Combinatorics and Reasoning: Representing, Justifying and Building Isomorphisms

Combinatoricsand Reasoning

Mathematics Education LibraryVOLUME 47

Managing Editor

A.J. Bishop, Monash University, Melbourne, Australia

Editorial Board

M.G. Bartolini Bussi, Modena, ItalyJ.P. Becker, Illinois, U.S.A.

B. Kaur, SingaporeC. Keitel, Berlin, Germany

F. Leung, Hong Kong, ChinaG. Leder, Melbourne, AustraliaD. Pimm, Edmonton, Canada

K. Ruthven, Cambridge, United KingdomA. Sfard, Haifa, Israel

Y. Shimizu, Tennodai, JapanO. Skovsmose, Aalborg, Denmark

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

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

Combinatoricsand Reasoning

Representing, Justifying and Building Isomorphisms

123

EditorsDr. Carolyn A. MaherRutgers UniversityGraduate School of Education10 Seminary PlaceNew Brunswick, NJ [email protected]

Dr. Elizabeth B. UptegroveFelician CollegeDepartment of Mathematical Sciences223 Montross AvenueRutherford, NJ [email protected]

Series Editor:Alan BishopMonash UniversityMelbourne [email protected]

Dr. Arthur B. PowellRutgers UniversityDepartment of Urban Education110 Warren StreetNewark, NJ [email protected]

ISBN 978-0-387-98131-4 e-ISBN 978-0-387-98132-1DOI 10.1007/978-0-387-98132-1Springer New York Dordrecht Heidelberg London

Library of Congress Control Number: 2010926950

© Springer Science+Business Media, LLC 2010All rights reserved. This work may not be translated or copied in whole or in part without the writtenpermission 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 inconnection 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 arenot identified as such, is not to be taken as an expression of opinion as to whether or not they are subjectto 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 studentswho participated in the longitudinal study andfrom whom we continue to learn so much. Wethank you for your continuing commitment,abundant trust, and generous sharing of howmathematical ideas and ways of reasoning arebuilt.

Preface

Our research project on mathematical learning focuses on the accomplishments ofa cohort group of learners from first grade though high school and beyond, concen-trating on their work on a set of combinatorics tasks. We describe their impressivemathematical achievements over these years. We illustrate in detail the processesby which students learn to justify solutions to combinatorics problems that werechallenging for their age and grade level. Based on transcribed video data and learn-ers’ inscriptions, we provide a careful and detailed analysis of the process by whichmathematical ideas are developed, discussed, modified, expanded, and justified.

Our work underscores the power of attending to basic ideas in building argu-ments; it shows the importance of providing opportunities for the co-construction ofknowledge by groups of learners; and it demonstrates the value of careful construc-tion of appropriate tasks. Moreover, it documents how reasoning that takes the formof proof evolves with young children and it discusses the conditions for supportingstudent reasoning.

We present in this book strong and compelling evidence that under appropri-ate conditions and with minimal intervention, learners can develop sophisticatedideas about proof and justification, generalization, isomorphism, and mathematicalreasoning at an early age and can continue to refine and expand those ideas overtime, developing increasingly sophisticated presentations and representations. Wealso describe an extension of this work with groups of undergraduate students, not-ing similarities and differences between the reasoning of the original cohort groupof younger students and that of the college students.

We include a detailed discussion of all the mathematical tasks, which can be usedin classrooms from elementary school to the graduate college level.

vii

Acknowledgements

We are deeply grateful for the many colleagues who have made this book possibleand would like to acknowledge their contributions.

We thank Fred Rica, Principal of Harding Elementary School, Kenilworth NewJersey, whose vision made the study possible, and the faculty and administration atKenilworth who gave their support and encouragement to students and researchers.

We thank the many graduate students, video crew and videographers, especiallyRoger Conant, Ann Heisch, Lynda Smith, and Elena Steencken, as well as the dedi-cated 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, AmyM. 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, aswell as his invaluable help with task design.

We thank the Sussex County Community College students who participated inthe 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 Sigleyfor his overall knowledge and management of the collection as well as his invaluabletechnical help, Patricia Crossley for her organization of the data for the studies, andManjit K. Sran and Dirck Uptegrove for their wonderful illustrations and artwork.

We are grateful for support for the longitudinal study by: (1) the NationalScience Foundation with grants: MDR-9053597 (directed by R. B. Davis andC. A. Maher) and REC-9814846 (directed by C. A. Maher), and (2) from the NewJersey Department of Higher Education, the Johnson and Johnson Foundation, theExxon Education Foundation, and the AT&T Foundation. Any opinions, findings,and conclusions or recommendations expressed in this book are those of the authorsand do not necessarily reflect the views of the funding agencies.

ix

Contents

Part I Introduction, Background, and Methodology

1 The Longitudinal Study . . . . . . . . . . . . . . . . . . . . . . . . 3Carolyn A. Maher

2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Carolyn A. Maher and Elizabeth B. Uptegrove

Part II Foundations of Proof Building (1989–1996)

3 Representations as Tools for Building Arguments . . . . . . . . . . 17Carolyn A. Maher and Dina Yankelewitz

4 Towers: Schemes, Strategies, and Arguments . . . . . . . . . . . . 27Carolyn A. Maher, Manjit K. Sran, and Dina Yankelewitz

5 Building an Inductive Argument . . . . . . . . . . . . . . . . . . . 45Carolyn A. Maher, Manjit K. Sran, and Dina Yankelewitz

6 Making Pizzas: Reasoning by Cases and by Recursion . . . . . . . 59Carolyn A. Maher, Manjit K. Sran, and Dina Yankelewitz

7 Block Towers: From Concrete Objects to Conceptual Imagination 73Robert Speiser

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

8 Responding to Ankur’s Challenge: Co-construction ofArgument Leading to Proof . . . . . . . . . . . . . . . . . . . . . . 89Carolyn A. Maher and Ethel M. Muter

9 Block Towers: Co-construction of Proof . . . . . . . . . . . . . . . 97Lynn D. Tarlow and Elizabeth B. Uptegrove

10 Representations and Connections . . . . . . . . . . . . . . . . . . . 105Ethel M. Muter and Elizabeth B. Uptegrove

xi

xii Contents

11 Pizzas, Towers, and Binomials . . . . . . . . . . . . . . . . . . . . . 121Lynn D. Tarlow

12 Representations and Standard Notation . . . . . . . . . . . . . . . 133Elizabeth B. Uptegrove

13 So Let’s Prove It! . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145Arthur B. Powell

Part IV Extending the Study, Conclusions, and Implications

14 “Doing Mathematics” from the Learners’ Perspectives . . . . . . . 157John M. Francisco

15 Adults Reasoning Combinatorially . . . . . . . . . . . . . . . . . . 171Barbara Glass

16 Comparing the Problem Solving of College Students withLongitudinal Study Students . . . . . . . . . . . . . . . . . . . . . 185Barbara Glass

17 Closing Observations . . . . . . . . . . . . . . . . . . . . . . . . . . 201Arthur B. Powell

Appendix A Combinatorics Problems . . . . . . . . . . . . . . . . . . . 205

Appendix B Counting and Combinatorics Dissertationsfrom the Longitudinal Study . . . . . . . . . . . . . . . . . . . . . . 213

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

Introduction

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

The theoretical foundation for the program of research on which this book is basedcomes from recognition that individual learning takes place within a community.The members of that community have access to and are influenced by the ideasof others. Individual learners are interconnected with other members of the com-munity; engagement with others opens up possibilities for sharing and comparingrepresentations of ideas and for revising existing schemes and building new ones.In the activity of problem solving, learners bring forth, communicate, and compareideas. They explore whether the ways that others represent ideas correspond withtheir own representations, thereby extending their personal repertoires of tools fordealing with new ideas. In this way further learning takes place and understandingdeepens (Davis & Maher, 1997; Maher, Martino, & Alston, 1993; Maher & Davis,1990).

The data for this book come from a long-term program of research detailing thecollective building of mathematical ideas, which we call the longitudinal study. Inthis book, we explore student work in one of the mathematics strands of the longi-tudinal study: counting and combinatorics. It investigates how students’ reasoningevolved 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 participantsmake sense of the complexity of problems, we trace the representations they share,the heuristics they invent and apply, and the modifications they make in buildingarguments and in offering justifications for solutions.

The authors of the constituent 17 chapters relate how an ordinary group of schoolchildren manifest over a 12-year period an extraordinary array of mathematical ideasthat they discursively build and how – with time – their ideas modify and mature asthey reason and justify their ideas. The book reports episodes from a long-term studyof 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, USAe-mail: [email protected]

xiii

xiv Introduction

data than can be presented here. However, we have selected narratives that featurethe voices of several children, as interpreted by a variety of researchers, to weavetogether a bigger story about how students can educate us about the multifacetednature of mathematical development. In an important sense, the really big story isstill being written as our work in preserving and further analyzing those 4,500 h ofvideo through the Video Mosaic Collaborative1 continues to reveal new narratives.We invite readers to view the videos at http://www.video-mosaic.org/. Along withthe narratives offered in the book chapters that follow, these videos enable readersto trace in detail the development of counting/combinatorics ideas and ways of rea-soning of learners over more than a decade. Thus, while only some of the children’svoices appear in this book, we are indebted to all of them for sharing their devel-oping mathematical ideas over time and in divergent contexts, which we continueto study and consider how these children’s extraordinary mathematical reasoningmay inspire the fields of mathematics education, teacher education, and the learningsciences.

To structure a story that emerges from the chapters, the editors have divided thisbook into four parts. The two chapters of the first part, respectively, provide histor-ical background of the research study from which the details of the later chaptersemerge and describe the design of the study. The first chapter describes the study andthe purpose of the research, how the study began, and the conditions under which theresearch was conducted. It also briefly describes the mathematical ideas and ways ofreasoning that emerged from the study. (The details are presented in later chapters.)The second chapter presents the method of the study, its design, including selectionof participants, data collection, and analysis, as well as the strand of tasks on whichparticipants were invited to work. The chapter also discusses the importance of thetask design for helping learners to develop ways of reasoning.

The second part of the book contains five chapters. These chapters chroniclethe work of the study’s participants over a 7-year period from grades 2–8, tracingthe development of their mathematical ideas, heuristics, and forms of reasoning.In particular, the reader will learn how the participating children represented theirideas; developed schemes and strategies; reasoned in specific ways; built induc-tive arguments; reasoned by cases and by recursion; and connected numbers inPascal’s Triangle to results of previous problems. The authors of Chapter 3 dis-cuss how young children use representations to express their mathematical ideaswhile 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 NationalScience 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 videocollection amassed by The Robert B. Davis Institute for Learning at Rutgers University throughtwo decades of research with over four millions dollars of grant funding from the NSF (awardsMDR-9053597, REC-9814846, REC-0309062 and DRL-0723475). In addition to preserving thevideo collection, new tools are being developed for conducting design research and an empiricalstudy that use the videos in the context of teacher education. The editors gratefully acknowledgethis considerable support from the National Science Foundation and wish to clarify that all viewsexpressed in this book are those of the authors are not necessarily those of the NSF.

Introduction xv

problem, described fully in Appendix A, along with all combinatorics problems dis-cussed herein). They show how children structure their representations in responseto requests to justify their problem solution and build convincing arguments toearly counting problems. The authors of Chapter 4 and 5 discuss students’ workon different versions of the towers problems (which involve determining how manytowers can be built of various heights when selecting from cubes of various num-bers of colors). They show the emergence of different forms of reasoning (cases,contradiction, recursion, and induction) and how, motivated by the need to findthe sample space for a basic probability exploration, students revisit the inductiveargument for building towers. Chapter 6 discusses how participants collaborativelybuild representations that help them use reasoning by cases and by recursion todevelop justifications for their solutions to classes of pizza problems. (Pizza prob-lems involve determining how many pizzas it is possible to make when selectingfrom various numbers of toppings and under various other constraints.) Completingthis part of the book, Chapter 7 presents the results of an interview with 13-year-old Stephanie, who discusses the relationship between the towers problems and thebinomial expansion, including how the towers answers can be found in Pascal’sTriangle.

The six chapters of the book’s third part closely examine the mathematical workof the research participants during their high school years. It shows how the studentsbuilt important connections using sophisticated mathematical reasoning. In thesechapters, the story revolves around the students’ proof making, use of representa-tions, acquisition of standard notation, and forging of conceptual connections amongisomorphic problems. Specifically, Chapter 8 shows that as they revisit their repre-sentations and arguments, students refine representations and clarify arguments. InChapter 9, students working in groups on towers problems are seen to find and gen-eralize formulas, using methods including controlling for variables, justification bycases, and induction. Chapter 10 shows how a tenth-grade student’s binary nota-tion helped his group form connections among the pizza and towers problems, thebinomial expansion, and Pascal’s Triangle.

Chapter 11 details how representations are a source for making connections insolutions to pizza and tower problems, resulting in the students mapping the struc-ture of the solution of these problems to Pascal’s Triangle and how their increasinglysophisticated use of representations led to further development of mathematical rea-soning and justification. Chapter 12 discusses how students moved from personalto standard notations in order to express in general form their understanding ofsolutions to the pizza and towers problems and to extend their understanding increating an isomorphism from the numerical results in those problems to Pascal’sTriangle. The chapter also shows how the students’ understanding of extensions ofthe pizza and tower problems led to their understanding of the addition rule forPascal’s Triangle. The final chapter of Part III, Chapter 13, reveals how as highschool seniors, days before graduation, the students used their understanding ofrelationships between the pizza and tower problems and Pascal’s Triangle to solve athird isomorphic problem – the Taxicab Problem. (This problem involves finding thenumber of routes from the starting point – the taxicab stand – to various points on

xvi Introduction

a rectangular grid.) They recognized the isomorphism, used it to make conjecturesabout the new problem, saw the need to prove their conjectures, and provided a con-vincing argument. This chapter concludes by examining some of the extraordinarymathematical accomplishments of the cohort group of students.

The last part of the book, consisting of four chapters, takes stock and looks for-ward. Chapter 14 examines the epistemological growth of the students, viewed fromtheir own perspectives. Students’ reflections on their learning over the years chal-lenges common views about student engagement in learning, and gives insight intohow 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 whenadult college students are asked to justify ideas and make convincing arguments, anunderstanding of mathematical reasoning, proof, and generalization can emerge. InChapter 16, Glass compares the strategies developed by children and older learn-ers for solving the combinatorics problems and discusses the implications for adultlearning.

In closing, Chapter 17 presents the epistemological and methodological contribu-tions of the book. We argue that students must be actively and purposely engaged intheir learning so as to take ownership of and be proud of their accomplishments.Mathematics educators and teachers need to create opportunities for students toengage in ways similar to those described in this book. We have shown that in aprogram of carefully selected tasks, with minimal intervention by educators whopay careful attention to students’ arguments and justifications, students can performmathematically at high levels. In addition to developing mathematical competency,students who participated in the study gained confidence and a sense of empower-ment and were successful in their career choices. They learned to trust their ownmathematical ability and they did not rely on outside authority for validation. Thisconfidence, sense of empowerment and propensity to reason carefully has been car-ried over outside their mathematical work; these students found that the knowledgeand ways of working that they gained through their participation in the longitudinalstudy continues to help them in many other areas of study and employment.

Contributors

John M. FranciscoSecondary Mathematics Education, Department of Teacher Education &Curriculum Studies, University of Massachusetts Amherst, Amherst, MA 01003,USA, [email protected]

Barbara GlassSussex County Community College, Newton, New Jersey, USA,[email protected]

Carolyn A. Maher Graduate School of Education, Rutgers University, NewBrunswick, NJ, USA, [email protected]

Ethel M. Muter5280 Antioch Ridge Drive, Haymarket, VA 20169, USA, [email protected]

Arthur B. Powell Department of Urban Education, Rutgers University, Newark,NJ, USA, [email protected]

Robert Speiser799 E 3800 N, Provo, UT 84604, USA, [email protected]

Manjit K. SranMathematics Department, Monroe Township High school, 1629 Perrineville Road,Monroe Township, NJ 08831, USA, [email protected]; College ofBusiness and Management, DeVry University, 630 U.S. Highway One, NorthBrunswick, NJ 08902, USA, [email protected]

Lynn D. TarlowDepartment of Secondary Education, The City College of the City University ofNew York, New York, NY 10031, USA, [email protected]

Elizabeth B. Uptegrove Department of Mathematical Sciences, Felician College,Rutherford, NJ, USA, [email protected]

Dina YankelewitzDepartment of General Studies, The Richard Stockton College of New Jersey,Pomona, NJ, USA, [email protected]

xvii

Part IIntroduction, Background,

and Methodology

Chapter 1The 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 retrievaland modification of representations of existing ideas. The representations that alearner builds for a mathematical idea or procedure can take different forms – phys-ical objects or actions on objects, words, and symbols, for example. As the learner’sexperience increases, old representations become elaborated, extended, and linkedto 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 opento extension and generalization. The connections that the learner makes when ana-lyzing and developing solutions to these problems provide further opportunity forgrowth in knowledge. Students are encouraged to revisit earlier problems becauserequirements to justify and generalize solutions can help students to see underlyingmathematical structure. It is a widely accepted view that when learners under-stand the fundamental structure of a subject, the gap between “elementary” and“advanced” knowledge is reduced (Bruner, 1960). There is increasing evidence thatlearners, under certain conditions, can build meaningful, mathematical relationshipsand understand the structure of mathematical problems at an early age. For exam-ple, a study of Norwegian children indicated that even as young as Grade 3, learnersare 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 commu-nicate their ideas. Sfard (2001) suggests that students learn to think mathematically


3C.A. Maher et al. (eds.), Combinatorics and Reasoning, MathematicsEducation Library 47, DOI 10.1007/978-0-387-98132-1_1,C© Springer Science+Business Media, LLC 2010

4 C.A. Maher

by participating in discourse about ideas – arguing, asking questions, and antici-pating feedback. We have emphasized that justifying ideas in problem solving is anessential 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 represen-tations public, they have an opportunity to talk further about them, compare them,and later revisit them. Similarities and differences in ideas naturally emerge. Whenlearners try to convince others that their answers are correct, they can reorganizeand reformulate their representations so as to make convincing arguments. In sum-mary, students learn mathematics by engaging in the process of building their ownpersonal representations, communicating them as ideas, and then providing supportfor those ideas by reorganizing and restructuring representations. Our view is thatthis process is a necessary prerequisite both for developing the idea of mathematicalproof and for making suitable connections between problems of equivalent structureby building isomorphisms.

In this book, we discuss how a group of students developed new and increasingsophisticated mathematical ideas by revisiting, reviewing, extending, and connect-ing old ideas that they had begun developing in first grade. They developed andmodified representations that became increasingly elaborated and extended. Theyparticipated in serious mathematical discourse. And ultimately they built a strongand durable understanding of the solutions to a set of mathematical tasks. Our lon-gitudinal work is important because it reveals the processes that these learners usedto 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 dur-ing a time when behaviorism mainly governed mathematics instruction. It was atime before the reform movement in the United States emphasizing conceptualunderstanding had made its entry. The K-8 Harding Elementary School in theworking-class community of Kenilworth, New Jersey, was typical of others at thattime. Half-hour sessions were devoted to mathematics, and mathematics instructionwas mainly rote. The rule was drill and practice for carrying out memorized proce-dures. For the most part, even the brightest students from the school did not excelwhen they moved on to high school mathematics classes, only doing average work.Most members of the community and most teachers had rather low expectations forstudent advancement.

But Fred Rica, principal of the Harding Elementary School, had higher expecta-tions for the students in his school. Formerly an elementary grade classroom teacherin Kenilworth, Fred Rica knew his staff and students well. Like other concerned edu-cators, he knew when the system was not serving its student population. He turnedto Rutgers University for help with instruction, first in language and literacy andthen in mathematics. It was shortly after this professional development work thatFred Rica and Carolyn Maher created a partnership between the Kenilworth Public

1 The Longitudinal Study 5

Schools and Rutgers University. It should be noted that the Rutgers–Kenilworthpartnership, with its focus on students building meaning of mathematical ideas andworking collaboratively with each other, began long before the National Council ofTeachers of Mathematics published its reform standards.

Initially, the project began as a teacher development intervention in mathemat-ics. The Rutgers University team of researchers and graduate students worked for3 years to help teachers build an understanding of the mathematics they wereexpected to teach and to learn to be attentive to the developing understanding oftheir students. (See Davis & Maher, 1993; O’Brien, 1994, for a detailed study of theteacher development project.)

The project could not have survived the early years without the full support andactive participation of the Kenilworth school administration. In particular, principalRica actively participated in the teacher-training sessions, encouraged teachers tobecome involved, and made sure that students who were involved in the study wereavailable to the researchers. Original financial support for the partnership came fromthe Kenilworth school district and through volunteer efforts of the Rutgers team. TheKenilworth school district continued to fund the study for several years as a compo-nent of its mathematics teacher development mission. The Rutgers research groupreceived outside funding for the research from two National Science Foundationgrants. The first grant awarded to Principal Investigators Robert B. Davis andCarolyn A. Maher was when the students were in Grade 4; the second grantawarded to Principal Investigator Carolyn A. Maher was when students were in highschool.

1.2.1 Teacher Development Component

It is not surprising that the teachers at the Harding Elementary School were notprepared to teach mathematics with understanding. What is surprising was theexpectation of principal Fred Rica that the teachers were capable, with some profes-sional development and classroom support, of understanding the mathematics theywere expected to teach. In fact, this view was remarkable for its time.

