adding a red, yellow, or blue cube to the bottom of each of the six three-tall towers. He noted that this would give all towers with two of the same color on the bottom and the other colors in all possible positions. Then Bob added a red, yellow, or blue cube to the top of the six three-tall towers, giving all towers with the of the same color on the top and the other colors in all possible positions. But with this procedure, Bob missed the towers with the duplicated color in the middle. When the instructor asked him to justify his solution, Bob abandoned this method and returned to a proof by cases. Bob’s solution by cases is discussed earlier in this chapter.

16.3.4 Frances’ Solution (G) Frances used a tree diagram as shown in Fig. 16.15, with red, yellow, and blue cubes (R, Y, and B). Her strategy was a combination of proof by cases and induction. Case I was to start with color R, Case II was to start with color Y, and Case III was to start with color B. Describing Case I on the tree diagram, she said, “(1) I started on the first block as R (color 1). Then the second could be R (color 1), Y (color 2), or B (color 3). (2) If the second is R, the third could only be Y or B. If the third is Y, then the fourth must be B. If the third is B, the fourth must be Y [2 towers]. (3) If the second is B, then the third could be R, Y, or B. If R, the fourth could only be Y. If B, the fourth could only be Y. If Y, the fourth could be R, Y, or B [5 towers]. (4) If the second is Y, then the third could be R, Y, or B. If R, the last could only be B. If Y, the last could only be B. If B, the last could be R, Y, or B [5 towers, for a total of 12].” Frances made the same inductive arguments for Cases II and III, for a total of 36 towers.

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Fig. 16.15 Frances’ tree diagram showing the generation of towers for three cases

16.4 Elimination Arguments Four students (two undergraduates and two graduate students) used elimination arguments. All started with the fact that 81 (34 ) towers can be made when building four-tall towers and selecting from three colors. Then they eliminated the towers that did not meet Ankur’s criteria. Penny listed all the towers and crossed off the ones that did not meet the criteria. Robert, Liz, and Mary calculated the number of towers that did not have all three colors and subtracting them from 81. Robert, Liz, and Mary also used formulas; they used the fact that the number of n-tall towers when selecting from three colors is 3n .

16.4.1 Penny’s Solution (U) Penny’s argument was a combination of inductive reasoning and elimination. She used a tree diagram to list all 81 four-tall towers with three colors, and then she crossed off the towers that did not meet Ankur’s criteria.

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16.4.2 Robert’s Solution (U) Robert used colors red, blue, and green. Robert started with the 81 four-tall towers that can be built when selecting from three colors; then he subtracted the towers that did not meet Ankur’s criteria; these are the towers with exactly one color and the towers with exactly two colors. Robert gave the number of towers with exactly one color as 3×14 , or three (one all red, one all blue, and one all green). He found the number of four-tall towers with exactly two of the colors by calculating the number of all possible four-tall towers (which is 24 ) and subtracting the towers with only one of the colors (which is 2×14 ). Since there are three combinations of two colors (red/blue, red/green, and blue/green), number by three. two-color he multiplied the His complete calculation is: 34 − 3 24 − 2 14 − 3 14 = 81 − 45 = 36 Refer to Fig. 16.16 for Robert’s written work.

Problem: Want to build a tower 4 high with 3 uniquer colors (I chose red, green and blue) with the restriction, that at least 1 of each color is used. I started by first calculating the total number of towers that are 4 tall w/3 colors and restrictions. So I drew a tower and mapped out possiilities 3

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Which I found to be 3, but decided to write as 3 (14). Which helped me see something better. Then I decided to subtract the ones that are 4 tall, 2 colors, at least one of each which 24. But I had to subtract 2 (or2(14)) because that left possibility of getting all one color or another. Since there are three combinations of two colors (red/blue, red/green, blue/green) I multiplied by 3 giving total number of 2 of 3 colors, 3 high, at least one of each color to 3(24–2(14)). That gave me the total number of 4 tall, 3 colors, at least 1 of each color to 34 –3(24 –2(14)) – 3(14) = 34 – 45 = 36

Fig. 16.16 Robert’s written justification

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16.4.3 Liz’s Solution (G) Liz’s strategy was similar to Robert’s; she started with the 81 four-tall towers that can be built selecting from three colors, and then she subtracted the towers that do not meet Ankur’s criteria, which are the towers with at most two colors. She observed that there are 16 towers with just red and green cubes (including the all red and all green towers). Similarly, there are 16 towers with just red and blue cubes (including the all red and all blue towers), and 16 towers with just blue and green cubes (including the all blue tower and the all green tower). Since the three towers of all one color each appeared twice in the three groups of 16, Liz concluded that there are 3×16 – 3 or 45 towers that do not fit Ankur’s criteria. Subtracting 45 from 81 gives 36, the number of towers with at least one of each color. Liz wrote: There are 34 = 81 towers four-tall when choosing from three colors. But not all of them have at least one of each color. Subtract out the ones that don’t have at least three colors. These are the ones that have at most three colors. Say the colors are r, g, b. – There are 24 = 16 with just r & g, including 1 all r, 1 all g. – There are 24 = 16 with just r & b, including 1 all r, 1 all b. – There are 24 = 16 with just b & g, including 1 all b, 1 all g. There are three duplicates here There are 3 × 16 – 3 = 45 with at most two colors. 81 – 45 = 36 with at least one of each color.

16.4.4 Mary’s Solution (G) Like Liz, Mary started with the fact that 81 four-tall towers can be built when selecting from three colors, and she also calculated that there are three sets of 16 four-tall towers that can be built when selecting from two colors. She subtracted 48 from 81, giving her 33 towers that she though fulfilled Ankur’s condition. Then she noticed that she subtracted three too many (the three single-color towers), and she added back the three, giving 36. Figure 16.17 shows Mary’s written explanation.

16.5 Analytic Method Only one student used a completely analytic method, relying exclusively on formulas, although others used formulas in individual pieces of their solutions.

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Fig. 16.17 Mary’s written work

16.5.1 Leana’s Solution (G) Leana used her knowledge of combinatorics formulas to calculate that the number of ways to arrange two A’s, one B, and one C is 4 factorial (4!) divided by 2 factorial (2!), which is 12. The same formula applies when B is the repeated color and again when C is the repeated color, giving 36 towers fulfilling Ankur’s criteria. Figure 16.18 shows Leana’s written analysis.