The teacher development team was made up of mathematics education doctoralstudents who had considerable experience in schools; its first members were AliceS. Alston and Judith H. Landis. The team worked closely with Fred Rica andhis teachers to establish a program of activities that involved not only videotapedteacher workshops and classroom sessions, but also study of those workshops andsessions. The Rutgers team worked directly with students and with their teachers,first observing classroom sessions and later collaborating with the teachers in thedesign and implementation of lessons. Alice Alston also worked in the classroomsalongside the teachers.

Principal Rica obtained school funding to support teachers’ summer work torevise the existing curriculum. Two years of summer professional developmentassisted by John O’Brien and Alice Alston resulted in a movement from a “drilland kill” approach to one in which students’ building of mathematical understanding

6 C.A. Maher

was central. Curriculum revisions included the use of more engaging and thoughtfullessons for the students and the introduction of manipulatives that allowed studentsto build models of their solutions.

Some Kenilworth teachers who participated in the teacher development programsalso became involved in classroom action research. As teachers were introducedto new resources and tools, they developed new units and piloted them during theschool year. Through course work opportunities at Rutgers, some teachers studiedthe 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 couldlearn with minimal intervention from teachers. Classrooms were organized so thatchildren might work together and collaborate on problem tasks. Children wereencouraged to use each other as resources in their investigations, to construct mod-els of solutions with available tools, and to revisit tasks and discuss their strategiesand solutions. An important observation during the first 3 years was that studentsproduced arguments that took on a variety of forms of reasoning to support theirsolutions to the problems. By Grade 4, it became increasingly clear to researchersthat students’ reasoning, in a natural way, took the form of proof. Children begantheir investigations by searching for patterns, organizing solutions, searching forcompleteness, deriving strategies for keeping track and checking, and then reorga-nizing justifications into arguments that were proof-like in structure. Using eachother as resources, children freely shared ideas, questioned each other, argued aboutthe reasonableness of ideas, and became comfortable in sharing and communicatingwith each other.

What encouraged both the school staff and the university collaborators was theenthusiastic feedback from students. The children enjoyed talking about their ideas;they engaged with each other with energy and enthusiasm, becoming increasinglymore comfortable making their ideas public. Their way of working underscored ademand for sense making, which then evolved as a cultural norm.

This book explores student work for one of the mathematics strands of the longi-tudinal study: counting and combinatorics. It investigates how students’ reasoningevolved over the course of the longitudinal study that continued from elementaryand high school years to college.

1.3 Longitudinal Study: Grades 1–3

In order to study the effectiveness of the intervention, the Rutgers team decidedto follow a class of students throughout their elementary grades as they workedon mathematical investigations that were not part of the school curriculum. Thestudy began with a class of 18 first-grade students from the Harding School. Thesechildren, randomly assigned to one of three first grades, became the initial focus

1 The Longitudinal Study 7

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

1.4 Longitudinal Study, Grades 4–8

After Grade 3, the students were distributed among different classrooms, accord-ing to school policy. However, the principal worked with Rutgers researchers tofacilitate maintenance of a focus group of 12 students for research purposes. Whenfamilies moved and new families entered the district, the composition of the focusgroup changed, but an attempt was made to maintain a group of comparable back-ground 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 stu-dents joined. During middle school, the school arranged for the cohort group tocontinue working with researchers during school hours, 4–6 times a year in two90-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 becamepart of a regional system. The community joined forces to protest the merger andsucceeded after 1 year. Hence, the first year of high school (ninth grade) proveddisruptive for the students, although some math problem-solving sessions wereconducted with small groups of students during that year in local homes, usu-ally on Saturdays. After Kenilworth de-regionalized and the students returned toKenilworth for the remaining 3 years of high school, groups of students resumedparticipation in the longitudinal study in informal, after-school sessions that wereheld during the year, usually on Friday. While students no longer met withresearchers during regular class hours, 14 students (some from the original groupof first graders and others who had joined the study at various times during middleschool and high school) made time in their schedules to meet after school about4–6 times a year for problem-solving sessions that lasted 1–2 h or longer. Thisgroup included ten students who had been with the study since Grade 1, two stu-dents 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 wereaccepted – a remarkable achievement for the district. However, not all studentsattended Rutgers; they attended a variety of universities, public and private; besides

8 C.A. Maher

Rutgers, these included Cornell University, Kean University, St. John’s University,and the University of Pennsylvania. Majors included accounting, American stud-ies, 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 dur-ing and after college. They do not generally work on problems (although sometimesold problems are revisited), but they talk about how being in the study has affectedthem, their academic careers, and their future plans.

In the next chapter, we detail how the study was conducted and we discussselected problems that formed the cornerstone of the student investigations overthe years.

Chapter 2Methodology

Carolyn A. Maher and Elizabeth B. Uptegrove

2.1 Introduction

In this chapter, we discuss how data were collected and analyzed, and we brieflydescribe some results, which will be more fully explored in later chapters. Wesummarize student work on fundamental problems and note how this work led toexceptional growth in the students’ mathematical understanding.

Researchers (professors at the Rutgers University Graduate School of Educationand their students) conducted all problem-solving sessions with the students; thesessions were always videotaped with one or more cameras. Researchers observed,described, and coded the videotape data, and they kept written and electronic filesof the emerging theoretic, analytic, and interpretative ideas about the students’mathematical behaviors. Researchers paid careful attention to children’s use ofinscriptions, the connections they made between and among codes, and their emerg-ing and extended ideas and ways of reasoning. Critical events in children’s reasoningwere flagged and transcribed and transcripts were coded according to the researchquestions. The connected series of events that formed a trace led to the emergenceof a narrative (Maher & Martino, 1996a; Powell, Francisco, & Maher, 2003).

The videotapes, researcher notes, and student notes did not capture every interac-tion 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 cameto light some time later in a different situation. Therefore, although we can makeinferences about what is observed, we cannot assume that a student who is quietdoes not understand.

By videotaping children as they worked together on mathematical tasks over longperiods of time, we were able to trace the origin and development of their mathe-matical ideas. We observed what children said to one another and showed to oneanother. We used videotapes and transcripts to study the meanings that children gave



10 C.A. Maher and E.B. Uptegrove

to mathematical situations and to note the different representations they made pub-lic. A detailed analysis of data made it possible to trace the origin and evolution ofchildren’s arguments. Our data indicate how children expressed their ideas throughspoken and written language, through the physical models they built, through thedrawings and diagrams they made, and through the mathematical notations theyinvented.

2.2 Theoretical Perspectives

Guiding our work is the view that children come to mathematical investigationswith theories they can modify and refine. We observe them do so in settings thatcombine personal exploration and suitable social interaction. The theories we con-sider can include criteria to decide (1) what, at some given moment, needs to beinvestigated, (2) how to conduct such an investigation, (3) what key features needto be explored in detail, (4) when useful progress has been made, and, given suchprogress, (5) if further investigation might be needed. We have found that theoriesof this kind often empower striking and effective ways for children to work concep-tually with mathematical ideas, often using concrete objects as specific anchors fortheir thinking.

2.3 Selected Problems

Mathematics arose from the need to count, measure, and calculate, but the disciplineevolved to include abstraction, logical reasoning, and the search for and analysis ofpatterns. Good mathematical problems are therefore those which give rise to theneed for abstraction, systematization, and pattern recognition. A focus of the studywas therefore to select problems that would give rise to these needs.

Another focus of the longitudinal study was on doing problems that were not partof the regular curriculum, because it was important for the students to come to theproblems fresh, without pre-taught algorithms. A major strand of the longitudinalstudy therefore consisted of problems in combinatorics, because in working on theseproblems, students can find the need to organize their work systematically, look forpatterns, and generalize their findings; also, counting problems were at the time out-side the regular elementary school curriculum and therefore unfamiliar to students.In addition, these problems lend themselves to the use of multiple personal repre-sentations that can be shared. Freudenthal (1991) cites the study of combinatoricsas “a most important matter for reinvention” (p. 53), specifically because combi-natorics can be learned through paradigmatic examples and because problems incombinatorics give rise to the need for convincing proof, including mathematicalinduction.

Another purpose of the longitudinal study was to provide an environment inwhich certain socio-mathematical norms could be established to elicit in children

2 Methodology 11

sense making, argumentation, and justification in mathematics. As Yackel and Cobb(1996) and Cobb, Wood, Yackel, and McNeal (1992) suggest, an appropriate socialcontext must be created to encourage students to try to convince others of the truthof the mathematical ideas that they build. The longitudinal study was structuredto investigate and track the nature of the schemes that students developed and themethods that the students used to build and retrieve representations to solve math-ematical tasks. In addition, the study attempted to trace how students shared ideasand how these ideas were adapted and assimilated by other students.

Appendix A describes all the combinatorics problems that students worked onover the years. We summarize here some example problems, along with briefaccounts of strategies and representations used by students and forms of reasoningthat developed.

2.3.1 Shirts and Jeans

Students worked on the shirts and jeans problem at the end of second grade andagain 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 anda pair of white jeans. How many different outfits can he make?

During second grade, most students drew pictures of outfits; some drew linesbetween shirts and jeans, and others made lists of outfits. Notational choicesinfluenced the way they reasoned about the data. For example, Stephanie used “blue-white” to stand for the white shirt/blue jeans outfit, and also for the blue shirt/whitejeans outfit. Contextual issues also played a role in the problem solving. For exam-ple, Dana discarded the white jeans/yellow shirt outfit on grounds that the resultingoutfit did not match and was thus not fashionable. That different students got differ-ent answers was not problematic for the children; in second grade, students seemedcomfortable with the notion that answers varied between three and seven outfits.They willingly shared their interpretations and strategies and talked to each otherabout their findings. In third grade, when the children were again presented withthis problem, they did not remember how they had solved the problem earlier, nordid they remember their earlier answers. Of particular interest is that evidence of fur-ther elaboration of earlier strategies emerged. Students used and built on strategiesof their second-grade partners. For example, Stephanie indicated different outfits bydrawing lines between drawings of shirts and jeans, as Dana had done in secondgrade.

By third grade, techniques for checking and for keeping track, such as controllingfor variables, were complete. Earlier ideas and strategies were refined to producecomplete, elegant solutions rather quickly. Students built on their heuristics to solvemore 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 stu-dents built on earlier ideas and, without intervention or approval from researchers,


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

2.3.2 Towers

Early in third grade (1990), students were given the four-tall tower problem for thefirst time:

Your group has two colors of Unifix cubes. Work together and make as many different towersfour cubes tall as is possible when selecting from two colors. See if you and your partnercan plan a good way to find all the towers four cubes tall.

The definition of a tower is an ordered sequence of Unifix cubes, snappedtogether. 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 iftheir colors match, block by block, from top to bottom. Unifix cubes are interlockingcubes 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 infifth grade (1992), they revisited the four-tall version. In 10th and 11th grades, theywere asked to provide a justification for the n-tall tower problem. Students discussedvariations and generalizations of the solution and they used their organization oftowers by cases and knowledge of the binomial expansion to build an understandingof how Pascal’s triangle grows.

Their work on the towers problems also illustrates how their representationschanged 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 redand yellow cubes. In some cases, a more general code emerged; some studentswould use X and O or 0 and 1 to indicate any two colors. More details on theseemerging 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 studentsreasoned with an isomorphic problem, the researchers introduced the set of pizzaproblems. When the students were first given the problem in fifth grade, they inter-preted the task as allowing different toppings on each half of the pizza, an alternativethat they knew that was available in some pizza restaurants. In response to their inter-est in counting the varieties allowing toppings on half a pizza, the researcher askedthem to solve it with only two toppings available. This pizza with halves problem isas follows:

Kenilworth Pizza has asked up to help them design a form to keep track of certain pizzasales. Their standard plain pizza contains cheese. On this cheese pizza, one or two toppingscould be added to either half of the plain pie or the whole pie. How many choices do cus-tomers have if they could choose from two different toppings (sausage and pepperoni) that

2 Methodology 13

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

The strategy that the students developed for the solution established the heuristicthat was applied later when there were five toppings available, again, allowing someor 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 solvedsuccessfully:

The local pizza shop offers a plain cheese pizza. On this cheese pizza, you can place up tofive different toppings. How many pizzas is it possible to make?

Pizza with Halves was the first of several variations of the pizza problem that thestudents worked on over the years. It illustrates a basic philosophy of the study – wedid not start students off with easier problems and then progress to the more diffi-cult ones. Instead, students began with the more difficult versions of the problems,which required them to tackle several challenges at once – organization (makingsure no pizzas were repeated and none were omitted), notation (how to distinguishbetween pepperoni and peppers, for example), and forming a valid argument – howto convince the researchers (and themselves) that they had the right answer.

Looking at students’ answers to the pizza problems over the years, we see growthin organization and in representations. At first, students drew fairly accurate rendi-tions of pizzas; they drew circles to indicate pizzas, and inside those circles werewavy lines to indicate sausages and smaller circles to indicate pepperoni, for exam-ple. When they had to answer a question involving half pizzas, they drew linesdown the middle of their pizza circles to show both halves, and they listed allthe pizzas using full words (for example, “whole plain, half sausage half plain”).Eventually, they turned to codes, starting with single letters or combinations of let-ters (to distinguish between peppers and pepperoni, for example) and then moving tomore abstract symbols such as 0s and 1s. These representations and organizationalstrategies 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 possiblepizzas grows as new toppings become available. In an extraordinary session last-ing over 2 h one evening in 1999, students generated a slightly nonstandard butmathematically correct equation for Pascal’s identity using standard combinatorialnotation:

(NX

)+

(NX + 1

)=

(N + 1X + 1

)

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


2.3.4 Taxicab

The group that had generated Pascal’s identity was introduced to the taxicabproblem in 12th grade (2000):

All trips originate at the taxi stand, in the upper-left corner of a grid. The problem is tofind the shortest route to three specific points on the grid and to determine the number ofshortest routes to each point.

Their work on this problem is discussed in greater detail in Chapter 13. It is inter-esting to note that by this time, the students, without prompting, solved the generalproblem, 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 con-nection 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 ofeliciting justifications from each other. Their pursuit of explanations that made senseand 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 topicsin combinatorics or other areas of mathematics. Instead, the aim was to establisha culture where the correctness of an answer came from the sensemaking of thestudents, rather than from the authority of the researcher. We asked students ques-tions about what was convincing, what made sense, and how they developed theiranswers. In justifying their answers, students usually exceeded our expectations.We were impressed by the seriousness with which students approached the prob-lems and the collegiality of their work, as well as by the forms of reasoning theydeveloped. In the early years of the study, children began to use inductive reason-ing, to organize work by cases, and to think about justification through contradiction.By middle school, these forms of reasoning were more sharply defined, and otherforms of reasoning emerged, such as controlling for variables. In high school, stu-dents began the process of building isomorphisms, using their own notation as wellas standard notation to describe how some problems were related to each other andultimately to Pascal’s triangle.

In the following chapters, we provide details on the specific problems, the spe-cific strategies and representations used by the students, and the specific resultsthey generated. In the next chapter, we discuss the students’ earliest work oncombinatorics problems, the second- and third-grade work on shirts and jeans.

Part IIFoundations of Proof Building (1989–1996)

Chapter 3Representations as Tools for BuildingArguments

Carolyn A. Maher and Dina Yankelewitz

Date and Grade: May 30, 1990 and October 11, 1990; Grades 2 and 3Tasks: Shirts and JeansParticipants: Dana, Jaime, Michael, and StephanieResearchers: 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 anda pair of white jeans. How many different outfits can he make? Convince us that you havethem all.

In examining their problem solving, we focus on the children’s early use of per-sonal representations. When introducing mathematics to children, it is important toinvite them to use their personal representations to express their ideas and ways ofreasoning (NCTM, 2000). These representations are the basic elements that chil-dren draw upon to express their ideas as they begin to engage in more abstractand logical reasoning. Children’s representations and how they are connected andrelated are foundational building blocks toward more sophisticated processes thatlead to the creation of new mathematical ideas. When children are provided withopportunities to “reinvent” mathematics, they are in a better position later on torecognize their own need for abstraction, generalization, and logical reasoning (seeChapter 13).



18 C.A. Maher and D. Yankelewitz

3.2 Representation as a Tool for Problem Solving

Davis (1984) has written extensively about the role of representation in mathemat-ical thinking. According to Davis, representations are mental models that allow forthe 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 explorewhat these intangible representations are, we are only approximating their truenature.

Davis stated that doing mathematics involves a series of steps, similar to thoseof a computer executing a program, through which the student must cycle one ormore times. First, in an attempt to make sense of the problem, the student buildsa representation for the input data. Second, the student searches his memory forknowledge that will assist in solving the problem. Finally, the student maps the datarepresentation with the knowledge representation. When the mapping seems accu-rate enough to tackle the problem at hand, the student uses techniques associatedwith the knowledge representation to solve the problem.

Students use representations that they build to make sense of and attribute mean-ing to the mathematics that they are doing. They use mathematical tools, which,according to Davis and Maher (1997) include mathematical notation, spoken andwritten language, physical models, drawings, and diagrams.

According to Maher and Martino (1996a), students who are encouraged to buildand use multiple representations as they work on problems become sense-makersand active members of the mathematical community. The use of different toolsto build and express ideas allows students to make connections between differ-ent representations and understandings and to better understand the mathematicsthat they are learning. In addition, when students build and express multiple exter-nal representations, this allows observers (such as teachers, researchers, and fellowclassmates) to better understand the students’ ideas. Using representations to makesense of problems and using representations to communicate ideas are therefore thebuilding 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 toprovide the students with mathematical problems for which they had no algorithmsand which would afford them opportunities to find patterns, be systematic, and gen-eralize findings. Combinatorics problems were well suited to these goals. In thesections that follow, we will consider the specific mathematical ideas, fundamentalboth to combinatorics in particular and to mathematics in general, that are elicitedby 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.

3 Representations as Tools for Building Arguments 19

Fig. 3.1 A diagram and anorganized list for displayingthe shirts and jeans solution

In solving this problem, students may abstract the mathematics underlying thereal-world situation; they may come to realize that the number of combinations ofshirts and jeans is equivalent to the product of the number of shirts and the num-ber of jeans. This type of multiplication is generally the most difficult for childrento model, comprehend, and apply (Cathcart, Pothier, Vance, & Bezuk, 2006). Thisproblem also introduces the need for notation or symbols to represent the real-worlditems described in the problem. So this problem creates a need for a bridge betweenthe real-world situation presented and the mathematical ideas that will provide asolution. The problem also provides students with a chance to realize that an organi-zation of the facts (by means of a diagram or an organized list) can help them to finda solution; this need for structure is fundamental to mathematics. The problem alsorequires students to think about how to justify their solution to others and convinceothers that they have found all the combinations. It has the potential to give rise tothe 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 arelisted on the left and the jean colors are on the right. The blue shirt can be combinedwith 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 wellas for the yellow shirt. Therefore, there are 2 × 3 or 6 possible combinations. Allpossible combinations are accounted for, since any other attempted combinationswill be duplicates of the ones listed above. For example, there cannot be a thirdoutfit 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 thirdgrades. 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


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

Maher and Martino (1996). In addition, selections of the video were includedin the Private Universe Project in Mathematics (Harvard-Smithsonian Center forAstrophysics, 2000), and the mathematical thinking and representations of thesestudents are discussed there.

In the second grade, all three students drew pictures of shirts and jeans to rep-resent the items in the problem, and they used the pictures in their attempts tofind different combinations of the shirts and jeans. Stephanie drew three shirts andlabeled them “w” for white, “y” for yellow, and “b” for blue. She drew two pairsof jeans, similarly labeled “b” and “w.” She then began to make a list, writing theletter symbolizing the shirt directly above the letter that represented the jeans thattogether comprised an outfit. She then numbered the combinations that she found.However, when recording the fifth combination, she erased the “w” that she initiallywrote to make a combination of a white shirt and blue jeans, and, in its place, wrotea “y” to show the combination of a yellow shirt and blue jeans (see Fig. 3.3). Shethen told the researcher that she had found five combinations and she was convincedthat she had found them all.

Fig. 3.3 Stephanie’s grade 2written work


Fig. 3.4 Dana’s grade 2written work

Dana, early on, expressed verbally her understanding of the structure of the solu-tion when she indicated that each of the three shirts could be combined with eachpair 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 subsequentproblem-solving steps that she took that she had built a scheme that closely matchedthe problem solution. As Martino (1992) notes, “From her explanation it can beinferred that Dana possessed a key strategy for exhausting all possible combina-tions” (p. 48). Dana used a strategy of connecting her representations of shirts andjeans 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 yellowshirt 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, soDana also arrived at a solution of five combinations. However, Stephanie’s solutionlacked the combination of a yellow shirt and blue jeans, while Dana’s was missingthe combination of a yellow shirt and white jeans (the combination that did notmatch, according to Dana). We can conclude that she was aware of all possibleoutfits but her sense of fashion resulted in her rejecting the yellow shirt and whitejeans.

Michael’s strategy differed significantly from that of his classmates. He drewdiagrams of the different color shirts and jeans, but said that he had arrived at threecombinations: a white shirt with white jeans, a blue shirt with blue jeans, and ayellow shirt with yellow jeans (see Fig. 3.5).

Although Stephanie and Dana pointed out that the shirts and jeans did not haveto 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. Stephanieand Dana again worked together, and they immediately began to draw diagrams torepresent each item in the problem. Stephanie then suggested that they draw lines to


Fig. 3.5 Michael’s grade 2written work

show each combination, and they arrived at a total of six combinations. When ques-tioned about why they drew lines to show the combinations, Stephanie explainedthat that was to ensure that they do not make any duplicate combinations. Figure 3.6shows 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 shirtsand jeans outfits as indicated in Fig. 3.7.

Fig. 3.6 Stephanie’s grade 3written work


Fig. 3.7 Dana’s grade 3written work

Fig. 3.8 Michael’s grade 3written work

Michael worked on this task with another student, Jaime. This time, Michaelused the strategy of connecting lines between the shirts and jeans to represent thepossible outfits, but, unlike Stephanie and Dana, he drew lines between the wordsin the problem, rather than between drawings of the shirts and jeans as indicated inFig. 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 alsoarrived at a solution of six combinations. He used a strategy similar to that usedby Stephanie in the second grade: he listed the combinations by writing the letterrepresenting 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 ofoutfits, although the way they solved the problem gave evidence that they were


building schemes that could account for some or all of the outfits. All three studentsdrew pictures in order to model the shirts and jeans, and all three used notation (firstletter abbreviations) to indicate the colors of the shirts and jeans. Their represen-tations of the problem included letters and line diagrams. It is important to notethat Dana showed evidence of building the scheme for controlling for variables inthe second grade. If not for her sense of style, Dana would have arrived at the cor-rect answer of six outfits in the second grade. In the third grade, all three studentsarrived at a correct solution and none recalled that their correct solution was differentfrom 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 thatthey accounted for all possibilities and also that they had not counted any combina-tion more than once. Stephanie also used a system of counting that enabled her tokeep track of her outfits. The children’s representations show that they are aware ofthe 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 stu-dents to bring forth personal representations and it offers opportunities for sensemaking, so that students can begin to discuss how they arrived at their solutions andhow they know their solutions are correct. Also, different ways of reasoning can beexplored while students can learn how to formulate organizational schemes that canhelp them solve other problems in mathematics. In addition, the real-world settingof the problem shows the direct connection between the mathematical ideas and theworld that students know. It also gives students a chance to think about the influencethat real-world considerations have on mathematics, as can be seen from Dana’ssense of fashion and her insistence that one combination of shirts and jeans cannotconstitute an outfit. Thinking about real-world considerations (which are impor-tant and which can be ignored) is a necessary step on the journey to mathematicalunderstanding and abstraction.

It is interesting to note the variety of strategies and approaches that were used inthis problem. For example, in second grade, the three children used three differentstrategies to solve the problem. Although none of the strategies produced a correctanswer, 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 studentsall responded that they found six outfits.

For the researchers, the goal of the sessions was not for the children to give thecorrect answer; we were confident that they would eventually succeed. Our primarygoal was to engage the children in thoughtful problem solving that could triggergrowth in schema. We wanted solutions to be meaningfully constructed by the stu-dents. As the following chapters will reveal, a pattern of working with the studentsin the longitudinal study was to revisit problems and solutions in cycles so that ear-lier ideas could be built upon and new representations could be revealed. As we


studied the progress made by the students over the years, we gained insight intotheir developing ideas and ways of reasoning. We observed the heuristics they usedand the schemes that were later retrieved and modified.

Over the months, there is evidence of durable learning. Also, there is evidencethat children learned from each other as revealed in elements of one another’s strate-gies 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) differedsignificantly from the way that Stephanie and Dana used lines and notation. Thissuggests the importance of encouraging the use of students’ personal representationsin building solutions.

These episodes also demonstrate some benefits of group work. The contributionof each student to the cumulative body of knowledge enables students to arrive attheir own solutions while simultaneously benefiting from the knowledge of others.

As the data indicate, the children built durable schemes to solve the mathemat-ical tasks that they were given. They used personal representations, as was seen inDana’s and Stephanie’s second-grade work, to make meaning of the problem situa-tion 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 byindividual students were used to effectively communicate to others the schemesupon which they had been built. This can be inferred from the students’ work inthe third grade, when these schemes and representations were assimilated into thesolution strategies and representations of classmates.

In this chapter, we have seen that, at an early age and at the beginning of theirinvestigations, students worked to make sense of the mathematics; they readilycommunicated their ideas and built on others’ ideas in forming their solutions. InChapter 4 , we follow two of these students – Dana and Stephanie – along with athird student, Milin, as they work on more problems designed to help them explorefundamental ideas in algebra and combinatorics.

Chapter 4Towers: Schemes, Strategies, and Arguments

Carolyn A. Maher, Manjit K. Sran, and Dina Yankelewitz

Date and Grade: 1990–1992; Grades 3 and 4Tasks: TowersParticipants: Dana, Jeff, Michelle I., Milin, and StephanieResearchers: Carolyn A. Maher and Amy M. Martino

4.1 Introduction

In the previous chapter, we examined the representations, strategies, andproblem-solving schemes used by four second- and third-grade students to buildtheir solution to the shirts and jean problem (which was to determine how manyoutfits could be formed from three different shirts and two different pairs of jeansand to provide a convincing argument of the solution). In their effort to make senseof the components of the problem and to monitor their work, the students developedvarious 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 studybuild on those representations and strategies in their work on some towers prob-lems. (A towers problem involves determining how many towers can be built of agiven height from a specified number of colors of Unifix cubes, small plastic cubesthat can be stacked together. Because Unifix cubes have a vertical orientation – theyhave a top and a bottom – so do towers. An n-tall tower is one that was built from nUnifix cubes. Appendix A provides an analysis of solutions to the towers problems.)

In this chapter, we examine the representations and strategies such as looking forpatterns, guess and check, and controlling for variables that were used by studentsas they worked on the towers task. We trace students’ use of heuristics and ways ofreasoning that were exhibited in their earlier problem solving with shirts and jeans.Finally, we trace the growth in students’ mathematical reasoning as their argumentsand solutions took on proof-like forms.



28 C.A. Maher et al.

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

4.2 Stephanie

We discuss here Stephanie’s emerging strategies as she worked on the towersproblem 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 combinationsof 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 anddiscussed in Martino (1992). We present here a discussion of Stephanie’s strategiesas she worked on the task in the third grade (see Fig. 4.1).

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

4 Towers: Schemes, Strategies, and Arguments 29

RED BLUE

Fig. 4.2 Stephanie’s four-tallopposites

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 towerof the same height with the second color in the corresponding position. Her first fivetowers included two sets of “opposites,” such as the tower with four blue cubes andits opposite, the tower with four red cubes (see Fig. 4.2).

Dana also initially built ten towers, including two pairs of opposites. Then thetwo girls decided to combine efforts, and Stephanie took each of Dana’s towers inturn and checked it against her own to see if it was a duplicate.

STEPHANIE: Everything we make, we have to check. Everything we make. . .Let’s make a deal. Everything we make, we have to check.

DANA: All right. I’ll always make it and you’ll always check it.STEPHANIE: 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. Stephaniesuggested that Dana build new towers while she checked each new tower againstthe existing ones to ensure that it was not a duplicate. They finally eliminated allduplicates, and after attempting to find more combinations but not succeeding, theyconcluded that there were only 16 combinations, since they had checked many timesand could not find new towers.

This activity was marked by a number of emerging strategies. First, Stephanieand Dana used trial and error to find as many towers as they could. In addition, boththought of finding a tower and its opposite in an attempt to generate as many towersas possible, but neither used this strategy extensively or consistently. Further, thetwo decided to compare results and eliminate duplicates, and ultimately used thisstrategy of elimination to find the remaining tower combinations.

Stephanie and Dana’s attempt to prevent duplication of combinations as theyworked on the towers task is reminiscent of their strategy for solving the shirts andjeans 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 theycounted each combination of clothing once and only once. They explained to theresearcher:


RESEARCHER: What are these lines that you drew? You drew lines between theshirts and the pants.

STEPHANIE: So that we could make sure; so instead of we didn’t do thatagain and say, “Oh, that would be seven, eight, nine, 10.” Wejust drew lines so that we can count our lines and say, “Oh wecan’t do that again, we can’t do that again.”

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

RESEARCHER: How could you be sure that you haven’t made any of them twiceor that one of you got them all? Is there a way that you could besure?

STEPHANIE: Well, there is a way. You could take one, like say we could takethis 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 – thatdon’t match, put back; and the ones that do match, eliminate.

Stephanie and Dana were then asked to predict how many three-tall towers theycould 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 four-tall towers. Upon experimentation, they found that removing one cube from each oftheir four-tall towers resulted in duplicates, or “pairs,” and they concluded that therewere only eight combinations of three-tall towers (see Fig. 4.3).

During an interview the next day, Stephanie explained why there were fewerthree-tall towers than four-tall towers.

RESEARCHER: What do you think you learned from what you did?STEPHANIE: Well, we learned that . . . you might think there’d be more

because there are less blocks so there’s more combinations youcan 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 thesame.

RESEARCHER: Oh . . . So then you don’t have more. You have-STEPHANIE: Have less.

RED BLUEFig. 4.3 From four-tall tothree-tall


Stephanie and Dana began to make conjectures and inferences based on theirprevious knowledge; both went on to suggest that there would be more five-talltowers 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 possiblefive-tall towers, selecting from two colors (in this case, yellow and red). Stephanieand Dana began to make towers along with their opposites (see Fig. 4.4); and theychecked their work as they progressed to prevent duplication.

DANA: And then I got another idea.STEPHANIE: Well, tell me it so I can do the opposite.DANA: I’m going to do the red – this, that-STEPHANIE: Show 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 downto create a new tower. Dana called this new tower “cousin” (see Fig. 4.5). They usedthis strategy to find more possible arrangements. After forming as many towers asthey 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 shereferred to as “families.” An example of Dana’s “family” is the elevator pattern con-sisting 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 onlygoes up to five blocks.” Her reasoning indicates an argument by contradiction of the

Fig. 4.4 Stephanie andDana’s group work


RED YELLOW

Fig. 4.5 Dana’s tower andcousin

RED YELLOW

Fig. 4.6 Dana’s family ofone red cube and four yellowcubes

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

RESEARCHER: Are there any other members of this family?DANA: No.RESEARCHER: Why not?DANA: 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, youcan make three. With two [yellow cubes] in between, you can make two. With three [yellowcubes] in between, you can make one. But you can’t make four in between or five in betweenor . . . anything else because you can only use five blocks.


RED YELLOW

Fig. 4.7 Families of five-talltowers with exactly two redcubes

By the close of this whole class activity, Stephanie and Dana had begun to explorean 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 onthe towers task.

4.2.3 Stephanie: Grade 4, Interviews

Stephanie’s work on problems involving towers continued throughout the fourthand 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 mathematicalproof is further documented by Martino and Maher (1999) and Maher and Speiser(1997b). Data from these episodes are presented here with attention to the emergentstrategies that Stephanie used while working on the tower tasks.

In an interview following the class session described above, Stephanie, usingred and blue cubes, extended her family organizations of opposites, cousins, andelevators, to include a new organization, the “staircase” pattern (see Fig. 4.8). Shediscovered that introducing additional patterns sometimes resulted in duplicate tow-ers that needed to be eliminated by checking. She said, “Yeah, we kept – we keptfinding different patterns, but we didn’t check it with the other patterns.”

RED BLUE

Fig. 4.8 Stephanie’s use ofdifferent patterns resulting induplicates


The interviewer asked Stephanie if there was a way she could be sure of howmany towers of a specific type could be made.

STEPHANIE: I guess . . . a very lucky guess.RESEARCHER: Is there anything else possible for towers with exactly one blue?STEPHANIE: No.RESEARCHER: Why are you convinced?STEPHANIE: Because if there are towers of five, you can only build that many

[with one blue cube]. You can’t really be convinced for every-thing because there’s no absolute way . . . you can’t go and say“I’m right.”

RESEARCHER: [referring to the set of towers with one blue cube that Stephaniehad shown] Well, this is an absolute way.

STEPHANIE: Yeah, this is one of the absolute ways.RESEARCHER: This absolute way is when you looked at only one blue and I

wonder if you could find absolute ways for looking at maybetwo blues, three blues, or four blues.

STEPHANIE: You could. Yeah, it is possible to have a certain number and getit right.

With this exchange, Stephanie demonstrated that the elevator pattern provided aconvincing 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 twoof a color, could be convincing. She began to consider families of towers as belong-ing to cases that could be justified individually to create the mutually exclusive andexhaustive 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 twocolors. 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, thetop two rows in Fig. 4.9 and the bottom two rows in Fig. 4.10. Notice, also, in bothfigures Stephanie applied her elevator pattern while holding both rows constant.

Fig. 4.9 Towers with top tworows constant


Fig. 4.10 Towers withbottom two rows constant

BLACK WHITE

Fig. 4.11 Stephanie’s global organization

In a subsequent interview, Stephanie shared with her classmates the strategy ofcontrolling for variables, that is, keeping the color of a cube in a particular positionconstant. This method of controlling for variables was useful to her in keeping trackof larger number of towers (Maher & Martino, 1996a).

During an individual interview on March 6, 1992, Stephanie presented a com-plete argument by cases. She was able to produce a global organization for four-talltowers. In her justification, she focused on number of white cubes yielding five cat-egories 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, andtowers 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-talltowers 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 towerusing trial and error and then making an opposite for each tower. Michael and Milin


RED YELLOW

Fig. 4.12 Milin’s cousin andopposite ways of makingpairs

RED YELLOW

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

Fig. 4.14 Class discussion and sharing of solutions

always paired their towers. Milin sometimes used the “opposite” strategy to makea pair and other times he utilized the “cousin” pairing, by inverting a tower. At theconclusion of the group work, the boys found all 32 towers. The strategies used bythe two students on February 6, 1992, included: trial and error, building an oppositetower to complete a pair by switching the color of each cube, building an oppositetower to complete a pair by inverting the original tower, and monitoring work bychecking for duplicates by comparing to previous towers (see Fig. 4.12).

Milin noted that there were three possible combinations of towers in which thered cubes were separated by one yellow cube, two in which the red cubes wereseparated by two yellow cubes, and one in which the red cubes were separated bythree yellow cubes (see Fig. 4.13).

During the sharing session (see Fig. 4.14), the children were attentive as theylistened to the findings and strategies used by their classmates.


4.3.2 Milin: Grade 4, Interviews

In an interview on February 7, the day after the class session, Milin made sets oftowers using the elevator method for moving a cube of one color to each floor ofthe tower and by moving two cubes of one color the same way. He then found theremaining combinations by trial and error, and by grouping towers together withtheir opposites. Although Milin believed that he had found all combinations, he wasonly able to provide a convincing argument for his elevator patterns and his solidtowers (see Fig. 4.15).

Later during this interview, Milin began to consider simpler cases, and he saidthat there were four towers that could be built that were two cubes tall, and twothat could be built that were one cube tall. Milin continued exploring simpler casesafter this interview and brought the cubes home to further explore his idea. Duringthe second interview 2 weeks later (on February 21), Milin reported that there were16 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 werepossible for each (see Fig. 4.16).

RED YELLOW

Fig. 4.15 Milin’s partialorganization by cases andopposites

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


BLACK LIGHT BLUE

Fig. 4.17 Milin’s inductivereasoning with “families”

Then, in a third interview (on March 6), he showed that larger towers could bebuilt from smaller ones. For example, one can build four two-tall towers from thetwo one-tall towers (a blue cube or a black cube) by placing either a blue or a blackcube on the blue cube and then placing either a blue or a black cube on the blackcube. Milin showed that groups of larger towers could be included in the family ofthe smaller tower from which it was built (see Fig. 4.17).

Later during the March 6 interview, Milin suggested that his rule for generatingtaller towers from shorter towers breaks down after five-tall towers. Toward the endof this interview, he retracted this claim, and he suggested that there were 64 possiblecombinations of six-tall towers. When asked if his pattern would hold for towerstaller than five, he said it should, indicating, “We followed the pattern till five. Whycan’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 groupof four children for accounting for all possible towers, three-tall, selecting from twocolors. This group sharing is referred to as “The Gang of Four” (see Fig. 4.18). Itwas conducted so that the children could share their strategies and arguments for

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


building towers of a variety of heights in earlier investigations. A simpler version ofthe problem was chosen deliberately for this session, as the evaluation was intendedto identify the forms of reasoning and methods of justification that the children usedto 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 six-tall towers could be built. Milin answered, “probably 64.” He was asked toexplain why, and he described his inductive rule: multiply the previous answer bytwo.

MILIN: Well, because there was a pattern.RESEARCHER: What’s that?MILIN: You just times them by two.RESEARCHER: Times what by two?MILIN: 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 problembeyond the three-tall case given to the group. He said that there were two one-talltowers, four two-tall towers, and eight three-tall towers. He was asked to re-explainhow he progressed from four to eight towers. In this clip, he noted that a cube ofeach color could be added to the top of the shorter tower to build the taller tower.

RESEARCHER: Why eight? That’s what Jeffrey asked about.MILIN: I know.RESEARCHER: Go ahead. Let Milin persuade Jeff.MILIN: If you do that, you just have to add – for each one of those you

have to add-RESEARCHER: Each one of what? These four?MILIN: Yeah. You have to add one more color for each one.RESEARCHER: Which way are you adding it? Where are you putting that one

more color, Milin?MILIN: No, two more colors for each one. See-RESEARCHER: So this one with red on the bottom and blue on the top.MILIN: You could put another blue or another red.

Later, Milin explained the logic behind the leap from two-tall towers to three-talltowers using inductive reasoning. He was able to demonstrate his doubling rule witheach individual tower. In his own words, he noted that there were two possible cubesto be added to each three-tall tower to make a four-tall tower: “This was for three, soyou could add two for each one of the three.” Milin explained this doubling rule bydrawing two three-tall towers from a two-tall tower by adding a different color cubeon top. He took his first two-tall tower with a blue cube on the bottom floor and ared cube on the top floor and generated two three-tall towers by first adding a redcube 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 actualcubes as he had during his individual interview.


Fig. 4.19 Milin’srepresentation showing hisdoubling rule

Fig. 4.20 Stephanie’srepresentation of an argumentby cases

In this session, Stephanie presented an argument by cases to account for buildingall possible towers, three-tall. She represented the towers in a grid using letters Band R, for blue and red cubes as indicated in Fig. 4.20. Details of the session aredescribed in Maher and Martino (1996b). She showed that there was only one way toform a tower without any blues. Then she showed that there were three combinationsof two red cubes and one blue cube using the staircase pattern.

She then used an argument by contradiction to show that this pattern could beused 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’dhave to have another block on the bottom. But then you have with three blues – well, notwith three blues. I’ll go like this.

Stephanie used the staircase pattern to argue by contradiction that there could notbe a fourth arrangement of two red cubes and one blue cube. What is of interest hereis that Stephanie felt the need to prove that her argument by cases was complete andconvincing, even though no one had challenged her answer. Stephanie continued herargument by cases by describing all the possible combinations of two blue cubes andone red cube. Stephanie’s organization was interesting in that she separated the caseof two blue cubes into sub cases: two adjacent blue cubes and two non-adjacentblue cubes. When her classmates pointed out that these two cases could be groupedinto one broader group, Stephanie insisted on continuing her explanation as she hadoriginally presented it. The entire conversation follows, starting with Stephanie’sdescription of the “all red” tower.

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


STEPHANIE: Blue, red, red; or red, blue, red; or red, red, blue. All right. Youcould put blue, blue, red; you could put red, blue, and blue.

MILIN: You could put blue, red, and blue. You could put . . .

STEPHANIE: Yeah, but that’s not what I am doing. I’m doing it so that they’restuck together.

RESEARCHER: Okay.JEFF: 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’sone with three reds and then . . .

MILIN: Ah, but see – you did the same thing, but there’s the blue.JEFF: 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.STEPHANIE: Well, that’s not how I do it.RESEARCHER: Let’s hear how Steph – we’ll hear that other way; that’s

interesting. Okay, now, so what you’ve done so far is –STEPHANIE: One blue, two blue.RESEARCHER: Okay, no bluesSTEPHANIE: One blue, two blue.RESEARCHER: One blue, and two blues, but Milin just said you don’t have all

two blues, and you said that – why is that?STEPHANIE: All right, so show me another two blues. With them stuck

together, because that’s what I am doing.MILIN: In that case, no.RESEARCHER: Okay, so now what are you doing, Stephanie?MICHELLE: What if you just had two blues and they weren’t stuck together,

you could –STEPHANIE: But that’s what I’m doing. I’m doing the blues stuck together.RESEARCHER: Okay.STEPHANIE: Then we have three blues, which you can only make one of.

Then you want two blues stuck apart – not stuck apart; tookapart.

RESEARCHER: Separated?STEPHANIE: Yeah, separated. And you can go blue, red, blue right here.

Although Stephanie insisted on explaining her method of using two categoriesof towers with two blue cubes during this session, she later indicated (in a writtenassessment) that she understood the arguments of Milin and Michelle. At that time,Stephanie organized her cases as her classmates had suggested, producing a moreelegant proof by cases (Maher & Martino, 1996a).

Toward the end of the session, the students used Milin’s argument by induction asa stepping-stone to generalize the solution to the towers problem. Their progressionof this understanding is documented by Maher and Martino (1997, 2000) and Maher(1998).


Stephanie, during interviews preceding “The Gang of Four,” noticed a relation-ship between the height of towers and the total number produced and conjectureda “doubling rule.” During the “Gang of Four” session, she made reference to herdoubling pattern and offered that there would be 1,024 ten-tall towers that could bebuilt selecting from cubes of two colors. However, during that session, she choseto 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, “GuessMy Tower,” we see Stephanie learn why the doubling rule works as she investigatesMilin’s inductive argument.