16.6 Discussion We placed the forms of reasoning displayed by these students into four major categories: elimination, inductive, controlling variables, and recursive. However, there is

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4tall 3 colors at least 1 of each color Colors A B C CASE I

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Fig. 16.18 Leana’s analysis of Ankur’s Challenge

considerable variation within each category as well as overlap among the categories. All but one of the students who used an elimination method used formulas to calculate the total number of towers. That one student (Penny) used an inductive method to generate her list of all 81 towers. All but two of the students who chose to do a justification by cases did so by controlling for variables. The other two students (Marie and Bob) instead used a recursive argument in which they focused on a fixed cubed and rotated it exhaustively for particular cases. The approaches to arguing by cases varied. Students chose different cases into which to separate the towers and different variables for which to control as they built their justification. There were also variations within the other approaches. For example, students started their inductive argument at different tower heights. Errol and Francis started at height one and Christina started at height two. Bob started at height three, but he missed some of the towers as a result. He eventually resolved the discrepancy when he used a

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solution by cases. Bob was the only student who used two different methods, and that strategy helped him to find the towers that he missed when using induction. The sharing of ideas was an important component in students’ problem solving. It provided them with the opportunity to review their work, reflect on their ideas, and sometimes to modify their results. While the written work does not show the interchange of ideas that came about as students discussed their work with others, the invitation to students to share their ideas resulted in a more careful review of the work and thus a greater confidence in the reasoning offered. For example, it was only after sharing her justification with the instructor that April became confident that she had indeed found all possible towers. Also, Mary found her error (subtracting out some towers twice) when reviewing her work for presenting to others. In some cases, the discussion revealed to students flaws in their reasoning, resulting in a reexamination of the solution. As an illustration, the process of justifying that he had the correct number of towers enabled Bob to realize that his inductive method had caused him to miss several combinations. In this context, we can observe how the process of justifying their answers can enable students to reflect upon what they have done and on whether their answer is reasonable. While the forms of reasoning generally fell into the four categories, the distribution of correct solutions was not uniform according to category. Few students (mostly graduate students) used formulas, and most of those students also used an elimination argument. Also, only graduate students and one senior undergraduate student used formulas correctly. Undergraduates successfully used arguments by cases and induction, and their primary method of solution was reasoning by cases. Rich problems can be challenging and engaging for students at a wide range of levels. Ankur’s challenge, a problem initially proposed to a group of high school students, has turned out to be of interest to students at many levels, and it has resulted in multiple kinds of thoughtful arguments. An important feature of this problem was that students were required to account for all of the towers and then to build arguments that were convincing to themselves and to others. It may be that problems that call for explanation and justification trigger sense making in students. We suggest that multiple opportunities for students to express ideas, revise them, and share them both in writing, and verbally are important contributors to students’ sense making. Therefore we recommend that problems that invite students to explain and justify their ideas in writing and in the verbal sharing of results be included in mathematics courses at all levels.

Chapter 17

Closing Observations Arthur B. Powell

In the previous 16 chapters, we have witnessed ordinary students develop extraordinary mathematical ideas, forms of reasoning, and heuristics. Extraordinary are these students’ accomplishments since their mathematical behaviors emerged not from quickly parroting rules and formulae but rather from deliberately engaging their own discursive efforts. As Speiser (Chapter 7, this volume) notes, these students built fundamental mathematical understanding, over time, through extended task-based explorations. They created models, invented notation, and justified, reorganized, and extended previous ideas and understandings to address new challenges. That is, they performed mathematics: created mathematical ideas and reasoned mathematically. These behaviors – ideating and reasoning – are fundamental human activities and how they occur in the realm of mathematics, specifically elementary combinatorics, is what this book contributes. Internationally, a community of mathematics education researchers has recognized this how question as substantially important. In January 1983, David H. Wheeler (1925–2000), the founding editor of the international journal, For the Learning of Mathematics, sent a letter to 60 or so mathematics educators inviting them to engage a daunting task: “suggest research problems whose solution would make a substantial contribution to mathematics education” (Wheeler, 1984, p. 40). The varied and thought-provoking responses of more than 15 educators were published, some in each of the three issues of the fourth volume of the journal. On Wheeler’s mind was the famous example of the 23 problems from various branches of mathematics that David Hilbert (1862–1943) announced in an address delivered to the Second International Congress of Mathematicians in 1900 at Paris (p. 40) and predicted that “from the discussion of which an advancement of science may

A.B. Powell (B) Department of Urban Education, Rutgers University, Newark, NJ, USA e-mail: [email protected]


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be expected” (Hilbert, 1900, p. 5).1 Of all the published responses to Wheeler’s challenge, Tall (1984) offered the briefest list of what he considered to be “the central questions”: (1) how do we do mathematics? and (2) how do we develop new mathematical ideas? (p. 25, emphasis added). Hilbert’s 23 problems contributed to more than a century of vigorous, fruitful research activity in physics and mathematics.2 Similarly, considered responses to Tall’s two questions require substantial research efforts in different environments over extended periods of time. For researchers in mathematics education to entertain these questions, we must find ways to observe what learners do as they do mathematics as well as to describe and analyze how they develop their mathematical ideas. It bears noting that 8 years after Tall issued his central questions, Davis et al. (1992) similarly challenged mathematics education researchers to study the emergence among learners of what lies at the core of mathematics: mathematical ideas. Expanding on Tall’s second question, Davis noted that “very little research in mathematics education has focused on the actual ideas in students’ minds or on how well teachers are able to identify these ideas, interact with them, and help students improve on them” (p. 732). The chapters of this book have presented rich descriptions and analyses of actual ideas that students built in the realm of combinatorial reasoning. This work has implications for teaching both in the design and sequence of effective tasks and in demonstrating how teachers could productively interact with student ideas. The global picture depicted in the chapters of this book underscores the need for time to think deeply and discursively. A special issue of Educational Studies in Mathematics collected several analyses concerning discourse in mathematics classrooms. Commenting on these studies, Seeger (2002) wonders about the possibility of a grand, panoramic theory of learning. In arguing for a comprehensive theory of mathematics education, he suggests that such a theory needs to embrace four metaphors of learning that form the axes “social – individual” and “construction – acquisition,” and represents them in a two-by-two grid (p. 289).3 In addition, Seeger further suggests that “theoretical work has to be balanced by the systematic development of focal problems for practice, theory, and research in mathematics education”