4.4 Summary of Strategies and Justifications

Figure 4.21 outlines the strategies, representations, and forms of justification usedby Stephanie and Milin during the five sessions on the towers problems. BothStephanie and Milin began by using trial and error and justifying their solutionempirically. They both progressed to more sophisticated strategies and forms of jus-tification. Stephanie looked for patterns and controlled for variables to eventuallyformulate her justification using cases. Milin considered simpler cases and then rec-ognized the recursive nature of the problem, arriving at his inductive justification.Both Milin and Stephanie arrived at a complete justification of their solution duringThe Gang of Four session. In addition, both students chose not to use the Unifixcubes to represent their towers but instead used notations in a grid (Stephanie) anddrawings (Milin) to represent the different tower combinations.

STEPHANIE MILIN

Strategies Tools Justification Strategies Tools Justification

ClassSession

Trial and error; Opposites; cousins, Staircase

Patterns, partial cases

Trial and error; Opposites; Staircase

Partial cases

Interview 1 Drawings and symbols

Patterns; partial cases; simpler problem

Partial Cases

Interview 2 Controlled for variables

Drawings and symbols

Considered simpler cases

Partial induction

Interview 3 Staircase; Controlled for variables

Drawings and symbols

Emergent cases

Inductive pattern recognition

Emergent induction

Gang of Four

Staircase, Controlled for variables; Pattern recognition

Grid with symbols

Case argument

Inductive pattern recognition

Drawing and symbols

Inductive argument

Pattern recognition Partial cases

Partial cases

Unifix cubesUnifix cubes

Unifix cubes

Unifix cubes

Unifix cubes

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 reasoningby Milin and Stephanie in their justification of their solutions to the towers task. Thestudents built and refined their representations over a period of time in which they


had the chance to reflect upon the task, recognize emergent patterns, and chooseschemes that best matched the representation that they had formed. Stephanie usedsymbols within a matrix to organize the towers by cases; Milin used drawings oftowers to explain how they grew. The call for justification of the three-tall towertask enabled Stephanie and Milin to make public the schemes that they had builtearlier.

There are some similarities in Stephanie’s early use of representations for boththe towers task and shirts and jeans tasks. In the second grade, Stephanie listed theoutfit combinations by using the initials of each color and recording the combina-tions in a vertical format (see Fig. 3.2). When working on the towers task, she againused initials for the colors of cubes using a grid organization to show the differenttowers. Stephanie also used the heuristic of controlling for variables as she orga-nized her tower combinations, a strategy that her partner Dana had used in the shirtsand jeans task.

Milin’s strategies of considering simpler cases and pattern recognition were pow-erful 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 backto reflect on their earlier work to make sense of more complex combinatorial tasksin later grades.

Importantly, these data show the advantage to revisiting tasks, group discussionsabout ideas, and sharing strategies. All of these components play a key role in theformulation and refinement of justifications. Stephanie and Milin, after having hadmultiple opportunities to think about and justify their ideas, presented a compellingargument to classmates during the group evaluation setting. As is evidenced in lateryears, unique aspects of the discussions that continued among this community oflearners further triggered the development of more complex cognitive structures,triggered by the students’ need to produce justifications for combinatorial tasks ofever-increasing complexity.

In Chapter 5, we follow Stephanie and other classmates as they continue to workon understanding Milin’s inductive argument for building towers as they retrieveearlier frames and cognitive structures revealed during the Gang of Four work.

Chapter 5Building an Inductive Argument


Date and Grade: February 26, 1993; Grade 5Task: Guess My TowerParticipants: Bobby (Bobby is called Robert in later chapters), Matt, Michelle

I., Michelle R., Milin, and StephanieResearchers: Carolyn Maher, Alice Alston, and Amy Martino

5.1 Introduction

In the previous chapters, we followed the strategies, schemes, and arguments builtby 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 ofthe inductive method of generating towers. This strategy was originated by Milin,but it was eventually adopted by many other students. We attempt to identify themoments at which individual students gained ownership of the inductive argumentand explained their new understanding to others.

5.2 Early Ideas

In third and fourth grades, the children continued to build powerful strategies andschemes as they worked on the tower problems (see Chapter 4). To support theirsolutions, students followed two different approaches. Stephanie and others madeextensive 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 showhow to generate the number of combinations of towers of any height. Stephanie also




observed this doubling rule (that the number of towers of height n was double thenumber of towers of height n – 1). Stephanie conjectured a doubling rule from hersuccessful 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 whethershe had thought about the problem since the small-group session, she showedthe interviewer a sheet of paper on which she recorded the “doubling method” offinding towers of a specific height. The interviewer introduced the idea of usinga tree diagram to show a recursive pattern that could be used to generate towersand to organize the tower combinations, similar to Milin’s inductive scheme. Theresearcher showed Stephanie the first two levels of the tree diagram and then askedStephanie to extend the tree to include three-tall and four-tall towers. Stephanieresponded by producing a partial extension of the tree organization as indicated inFig. 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 writtenassessment (see Fig. 5.2).

The children worked in pairs on this assessment to provide a convincingjustification of the towers task. Stephanie and Milin were partners and providedindividual written work of their solution. (See Fig. 5.3 for Stephanie’s written workand Fig. 5.4 for the work of Milin.) In her letter, Stephanie gave an elegant argumentby cases to show that she found all the towers, and then used a doubling pattern topredict taller towers, offering a general method. She said, “All you have to [do] isfind 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 gridof towers three-tall using letters “B” and “G” to represent the two colors. Then hepaired the two sets of “opposites” and two sets of “cousins” (see Fig. 5.4). He alsodemonstrated an understanding of the doubling rule in his written work. Notice, too,

Fig. 5.1 Stephanie’s partialextension of the tree diagram

5 Building an Inductive Argument 47

Fig. 5.2 Assessment June 15, 1992

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


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 three-tall 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 October25. 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 usingthe “doubling” method (see Fig. 5.6). It’s interesting that again Stephanie relied on

Fig. 5.5 Assessment October25, 1992


Fig. 5.6 Stephanie’s October25 written assessment

her case argument to justify her solution. (Refer to Maher & Martino, 1996a, forfurther details.)

Milin, in his October 25 letter, again explained his doubling rule and drew alleight 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


5.3 Investigating Inductive Reasoning

Further activities with towers led several students to make advances in understand-ing of inductive reasoning previously introduced by Milin to some classmates. Wediscuss below the traveling of ideas within a small community of students, initiatedfirst, by Milin; then, from Milin to Michelle I.; then from Michelle I. to Matt; thenfrom Matt to Stephanie, Bobby, and Michelle R.; and then from Stephanie to theentire 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 prob-lem called “Guess My Tower,” in which they were asked to decide what kind oftower was most likely to be selected at random from a box containing first allpossible 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 tomonitor the durability of students’ strategies and arguments. Since solving the GuessMy Tower problem required the building of a sample space for all possible events, itrequired that the students revisit the question of finding the total number of four-talltowers. We have seen that earlier, Stephanie had already built and justified by casesall possible four-tall towers and used the doubling method for determining the totalnumber of towers. She was exposed to Milin’s inductive argument during previoussessions on March 10 and June 15.

Stephanie and Matt worked together on this problem first, using paper and penciland 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 towersthey would find. Stephanie remembered the pattern that she had noticed the previousyear. She said, “Oh, I remember the way that you could make sure how many. It was

Fig. 5.8 Stephanie andMatt’s towers


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

STEPHANIE: Well, a couple of us figured out a theory because we used to seea pattern forming. If you multiply the last problem by two, youget the answer for the next problem. But you have to get all theanswers. See, this didn’t work out because we don’t have all theanswers here.

MATT: I thought we did.STEPHANIE: 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 mypapers at home that say it works. I know that you had to multiplyit [the total number of towers of a given height] by something.Maybe it wasn’t two because I know it worked. Maybe it wasadding two.

Stephanie and Matt continued to attempt to find more four-tall towers, but theycould not find more than 12. Stephanie continued to assert that the total would haveto be 16.

STEPHANIE: I don’t know how it worked. I know it worked. I just don’t knowhow to prove it because I’m stumped.

MATT: Steph! Maybe it didn’t work!STEPHANIE: 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 Igoofed, that I messed up because I know that . . .

Stephanie’s memory of the doubling pattern and the fact that it had been used toverify her solution to the towers problem in the past was sufficient for her to remainconvinced 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 convincehis partner Michelle I. that the inductive method of generating towers was a validway of accounting for the total number of towers of a given height. Recall thatboth Milin and Michelle I. participated in the Gang of Four session almost a yearearlier on March 10, 1992, in which Stephanie gave her argument by cases and Milinintroduced his inductive argument. (Refer to Chapter 4 for details.) Michelle I. wasintroduced to Milin’s argument the previous year in the group discussion and gaveevidence 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


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 howhe generated new towers using an inductive argument. Milin then explained hisreasoning for a second time (see Fig. 5.9).

Once Milin finished explaining his reasoning for building three-tall towers byadding 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 shehad come to her own understanding of Milin’s method and was able to extrapolatethe 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 Stephanieand 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 gen-eration of towers. As Michelle I. continued to generate taller towers, Matt joined inthe explanation, commenting “and so it’s a family tree” (see Fig. 5.12). Matt thenproposed that Milin’s method might be connected with the doubling pattern that


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 oppor-tunity 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 ofany height (see Fig. 5.13), by using a tree diagram to build towers in an organizedfashion, explaining.

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


Stephanie and Matt moved on to talk to Bobby and Michelle R., who showedthem that they, also, had found 16 towers. Stephanie commented, “When we multi-plied it out we got 16. . .. But we weren’t able to find that many. We were only ableto find . . . like 12.” After Matt, joining Michelle I., had demonstrated his under-standing of the inductive pattern, the researcher asked Stephanie to explain Matt’sreasoning to Michelle R. who still was not convinced about the inductive method.

RESEARCHER: I want to know how you are going to get to two-high.STEPHANIE: 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 nextnumber. And the next number is two. Okay? So you have fourof two.

RESEARCHER: That’s a big jump for me, Stephanie. You’re jumping too fastfrom four to two. I don’t know how they came. I don’t knowhow they grew.

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

MATT: All right. Now, from here, you add an opposite, an opposite, anopposite or the same color on. So then you add the yellow andthe red on to the last one. So you have . . .

RESEARCHER: What do you think of that, Michelle and Bobby? . . . Do youunderstand what he is saying?

MATT: . . . So you have a red on the bottom . . .. You add a red or ayellow on top. You have the same yellow on the bottom, butyou add a red or a yellow on top. Then you have . . .

RESEARCHER: Is that all you can do?MATT: 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-talltowers, Stephanie joins Matt and explains how two four-tall towers can be generatedfrom each three-tall tower. A confident and elated Stephanie (see Fig. 5.14) offersto 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-talltowers and, counting the towers, commented

STEPHANIE: I understand. I’m just very happy that my rule worked.RESEARCHER: Your rule worked. But what . . . you know what I think is really

valuable . . . for people to understand is to know why that ruleworked.

STEPHANIE: Well, I know what it is now. I, I figured it out! But I’m justhappy that it worked.


Fig. 5.14 I knew it!

RESEARCHER: Because you see how you can forget a rule, but if you know whyit worked . . .

STEPHANIE: Yeah, yeah.

5.3.4 Stephanie’s Sharing Milin’s Family Tree

Finally, during a whole class discussion, Stephanie confidently explained thereasoning 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 theseyou 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 forthe yellow you can add on a red to the yellow and a yellow to theyellow, okay?

Fig. 5.15 Stephanie explainsto the group


MICHELLE I.: So you don’t have to look for duplicates.STEPHANIE: Then each one of these has two, like, okay? If this is like a family

tree . . . the mother, the parents . . . the mother, the parents . . . andthen six kids, okay? Well, no. Actually eight kids . . . then theyhave 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 . . . .

ALL: And on, and on, and on, and on.STEPHANIE: ’Cause each one of them is different . . . you keep adding on. And

then here you can add the exact same pattern.

5.4 Discussion

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

Eve

nt

Dat

e

Inte

ract

ion

Typ

e

Pri

mar

y P

arti

cipa

nt

Oth

er

Par

tici

pant

s

Arg

umen

t

1 Michelle I.,

Michelle I.

Michelle I.Michelle I.Michelle I.

Stephanie, Jeff Inductive

2 Milin, Stephanie, Jeff

Extension of Stephanie’s cases to doubling

3 Milin, Michelle I., Jeff

Argument by cases, noting of the doubling pattern

4 Doubling pattern; organization by cases

56 Stephanie Cases; doubling; 3 colors 78 Organization by cases, verification using

the doubling pattern

9 Explains inductive pattern

10 Illustrates inductive pattern 11

Stephanie

Stephanie

Stephanie

Stephanie, Bobby,

Presents inductive pattern

12Michelle R.

Presents inductive pattern

13 Michelle R. and Bobby

Leads presentation of inductive pattern

14 Connects inductive pattern to doubling

3/10/1992

3/10/1992

3/10/1992

5/15/19926/15/1992

6/15/199210/25/1992

SG (E)

SG (E)

SG (E)

RT (I)

Milin

Stephanie

StephanieDoubling rule Milin

Milin

Milin

Matt

MilinMilin, Matt,

WA

WAWA

10/25/1992

2/26/19932/26/19932/26/1993

2/26/1993

2/26/1993

WA

PT

2/26/1993 PT

SG

SG

SG

PT, RT

Induction, doubling, Stephanie

15 2/26/1993 Presents inductive pattern to group StephanieWC

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


shows a trace by which the members of the group built an understanding of theinductive argument for the growth of the towers.

Gaining ownership of a mathematical idea involves a process by which thelearner takes responsibility for knowing. Faced with a conflict in one’s understand-ing, a learner can work with others to express what may be clear or unclear intheir understanding so that an obstacle can be resolved. Faced with a conflict, stu-dents can be motivated to find resolution. Value added is that there is personalgain and confidence in one’s ability as a problem solver and achieve understand-ing. Children, given the opportunity to share ideas, can contribute to each other’sgrowth in understanding.

In this chapter we have shown how sharing and discussion helped students attainownership of mathematical ideas. In the next chapter, we show how students madeuse of the strategies developed here in solving new problems in counting.

Chapter 6Making Pizzas: Reasoning by Casesand by Recursion


Date and Grade: March and April 1993; Grade 5Tasks: Pizzas with halves; PizzasParticipants: Amy-Lynn, Ankur, Bobby (Robert), Brian, Jeff, Matt, Mike,

Michelle I., Michelle R., Milin, Romina, and Stephanie.Researcher: Carolyn Maher

6.1 Introduction

In previous chapters, we followed the students in Kenilworth as they worked on theshirts and jeans task and the towers problems in the second through fifth grades. Inthis chapter, we discuss five sessions during which these students worked to makesense of the pizza problems. These problems presented new challenges and requiredthe students to adapt the representations and solution strategies that they had previ-ously formed to meet the needs of these tasks. (Portions of the data analyzed hereare described in Bellisio (1999), Muter (1999), and Tarlow (2004).)

As we trace the students’ problem-solving attempts, we will identify the forms ofjustification and reasoning that were used by the students, the methods of notationthat they used, and the heuristics and strategies that were developed as the studentsworked to resolve the complexity of the problems.

During the first four sessions, the students worked on the pizza with halvestask:

A local pizza shop has asked us to help them design a form to keep track of certain pizzasales. Their standard “plain” pizza contains cheese. On this cheese pizza, one or two top-pings could be added to either half of the plain pizza or the whole pie. How many choicesdo 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 possibil-ities. Show your plan for determining these choices. Convince us that you have accountedfor all possibilities and there could be no more.

During the fifth session, the students were presented with three related tasks; theyused the strategies that had been developed for the more complex pizza with halvestask 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 firstpizza explorations. We will trace the discussion of the two groups separately, thoughthey 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 set-tled their differences in notation as they began to talk about what their drawingsrepresented.

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

Just as Stephanie and Dana had discussed their method of notation with eachother 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 beforethey began to discuss their findings. This group of students used notation as well asdrawings to identify their pizza combinations. Matt’s suggestion to use a notationof HP to signify half a pie signaled the need to clarify what would be symbolizedusing letters and what would be drawn.

6 Making Pizzas: Reasoning by Cases and by Recursion 61

In addition to discussing their method of notation, they discussed which pizzacombinations were distinct and which were identical and could only be listedonce.

MICHELLE I.: Then we can switch the side so that the sausage-JEFF: No, it’s not going to matter which side. It’s going to be the same

pie no matter what side sausage is onMICHELLE I.: No, it’s a different side, though. It’s the same pie but it’s a

different side.JEFF: 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.]STEPHANIE: 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 andcheese/sausage) were not different in practice.

The students worked to find and justify their pizza combinations. After somediscussion, four of the five agreed that there were ten possible pizzas, while Milinwas still unsure that they had found all combinations.

The group of students discussed their ideas with each other and lookedat their own work. Matt found that he had named one pizza S and anotherone 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 piz-zas. Matt made a chart with C, P, and S as headings, listed all the possible pizzacombinations for each column, and then eliminated the duplicate combinations asindicated in Fig. 6.2. Stephanie then showed him a different way of organizingpizzas:

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 withone 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 pictureof half pizzas


Fig. 6.2 Matt’s organized listof pizza combinations

Stephanie’s method of organization is of note here. When Stephanie worked onthe towers problem (see Chapters 4 and 5), she organized the towers by cases. Shefirst considered towers containing no cubes of a color, then one of that color, thentwo of that color, three of that color, and so on. Now when she began to organizeher pizza combinations, she did the same, first considering pizzas with only onetopping, 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 inorganization. We will now contrast the work of the second group of students andidentify key differences in their strategies and representations.

6.2.2 Group 2

Amy-Lynn, Ankur, Bobby (Robert), Brian, Mike, Michelle R., and Romina werein the second group. As the students began their investigation, Ankur, assisted byBrian, took a leading role. He verbally listed six possibilities; when asked to showthat he had all the possibilities, he directed the others to write down the choices as hedictated them. Five of the others worked with Ankur and also contributed by callingout possible combinations. Each student wrote the list of pizzas that was compiledby the group (see Fig. 6.3).

Ankur’s list, like the lists of the other members of the group, used words todescribe the pizza combinations. However, his drawings at the end of the list showed

Fig. 6.3 Ankur’s writtenwork on March 1, 1993


his move toward using letters to symbolize toppings. Of note as well is his divisionof the pizzas into halves as well as fourths, which may have been a factor thatresulted in the students’ confusion about the total number of combinations.

Mike worked alone. From the start, he used drawings of circles to represent piz-zas, 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 methodmay have contributed to his confusion and duplication of pizzas.

After a few minutes, the students decided to discuss the possibility of havingmore 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 setof pizza pictures, as Amy-Lynn labeled them (see Fig. 6.5).

After this phase, some students had a list of 11 pizzas, while Brian conjec-tured that there were 12. Amy-Lynn found that there was a duplicate in her list of11 pizzas and pointed out to Mike that he also had one repeated combination.

ROMINA: It’s the same thing if you turn it around.ANKUR: No it’s not because of, half of it is mixed and the other half of it is

pepperoni.AMY-LYNN: We have that. We have half plain, half pepperoni sausage.BRIAN: Mixed.

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

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


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

In earlier years, as students worked on the shirts and jeans task and the towersproblem, they were careful to eliminate duplicate combinations. As the studentsworked on the pizza with halves task, they did the same, and like the other groupthey had lively discussions about what was considered a duplicate. Their differencesin their definition of mixed pizzas, however, were not resolved until the next session.

The students carefully color-coded the pizza representations, but the first sessionwas 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 returnedthe next day and continued working on the problem. However, the students fromthe two groups had discussed the problem between the sessions and shared theirsolutions with each other.

6.3.1 Group 1

Michelle I. began the discussion by commenting that the other group had foundmore than ten combinations, and that this group’s solution from the previous sessionwas therefore probably incomplete. Stephanie insisted that their original solutionwas 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 categorizedaccording to the number of distinct pizza toppings on each pizza as indicated inFig. 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: She’s wrong. C P . . . Look, C.MICHELLE I.: Yeah, cheese, pepperoni, sausage.MILIN: 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 pizzacombinations


Fig. 6.7 Michelle I.’s list ofpizza combinations

MICHELLE I.: Yes it does because cheese and pepperoni are mixed together. Onthe other side is just cheese.

JEFF: But that would be the same as that, though. Think about it. Watch.MATT: It has to be halves.MILIN: It already has cheese.MATT: We’re only working with halves, not quarters.

Stephanie then asked the group why they wrote a C for cheese at some times andnot at others. Jeff insisted that cheese was not a topping.

JEFF: Cheese is another way of saying plain. You get a plain, cheesepizza. [Jeff writes C = plain.]

STEPHANIE: Then why do we put cheese here and here and here?JEFF: If we put plain that would get confused with that.STEPHANIE: Why couldn’t we jut put like that?JEFF: What does that mean?STEPHANIE: Half of the pizza.MICHELLE I.: One side sausage and one side plain.JEFF: 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 asecond time, they concluded that there were ten combinations. However, they firstdiscussed whether a pizza with sausage and pepperoni mixed together was the sameas or different from the pizza that had sausage on one half and pepperoni on theother.

The researcher asked the students to explain their solution and justify it. Shereviewed their drawings and lists and asked Matt and Milin why and how theirlists differed from one another’s. They reported that Matt had organized the pizzasaccording to the number of toppings, while Milin had three categories: whole pizzas,half pizzas, and mixed pizzas.


6.3.2 Group 2

As part of the research protocol, researchers had photocopied the students’ workfrom 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 photo-copies did not show their color-coding. They were joined by Researcher 1, whoasked them to think about organizing their work before taking the trouble to drawelaborate diagrams. They decided to check whether all students had the same listof 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 asthe “half pepperoni half sausage plain and half pepperoni-plain half sausage.” Aquestion arose as to whether a half pepperoni/half sausage and pepperoni pizza wasallowed, and the group decided that it was. They again came up with a list of tenpizzas.

As they tried to justify this answer, Amy-Lynn made a list of the pizza com-binations. The students were convinced that there were only four whole pizzacombinations, 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 ofpizzas. 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 anymixture of two toppings. They then listed the pizzas that contained topping mix-tures 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 threehalf pizzas and the remaining four.

RESEARCHER: Okay, what was special about these first three halves? Theywere different from the others.

BRIAN: They weren’t mixed.BOBBY: 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 pepperonihas 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, becausethat 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 accountedfor. After further discussion, they concluded that all the toppings were used in allpossible combinations for each case, and so they reported that they felt they couldjustify their solution.


6.4 Third Session: Getting the Right Answer

At the third session, the researcher discussed with the students the different kindsof arguments that they had used. When they separated the pizzas into categories ofwhole, half, and mixed, they were using an argument by cases. In previous inves-tigations with towers, they had also used an inductive argument. They argued bycontradiction when they had stated that no more “whole” pizzas were possiblebecause all toppings were used. Researcher 1 also discussed the importance of usinga notation that clearly represented their ideas and communicated the meaning of theideas clearly to others. The session ended with the following discussion:

ANKUR: Did we get that problem right?RESEARCHER: What do you think? Is this a question? Ankur? Did you get

the problem right? How many of you believe that you got theproblem right?

JEFF: Yeah, can you tell us the answer to it?RESEARCHER: What is the answer to it?CHORUS: Ten. Ten.RESEARCHER: Have you proved it?CHORUS: Yes!RESEARCHER: Then why are you asking me?JEFF: ‘Cause maybe, you’re the one-RESEARCHER: It’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 thepizza 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 isshown in Fig. 6.8.

Amy-Lynn’s written explanation is of note in the method she described. Shewrote, “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 inthe third, fourth, and fifth grades. (See Chapters 4 and 5.) During those activities,the students worked on finding tower combinations, and they consistently searchedfor towers and their opposites, which were the towers with colors reversed. Amy-Lynn’s reference to this strategy indicates that the students used similar heuristicsas they tackled two problems that, on the surface, seem to need very differentproblem-solving tactics.


Fig. 6.8 Mike’s letter onMarch 5, 1993

6.6 Fifth Session: Additional Justifications

This session, lasting 2 h and 30 min, provided an opportunity for students to work onfour different pizza problems and to expand their repertoire of justifications. Theycontinued the pizza with halves problem, and they also worked on variations ofthe four-topping pizza problem, including thin and thick crust and the four-toppingpizza with halves problem.

6.6.1 Problem 1: Pizzas with Halves

At the beginning of this session, the students spent about 40 min discussing thethree “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 aprecise definition:

ANKUR: . . . ‘cause if you take one slice, they’re all gonna be the same withthe mix.

BRIAN: If you take one slice and you take another slice and you compare it, itwould have sausage and pepperoni on both.

ANKUR: They would both have the same.


Fig. 6.9 Final decision oncases for the pizzas withhalves problem

This definition clearly marked the difference between “whole” pizzas and otherpizzas. The students continued by placing the pepperoni/sausage “whole” pizza ina subcategory of its own and also placing the three “mixed” pizzas as a subcategoryunder the “half pizzas” category. Their final set of cases is given by the table inFig. 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 followingtoppings: peppers, sausage, mushrooms, and pepperoni. How many choices for pizza doesa customer have? List all the possible choices. Find a way to convince each other that youhave accounted for all possibilities.

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

ANKUR: Okay. You start with the first one, that’s P. And you mix it withthe second one. That’s P slash S. And then you start with the firstone 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 fourthone, PE. And then you start with the S since that’s the . . . ‘causeyou can’t use plain. We start with S and mix it with M.

RESEARCHER: Where’s that?ANKUR: 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.]RESEARCHER: Okay. So why is it you can’t go M with P?ANKUR: 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 ques-tioned by the researcher about their solution, Brian explained that they were sure


that they had found all possibilities because “we have an order.” The students werevery confident about their solution. As Brian and Ankur told the members of anothergroup:

ANKUR: Sixteen. And we can prove it.BRIAN: Sixteen.ANKUR: And you guys can’t prove it.BRIAN: And we can prove it.

6.6.3 Problem 3: Another Pizza Problem

Following the four-topping pizza problem, the students considered a problem calledAnother Pizza Problem:

Kenilworth Pizza was so pleased with your help on the first problem that they have askedus to continue our work. Remember that they offer a cheese pizza with tomato sauce. Acustomer can then select from the following toppings: peppers, sausage, mushrooms, andpepperoni. 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 wayto convince each other that you have accounted for all possible choices.

Within less than a minute, without writing anything, students were proclaimingthe 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’ssixteen. 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 byrecursion.

6.6.4 Problem 4: The Final Pizza Problem

The session concluded with the Final Pizza Problem, which included componentsof all the previous problems:

At customer request, Kenilworth Pizza has agreed to fill orders with different choices foreach half of a pizza. Remember that they offer a cheese pizza with tomato sauce. A customercan 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 choicesfor pizza does a customer have? List all the possible choices. Find a way to convince eachother 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 reasonswhy the numbers would be an answer to the question. When Matt proposed a methodbased on Ankur’s previous method, they began to focus on finding a justified solu-tion. 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


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

Fig. 6.11 Matt’s writtenwork on April 2, 1993

combinations on the other half of the pizza. The next pizza had peppers on one halfand each of the remaining 14 combinations on the other half. Matt continued thisrecursive procedure to arrive at an answer of the sum of 1–16, times 2 (to account forthe two different choices of crust). Refer to Fig. 6.10 for Matt’s work. Although hisprocedure was correct, Matt’s answer was incorrect, due to an arithmetic mistake.

Matt was unable to convince the other students to consider his procedure whichis 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 anew categorization procedure:

First you’re gonna have your wholes and then you’re gonna have half with one topping onone side and two toppings on the other side. Then you’re gonna have one topping, threetoppings. Four top- I mean, one topping with four toppings. And then you’re gonna go to


[sighs] here. I’ll write it down. You’re gonna have a whole first. And then you’re gonnahave all your wholes, which equals up to, I think, five. And then you’re gonna have half of itwith one topping, half of it with two. Then you’re gonna keep on doing that until you cometo 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. Thistime they agreed that his method would count all the possible combinations and notinclude any duplicates, and they accepted the recursive method as a valid strategyof justifying the solution to this complex task. The children, pleased with their workand solutions, expressed jubilation at the end of the session.

6.7 Discussion

During these five sessions, the students used two kinds of justifications: proof bycases and recursive arguments. These forms of reasoning had been seen in theschemes displayed in students’ earlier work on the towers problem. Now we seehow the students retrieve, build upon, and extend earlier schemes to reason aboutpizza problems, despite differences in surface features. As the students folded back,activated, and drew on previously built cognitive structures, they thought about newways of applying these strategies and they worked on settling differences in under-standing and notation. They were animated and engaged in these interactions. Inaddition, we see students adopting some strategies of others as they worked on newproblems; we see the students sharing ideas with their group members and attempt-ing to convince them of the validity of their methods. In this way, ideas traveledacross the learning community and were applied and adapted to new challenges.The episode shows the continual development of cognitive structures that were builtearlier.

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 dis-cussed here and in Chapter 5 and explored connections between these problems andPascal’s triangle, showing further elaboration of her cognitive structures.

Chapter 7Block Towers: From Concrete Objectsto Conceptual Imagination

Robert Speiser

Date and Grade: March 13 and 27, April 17, 1996; Grade 8Tasks: Binomials and towersParticipant: StephanieResearchers: Carolyn A. Maher and Robert Speiser

7.1 Introduction

In previous chapters, we looked at the development of various forms of reasoningin students working in a classroom in small group settings. In this chapter, we focuson an individual student – we examine Stephanie’s development of combinatorialreasoning. In previous chapters, we saw how Stephanie, working with others andon her own, made sense of the towers and pizza problems. In this chapter we seehow Stephanie extended that work. In her examination of patterns and symbolic rep-resentations of the coefficients in the binomial expansion, using ideas from earlierexplorations with towers in grades 3–5, she examined several fundamental recursiveprocesses, including the addition rule in Pascal’s Triangle.

This chapter centers on how children can build fundamental mathematical under-standing, over time, through extended task-based explorations. They create models,invent notation, and justify, reorganize, and extend previous ideas and understand-ings 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 ingrade 2, with the shirts and jeans problem (refer to Chapter 3). Even at this earlystage, she would validate or reject her own ideas and the ideas of others, based onwhether they made sense to her or not. Stephanie would monitor and often refer toideas and conclusions of other group members and would often integrate the ideas

R. Speiser (B)799 E 3800 N, Provo, UT 84604, USAe-mail: [email protected]


74 R. Speiser

of partners into her work and discourse. This constant, extended process of evalua-tion and revision helped her to keep track of data and to reconsider, strengthen, andextend her explanations (Davis, Maher, & Martino, 1992; Maher & Martino, 1991,1992a, 1992b).

In grade 3, Stephanie was introduced to investigations with block towers (seeChapter 4) that enabled her to build visual patterns of her ideas, such as the localorganization within specific cases, based on ideas like “together,” “separated,” “howmuch separated.” She recorded tower arrangements first by drawing pictures oftowers and placing a single letter on each cube to represent its color, and later byinventing a notation of letters to represent the colored cubes. Stephanie’s workingknowledge about towers, gained over long periods of time through very concreteexplorations, led as, we shall see, to powerful and personally meaningful new waysto work with mathematical ideas.

7.2 Theoretical Perspectives

We believe that children come to mathematical investigations with theories that canbe built upon, modified, and refined. In turn, children’s theories and their ways ofworking with these theories help us, as researchers, to constitute our own concep-tions of children’s emerging work and thought, and so affect the way we build thediscourse, day by day, that we will share with them. In the task-based interviewsthat we report, we, too, will seek to build a theory. Our emphasis on building theoryinforms directly how we structure research interviews. Initially, one interviewer willengage the child in a specific exploration, seeking to estimate the working theorythat might guide the child’s thinking. Later, in the same interview or in a subsequentfollow-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, wefrequently begin with very concrete discussions, followed by what might be calleda “teaching phase” intended to investigate deeper connections. In such interviews,children will sometimes make powerful connections early and so break the flow wemight naively have imagined. We have come to view such “unique outcomes” aspotential opportunities to gain important insights from the children that we study.Therefore, when a child’s connection appears to break the flow, the interviewer, onprinciple, will invite more detailed explanation.

In Mindstorms (1980), Seymour Papert reflects on how he built his personalmathematical understanding – an understanding that inspired his later work – basedon 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 activ-ities, in the early grades, when she used block towers to investigate conceptuallyimportant counting problems.

The specific arguments that Stephanie investigates below were first developedand explained in Speiser’s paper (1997), where block towers underpin a concretemicroworld for productive exploration. These arguments, shaped specifically within

7 Block Towers: From Concrete Objects to Conceptual Imagination 75

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

7.3 Setting

Stephanie participated in the longitudinal study starting in first grade. Stephanie andher classmates were challenged in their mathematics classrooms to build solutions toproblems and construct models of their solutions. This setting, which for Stephaniecontinued to grade 7, encouraged differences in thinking that were discussed andnegotiated. In fall 1995, Stephanie moved to another community and transferred toa parochial girl’s school. Her mathematics program for grade 8 was a conventionalalgebra course. Stephanie continued to participate in the longitudinal study througha series of individual task-based interviews. A subset of these interviews providesthe data for this chapter.

7.4 Guiding Questions

The following questions guide our analysis in order to consider, systematically, theways in which Stephanie’s past experience is drawn upon: (1) How does Stephaniework with towers in building images and understandings for higher mathematicalideas? (2) What is the role of past experience in building new ideas? (3) How areher ideas modified, extended, and refined over time?

Data come from two of eight individual task-based interviews of Stephanie. Theinterviews were videotaped with two cameras, positioned to capture in detail whatwas 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 andanalyses of the interviews were made and verified by a team that included severalgraduate students in addition to both authors. Stephanie’s written work preparedfor the interview, and several observers’ notes, provide further data. The teachingexperiment 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 typi-cally begin with inquiries about the mathematics that Stephanie currently studied ineighth-grade algebra, both to open opportunities to talk about that mathematics andto explore her thinking about fundamental mathematical ideas.

76 R. Speiser

7.5 Results

To introduce each data segment, we provide a brief discussion of the mathematicsStephanie was invited to explore. On this basis, we can more clearly understandeach 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 hersymbolic representation of the expansions of (a + b)2 and (a + b)3.

STEPHANIE: So there’s a cubed [a3].RESEARCHER: That’s 1.STEPHANIE: And there’s three a squared b [3a2b] and there’s three a b

squared [3ab2] and there’s b cubed [b3]. [Interviewer writes 1 33 1 under 1 2 1 as Stephanie speaks.] Isn’t that the same thing?

RESEARCHER: What do you mean?STEPHANIE: As the towers.RESEARCHER: Why?STEPHANIE: It just is.

Stephanie asserts (in her own way) that each three-high tower gives a non-commutative monomial of degree 3 in a and b, and she has indicated that thesenon-commutative monomials, indexed by the corresponding towers, collect to givethe coefficients for the commutative monomials that appear in her expansions of(a + b)2 and (a + b)3. Our interpretation, therefore, is that Stephanie visualizestowers (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 reasonsabout polynomials based on her mental images of towers.

Working at home before the interview, Stephanie had written out the first sixpowers of the binomial a + b, and brought her written calculations to the inter-view. The interviewer covered Stephanie’s paper, guessed the coefficients for thesixth-power expansion, and wrote down the terms in full. Her coefficients were thesame as Stephanie’s, although one monomial was slightly different. Several minutesfurther in the conversation, Stephanie gives further evidence, that she proceeds byvisualizing towers and then reasoning based on her mental images.

RESEARCHER: So you have two factors of a. Right?STEPHANIE: Um hm.RESEARCHER: You have one of those. One thing with two factors of a. One

thing with two a’s in it.STEPHANIE: Um hm.RESEARCHER: I don’t want to think of a’s. I want to think of red.STEPHANIE: Okay [laughing].RESEARCHER: Can you switch that a minute?STEPHANIE: Yeah.RESEARCHER: So now I have one thing with two reds. What thing can I be

thinking of with two reds?STEPHANIE: That’s a tower that’s two high.


RESEARCHER: Okay. And here I’m talking about two things.STEPHANIE: Um hm.RESEARCHER: One is.STEPHANIE: Red and . . .

RESEARCHER: one is . . .

STEPHANIE: . . . one is yellow.RESEARCHER: Is that possible in two high?STEPHANIE: Yeah.RESEARCHER: Having one red and one yellow? There are two of them?STEPHANIE: Yeah.RESEARCHER: Which two?STEPHANIE: ‘Cause the one is the red could be on the top or the bottom, with

the yellow the same thing.RESEARCHER: What about b squared?STEPHANIE: Um. Two yellow.

In a March 27, 1996, interview, Stephanie is invited to explain to a secondinterviewer (unfamiliar with her recent work) what had happened in the March 13interview described above. Here Stephanie begins with towers, then reviews thebinomial coefficient notation C(n, r), working through a sequence of examples withincreasing n. Stephanie remarks that “r is a variable,” which she understands canrange from 0 to n. This observation shifts the level of abstraction upward from spe-cific 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 bothvary. This richer context triggers, with encouragement from Interviewer 1, a confi-dent, detailed, and carefully presented recapitulation by Stephanie of the recursiveconstruction of the towers of height n from the towers of height n – 1, as it had beenintroduced by classmate Milin in grade 4 and revisited in grade 5 (see Chapters 4and 5).

During a previous interview, on March 13, Stephanie also referred to Pascal’sTriangle, in particular to its addition rule, to make similar predictions, but she haddone so in a conceptually quite different domain: to predict, in effect, the numbersof n-tall towers in each given case (of r red blocks, say, for given r) for new values ofthe height. Stephanie’s choice to center, in the present interview, directly on binomi-als strongly suggests that Stephanie now grasps the isomorphism between Pascal’sTriangle, which she had built, at first, to summarize her towers cases, and the arrayof 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 asconcrete anchors to establish formal facts about the C(n, r), viewed either formallyas binomial coefficients or as counts of combinations, or, more concretely, as thenumbers of specific kinds of towers.

Fermat’s recursion. One goal for the March 27 exploration was to offer Stephaniethe tools she’d need to construct a formula, originally due to Fermat (Weil, 1984),that expresses the relationship between two successive binomial coefficients. Insymbols, here is Fermat’s formula:

78 R. Speiser

C(n, r + 1) =(

n − r

r + 1

)· C(n, r) (7.1)

This equation, applied repeatedly beginning with the simple case r = 0, leadsdirectly to the standard formula for C(n, r).

To make sense of this formula, it seems especially helpful in this setting to inter-pret Equation (7.1) in terms of towers. For concreteness, take red and yellow for thecolors of the blocks available. On the right side, C(n, r) counts the towers of heightn that have exactly r red blocks, hence n – r yellow blocks. Call these the originaltowers. On the left side, C(n, r + 1) counts the towers of height n with exactly r +1red blocks. Call these the new towers. In concrete terms, Equation (7.1) tells us thatthe number of new towers can be found by multiplying the number of old towers bythe number of yellow blocks in each and then dividing by the number of red blocksin a new tower.

In the data below, the interviewers will fix n, the height of a tower, and thenvary r, beginning either with r = 0 or r = 1, for which C(n, r) is either known toStephanie or easily determined by inspection. For each r, the interviewers will theninvite Stephanie to construct new towers from a given set of original towers andexplore with her what she has found. The construction process Stephanie exploreswill work for any height n and any r < n.

For concreteness, we explain this process when n = 4 and r = 1. In this case, wehave 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 itsthree yellow blocks with a red block. For each of the four original towers, we cantherefore build three new ones. Working in this way (we’ll say by day) we obtainfour groups of three new towers. The 12 towers constructed in this way clearlyinclude 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 inturn 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 rightside of Equation (7.1), but – this is a key point – the new towers we just built are notdistinct. In fact, each tower appears exactly twice among the 12, as the denominatorr + 1 predicts. To understand how duplicates emerge, consider a particular newtower. This tower has exactly two red blocks. Each of these two red blocks can bereplaced (working, we shall say, by night) with a yellow block, producing one oftwo 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)


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 redblocks. Because towers correspond to combinations, we have used the known resultC(4, 1) = 4 to show that we have C(4, 2) = 6. For example, consider Fig. 7.2. Bynight, begin with a new tower four blocks high (n = 4) with two red blocks. Weobtain two original towers, each with one red block, by replacing one of the two redblocks in the given new tower with a yellow block.

Stephanie began to explore the construction shown in Fig. 7.1 during the March27 interview, first with three-tall towers. Her blocks were blue and green. Continuingto four-tall towers, she next built the four towers with exactly one green block andguessed initially (but incorrectly, perhaps based on her experience with three-talltowers) that from each such original four-tall tower she could obtain two new ones.

RESEARCHER: I wonder why you get two of them.STEPHANIE: I don’t know. Maybe cause it’s bigger.RESEARCHER: What would that have to do with it?STEPHANIE: I don’t cause you have more room to build on.RESEARCHER: Tell me, can you explain to me?STEPHANIE: Oh, well, maybe it’s because like you already have one [green

block] that’s taking up space, so you only have three placesto move it.

RESEARCHER: I gotcha, okay.INTERVIEWER 2: Okay.RESEARCHER: So what would you predict if you were building towers five

high?STEPHANIE: You’d have four.

Here we see Stephanie revise and then explain her observations, starting from aset of four-tall towers that she had physically built. On this basis, she extends herobservations to a set of five-tall towers she has just imagined. So far, she knows thatduplicates appear in the construction she discusses, but has not yet explored in detailhow 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) hercolors 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 originaltowers, with r = 0, 1, 2, and 3 in succession, the researcher invites Stephanie to

NightFig. 7.2 By night, replaceeach of the red blocks(shaded) by a yellow block(unshaded)

80 R. Speiser

predict how many duplicates she would expect for towers of height 5 and helps herbuild the towers that she needed as they proceed. They begin with a tower with fiveyellow blocks (the case r = 0).

STEPHANIE: The first one’s one.RESEARCHER: There’s one of thoseSTEPHANIE: times five.RESEARCHER: Why five?STEPHANIE: ‘Cause there’s five positions.RESEARCHER: Okay.STEPHANIE: Divided by one, ‘cause they come in groups of one.RESEARCHER: Um, hm.STEPHANIE: Five.RESEARCHER: Okay. So that’s five things taken one at a time.STEPHANIE: Yes. The second oneRESEARCHER: Why don’t you write that down? Five things taken, equals five

things taken one at a time. [Stephanie writes.]STEPHANIE: 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 groupsof – I don’t know, um, that’s what we don’t know though.

RESEARCHER: All right. So. Let’s – can we make these five? Just, here.STEPHANIE: Well, maybe they might come in groups of two?RESEARCHER: One. Let’s think about at least one of these.STEPHANIE: 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 four-tall towers. At this point, the interviewer arranges the five towers they have built infront of Stephanie and offers Stephanie the tower with its one red block in the topposition.

RESEARCHER: Okay. So what you’re saying here – move some of this aside –um, okay. Let’s think of that one.

STEPHANIE: Okay.RESEARCHER: There are five.STEPHANIE: [Builds a tower with a second red block just below the first.]