1 According to Gray (2000), Hermann Minkowski (1864–1909), whose metric concept (order-p geometry) provided the theoretical foundation for non-Euclidean, taxicab geometry, was a close friend of Hilbert and urged him to accept the invitation to speak at the Congress: “Most alluring would be the attempt to look into the future, in other words, a characterisation of the problems to which the mathematicians should turn in the future. With this, you might conceivably have people talking about your speech even decades from now” (as quoted in Gray, 2000, p. 1). 2 See Grattan-Guinness (2000) for a critical appraisal of “the range of Hilbert’s problems against the panoply then evident in mathematics.” 3 Here Seeger (2002) differs from Sfard’s (1998) theorization in which she argues for two metaphors – acquisition and participation or construction – that conceptualizes perspectives on learning and in which she claims that though complementary they are mutually not amenable to critique.

17

Closing Observations

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(p. 289). He proposes two focal problems for mathematics education, one concerning ecological validity and representation and the other referring to the question of time and change. Besides epistemological concerns, the question of time and change also concerns methodological issues. Building ideas and understanding are certainly temporal and unbounded. Consequently, there are complex judgments an investigator has to make when inquiring into what learners build, understand, or acquire from a discussion or lesson on a particular issue. When does an investigator examine what learners say, do, and write? Should these actions be examined in the immediate proximity of the discussion or lesson, in some other, more distinct time, or in some combination of these times? Ball and Lampert (1999) raise somewhat similar questions in a study of teaching practice. Epistemologically and methodologically, this book contributes to understanding the relation between time and development. As outcomes of individual and collective constructive actions over the course of the longitudinal study, the participants build ideas, reason, and employ heuristics to resolve various tasks. They reveal and make salient the important relationship between time and development. The processes by which the participants build their ideas evidence an epistemological reality: knowledge construction is often a slow process. Mathematical ideas do not develop instantaneously and robustly but rather emerge slowly and in their nascent state are rather fragile. Ideas dawn and mature over time. To loose fragility, among other things, ideas need to be reflected on deeply, presented publicly, submitted to challenge, available for negotiation, and subject to modification. That is, the essence of developing and understanding mathematical ideas is often a protracted, iterative, and recursive phenomenon, occurring over more time than is usually appreciated or acknowledged in practice in classrooms and in reports in the literature (Pirie & Kieren, 1994; Seeger, 2002). If learners are to develop deep understandings that are less fragile and more durable than is often witnessed by teachers in schools, they need to be offered extended periods of time to wrestle with a problem as well as to debate and negotiate heuristics, to articulate and justify their results, and to have their ideas challenged and then defend or modify their ideas. If we agree that students must be actively and purposely engaged in their learning so that they can take ownership and be proud of their accomplishments, we need to create opportunities for this to occur. For example, strands of investigations can be integrated into the regular curriculum as enrichment. When we eliminate the pressure of testing and grades, students can invest in thinking and reasoning for its own sake and for the intrinsic rewards that knowing deeply entails. Perhaps every several weeks, within particular strands, investigations can be revisited, and students can bring their more recent, accumulated knowledge to a more sophisticated examination of earlier solutions, and thereby extend their knowing. A focus on justification as a strand of school mathematics has great potential for building a solid foundation for the later study in many fields and certainly of mathematics. A focus on reasoning and sense making is an important requirement for a productive, responsible citizenry. Questioning, challenging, analyzing, revisiting all lead to better ways of knowing. Can we as educators meet the challenge of educating thoughtful students

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who are motivated by sense making and the critical review of ideas? The challenge awaits us. The Video Mosaic Collaborative at Rutgers University provides a mechanism for the ongoing building and sharing of knowledge. We invite readers to visit our website at http://www.video-mosaic.org/, view the videos and accompanying objects that provided the data for this book and join our expanding community of researchers by providing additional study and analysis.

Appendix A Combinatorics Problems

Listed here are the major combinatorics problems the students encountered from elementary school through high school, along with brief discussions of solutions. 1. Shirts and Jeans (May 1990, Grade 2; October, 1990, Grade 3) – Stephen has a white shirt, a blue shirt, and a yellow shirt. He has a pair of blue jeans and a pair of white jeans. How many different outfits can he make? He can make six different outfits; each of the two pairs of jeans can be matched with each of the three shirts. The outfits are: blue jeans/white shirt, blue jeans/blue shirt, blue jeans/yellow shirt, white jeans/white shirt, white jeans/blue shirt, and white jeans/yellow shirt. 2. Shirts and Jeans Extended (October, 1990, Grade 3) – Suppose Stephen had another pair of jeans, a black pair. How many different outfits can he now make? He can make 12 different outfits: number of shirts times number of jeans. 3. Four-Tall Towers (October 1990, Grade 3; December 1992, Grade 5) – Your group has two colors of Unifix cubes. Work together and make as many different towers four cubes tall as is possible when selecting from two colors. See if you and your partner can plan a good way to find all the towers four cubes tall. At each position in the tower, there are two color choices. Therefore, there are 2×2×2×2=16 possible towers that are four cubes tall. This can be generalized to an n-tall tower with two colors to choose from; there are 2×2×2. . . ×2=2n possible towers that are n cubes tall, when there are two colors to choose from. This can also be generalized to an n-tall tower with m colors to choose from; there are m×m×m. . . ×m=mn possible towers that are n cubes tall with m colors to choose from. In the following discussions, we will call the first generalization (the n-tall tower with two colors) the towers problem, and we will call the second generalization (the n-tall tower with m colors) the generalized tower problem. 4. Cups, Bowls, and Plates (April 1991, Grade 3) – Pretend that there is a birthday party in your class today. It’s your job to set the places with cups, bowls, and C.A. Maher et al. (eds.), Combinatorics and Reasoning, Mathematics Education Library 47, DOI 10.1007/978-0-387-98132-1, C Springer Science+Business Media, LLC 2010