You have one like that. [She builds another tower with a secondred block two spaces below the first.] One like that.

RESEARCHER: Well, can you predict before you do it?STEPHANIE: Yeah, there’s going to be four from each.RESEARCHER: Four from each.STEPHANIE: Yeah.RESEARCHER: Okay. So – and what’s the each? How many make up each?STEPHANIE: How, wh, what do you mean?RESEARCHER: You’re saying, it’s four from this.STEPHANIE: Yeah, four from


RESEARCHER: What does-STEPHANIE: One.RESEARCHER: -each mean in this case?STEPHANIE: 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.RESEARCHER: Okay. So how many eaches?STEPHANIE: There’s five.RESEARCHER: Five eaches. Okay.STEPHANIE: Yeah.RESEARCHER: All right. So that, you say, five times four.STEPHANIE: Yes. I have that. I just don’t know what the-RESEARCHER: Right.STEPHANIE: -bottom part – itRESEARCHER: So – and by the groups, you mean. The groupings you mean.STEPHANIE: Groups like – one after we’ve put them all out. Like how many

groups, they’re going to come in-RESEARCHER: I don’t know. I’mSTEPHANIE: duplicates?RESEARCHER: I’m wondering. When you say you divide bySTEPHANIE: Oh! ‘Cause that’s the number of duplicates – that there are.

Again, working by day, with towers on the table, Stephanie corrects her guess, butthen – 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, orat least to find the number of such groups. In effect, she has proposed the key stepby herself: to find how many duplicates each new tower has.

RESEARCHER: But how do you know beforehand? Do you think there’s a way?STEPHANIE: [Building towers.] Oops.RESEARCHER: So if this, um, is going to be a pattern to this – the five times

four – what do you think you would divide by?STEPHANIE: Five times four – what do you think I’d – um – maybe two.

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

STEPHANIE: Ten. So it would be ten times three and you divide by three.[Writes as she speaks.]

RESEARCHER: And it worked?STEPHANIE: 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 putit – it’s five times one divided by five equals one.

82 R. Speiser

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 ofEquation (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 redblock.

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 fol-low a numerical pattern that she has observed empirically. At this point, Stephaniedoes not yet seem able to explain why the number of duplicates that she observesmust necessarily be r + 1. Nonetheless, we remain astonished, after 12 years, by thedepth and strength of the connections Stephanie has made, based on her familiaritywith towers.

The addition rule in Pascal’s Triangle. By March 1996, as noted above, Stephaniealready 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 fromthe row before it, by adding each pair of successive entries in the row above. Toconnect this formula to combinations, and in this way make sense of it, we will readeach term as a count of towers. Specifically, the first term on the right counts thetowers of height n that have exactly r – 1 red blocks, while the second term countstowers of the same height, but with one additional red block. So interpreted, theright side of Equation (7.2) at least suggests that every tower of height n + 1 thathas exactly r red blocks can be constructed from suitable shorter towers of heightn, either by placing a red block on top of a tower of height n with r – 1 red blocksor 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 towersof height 3 (n = 3): one tower with no red block (r = 0) and three towers with onered block (r = 1, shaded). To accomplish the recursion, attach a red block (R) on

YR

Fig. 7.3 A specific exampleof the addition rule forPascal’s Triangle


top of the single tower in the first set, and a yellow block (Y) on top of each towerin 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 + 1that has exactly r red blocks, it’s clear that all such towers have been counted on theright 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 num-bers 1 and 3 that correspond to the towers shown in the top row of Fig. 7.3. She isjust about to write the number 4 below them, and then draw two diagonal lines, toassociate the numbers 1 and 3 to the 4.

RESEARCHER: Okay. Um. Let’s explore, um – which one should we explore?[Draws lines as above.] Let’s do this one.

STEPHANIE: Um, hm.RESEARCHER: Do you know what this one means? If you had to build this one,

what would that tower look like?STEPHANIE: That one?RESEARCHER: What would that one look like? What would those two look

like?STEPHANIE: [A pause, while Stephanie builds towers.] I think that one would

be like this. [Stephanie has built the tower of height 3 with allyellow blocks, and she indicates the one that Interviewer 1 hasdrawn] and that one.

RESEARCHER: Three high, no red.STEPHANIE: Like this. [She has just built the first two towers of height three

with one red, as in Fig. 7.3.]RESEARCHER: Okay. Three high, exactly one red.STEPHANIE: Yes.RESEARCHER: Okay.STEPHANIE: Oh! Wait! [Builds the remaining tower.]RESEARCHER: 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 believeyou’re gonna do it. Okay. Now. When we’re doing this [pointsto the 1, the 3, and the 4 that she has written].

STEPHANIE: Um, hm.RESEARCHER: What’s different about these and this tower here [taps the

number 4 in Pascal’s Triangle] that I call four? There.STEPHANIE: Well – it’s four high.

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

84 R. Speiser

RESEARCHER: I want to know from hereSTEPHANIE: Uh, hm.RESEARCHER: what you do to these [the towers of height 3]-STEPHANIE: Well.RESEARCHER: -to get me, to get me-STEPHANIE: Well, I’d build them higher.RESEARCHER: Well, don’t do it yet. Just think about it for a minute. Remember

what they’re going to look like.STEPHANIE: Yeah.RESEARCHER: There’s going to be exactly one red.STEPHANIE: This would go here [she moves the all-yellow tower of height

3] and there would be red.RESEARCHER: 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 whenyou’re all done-

STEPHANIE: Um, hm.RESEARCHER: -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-hightowers with exactly one red block, and she can easily imagine them. But does sheunderstand, working by day, how those towers can be built from the four towers ofheight 3 she has in front of her?

STEPHANIE: I’m going to put a yellow here [points to the first tower on theright in the top line of Fig. 7.3],

RESEARCHER: Okay.STEPHANIE: I’m going to put a yellow here [points to the next tower],RESEARCHER: Right.STEPHANIE: 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 theall-yellow tower].

RESEARCHER: Okay. So how many ways – how many do you end up with?STEPHANIE: Four.RESEARCHER: Four. So from the one three tall with no redsSTEPHANIE: Um, hm.RESEARCHER: And the three three-tall with one red, right?STEPHANIE: Yes.RESEARCHER: You end up four four-tall with one red.STEPHANIE: Yes.RESEARCHER: Isn’t that neat?STEPHANIE: Yeah.

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


Fig. 7.4 Stephanie’s towerexploration, grade 8

A conceptual reflection. In these data, we see Stephanie refining and revising newideas that she has built from raw materials she draws from prior experience withtowers and combinations. This prior experience includes a variety of proofs (first bycases, later by induction), expressed concretely with block towers and more formallythrough language and notation that she and peers have personally developed andrefined throughout their long collaboration.

This process of revision and refinement, which we emphasize throughout, mightbe most clearly visible across the data we see here as a progressive movement fromsets of towers that Stephanie built physically (see Fig. 7.4) toward sets of towersthat Stephanie comes to imagine. These imagined towers (such as the final set offour above) are not simply visualized as static images from prior tasks; indeed, thenew towers have been constructed based on new conceptual ideas that Stephanie hasbegun 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 usedthe metaphor of text to state the following conclusions (Maher & Speiser, 1997b,p. 131):

Images, patterns, and relationships have become mathematical objects that Stephanie seesand works with mentally to build abstractions. Our conversations with her elicited bothspoken and written texts. These texts, together with our interpretations, anchor an analyticnarrative of the development of certain mathematical ideas. Such texts (which we proposeto view as work in progress) extend through time and serve as records of particular eventsupon which later texts can comment. Further, they can serve as raw material from whichnew texts can be composed.

We revise our texts, and so does Stephanie, as our experiments proceed through detailedinteractions with each other. Hence, as Stephanie’s developing judgment enters the discus-sion, 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 thatStephanie pursues.

86 R. Speiser

After the interviews we have considered here, events that neither Stephanie,nor her peers, nor the researchers could foresee in 1996, would offer opportunitiesfor everyone involved to deepen, reconsider, refine, and extend their previous per-spectives and conclusions. In the discussion we have just presented, perhaps moststriking is the heightened prominence we see of personal, conceptual imagination toaddress 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 whoalso build on previous explorations to make sense of Pascal’s Triangle and rules forits generation.

Part IIIMaking Connections, Extending,

and Generalizing (1997–2000)

Chapter 8Responding to Ankur’s Challenge:Co-construction of Argument Leading to Proof

Carolyn A. Maher and Ethel M. Muter

Date and Grade: January 9, 1998; Grade 10Tasks: TowersParticipants: Ankur, Brian, Jeff, Mike, and RominaResearcher: Carolyn Maher

8.1 Introduction

In previous chapters, we saw elementary students work in classrooms on countingproblems presented by researchers. In this chapter, we observe a group of fivehigh school students working under different circumstances. When the students inthe longitudinal study entered high school, they no longer worked on problems inclass. Instead, the students who remained in the study worked on problems in after-school sessions scheduled by the researchers and for which the students rearrangedtheir after-school schedules in order to attend. In addition, this session is uniquebecause the students worked on a problem proposed by a fellow student. In this ses-sion, Ankur and Mike were invited to propose and solve their own problem. Ankurproposed 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 threedifferent 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 towersbuilt from three colors and then subtract the number that did not fulfill the statedcriteria. They started by writing combinations of towers, using the numbers 1, 2, and3 to represent colors red, blue, and yellow, respectively, and using a 0 to representthe duplicated color. At first, they omitted some towers in their count, but later inthe session they discovered the missing set.



90 C.A. Maher and E.M. Muter

As Mike and Ankur proposed a variety of numerical answers (72, 54, 45, and36) 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 listhad 37 towers, but he thought that it included a duplicate.) Although they workedseparately, the two groups periodically exchanged comments and suggestions. Thefirst firm finding on which all agreed was that the total number of four-tall towersthat 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 Mikeand Ankur, who were working on developing a justification based on cases. Briansat quietly, sketching out ideas on a sheet of paper. Romina worked on her own, attimes thinking out loud, as illustrated here:

You know, it might be 36. ‘Cause I’m working with sixes now. And okay, you put them, likeyou pair ‘em up. ‘Cause you’re only gonna have . . .

At this point, Romina put up her hands and indicated that she needed to collecther thoughts. She continued:

Let me think first, organize my thoughts a little. We’re gonna have them together. Togetherlike over here [indicating her list]. These are together. These are together. These aretogether. Like two of the same color together. And then, in like a pattern, like, we’ll putthem somewhere and then we’ll switch them around, so I’m up to 24 now and I’m going toput them the same way here and here. So then that’s 30. And I put the same ones here andhere. 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 thethree 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), andthen she moved on to the next possible arrangement. This was the beginning of ajustification based on cases. She explained that by using this method she had found24 towers so far and that there were two additional groupings that she had yet tocomplete. Counting all of the towers in her list would mean that there were 36possible towers as the answer to Ankur’s Challenge. At this point, Romina began togeneralize her solution. Instead of listing all of the combinations, she listed only thecubes that were duplicated in each tower.

As Romina tried to get the attention of Jeff, Mike, and Ankur in order to explainher solution to them, Brian interrupted and pointed out that she had duplicatedone of the rows in her list. Undaunted, Romina reached for a clean piece of paperand began to redraw her table. Brian watched carefully as she worked, offeringhelpful suggestions as she created this version of her solution. She drew boxesshowing the duplicated cubes in five of the six possible positions. With Brian’s

8 Responding to Ankur’s Challenge 91

Fig. 8.1 Romina’s firstattempt to write out thetowers of Ankur’s Challenge

assistance, Romina reviewed what she had already written and included the sixthrow (Fig. 8.2).

She was then ready to present her thoughts to Jeff, who had moved over to seewhat she and Brian were doing. She counted her rows to verify that she had therequired 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, whichstands for putting these and these, right? And you’re gonna have them in, and then therest you fill up, right? And then you’re gonna have the . . . And there’s only two other onesyou could have. So this, you have this one, you have to multiply it by two. Well, one, two,

Fig. 8.2 Romina’sgeneralization


three, four, five, six. . . . You multiply this by two [indicating each row in turn], multiplythis by two, multiply this by two, by two, by two, and by two. And then, one, two, three,four, five, six.

This attempt to convey the idea that each row represented a four-tall tower thathad to contain two cubes of one color with one of second color and one of the thirdcolor was not convincing to Jeff. Romina did not explain the reason for multiplyingby two.

Romina and Brian decided that Romina would write her ideas more neatly, andshe produced a third version of her diagram. As Ankur and Mike began talking tothe researcher about their solution, Jeff and Romina interrupted with the informationthat they had found a solution of 36, and they could prove that it was correct. Theyasked Mike to pay attention to their explanation and, although he agreed, it becameclear that he was still thinking about his own ideas for finding the complement toAnkur’s Challenge.

Romina’s presentation improved as she related her ideas this second time. Thediscussion follows.

ROMINA: 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 themseparated by one. You have them on the ends, in the middle, two andfourth 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. Likethese are three different colors – an o and an x and an x and an o.Right?

ANKUR: Yes.ROMINA: So that’s six.ANKUR: Yes.ROMINA: So you have to multiply each of these six by two.JEFF: But you couldn’t have like x x because that wouldn’t fit the require-

ment. So you multiply each one by two. So that would give you 12.Correct? ‘Cause that means you could have like this . . .

BRIAN: 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 pos-sible positions for the duplicated color. She explained how she placed the remainingtwo colors in her diagram and said why she multiplied each row by two (to accountfor the fact that the two non-duplicated colors could switch positions). Jeff and Brianadded clarification of the reason for multiplication by two.

Ankur accepted Romina’s solution. Mike, however, remained fixed on his earlierstrategy to justify that the remaining towers, those that formed the complement ofAnkur’s Challenge, numbered exactly 45. He was not willing to accept the num-ber 45 as the difference between 81, the total number of towers that they hadagreed on at the outset, and the 36 combinations that Romina, Brian, and Jeff hadfound. Although Ankur and Jeff claimed that Romina’s proof should be sufficient toprove that 45 was the solution to the complement problem, Mike wanted an explicitproof:


Fig. 8.3 Romina’schalkboard version of herproof of Ankur’s Challenge

ANKUR: The only way you could prove you were right is to prove the other side.JEFF: We proved the other.MIKE: 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 thathe 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 andthe diagram she utilized as she spoke while writing it on the chalkboard. Refer toFig. 8.3.

At the group’s next meeting, Romina brought a written copy of her proof ofAnkur’s Challenge. Notice her further refinement in showing 36 possibilities in herwritten work shown in Fig. 8.4.

Fig. 8.4 Romina’s secondversion of her proof ofAnkur’s Challenge


In Romina’s written explanation, she indicated a realization that there were threedifferent colors for selecting the blocks and four positions on a tower to place ablock. She indicated, also, that there would be a double of one of the colors and asingle 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 wrotethat placing the red cubes in all possible positions would produce six towers. Shepointed out that there were only two possibilities for the remaining colors, therebyproducing 6 × 2 or 12 towers with double red blocks. She then concluded that byconsidering the remaining 3 colors, 3 × 12 or 36 towers would yield all possiblecombinations, thereby producing an elegant justification for the solution to Ankur’sChallenge.

8.3 Discussion

In their work on Ankur’s Challenge, Romina, Brian, Jeff, Ankur, and Mike demon-strated how they worked as problem solvers. Ankur posed an interesting problem,which he and Mike partnered to solve. They started by applying previous knowl-edge (how to find the number of towers when selecting from two colors) to anew situation (three colors are now available). They successfully found the totalnumber and proceeded to use a subtraction strategy. They listed exceptions bycase; when the notation they originally chose proved inadequate, they introduced anew notation.

Later, when provided with an answer derived from a different approach, Mikecontinued to work on Ankur and his initial strategy. Although he accepted the directapproach explained by Romina from the other group, he was unsatisfied that Ankur’sand his approach, seemingly reasonable, did not work.

Romina and her group used a direct approach. They brought together ideas andnotations from the past as they constructed a solution for Ankur’s Challenge. Theyused a variation of the binary coding scheme that Mike had introduced the previ-ous month. They profited from the strategy that Ankur had presented in the fifthgrade of fixing one of the variables and then considering the possibilities that satis-fied that case. As Romina explained her strategy to Brian, she formulated differentways to express her thoughts; Brian assisted by pointing out additional cases. AsRomina tried to communicate her ideas so that Jeff would follow, she revised herrepresentations.

Mike, ready to hear about the solution of the other group, asked Romina to repeather explanation. In response, she presented her work at the blackboard with fur-ther refinement of their representations. Her final written summary provided anotheropportunity for detail, refinement, and generalization. In summary, we can say thatRomina and her group profited by using their personal representations, communi-cating them as ideas, and then providing support for those ideas by reorganizing andrestructuring representations. Further, in each iteration of the argument, Rominamade refinements and clarified her reasoning. This suggests advantages for students


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 formultiple ways to represent ideas.

Romina, 1 week later, shared a written solution to the problem, indicating aninterest in refining her explanation and demonstrating her motivation for furtherthought and reflection.

In the following chapters, we return to the pizza and towers problems and see howgroups of students continue to refine their representations, clarify their reasoning,and extend and generalize their understanding of mathematical ideas by revisitingold ideas, communicating their findings, and listening to the findings of others.

Chapter 9Block Towers: Co-construction of Proof

Lynn D. Tarlow and Elizabeth B. Uptegrove

Date and Grade: November 13, 1998; Grade 11Tasks: TowersParticipants: Ali, Angela, Magda, Michelle, Robert, and SherlyResearchers: Carolyn Maher, Alice Alston, Susan Pirie

9.1 Introduction

In previous chapters, we observed elementary school students working to makesense of the towers problems by building representations, formulating conjectures,and defending their solutions in discussions with classmates and researchers. In thischapter, we observe a cohort of high school juniors as they engage in explorationsand constructions in the towers problem. During this session, the students found andgeneralized formulas for solutions to the original towers problem (building towerswhen selecting from two colors of Unifix cubes) and extensions (with more thantwo colors of cubes), using methods including controlling for variables, justificationby cases, and inductive reasoning.

9.2 Building Towers

In the 2-h session, students worked in pairs on tower problems. They came upwith a general rule for the number of possible towers of height n when selectingfrom x colors (xn) and an explanation of that result based on an inductive argumentbased on generating all possible towers of a given height. Their arguments containedreasoning by cases, induction, and reasoning by contradiction. In addition, Robertproduced an equation for the number of towers having exactly two cubes of onecolor (when selecting from two colors), for a tower of any height.

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


98 L.D. Tarlow and E.B. Uptegrove

9.2.1 Angela and Magda

Neither Angela nor Magda had previous experience with the towers problem, asthey had both joined the longitudinal study in sixth grade. In this, their first expe-rience 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 fifthgraders that participated earlier in the study (see Chapter 4). The girls organizedtheir work by cases: (1) one blue, (2) two blues, (3) three blues, and (4) four ofthe same color. The two single-color cases consisted of one tower each; the one-blue-cube and three-blue-cube cases exhibited a local organization; they built thosetowers by moving the single cube of one color into each of the four possible posi-tions. When asked how they knew that they had all the towers with one blue cube,they described their organization:

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

Initially, they had no support for accounting for the towers in the two-blue-cubecase; they explained that they were unable to find any more. However, after theyfound four of the towers for this case, and they were asked how they knew theyhad them all, Angela alluded to a preliminary organization using controlling forvariables 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 thebottom, and the blue on top, yellow on bottom, then blue on top and bottom, and yellow ontop and bottom.

As she was saying this, Angela found the two towers for this case that weremissed. There are two towers with yellow on the top and blue on the bottom andtwo towers with blue on the top and yellow on the bottom; they had originally foundonly one of each of those pairs.

When asked to determine the number of three-tall towers, Angela and Magdamoved 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 inFig. 9.1. After thinking about their findings, they developed a general rule; accord-ing to what they called “Angela’s Law of Towers,” the number of n-tall towers whenyou have x colors to choose from is xn. Thus Angela and Magda not only pro-vided a solution to the specific four-tall towers problem posed, but they also poseda generalization from towers with two colors to towers with x colors.

Case 1

Y

Y

Y

B

B

BB Y Y

YB

B

Y

Y Y

B B Y

BB

B

B

Y B

Case 2 Case 3

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

9 Block Towers: Co-construction of Proof 99

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-gradestudents who, when first encountering the towers problem, used the strategy ofgrouping 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-talltowers using this strategy. When they were asked to predict the number of three-talland then five-tall towers, they made a prediction based on patterns. They conjec-tured that since the number of four-tall towers is 16 (4 times 4), there would be 9three-tall towers (3 times 3) and 25 five-tall towers (5 times 5). After realizing thatthere are only 8 three-tall towers, they revised their conjecture for five-tall towersto 24, which follows a pattern of multiples of 8. During the whole group discus-sion (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 oftheir work on five-tall towers, Sherly and Ali were asked to investigate the case offive-tall towers with exactly two blue cubes. They organized this group of towersby cases: (1) two blue cubes together; (2) two blue cubes separated by one yellowcube; (3) two blue cubes separated by two yellow cubes; and (4) two blue cubesseparated by three yellow cubes; using this organization they found all ten towersthat satisfied this condition. Refer to Fig. 9.2 for a diagram of the “two blues sep-arated by one yellow” case. During the discussion of this work, Sherly provided aproof by contradiction: When a researcher asked if the two blue cubes could be sep-arated by four yellow cubes, Sherly noted that this could not happen, “because therewould be six [cubes in the tower] then.” We see that even though Ali and Sherly hadnot previously been exposed to the way younger students in the longitudinal studyworked, they had by the end of the session adopted some of the methods that hadbeen developed earlier by their classmates.

B

B

Y

Y

Y

B

B

Y

Y

Y

B

B

Y

Y

Y

Fig. 9.2 Sherly and Ali’sorganization for “two bluesseparated by one yellow”

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 orga-nizational strategies. Michelle built towers randomly; she said it was without “anyset plan,” and she rearranged the towers into pairs that she called “twos.” (These


B

Y

B

Y

B

B

Y

Y

B

Y

Y

B

Y

B

B

Y

Y

Y

B

B

Y

B

Y

B

Case 1: blue on top Case 2: blue on secondposition from top

Case 3: blue on thirdposition from top

Fig. 9.3 Robert’s three casesof four-tall towers withexactly two blue cubes

were pairs of opposites like those built by Sherly and Ali, described above.) Robertimmediately built and organized the towers by cases, focusing on the blue cube; hisfive cases were zero, one, two, three, and four blue cubes.

When asked to show that he had all the possible four-tall towers with exactlytwo 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 thetop, while he moved the lower blue cube down one position each time he built anew tower. When the lower blue cube had been moved down to all of the possi-ble positions, the upper blue cube was moved down one position. The process wasrepeated until it was not possible to move either blue cube down. With these threecases, Robert demonstrated that there were six towers in the two-blue case. Refer toFig. 9.3 for a diagram of Robert’s organization.

Robert and Michelle next worked on three-tall towers. Robert showed that thethree-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 reasoningforeshadows Robert’s use of inductive reasoning later in the session. Robert andMichelle went on to chart their results for the total number of towers two-tall throughfive-tall. See Fig. 9.4.

Their entry for two-tall towers is incorrect (the actual number is 4), but the restof the numbers are correct. They found a pattern that they believed applied to towersof height 3 and taller, and Robert used that pattern to predict that there would be 64six-tall towers. Their rule for the number of towers was 2h, where h is the height ofthe tower.

The researcher then asked them to consider the question of how many five-talltowers 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 andthen moving it successively lower in the tower. Robert and Michelle did more than

Height Number of towers 2 2 3 8 4 16 5 32 6 64 (prediction)

Fig. 9.4 Robert andMichelle’s chart for numberof towers for height 2–6


Height # of towers with exactly 2

blue cubes Difference 2 1 – 3 3 2 4 6 3 5 10 4 6 15 5 7 21 (prediction) 6 8 28 (prediction) 7

Fig. 9.5 Robert’s results forthe number of towers withtwo blue cubes

was asked, finding that for towers with heights two, three, four, five, and six, thereare 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 28towers with exactly two blue cubes.

Robert then looked for an explicit formula for the number of towers with exactlytwo blue cubes for any height. With his explanation for why a formula would beuseful, we observe Robert thinking like a mathematician:

What if someone just, they just did this problem for the first time, and they just came upwith like how many two yellow for fifteen, and they wanted to find out for fourteen withoutdoing it? How would they do that? . . . We are not going to sit down and write out one plustwo plus three plus four plus five . . .

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

( )( ) ( )( )1! .522 2 ! 2! 2

h hh h hhh

⋅ −−

−⎛⎝ ⎛

⎝

= ==

There was an extended discussion of this formula; the researchers wanted toknow where 0.5 came from and how Robert thought of it. He said, “I don’t know, itjust seems to work.” Again thinking like a mathematician, Robert wondered aloud ifthe formula would work for towers build from three colors. (He conjectured that the2’s in the formula would be replaced by 3’s.) Although Robert did not pursue thisthought, this led to a discussion of the general towers formula. A researcher askedthe students to illustrate how powers of 2 gave the numbers of towers for variousheights. In the following discussion, we see how Robert, initially confused aboutthe meaning of 21, comes to an understanding with the help of Michelle and goeson to describe the use of the general rule.


RESEARCHER: Show me what it would be for two to the first. What would thetowers be?

ROBERT: Uh, two blues. This and this. [Robert indicates one tower withtwo blue cubes, and one with two yellow cubes].

MICHELLE: It’s just ones, so it would just be one, one.ROBERT: Yeah, total combinations like-RESEARCHER: Two to the first would be, what would they be, show me.ROBERT: Right here.RESEARCHER: No.MICHELLE: No, no. That isn’t two to the first; this is two to the first.

[Michelle holds one single blue cube and one single yellowcube.]

ROBERT: Ah.RESEARCHER: Okay, so two to the first would these guys. Two to the second,

according to your theory would be how much?ROBERT: Oh, right here. Four. We have them right here. [Robert indicates

the four two-tall towers that were already built.] There’s four.RESEARCHER: How did you get from here [one-tall towers] to here [two-tall

towers]?ROBERT: We just built on top of them, I guess.

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

ROBERT: And then here [the set of four two-tall towers] you have all youcould 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 ablue and a yellow to the top. All the way through, and then –you know what I am saying?

RESEARCHER: Does that make sense?ROBERT: Yes, it does.RESEARCHER: Does it keeping going?

B

B

Y

Y

B

YB

Y

Y

B

Fig. 9.6 Robert generatesfour two-tall towers from thetwo one-tall towers


ROBERT: Yeah, I guess it would keep going on forever. That’s why thatthing 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 Magdaand Angela. They set up towers as shown in Fig. 9.6.

A portion of their discussion follows.

MAGDA: You kind of add that one on top of that color?ROBERT: 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 yellowand a blue and took that and added a yellow and blue like for all ofthem. [Robert indicates each two-tall tower in turn.]

ANGELA: Oh.ROBERT: 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.ANGELA: That’s very lovelyROBERT: She said it branches.MICHELLE: It branches.ANGELA: 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 (“Therewould be three little thingies. . . . It would be a lot bigger”), and she added, “Wouldyou like to see our theory? It is x to the n.” Robert responded, “Yeah, we have thatsame theory.” He directed Michelle to change their formula from 2h (the total num-ber of towers of height h when there are two colors) to xh, where x is the number ofcolor choices and h is the height of the tower.

Four students (Robert, Michelle, Magda, and Angela) worked as a group to writeup this result; Robert dictated, Michelle did the writing, and Magda and Angelaobserved. Robert said they had to carefully explain their notation so that any readerwould understand what they had written and so that all the students would be ableto explain it. Refer to Fig. 9.7 for their write-up.

Xh

X = # of colors

h = height

Tells all combinationsFig. 9.7 The group’s generaltowers result


9.3 Discussion

In this chapter, we discussed a session in which students working in small groupsand in a larger group gave specific and general answers, with justifications, tomathematical questions. They found formulas to answer specific towers questions;they generalized the formulas to handle general towers questions; they often gaveconvincing justifications for their answers, using methods including controlling forvariables, justification by cases, and induction; and they explained most of thepatterns they found by showing how the patterns made sense in the context of theproblems. Although Robert sometimes took the lead in moving further and creat-ing understandable explanations, all students (even those separated and new to thestudy) worked diligently on the problems, took care to explain their thinking, andmade convincing arguments and justifications.

In the next chapter, we observe another cohort of 10th-grade students working onthe towers and pizza problems. They too found ways to make sense of the problems,generalize answers, and provide convincing justifications for those answers. In addi-tion, their work led them to a deeper understanding of the binomial coefficients andPascal’s Triangle.

Chapter 10Representations and Connections

Ethel M. Muter and Elizabeth B. Uptegrove

Dates and Grade: December 1997 through March 1998; Grade 10Tasks: Towers and pizzasParticipants: Ankur, Brian, Jeff, Mike, and RominaResearchers: Carolyn Maher and Robert Speiser

10.1 Introduction

In the previous chapter, we viewed a cohort of high school students from the lon-gitudinal study as they explored the towers problems. In this chapter, we observe adifferent cohort of students also exploring the towers and pizza problems. In the fivesessions discussed here, spanning December 1997 through March 1998, the five stu-dents in this cohort were reintroduced to the towers and pizza problems, which theylast explored in elementary school as described earlier in Chapters 5 and 6. Theyfound general solutions to those problems and a way to organize their solution liststo prove that all solutions were present. They recognized that those problems wererelated to each other, to the binomial coefficients, and to Pascal’s Triangle. Theymade use of their understanding of the structure of those problems to form prelimi-nary ideas about the meaning of Pascal’s Identity. We show how their developmentand use of a sophisticated general representation scheme helped them make theseconnections 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-hafter school sessions that continued throughout high school. At their first meeting,



106 E.M. Muter and E.B. Uptegrove

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 studentshad enthusiastically tackled the problem by randomly generating combinations. Asdescribed in detail in Chapter 5, in approximately 15 min they found all 16 pos-sibilities by using an alphabetical coding scheme to represent each pizza as it wasgenerated.

In tenth grade, working on the three-, four-, and five-topping pizza problem, Jeffand Romina utilized an alphabetical coding scheme similar to the one that they usedin fourth grade (p for pepperoni, m for mushroom, etc.). Ankur and Brian used anumerical 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 ofthe combinations. The binary representation became a useful tool in many of theirfuture discussions of combinatorial problems.

As the first four students discussed the problem, they realized that they needed acommon notation and adopted the alphabetical model. They kept the plain pizza sep-arated, and so they found 7 possible pizzas for the three-topping case (plus plain) and15 pizzas for the four-topping case (plus plain). When they found 30 five-toppingpizzas (plus plain), they realized it did not fit the pattern. They hypothesized thata doubling rule might be involved and they decided to rethink their solution. Atthis point, Mike re-entered the discussion in order to introduce his binary codingscheme. He proposed that pizzas be represented by binary numbers; a four-toppingpizza would be represented by a four-digit binary number, with a 1 in the kth digitrepresenting the presence of the kth topping and a 0 representing the absence ofthe kth topping. For example, all one-topping pizzas are represented by all four-digit binary numbers with exactly one 1: 0001, 0010, 0100, and 1000. This codingscheme was more easily generalizable than the letter code scheme: to add anothertopping, just add another binary digit. After listening to Mike’s explanation of thebinary code, the group made a connection between the two representations (lettercodes and binary notation: use the letters that stand for toppings as column headersfor 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 todescribe the solution to the pizza problem gave him an insight into a generalizationabout the number of pizzas that can be created from n toppings; Mike hypothe-sized that the answer to the n-topping pizza problem is 2n. The group discussed thenumerical solution for some time – there was some confusion about whether thecoefficient might be n – 1 or n + 1, or whether the answer might be 2n – 1, possibly

M

1

0

0

1

0

0

0 onion pizza

mushroom pizza0

O P S

8 4 2 1

Fig. 10.1 Students’ tablelinking topping codes andbinary notation [annotationadded]

10 Representations and Connections 107

due to the uncertainty about how to count the plain pizza, but ultimately all agreedthat the solution was 2n.

As they were wrapping up the session, the researcher asked if the problemreminded them of other problems, and Brian mentioned towers: “Every thing weever do always is like the tower problem.” In order to investigate the possible rela-tionship between the two problems, the students worked on the three-tall towerproblem and concluded that the answer was the same as for the three-topping pizzaproblem; there are eight three-tall towers, just as there are eight possible three-topping pizzas. Because they were focused on relating the pizza toppings to thecubes’ 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-redtower, 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 possiblerelationship between pizza and towers problems. Although they were asked to con-sider only the two-color towers problems, they kept returning to the question of howto count the possible number of towers when there were cubes of three or more col-ors. Looking at this issue led them to the realization that when the height of thetowers is the only variable under consideration, the towers problem is identical tothe pizza problem. This time, they mapped the height of the tower to the number ofpizza topping choices (contrary to the previous week, when they attempted to mapnumber of colors to number of pizza topping choices). A portion of their discussionfollows.

JEFF: If the only variable we’re changing is height, it stays the same.MIKE: It would be the same as the pizza.JEFF: What would that be like changing on the pizza, though?MIKE: You could change the height, the number of toppings.JEFF: Changing the height would be like changing the number of toppings.ANKUR: Yes.JEFF: Changing the color would be like, what?ANKUR: Say what you just said again.JEFF: 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 twotoppings to three toppings.

ROMINA: Okay.ANKUR: Okay.

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


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

BRIAN: Two has to stand for something.MIKE: 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: Okay. For the pizza problem, the 2n [meaning 2n], the two representseither topping or no topping. Right?

MIKE: There’s two different possibilities for each.JEFF: That’s why there’s two. We didn’t know, I don’t think we explained

that last time, why it was two.JEFF: Topping or no topping, and that’s what the two is. Now the n, Romina.ROMINA: Is toppings.JEFF: The number of toppings.

Mike used binary notation again in this session, this time using it to represent thetwo colors of the towers problem. He noted that instead of relating binary digits tothe 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 boththe two-topping pizza problem and the two-tall towers problem. The column headers1 and 2 represent the two pizza topping choices and the two levels of the tower. The1 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 thepresence or absence of the topping. If the labels stood for positions in towers, thezeros and ones would represent the color of a cube; e.g., one would represent blueand zero would represent red.

During the second half of this session, at the researcher’s request, the studentsexplored geometric interpretations of the binomial expansion. Figure 10.3 repre-sents their drawing of (a+b)2. They went on to spend over half an hour working ondrawings and three-dimensional models for (a+b)3, although no model was entirelysatisfactory to them. These investigations can be seen as preparation for their laterwork describing the isomorphic relationship among the towers and pizza problemsand the binomial expansion.

2

1 0110

0110

010

1010

1 p m

Fig. 10.2 Mike’s listing oftwo-tall towers andtwo-topping pizzas


a2

b2

ab

bab

b

a

a

Fig. 10.3 Geometricinterpretation of (a + b)2

In this session, the students gave the researchers clear and convincing explana-tions of the isomorphism between the pizza and towers problems and of the meaningof the components of the formula, and it appeared that they had a firm grasp ofthe underlying structure of the problems. But a few months later (in sessions dis-cussed later in this chapter), we see them return to their focus on relating numberof pizza toppings to number of colors (instead of to the height of the towers), onceagain deciding that the problems were similar but not identical. This illustrates howimportant it is to revisit problems and re-examine solutions, to solidify and expandstudents’ understanding.

10.4 Session 3: Towers and the Binomial Expansion

When the students met in January 1998 after the holiday break, they returned to thetopic of towers. The researchers gave them a problem from fourth grade: When youare choosing from red and yellow cubes, how many five-tall towers can you buildcontaining exactly two red cubes? They immediately answered “ten,” and they werethen challenged to provide an explanation. Mike and Ankur provided a justificationin approximately 2 min, using Mike’s binary coding scheme, with 0 representinga 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 redcube fixed in the top position and then moving the second red block into successivelylower positions until it reached the bottom position. This process was repeated, hold-ing the red cube fixed in the second and then third and fourth positions. Figure 10.4shows their original list followed by the re-organized list of all ten towers. This

00011

00101

00110

01001

01010

01100

10001

10010

10100

11000

01001

00101

01010

10001

10010

10100

00011

00110

01100

11000

Fig. 10.4 Mike and Ankur’stwo lists of five-tall towerswith exactly two red cubes


illustrates the importance of revisiting and re-explaining answers; in the processof explaining their solution, they organized the list so as to make it clear that allpossibilities were accounted for and none were missing.

The other group (Jeff, Romina, and Brian) worked on their solution for approxi-mately 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 justi-fication based on cases; their cases were (1) all red: 1 tower, (2) one red and fouryellow: 5 towers, (3) two red and three yellow: 10 towers, (4) three red and twoyellow: 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 Ankurand Mike were done, Ankur proposed a problem that became known as Ankur’sChallenge. 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 ofcombinatorics. She told the students that asking how many five-tall towers haveexactly two red cubes is the same as asking how many combinations there are whenselecting two of five objects. She showed four different ways to write this, as shownin 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, shewrote the expansion of (a + b) to powers 0–3, drew Pascal’s Triangle, and then askedthe students to think about the relationship. (Refer to Fig. 10.6 for the researcher’sdrawings).

In the following excerpt, the researcher hinted about the relationships that thestudents were in the process of discovering.

RESEARCHER: The question is, what’s the relationship here? How could youmodel it? How could you show this relationship? And why doesit work? That’s the question. So that’s sort of the direction. Areyou interested in knowing that? I think you have the bits andpieces to put it together.

ANKUR: Some of the pieces are really small.RESEARCHER: They’re bigger than you think. You’ve been working on this for

a long time.

5C2 C5,252 C2

5⎛ ⎝

⎞ ⎠

Fig. 10.5 Notation for selecting two of five objects

Fig. 10.6 The binomialexpansion and Pascal’sTriangle


ROMINA: Is this what we did today though?RESEARCHER: You’ve been dealing with some of this today. So think about it.ANKUR: So are all of the things we learned for the past 8 years sort of

combined into one thing?BRIAN: Imagine that.

Immediately following that discussion, the researcher asked the students to makeconcrete the numbers in Pascal’s Triangle, by thinking about them in a “very realway” (linking them to towers problems).

RESEARCHER: When you first came in here today, you produced that numberten. [She refers to the first 10 in row 5 of Pascal’s Triangle – 15 10 10 5 1.] Right?

ANKUR: Yes.RESEARCHER: And what problem were you solving?ANKUR: Two were red and three something else.RESEARCHER: Okay. So you can think of that ten in a very real way, if you

want to, right?ANKUR: Yeah.RESEARCHER: Can you think of those other numbers in a real way? Does that

help?ANKUR: The 1 is, in 1 4 6 4 1, the 1 represents all red. The other 1

represents all yellow I guess . . .

RESEARCHER: All red and all yellow for what?ANKUR: Of four high.RESEARCHER: So this is four high. [The researcher points to row 4 of Pascal’s

Triangle.] And these are all red. [The researcher points to thefirst 1 in that row.]

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

10.5 Session 4: Pizzas, Towers, and Pascal’s Triangle

In this session, three students (Ankur, Jeff, and Romina), in a first meeting with visit-ing researcher Robert Speiser, explored the relationships among the pizza problem,


the towers problem, and Pascal’s Triangle, and (for the first time) they discussedPascal’s Identity in terms of operations on physical objects (adding cubes to towers).

When Professor Speiser asked the students about their recent work, the studentsdid not mention a relationship between the towers and pizza problems. Instead, asthey had initially done back in December, they maintained that the problems weredifferent. Romina, recalling Ankur’s earlier reasoning, said that red-blue cubes ona tower are different from blue-red cubes, but sausage–pepperoni is the same aspepperoni–sausage. Ankur added that a five-topping pizza problem is like a five-color towers problem. Jeff agreed, saying that a tower could have two of the samecolor but that a pizza could not have pepperoni–pepperoni. Although all three hadparticipated in the earlier, correct, discussion of the relationship between the tow-ers and pizza problems, they recalled now only their own original ideas. They saidnothing about the height of the tower being connected to the number of toppings, orhow on-the-pizza/not-on-the-pizza could be made to correspond to blue/red cubesvia the binary representations 0 and 1.

The researcher reminded the group about the combinatorics notation that hadbeen introduced a month earlier and she reminded them how the notation was relatedto the five-tall towers problem. She went on to demonstrate the binomial expansionand to ask explicit questions: What are the relationships, if any, among (a+b)5, thefive-tall towers problem, the five-topping pizza problem, and the fifth row of Pascal’sTriangle? This question is significant in terms of the students’ later work, as it rep-resents the first time the students were asked to think about a four-way link, amongthe binomial expansion, the two combinatorics problems, and Pascal’s Triangle. Thestudents were able to make the connection, evoking and expanding Ankur’s explana-tion from the previous month of how the four-tall towers problem could be found inrow 4 of Pascal’s Triangle. (This also anticipated their night session explanation ofentries in Pascal’s Triangle in terms of pizzas.) In the following excerpt, the studentslinked the binomial expansion to the towers problem.

RESEARCHER: What are the a’s and the b’s here?ROMINA: Colors. . . .

ANKUR: a and b is red and blue. . . .

RESEARCHER: What do you mean by red and blue?ANKUR: a is red and b is blue. That’s [red-blue tower] a b. So b a would

be a blue red.RESEARCHER: So how, if you have them in front of you, how would they look

different?ANKUR: 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-talltowers problem in row 5 of Pascal’s Triangle, and then Jeff and Romina locate thepizza answers in row 6. (Row 6 of Pascal’s Triangle contains the numbers 1 6 15 2015 6 1.)

ANKUR: This [1] is no red.JEFF: Yeah.


ANKUR: So there’s one with no red. There’s six with one red. . . . There’s fifteenwith two reds. Twenty with three reds. Six with five reds.

JEFF: And one with no-ANKUR: And one with no-ROMINA: No.ANKUR: No. Six reds.JEFF: One with six reds. . . . All right. Now. What does that have to do with

pizza?ANKUR: Just relate the tower problem to the pizza problem.JEFF: Well, we’re saying that this [1] is a pizza with just plain.ROMINA: Yeah. That’ll be the plain pizza.JEFF: Plain. This [6] is with all your six toppings.ROMINA: That’s with one topping.ANKUR: You can’t exactly relate these numbers to the pizza problem.JEFF: Well, we’ll try really quick.ROMINA: Yeah. You can. ‘Cause this [1] is plain, just plain pizza.ANKUR: And what will the other 1 represent?ROMINA: With everything on it.ANKUR: Okay.JEFF: So this is plain.ANKUR: Okay. Six with-JEFF: With one of each. Fifteen is with-ROMINA: Two toppings.JEFF: Just two toppings out of your six. Twenty is with three toppings.

Fifteen is with the four toppings. Six is with the five toppings.ROMINA: Five toppings.JEFF: And the other one is-ROMINA: And the one is with all of them.JEFF: Like the supreme.ROMINA: Is that good?ANKUR: Cool. We’re on fire today.