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plates. The cups and bowls are blue or yellow. The plates are blue, yellow, or orange. Is it possible for ten children at the party each to have a different combination of cup, bowl, and plate? Show how you figured out the answer to this question. Each of the two cup choices can be matched with each of the two bowl choices, and each cup-bowl pair can be matched with any of the three different plate choices. Therefore, there are 2×2×3 = 12 possibilities. Therefore, yes, it is possible for ten children at the party each to have a different combination of cup, bowl, and plate. 5. Relay Race (October 1991, Grade 4) – This Saturday there will be a 500-m relay race at the high school. Each team that participates in the race must have a different uniform (a uniform consists of a solid colored shirt and a solid colored pair of shorts). The colors available for shirts are yellow, orange, blue, or red. The colors for shorts are brown, green, purple, or white. How many different relay teams can participate in the race? There are four choices for shirts and four choices for shorts, so there are 4×4 = 16 ways to make uniforms. Sixteen different relay teams can participate. 6. Five-Tall Towers (February 1992, Grade 4; December 1997, Grade 10) – Your group has two colors of Unifix cubes. Work together and make as many different towers five cubes tall as is possible when selecting from two colors. See if you and your partner can plan a good way to find all the towers five cubes tall. There are 25 = 32 towers five cubes tall. 7. Four-Tall Towers with Three Colors (February 1992, Grade 4) – Your group has three colors of Unifix cubes. Work together and make as many different towers four cubes tall as is possible when selecting from three colors. See if you and your partner can plan a good way to find all the towers four cubes tall. Since there are three choices for each of four positions, there are 34 = 81 possible towers that are four cubes tall when selecting from three colors. 8. A Five-Topping Pizza Problem (December 1992, Grade 5; December 1997, Grade 10) – Consider the pizza problem, focusing on the number of pizza combinations that can be made when selecting from among five different toppings. There are 25 = 32 different pizzas. 9. Guess My Tower (February 1993, Grade 5) – You have been invited to participate in a TV Quiz Show and the opportunity to win a vacation to Disney World. The game is played by choosing one of four possibilities for winning and then picking a tower out of a covered box. If the tower you pick matches your choice, you win. You are told that the box contains all possible towers that are three tall that can be built when you select from cubes of two colors, red, and yellow. You are given the following possibilities for a winning tower:


207

All cubes are exactly the same color. There is only one red cube. Exactly two cubes are red. At least two cubes are yellow. Which choice would you make and why would this choice be better than any of the others? In order to decide which is the best choice, we need to find the probability of each choice. The total number of 3-tall towers is 8. The probabilities are: All cubes are exactly the same color: There are two ways (all red or all yellow). The probability is 2÷8 = 0.25. There is only one red cube: There are three ways; the red cube can be on the top, in the middle, or on the bottom. The probability is 3÷8 = 0.375. Exactly two cubes are red: This is the same as saying exactly one cube is yellow. The probability is the same as for exactly one red cube: 3÷8 = 0.375. At least two cubes are yellow: This is equivalent to saying that either exactly two cubes are yellow or exactly three cubes are yellow. As discussed above, the probability that exactly two cubes are yellow (the same as the probability that exactly two cubes are red) is 0.375. Since there is one way for exactly three cubes to be yellow, that probability is 1÷8 = 0.125. The probability of either event is therefore 0.375 + 0.125 = 0.5. (We can add because the two events are mutually exclusive.) “At least two cubes are yellow” is the most likely event. Assuming you won, you can play again for the Grand Prize which means you can take a friend to Disney World. But now your box has all possible towers that are four tall (built by selecting from the two colors yellow and red). You are to select from the same four possibilities for a winning tower. Which choice would you make this time and why would this choice be better than any of the others? The total number of four-tall towers is 24 = 16. The probabilities are: All cubes are exactly the same color: There are two ways (all red or all yellow). The probability is 2÷16 = 0.125. There is only one red cube: There are four ways; the red cube can be on the top, second from the top, second from the bottom, or on the bottom. The probability is 4÷16 = 0.25. Exactly two cubes are red: The number of ways to accomplish this is C(4,2) = 6. The probability is therefore 6÷16 = 0.375.

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At least two cubes are yellow: This means that exactly two cubes are yellow, exactly three cubes are yellow, or exactly four cubes are yellow. As discussed above, the probability that exactly two cubes are yellow (the same as the probability that exactly two cubes are red) is 6÷16 = 0.375. The probability that exactly three cubes are yellow is the same as the probability that one cube is red: 4÷16 = 0.25. Since there is one way for exactly four cubes to be yellow, that probability is 1÷16 = 0.0625. The probability of any one of the three events is therefore 0.375 + 0.25 + 0.0625 = 0.6875. “At least two cubes are yellow” is the most likely event. 10. The Pizza Problem with Halves (March 1993, Grade 5) – A local pizza shop has asked us to help them design a form to keep track of certain pizza sales. Their standard “plain” pizza contains cheese. On this cheese pizza, one or two toppings could be added to either half of the plain pizza or the whole pie. How many choices do customers have if they could choose from two different toppings (sausage and pepperoni) that could be placed on either the whole pizza or half of a cheese pizza? List all possibilities. Show your plan for determining these choices. Convince us that you have accounted for all possibilities and that there could be no more. With two topping choices, there are four possibilities for the first half pizza, because each topping can be either on or off that half of the pizza. The four choices are: plain (sausage off, pepperoni off), sausage (sausage on, pepperoni off), pepperoni (sausage off, pepperoni on), and sausage/pepperoni (sausage on, pepperoni on). Consider each of the four possibilities in turn. Case 1: Plain. There are four possibilities for the other half of the pizza, the four listed above (plain, sausage, pepperoni, and sausage/pepperoni). Case 2: Sausage. There are three possibilities for the other half of the pizza: sausage, pepperoni, and sausage/pepperoni. (We omit plain, because we already accounted for the plain-sausage pizza in Case 1.) Case 3: Pepperoni. There are two possibilities remaining for the other half of the pizza: pepperoni and sausage/pepperoni. (Plain and sausage are already accounted for.) Case 4: Sausage/pepperoni. There is only one possibility left for the other half of the pizza; that is sausage/pepperoni. There are 4+3+2+1 = 10 possible pizzas with halves. 11. The Four-Topping Pizza Problem (April 1993, Grade 5) – A local pizza shop has asked us to help design a form to keep track of certain pizza choices. They offer a cheese pizza with tomato sauce. A customer can then select from the following toppings: peppers, sausage, mushrooms, and pepperoni. How many different choices for pizza does a customer have? List all the possible choices.


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Find a way to convince each other that you have accounted for all possible choices. There are 2×2×2×2 = 16 possible pizzas. 12. Another Pizza Problem (April 1993, Grade 5) – The pizza shop was so pleased with your help on the first problem that they have asked us to continue our work. Remember that they offer a cheese pizza with tomato sauce. A customer can then select from the following toppings: peppers, sausage, mushrooms, and pepperoni. The pizza shop now wants to offer a choice of crusts: regular (thin) or Sicilian (thick). How many choices for pizza does a customer have? List all the possible choices. Find a way to convince each other that you have accounted for all possible choices. Each of the 16 four-topping pizzas has two choices of crust, so there are 32 pizzas. 13. A Final Pizza Problem (April 1993, Grade 5) – At customer request, the pizza shop has agreed to fill orders with different choices for each half of a pizza. Remember that they offer a cheese pizza with tomato sauce. A customer can then select from the following toppings: peppers, sausage, mushroom, and pepperoni. There is a choice of crusts: regular (thin) and Sicilian (thick). How many different choices for pizza does a customer have? List all the possible choices. Find a way to convince each other than you have accounted for all possible choices. The first half of the pizza can have 24 = 16 possible topping configurations, as described above. Consider each of those configurations in turn. Following the procedure described above for the two-topping half-pizza problem, we find that there are 16+15+14+. . . +3+2+1 possible pizzas; this sum is given by 16×17÷2. Since each pizza can have a thick or thin crust, we multiply by 2. The number of possible pizzas is 16×17÷2×2 = 272. 14. Counting I and Counting II (March 1994, Grade 6) – How many different twodigit numbers can be made from the digits 1, 2, 3, and 4? Each of four cards is labeled with a different numeral: 1, 2, 3, and 4. How many different two-digit numbers can be made by choosing any two of them? Counting I: Assuming that you are not permitted to reuse digits, there are four choices for the first digit and three for the second digit, giving 12 two-digit numbers. (They are 12, 13, 14, 21, 23, 24, 31, 32, 34, 41, 42, and 43.) Counting II: There are four choices for the first digit and four choices for the second digit. This makes 16 different two-digit numbers. (They are 11, 12, 13, 14, 21, 22, 23, 24, 31, 32, 33, 34, 41, 42, 43, and 44.) 15. Towers-Binomial Relationship (March 1996, Grade 8) – In an interview, Stephanie discusses the relationship between the towers problems and the binomial coefficients.

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Binomial coefficients arise in connection with the binomial expansion formula (a+b)n . The following can be shown by induction: (a + b) = n

n n r=0

r

an−r br

The coefficient of an–r br is given by: n! n = r r!(n − r)! This number is the rth entry in the nth row of Pascal’s Triangle, and it gives the number of towers with exactly r cubes of one color, when building towers that are n-tall and there are two colors to choose from. Hence, the binomial expansion and the towers problem are isomorphic, with the number of instances of a in the rth term being equal to the number of towers having exactly r cubes. 16. Five-Tall Towers with Exactly Two Red Cubes (January 1998, Grade 10) – You have two colors of Unifix cubes (red and yellow) to choose from. How many five-tall towers can you build that contain exactly two red cubes? You are selecting two items (the positions of the two red cubes) from five choices (the number of cubes in the tower); there are ten ways to do this: 5! 5 = 10 = 2 2!(5 − 2)! 17. Ankur’s Challenge (January 1998, Grade 10) – Find all possible towers that are four cubes tall, selecting from cubes available in three different colors, so that the resulting towers contain at least one of each color. Convince us that you have found them all. Suppose the colors are red, blue, and green. We are counting the towers in three cases: (1) those with two red cubes, one blue cube and one green cube, (2) those with one red cube, two blue cubes, and one green cube, and (3) those with one red cube, one blue cube, and one green cube. The following equation gives the number of ways of selecting m groups of objects of size r1 through rm :

n r1 , r2 , ..., rm

=

n! , where ri = n r1 ! · r2 ! · ... · rm !

So the number of four-tall towers containing exactly two red cubes, one blue cube, and two green cubes is:

4 2, 1, 1

=

4! = 12 2! · 1! · 1!


211

Similarly for the other two cases:

4 1, 2, 1

=

4 1, 1, 2

= 12

Hence the number of towers with the required condition is 12+12+12 = 36. 18. The World Series Problem (January 1999, Grade 11) – In a World Series, two teams play each other in at least four and at most seven games. The first team to win four games is the winner of the World Series. Assuming that the teams are equally matched, what is the probability that a World Series will be won: (a) in four games? (b) in five games? (c) in six games? (d) in seven games? The number of ways for a team to win the series (four games) in n games is the number of ways it can win three times in n–1 games (and then win the last game). This is given by C(n–1,3). The probability of any given set of outcomes for n games is 1÷2n (since there are two equally likely outcomes for each game). So the probability that one team wins the series in n games is given by C(n–1,3) ÷2n , and the probability of a win for either team is double that: C(n–1,3) ÷2n–1 . The probabilities are: (a) C(4–1,3) ÷24–1 = C(3,3) ÷23 = 1÷8 = 0.125. (b) C(5–1,3) ÷25–1 = C(4,3) ÷24 = 4÷16 = 0.25. (c) C(6–1,3) ÷26–1 = C(5,3) ÷25 = 10÷32 = 0.3125. (d) C(7–1,3) ÷27–1 = C(6,3) ÷26 = 20÷64 = 0.3125. 19. The Problem of Points (February 1999, Grade 11) – Pascal and Fermat are sitting in a café in Paris and decide to play a game of flipping a coin. If the coin comes up heads, Fermat gets a point. If it comes up tails, Pascal gets a point. The first to get ten points wins. They each ante up 50 francs, making the total pot worth 100 francs. They are, of course, playing “winner takes all.” But then a strange thing happens. Fermat is winning, eight points to seven, when he receives an urgent message that his child is sick and he must rush to his home in Toulouse. They carriage man who delivered the message offers to take him, but only if they leave immediately. Of course, Pascal understands, but later, in correspondence, the problem arises: how should the 100 francs be divided? We can list all the circumstances where Fermat gets two points before Pascal gets three points. He can do this in two flips, three flips, or four flips. (The game cannot proceed past four flips. As soon as both players get to nine points, the next flip will produce a winner. It takes three flips for this to happen.) (a) Two flips: Fermat wins both. Probability =1÷22 = 1÷4. (b) Three flips: Fermat wins one of the first two and the last one. Probability = C(2,1)÷23 = 1÷4.

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(c) Four flips: Fermat wins one of the first three and the last one: Probability = C(3,1)÷24 = 3÷16 Probability of any of these events = 1÷4 + 1÷4 + 3÷16 = 11÷16. Therefore Fermat should get 100 × 11÷16 Francs ≈ 69 Francs and Pascal should get 31 Francs. 20. The Taxicab Problem (May 2002, Grade 12) – A taxi driver is given a specific territory of a town, shown below. All trips originate at the taxi stand. One very slow night, the driver is dispatched only three times; each time, she picks up passengers at one of the intersections indicated on the map. To pass the time, she considers all the possible routes she could have taken to each pick-up point and wonders if she could have chosen a shorter route. What is the shortest route from a taxi stand to each of three different destination points? How do you know it is the shortest? Is there more than one shortest route to each point? If not, why not? If so, how many? Justify your answer.

Using Powell’s et al. (2003) notation to denote coordinates on the taxicab grid, (n,r) indicates a point n blocks away from the taxi stand and r blocks to the right. So the blue dot is at (5,1), the red dot is at (7,4), and the green dot is at (10,6). Taking the shortest route means going in two directions only (down and to the right). Finding the number of shortest paths from the taxi stand (0,0) to any point (n,r) involves the number of ways to select r segments of one kind of movement in a path that includes two kinds of movements; i.e., the number of shortest paths to (n,r) is C(n,r). For the specific cases given above, the shortest paths are: Blue: C(5,1) = 5. Red: C(7,4) = 35. Green: C(10,6) =210.

Appendix B Counting and Combinatorics Dissertations from the Longitudinal Study

Franciso, J. M. (2004). Students’ reflections on their learning experiences: Lessons from a longitudinal study on the development of mathematical ideas and reasoning. Unpublished doctoral dissertation, Rutgers University, Newark, NJ Glass, B. H. (2001). Mathematical problem solving and justification with community college students. Unpublished doctoral dissertation, Rutgers University, New Jersey. Kiczek, R. D. (2001). Tracing the development of probabilistic thinking: Profiles from a longitudinal study. Unpublished doctoral dissertation, Rutgers University, New Jersey. Martino, A. M. (1992). Elementary students’ construction of mathematical knowledge: Analysis by profile. Unpublished doctoral dissertation, Rutgers University, Newark, NJ. Muter, E. M. (1999). The development of student ideas in combinatorics and proof: A six year study. Unpublished doctoral dissertation, Rutgers University, Newark, New Jersey. O’Brien, M. (1994). Changing a school mathematics program: A ten-year study. Unpublished doctoral dissertation, Rutgers University, New Jersey. Powell, A. B. (2003). “So let’s prove it!”: Emergent and elaborated mathematical ideas and reasoning in the discourse and inscriptions of learners engaged in a combinatorial task. Unpublished doctoral dissertation, Rutgers University, New Jersey. Muter, E. M. (1999). The development of student ideas in combinatorics and proof: A six year study. Unpublished doctoral dissertation, Rutgers University, Newark, New Jersey. Sran, M. K. (2010). Tracing Milin’s development of inductive reasoning: A case study. Unpublished doctoral dissertation, Rutgers University, Newark, NJ. Steffero, M. (2010). Tracing beliefs and behaviors of a participant in a longitudinal study for the development of mathematical ideas and reasoning: A case study. Unpublished doctoral dissertation, Rutgers University, Newark, NJ. Tarlow, L. D. (2004). Tracing students’ development of ideas in combinatorics and proof. Unpublished doctoral dissertation, Rutgers University, New Jersey. Uptegrove, E. B. (2005). To symbols from meaning: Students’ investigations in counting. Unpublished doctoral dissertation, Rutgers University, New Jersey. 213

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Index

A Abstraction, 10, 17, 24, 77, 85 Addition rule, 13, 73, 77, 82, 111–114, 124–125, 129–130, 133, 136–138, 141, 152, 181 Addition rule of Pascal’s Triangle, 181 Adults, 171–183 Ali, 99–100 Alston, Alice S., 5–6, 35, 45, 67, 97, 121, 182 Amy-Lynn, 59, 62–63, 66–67, 121–122, 124–126 Angela, 97–98, 103, 121, 127–129 Angela’s Law of Towers, 98 Ankur, 59, 62–64, 66–71, 89–90, 92–94, 105–107, 109–119, 133–135, 138–140, 142, 157, 159–160, 162, 164–165, 167, 185 Ankur’s Challenge, 89–95, 110, 183, 185, 199–200, 210 Another Pizza Problem, 70, 209 Argument, 13, 27, 31, 33–34, 35, 37, 40–43, 45–57, 61, 66–67, 72, 75, 89–95, 97, 102–103, 175–176, 185, 191, 195, 200 B Beliefs, 157–159, 168–169 Binary notation, 106, 108, 116, 118–120, 135, 141, 143 Binary numbers, 106, 137 Binomial, 12, 14, 73, 76–77, 104–105, 108–112, 114, 116, 121–131, 134–136, 143, 152, 209–210 Binomial expansion, 12, 14, 108–112, 114, 116, 134–136, 210 Bobby (Robert), 45, 59, 62 Branches, 103, 201

Brian, 59, 62–63, 66, 68–71, 89–92, 94, 105–108, 110, 119, 133–135, 138–139, 142, 145–150, 153, 157, 159, 161, 163, 166–167 C Cases, 12, 14, 33–34, 36–37, 39–43, 45–46, 49, 51, 56, 62, 66, 68, 74, 77, 85, 90, 94, 97–100, 104, 110, 119–120, 127–128, 130, 143, 145, 153, 158, 174–175, 177, 182, 185–192, 194–195, 200 Choices, 11–12, 59–60, 69–71, 103, 107–108, 116, 122, 124, 127–128, 130, 143, 151, 177, 196 Choose, 12, 43, 59, 98, 107, 123, 127, 135–140, 142 Choose notation, 135–136, 140 Claims, 145–153, 164, 168, 202 Coefficient, 77, 106, 135 Collegiate, 183 Combinations, 13, 19–23, 25, 28–30, 33, 36–38, 40, 42–43, 45–46, 60–67, 69–71, 77, 79, 82, 85, 89–90, 92, 94, 101–103, 106, 110, 122, 124, 127–128, 130, 137, 172, 175–179, 182, 189, 192, 195–196, 200 Combinatorial reasoning, 73, 202 Combinatorics, 3, 6, 9–11, 14, 17–18, 24–25, 73, 110, 112–114, 120, 122, 131, 133, 140–141, 144–146, 169, 198, 201 Combinatorics problems, 11, 14, 17–18, 112–113, 131, 133, 144 Conditions, 3, 28, 142, 158, 168–169, 171, 183, 190–191 Conjecture, 99, 147 Connections, 3–4, 9, 18, 72, 74, 82, 105–120, 130–131, 141, 143, 171–172, 178–182 Consumer math, 173

221

222 Contemporary Mathematics, 172–173 Contradiction, 14, 31, 40, 66–67, 97, 99 Controlling for variables, 14, 35, 43, 97, 99 Convincing arguments, 4, 104 Counting I, 209 Counting II, 209 Counting methods, 172–173 Cousin, 31–33, 36, 42, 46 Critical events, 9 Cups, Bowls, and Plates, 205–206 D Dana, 11, 17, 19, 21–25, 27–32, 43, 60 Danielle, 171, 174 Discourse, 4, 74, 142, 146, 150, 159, 164–165, 202 Discursive, 150–151, 164, 167, 201 Donna, 171, 186–187 Doubling rule, 39–40, 42, 49, 56, 106, 130 Duplicates, 19, 29–30, 33, 36, 56, 61, 72, 78–82, 100, 122, 127, 197 Durability of ideas, 141 Dyadic choice, 151 E Educational Studies in Mathematics, 202 Elevator, 31, 34, 37 Empirical, 82, 152 Epistemological beliefs, 157–159, 168 Errol, 171, 175–176, 192, 199 Euclidean geometry, 150 Explanation, 21, 40, 51–55, 67, 70, 74, 92–95, 97, 101, 106, 109, 112, 115–117, 119, 123, 126, 135–136, 141, 172–175, 180–181, 197, 200 F Factorials, 122 Family tree, 52–53 Fermat, 77 Fermat’s Recursion, 77–78 Final Pizza Problem, 70, 209 Five-tall towers with exactly two red cubes, 33, 109, 119 Five-tall towers selecting from two colors, 31 Five-topping pizza problem, 13, 106, 112, 135, 174, 181 Forms of reasoning, 6, 11, 14, 39, 72–73, 185, 200–201 Formula, 77–78, 82, 103–104, 108–109, 122, 125, 128, 130, 141, 143, 178, 199 For the Learning of Mathematics, 201

Index Four-tall towers selecting from 2 colors, 28, 196–197 Four-tall towers selecting from 3 colors, 195 Four-topping pizza problem, 68–70, 123, 127, 174, 181, 208 Francisco, John M., 157–169 G Gang of Four, 38, 42–43, 51, 75 Generalization, 3, 17, 91, 94, 98, 106, 126 Generalize, 3, 10, 18, 41, 90, 95, 104–105, 133, 147, 151, 172 General rule, 97–98, 101, 124, 135, 141, 143 Geometric, 108–109 Glass, Barbara, 171–183, 185–200 Grade 1, 7 Grade 2, 20–22, 24, 73 Grade 3, 3, 7, 22–23, 28, 74 Grade 4, 5–6, 31, 33–42, 77 Grade 5, 45, 59, 77 Grade 6, 7 Grade 7, 75 Grade 8, 33, 73, 75, 85 Group work, 25, 31, 36, 70, 103, 121, 162, 165, 177 Guess My Tower, 42, 45, 50 H Harding School, 6 Heuristic, 11–13, 25, 27, 43, 59, 67, 147, 201, 203 High school, 4–8, 14, 89, 97, 105, 133–134, 142, 145–146, 157–158, 161, 169, 172–173, 182–183, 185–186, 200 Hilbert, David, 201 I Inductive argument, 42–43, 45–57, 67, 97, 102–103, 176–177, 182, 185, 191–195, 199 Inductive reasoning, 14, 38–39, 50–56, 97, 100, 195 Interlocution, 151 Isomorphic relationship, 108, 118, 120 Isomorphism, 4, 14, 77, 109, 126, 130, 133, 147, 151–153, 179–180 J Jaime, 17, 23 Jeff C., 171, 174, 176–177, 179–180 Junior year, 134 Justification, 11–12, 14, 24, 34, 39, 42, 46, 59, 67, 90, 94, 97, 100, 104, 109–110, 118–119, 126, 147, 168, 173, 175–177, 185–193, 196, 199–200, 203

Index Justification by cases, 97, 104, 119, 177, 185, 192, 199 Justify, 3, 12, 19, 39, 42–43, 49, 61, 65–66, 73, 92, 130–131, 146–153, 168, 172, 175–176, 183, 194, 200, 203 K Kenilworth, 4–7, 12, 59, 69–70, 161, 182 Kiczek, Regina, 121, 133 Knowing deeply, 203 L Learning environment, 131 Lecture classes, 118–120 Letter codes, 106, 122, 142 Liberal-arts, 171–173, 185 Linda, 171 Lisa, 171, 175, 180 Logic, 39, 172 Logical reasoning, 10, 17 Longitudinal study, 3–8, 10–11, 14, 18, 24, 27, 75, 86, 89, 98–99, 105, 130, 157–161, 167–169, 172, 177, 183, 185–200, 203 M Magda, 97–99, 103, 121, 127–130 Maher, Carolyn A., 4, 45–57, 59–72, 89–95, 105, 121, 133 Making sense, 131, 134, 141, 145 Martino, Amy M., 4, 9, 17–21, 27–28, 33, 35, 39, 41, 47, 49, 74–75, 178, 180, 182 Mary, 171, 175, 195, 197–198, 200 Mathematical beliefs, 157, 159, 168–169 Mathematical concepts, 6, 119, 161, 172–173 Mathematical ideas, 4–5, 9–11, 17–19, 24, 57, 74–75, 85, 95, 130–131, 133, 142, 145, 158, 162, 172, 201–203 Mathematical structure, 3, 147 Melinda, 171, 178, 180 Metaphor, 13, 85 Michael, 17, 19–21, 23, 35 Michelle, 27, 38, 41, 45, 50–54, 56, 59–60, 62, 64, 97, 99–103, 121–122, 127 Michelle I., 27, 45, 50–54, 56, 59, 64–65 Michelle R., 45, 50, 52–53, 56, 59, 62 Mike, 59, 62–64, 66–68, 70–71, 89–90, 92–94, 105–110, 114–120, 133–142, 145–152, 157, 159–160, 162, 165–167, 171, 177, 181–182 Milin, 25, 27, 35–41, 45–52, 55–56, 59–61, 64–65, 71, 77 Muter, Ethel M., 59, 89–95, 105–120

223 N National Council of Teachers of Mathematics (NCTM), 5, 17, 144, 168 National Science Foundation, 5 Night session, 112, 116, 118, 133, 135–143, 181 Notation, 13–14, 18–19, 24–25, 27, 42, 59–61, 65, 67, 72–74, 77, 103, 105–108, 110, 112, 116, 133–142, 201 n-Tall tower, 12, 27 N-tall towers selecting from r-colors, 42, 77 O Opposite, 29, 31, 35–36, 42, 46, 54, 67, 99–100, 150, 175 Order, 6, 12, 24, 70, 75, 89–90, 106–107, 113, 118, 124, 127–128, 130, 133, 140, 143–144, 153, 173, 175, 182, 189, 191, 202 Outfit, 11, 19–21, 23–24, 43 Over time, 73, 75, 160, 162, 168, 171, 201, 203 P Pantozzi, Ralph, 121, 161–162, 166–167 Papert, 74–75 Partial cases, 42 Pascal, 211–212 Pascal’s Identity, 13–14, 105, 111–115, 118, 120–121, 124–125, 127–130 Pascal’s Triangle, 12–14, 72–73, 77, 81–83, 86, 104–105, 110–117, 120–121, 123–126, 128–131, 133–138, 140–141, 143–144, 147, 151–153, 160, 181–182 Pattern, 6, 10, 18, 24, 27–28, 31, 33–34, 37–43, 46, 50–54, 73–74, 77, 81–82, 85, 90, 99, 101, 104, 106, 119, 123, 128, 130, 147, 174–175, 181–183, 190 Penny, 171, 177, 195, 199 Permutations, 172, 178, 182, 192 Pirie, Susan, 97, 158, 203 Pizza with halves problem, 12–13, 67–69, 174 plain, 12, 59, 106–107, 113, 117–118, 122–123, 125, 127, 141, 208 problem, 12–13, 59, 68–70, 72–73, 104–107, 111–113, 116–123, 125, 127, 129–130, 135–136, 141, 145, 169, 172–174, 177–178, 180–181, 183 Powell, Arthur B., 9, 145–154 Power of 2, 122, 124 Probability, 165, 172

224 Problem of Points, 211 Proof by cases, 41, 72, 119, 130, 175, 191, 194 by contradiction, 99 R Reasoning, 3–4, 6, 9–11, 14, 17, 24–25, 27, 31, 38–39, 42, 45, 50, 52, 54–55, 59–72, 75–76, 84, 94–95, 97, 100, 112, 145, 147–153, 158, 168, 171–183, 185, 188, 191, 195, 201–203 by contradiction, 97 Recursive argument, 72, 199 Relay Race, 206 Representation, 3–4, 10–14, 17–25, 27, 46, 51, 59, 62, 64, 73, 76, 94–95, 97, 105–120, 133–144, 168, 172, 203 Rica, Fred, 4–5 Rob, 171, 175, 177–178, 181–182, 187 Robert, 121–122, 124–127 S Samantha, 171, 179 Scheme, 21, 24, 94, 105–106, 109, 174 Second International Congress of Mathematicians, 201 Senior year, 145, 200 Sense making, 6, 11–12, 24, 159–162, 200, 203–204 Shelly, 121–127 Sherly, 99–100, 121, 127–129 Shirts and jeans, 11, 17–24, 27, 29, 59–60, 64, 73 Shirts and jeans extended, 205 Socially emergent cognition, 150 Sociomathematical norms, 153 Sophomore year, 119, 134–135, 140, 143 Specialize, 147–148, 151 Speiser, Robert, 33, 73–86, 105–112, 201 Sran, Manjit K., 27–43, 45–57, 59–72

Index Staircase, 33, 40, 42, 174–175 Standard notation, 133–144 Statistics, 172, 178 Stephanie, 11, 17, 19–25, 28–35, 40–42, 45–62, 71, 73–85, 121–127, 171, 178, 180, 182 Stephanie C., 171, 178, 180 Steve, 171, 175, 181 Strategy, 13, 21, 23, 29, 31, 35–36, 43, 45, 51, 67, 69, 72, 92, 94, 98–99, 100, 119, 147, 153, 177, 182, 192, 194, 197, 200 T Tarlow, Lynn D., 59, 97–104, 121–131 Taxicab geometry, 150, 202 Taxicab problem, 14, 144–147, 150–153, 212 Theory, 51, 74, 102–103, 175, 202 Tim, 171, 177, 190 Tower binomial relationship, 209 families, 31–32, 34 Tracy, 171, 178 Tree diagrams, 122, 127 Trial and error, 29, 31, 35–42, 51 U Understanding, 4–5, 9, 12, 18, 21, 24, 33, 41, 43, 45–46, 51–54, 57, 73–76, 95, 101, 105–106, 109, 121, 134, 140, 158, 160–164, 180, 201, 203 Unifix cubes, 12, 27, 42, 50, 97, 126 Uptegrove, Elizabeth B., 9–14, 97–104, 105–120, 133–144 W Wesley, 171, 175–176, 180 Y Yankelewitz, Dina, 17–25, 27–43, 45–57, 59–72

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