Thus Ankur, Jeff, and Romina used the two combinatorics problems they knew inorder to explain the numbers in Pascal’s Triangle. This was the first time they wereobserved connecting the pizza problem to Pascal’s Triangle. Although Ankur hadinitially been reluctant to attempt a definition of the relationship between Pascal’sTriangle and the pizza problem (“You can’t exactly relate these numbers to the pizzaproblem”), he still participated in the discussion and at the end expressed satisfactionwith their work. (“We’re on fire today.”) We noted earlier that the students receivedno special concrete rewards for participation in the study. Ankur’s remarks illustrateour belief that the intellectual enjoyment involved with solving difficult problemswas a factor in the students’ continuing involvement in the study.

After explaining the link between specific numbers in Pascal’s Triangle and thepizza and towers problems, the students described an instance of the addition rule interms of towers problems. They explained the instance of Pascal’s Identity shown in


Add blue

1 5 10

20 15 6 11561

10 5 1

Add red

All blueAll red

Fig. 10.7 One instance ofPascal’s Identity whichstudents linked to towersproblems

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

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

JEFF: The tens show-ANKUR: The tens show two of one color.ROMINA: And three of another.ANKUR: 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 either-JEFF: You could either put a red or like blue.RESEARCHER: The first ten in that row of five high has two reds and three

blues? We’re counting reds?ANKUR: Yes.RESEARCHER: And the second ten has-ANKUR: Three reds and-RESEARCHER: -three reds and two blues. Now coming down here, the twenty

is supposed to count the ones that have three of each.ANKUR: Three red. Three reds and three blues.JEFF: Right.RESEARCHER: So how do the two tens add to give the twenty?ANKUR: 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 ablue on top of each one.

This is the first instance where a connection was made between Pascal’s Identityand 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. Muchof the session was devoted to attempts by Ankur, Jeff, and Romina to explain toMike specific instances of Pascal’s Identity in terms of towers and the binomialexpansion. This session also provided another example of the importance of revisit-ing 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


1 3 3 1

14641

Fig. 10.8 Another instanceof Pascal’s Identity discussedin terms of towers problems

terms of towers: to create the 20 six-tall towers that have three red and three bluecubes, add a blue cube to the 10 five-tall towers with two blue cubes and add a redcube to the 10 five-tall towers with two red cubes. But this time, when asked toexplain 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 disassemblingand re-assembling existing towers, an approach that surprised and confounded theresearcher.

They correctly mapped the numbers 1 and 3 into tower groups (1 = white-white-white and 3 = blue-white-white, white-blue-white, and white-white-blue, as shownin Fig. 10.9).

But they tried to explain the 4 by breaking apart the tower representing 1 towerand distributing its cubes among the other three towers. After the researcher ques-tioned this method, Mike gave a different explanation for how to represent the 4.

JEFF: We’ve got this [the white-white-white tower, representing 1].And we’re saying how this goes together. [Jeff has assembledthe three towers each with one blue cube to represent the 3.Refer to Fig. 10.9.] We’re saying- [Jeff starts to dismantle thewhite-white-white cube.]

RESEARCHER: No. No. Don’t take that apart. Because-JEFF: Well, that’s why I made this. So I could.ANKUR: We made another one so we can take that one apart. . . . And

show you.RESEARCHER: You mean, you mean, you mean you get the four by taking

something apart?ANKUR: You’re not taking it apart.ROMINA: You’re not taking it apart; you’re just seeing how they go

together. . . .

MIKE: 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 fourblocks [towers] is going to have something added to them toequal the same thing.

ANKUR: Yeah.

1 3

B

B

B

W W

W

WWW

W

W

WFig. 10.9 Students mapspecific towers to 1 and 3 inrow 3 of Pascal’s Triangle


MIKE: These blocks [towers] are going to have, they’re going to have awhite block added to them. [Mike indicates the three three-talltowers with one blue cube.]

ANKUR: They’re going to have a b added to them.MIKE: And this one’s [the white-white-white] going to have a, a blue

added to it.ANKUR: An a added to it.MIKE: 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 ofMike and Ankur’s suggestions.]

The other students accepted Mike’s explanation and apparently this time theycomprehended the process, as evidenced by their later work in the night session (seeChapter 12) and subsequent interviews. In addition, Ankur reiterated a link notedin the February session, observing that the a’s and b’s in the binomial expansioncould be connected to the blue and white cubes, respectively, in the towers. Thesame connection was explained a year later during the night session.

Next, the researcher asked the students to relate the tower problems to binarynotation and the pizza problem; she said, “If you had to make up a pizza problemto model this row [row 2 of Pascal’s Triangle], what’s the pizza problem?” Ankurreiterated the position that he had taken in two previous sessions, that a peppers andpepperoni pizza is the same as a pepperoni and peppers pizza; it appeared that hehad not yet firmly established that there was a connection between the two choicesof cube colors for each cube in a tower and the two choices for each topping – onor off the pizza. Although the group noted that the nth row of Pascal’s Trianglecould be linked to the n-topping pizza problem, they did not propose an explana-tion about how to use the numbers in the nth row to enumerate pizzas. When theresearcher asked for clarification and Ankur insisted that there was no relationshipbetween colors and pizza toppings, Mike interrupted with his own explanation. It is

WW

31

B

B

B

B

B

B

B

W

W

W

W W W

W

W W W

W

W

W

W

W

W

W W W

Fig. 10.10 Mike and Ankurillustrate 1 + 3 = 4 by addingcubes to towers


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

RESEARCHER: Now wait. Now I’m lost again. What, what, what was this?. . . [The Researcher indicates the single white and blue cubesrepresenting row 2 of Pascal’s Triangle.]

ANKUR: The colors don’t, don’t look at the colors.MIKE: No. No. No.ANKUR: Just look at this [Pascal’s Triangle]. . . . But the colors don’t

specifically represent anything.ROMINA: Yeah.MIKE: Yes. It does.ANKUR: No, it don’t.MIKE: Topping. [Mike points to the blue cube.] Or no topping. [Mike

points to the white cube]. Just say like that. And if you look atit like this, you know.

ANKUR: So all of the whites are no topping?MIKE: 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.JEFF: All right.ANKUR: Okay.ROMINA: Okay.MIKE: This [the blue-white-blue tower] would be a pizza-ROMINA: Oh. With the one. Ooh.MIKE: -with two different toppings, without the other, third topping.ROMINA: That’s what I was asking.ANKUR: Okay.

Pizza withSecond Topping

Pizza withFirst Topping

Pizza withBoth Toppings

Plain Pizza

W W

W W

W

W B

B

Fig. 10.11 The students’ linkbetween two-tall towers andtwo-topping pizzas


JEFF: . . . Well, yeah. Well, if you’re just saying that this [the white-white-white tower] is the pizza with three no toppings, it’s plain.

ROMINA: It’s just a plain pizza.ANKUR: All right. All right. So that’s [blue-blue tower] two toppings.ROMINA: Yeah.JEFF: Yeah. All right. So.MIKE: That’s [white-white tower] . . . a choice of two, but you want it

plain.ANKUR: You have a choice of two toppings.JEFF: Yeah, so this is, this [blue-blue tower] is choice of two using

two. This [blue-white tower] is choice of two using one.ANKUR: Two using one.JEFF: This [white-blue tower] is choice of two using the other one.ANKUR: That’s using the other one. And that’s [white-white tower] using

nothing.ROMINA: Yeah.RESEARCHER: And that’s all the possibilities?ANKUR: Yes.ROMINA: Yeah.RESEARCHER: You like that?ROMINA: Oh, wow!

During this episode, the other three members of the group immediately acceptedand built upon Mike’s brief remarks. All he had to say was “topping” and “no top-ping,” and all three of the others began immediately to form connections betweenspecific individual towers and specific pizzas. This represented the fifth discussionof the pizza problem in 4 months, and at least three members of the group apparentlybegan 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 top-pings could be selected in any order). But it appears that this discussion helpedthem finally to make sense of the isomorphic relationship, because the pizza prob-lem was the one that the group selected during the night session a year later, toexplain Pascal’s Identity.

10.7 Discussion

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


solution of the pizza problem. Mike recalled an episode from an eighth-grade classand applied his previously constructed knowledge to a totally new situation. Hisintroduction of a coding system, the zeroes and ones of the binary system, to thejustification 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 prob-lems that had been explored in earlier years, retrieving earlier ideas and imagesas they built solutions and justifications. These ideas and images sometimes reap-peared just as they were formed in the prior occurrences. In the third session of thesophomore year, we see Mike and Ankur’s swift production of a justification for thenumber of five-tall towers with exactly two red cubes. They reproduced a justifica-tion by cases that had originally been built in their fifth-grade classroom, but usingMike’s binary notation. In addition, they offered a second justification, utilizing astrategy that depended on controlling for variable that was first introduced by Ankurwhile solving a pizza problem in grade five. His ownership of this strategy allowedhim to adapt it for use in the isomorphic block tower problem.

For the same problem, Jeff retrieved a strategy used during that same fifth-gradesession. Mike and Ankur had enthusiastically participated in the whole classroomdiscussion, 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 thetowers that he built. Jeff’s partners, Romina and Brian, also had more difficulty inproviding a justification during the session in their sophomore year. In grade five,while Mike and Ankur were active participants in the classroom discussion and Jeffwas quietly pursuing his own line of thinking, Brian and Romina were in anotherclassroom. Although they worked on the five-tall towers problem, their class didnot offer a convincing justification for the answer to that problem. The difficultiesexperienced by these students as they worked on the block tower problems as tenthgraders might be explained by the absence of some earlier experiences. They con-structed the images and representations to the block tower problem for the first timein this tenth-grade experience.

In the attempt to think about the potential connection between the pizza andblock tower problems, the students came to discuss many powerful mathematicalconcepts. While they were able at an early point to determine that the answer to then-topping pizza problem is 2n, they came to this number by recognizing the patternof {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 thediscussing the possible relationship between pizzas and block towers; then that theycame to see that the base 2 represented the presence or absence of a topping. Thisrealization came, not from working on the pizza problem, but instead as a result oftheir search for an answer to the three-color four-tall tower problem. Thus we seethat the opportunity to work on open-ended problems and follow paths determinedby the interest of the moment can lead to greater understanding of other problems. Inthis case, the opportunity to investigate an isomorphic problem provided the studentswith the tools necessary to complete the formulation of the imperfectly developedearlier idea.


In summary, the students investigated isomorphic problems in combinatorics andused them to explore how Pascal’s Triangle grows and to make sense of Pascal’sIdentity. Between December 1997 and March 1998, they first found general solu-tions to the pizza and towers problems, using letter and number codes and binarynotation to enumerate the pizzas and towers. Then they organized their lists of solu-tions, organizing the pizza problem solutions according to number of toppings andthe towers problems solutions according to the number of cubes of one color. Theselists not only provided a way to show that all cases were present, but they alsoprovided the means to associate those cases with the numbers in Pascal’s Triangle.In discussions with the researcher and other researchers, the students described theisomorphic relationship between the pizza and towers problems. Their extensiverepertoire of representations proved essential; in this process, they made use ofwords, written inscriptions, and concrete materials (as when Mike held up a bluecube and a white cube and said “topping” and “no topping”). The opportunity torevisit problems also proved crucial, as students often needed to have two, three, ormore discussions on the same topic before critical ideas were firmly established.

In the next chapter, we observe another cohort of students who also make senseof the relationships among towers, pizzas, and Pascal’s Triangle. They broughttheir own experience, their own representations, and their own ideas to the prob-lem, but they too used personal representations, communicated findings, and madegeneralizations that showed their increased understanding of these problems incombinatorics.

Chapter 11Pizzas, Towers, and Binomials

Lynn D. Tarlow

Date and Grade: March 1, 1999; Grade 11Tasks: Pizzas, Towers, and Pascal’s TriangleParticipants: Amy-Lynn, Angela, Magda, Michelle, Robert, Shelly, Sherly,

and StephanieResearcher: Carolyn Maher, Alice Alston, Regina Kiczek, Ralph Pantozzi

11.1 Introduction

In the previous chapter, we observed a cohort of tenth-grade students as theyinvestigated the connections among the pizza problems, the towers problems, thebinomial coefficients, and Pascal’s Triangle, leading to their increased understand-ing 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 sub-groups with different backgrounds (from students new to the study to students whohad been in the study from the start) worked together, sharing their ideas, comparingrepresentations, and discussing relationships among problems. Through their col-laborative work, they too came to discover generalized rules for the pizza and towersproblems, see how both problems were related to Pascal’s Triangle, and explain themeaning of Pascal’s Identity through the use of two different metaphors: solving thepizza problems and solving the towers problems.

The eight students at this session were organized into two groups of four. Eachgroup worked independently. The four students at Table A (Robert, Stephanie,Shelly, and Amy-Lynn) had participated in the tower and pizza investigations ingrades 3–5 through five. Only Robert had participated in the previous 11th-gradetower 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, USAe-mail: [email protected]


122 L.D. Tarlow

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 fromthe following toppings: pepper, sausage, mushrooms, and pepperoni. How many differentchoices for pizza does a customer have? List all the choices. Find a way to convince eachother 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 thatshe “just did this in school, combinatorics stuff,” but she was not able to remembera formula, although she thought it involved factorials. Her remarks showed that sheremembered 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 fivefactorial plus four factorial plus three factorial plus two factorial plus one factorial. . . . Ican’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 combina-tions, not of five factorials), although the solution is often seen as a power of 2: twochoices (on or off) for each of four toppings give 16 possible pizzas:

(40

)+

(41

)+

(42

)+

(43

)+

(44

)= 24 = 16

Nevertheless, Amy-Lynn agreed that factorials were appropriate, and so Shellytook the sum of 1! to 5! and got an answer of 153. Although no one questionedthis use of factorials, the group decided that the answer needed to be verified; andso they started to list the possible pizzas. Stephanie and Shelly discussed possiblestrategies:

STEPHANIE: Do we just want to, um, plot out the pizzas, like with shirts andpants or towers? Do you know what I’m talking about?

SHELLY: Yeah. The tree diagram type thing.STEPHANIE: 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 repre-sent 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 tobe combined with other toppings. After a discussion about real pizzas, they decidedthat 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 duplicatesthat resulted. They compared answers and, unprompted, decided to create a new

11 Pizzas, Towers, and Binomials 123

pl

peps m

pe

ms pe m pe spe

pl = plainpe = pepperss = sausagem = mushroomspe = peperoni

Fig. 11.1 Shelly’s tree of all16 pizzas when there are fourtoppings to choose from

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

SHELLY: One, four, wait a minute. One four six four one, so the next onewill be one. This is the-

STEPHANIE: The triangle.SHELLY: The triangle.STEPHANIE: Yeah. So the next one is one five ten ten five one.SHELLY: We’re done. [Shelly laughs.]STEPHANIE: But what does that mean? [All three girls laugh.]SHELLY: I don’t know. . . .

STEPHANIE: But what it, like, what does one four six four one. That meansnothing to me.

SHELLY: It means nothing to me either, but it’s the pattern we saw.STEPHANIE: So we have a pattern, but how do we apply it to getting sixteen

pizzas?SHELLY: That would be the problem.

Stephanie and Shelly followed up with a discussion of how the numbers in row4 could be matched to the answers to the four-topping pizza problem.

STEPHANIE: So, well, okay, let’s figure this is saying that we have one plainpizza.

SHELLY: Uh-huh.STEPHANIE: And then we have four pizzas with two toppings?SHELLY: With one. Because it’s the plain and then with the one topping.STEPHANIE: Okay. So we have four pizzas with one topping. And we have four

pizzas with two toppings. Oh, no, we have six pizzas with twotoppings, four pizzas with three toppings.

124 L.D. Tarlow

SHELLY: And one pizza with four toppings.STEPHANIE: Okay.

Then they used the next row of the triangle to determine the number of possiblepizzas with five available toppings, 32 (the numbers 1 5 10 10 5 1 representing thenumbers 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 rowsof Pascal’s Triangle from existing rows (Pascal’s Identity) in terms of pizzas. Thestudents had some trouble figuring out how that would work. Stephanie said, “I justcan’t get past the fact that you can’t make a pizza out of other pizzas. I think maybeif it was applied to something else I could look at it differently.” Robert suggestedthat they try relating the towers problem to Pascal’s Triangle. Robert started makingdrawing in order to follow this line of thinking, but the others went back to thinkingabout pizzas.

Stephanie and Shelly tried to use pizzas to try to explain one particular instanceof 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 Identitydid not make sense, “because, like, one is no topping, so adding one to three doesn’tmaterialize another topping.” She understood that, in general the nth row of Pascal’sTriangle represented all the possible pizzas that could be made with n toppings avail-able, but she could not see how one row generated the next one. She called overResearcher 4 and he suggested that she think of having made all eight three-toppingpizzas and then finding that another topping had become available. Then Stephanierealized that each existing pizza could either acquire the new topping or stay thesame. Therefore, the addition rule could be explained by thinking of each existingpizza either acquiring a new topping or staying the same. After Stephanie was sat-isfied that she understood how the addition rule worked, the researcher moved onto another topic with Robert. Robert had drawn rows 0–5 of Pascal’s Triangle, andto the right of each row he wrote the sum of the numbers and the sum expressedas a power of 2. (Refer to Fig. 11.3.) He gave the general rule: that the number ofpossible 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 com-binations 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

2

3 3

11

1

14641

11

1

Fig. 11.2 Two instances ofPascal’s Identity


1

1

1

1

1

1

1

11

1

15

4

32

3

4

5 10 10

20 = 1

21 = 2

22 = 4

23 = 8

24 = 16

25 = 32

6

Fig. 11.3 Robert’s diagramof Pascal’s Triangle andpowers of 2

The question of the role of the number 2 was temporarily deferred, as Researcher3 stopped by, and Stephanie demonstrated that she had made sense of the relation-ship between pizza problems and Pascal’s Identity by explaining two instances ofthe 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 withthe new topping. Okay, so this becomes, instead of one plain pizza, this is one pizza withone topping. Cause this one’s getting like the pepperoni thrown into it. And that producesthe 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 top-ping get an extra topping added on. So these become three pizzas with two toppings. Andthen 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; inone move the pizza remains the same and in the other move the pizza gets the newtopping. For example, a pizza with peppers could be moved to the left and remain apizza with peppers, or it could move to the right and become a pizza with peppersand mushrooms. Amy-Lynn then returned to the unanswered question about the roleof the 2 in 2n, Robert’s expression for the number of possible pizzas when n toppingsare available: “Maybe that’s where he got the two to the n; maybe that is where thetwo comes from.” Shelly and Stephanie agreed:

SHELLY: That makes sense. Yeah that’s where each of the twos comefrom.

STEPHANIE: That was good, that was really goodRESEARCHER: 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?STEPHANIE: Yeah, it drops down as a plain pizza.RESEARCHER: Because this one is the plain, I see. Okay, so your drop-down

idea is that it stays the same.STEPHANIE: It stays the same once. And it changes once. Where I guess Amy

got the two.RESEARCHER: Very interesting. Do you agree with this?ROBERT: Uh-huh.

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

126 L.D. Tarlow

Stephanie and Robert said that the numbers in row 3 (1 3 3 1) were for three-tall towers, and Robert added that the height was the same as the number ofavailable 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.” Theother students agreed; they wrote up their work as shown in Fig. 11.4. Stephaniethen described how part of the row in Pascal’s Triangle that would representn 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 num-bers in the 1 3 3 1 row of Pascal’s Triangle would represent in terms of towers, usingthe colors blue and red. Researcher 1 asked: “So why does one plus three give youfour? 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 visualizethe 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 thebottom.”

The group was asked to make a picture of Unifix cubes to illustrate the addition1 + 3 = 4 (the same process illustrated in Fig. 11.2 that they had explained earlier interms of pizzas). Stephanie drew diagrams of towers, using b for blue Unifix cubesand R for red, as shown in Fig. 11.5. Stephanie and Shelly then explained: Each ofthe three-tall towers gets a cube added on top to become a four-tall tower. The towerwith no red cubes gets a red cube (R), and the three towers that already have onered cube each get a blue cube (b). These towers represent the case of four four-talltowers with exactly one red cube in each tower.

Shelly said that it was easier to explain the “two thing” with the towers becausethere are only two colors. Stephanie agreed and added that with all the pizza top-pings, “it throws you off; you expect eight hundred pizzas.” Researcher 1 asked ifthere were another way to think about the pizzas. Robert supplied the explanation ofthe isomorphism: the number of available pizza toppings corresponds to the height

Fig. 11.4 Students’justification for theirgeneralization of 2n


bbb

Rb R

b

R

bbbR

bb

bbb

bb

R

bb

Rbb

Rb

b

Fig. 11.5 Stephanie’sillustration using towers for1 + 3 = 4

of the tower, and the two colors would indicate whether or not a topping was on thepizza. Researcher 1 pointed to the towers that Stephanie had drawn and asked whatthe b in a tower would mean if you were thinking of pizzas. Stephanie answeredthat it would mean that you either had or did not have the topping. Researcher 1then 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 indi-cated whether or not you would choose each of the four toppings. Stephanie addedthat it would be, for example, mushrooms “going all the way across,” and each towerthat had an R in that position would indicate that pizza did not have mushrooms onit. 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 prob-lem by making individual tree diagrams. This work was interrupted with a questionabout order: Angela asked whether a sausage and pepperoni pizza is the same as apepperoni and sausage pizza. Although the consensus was that this did represent thesame pizza, they decided to leave duplicates in the tree diagram and remove themlater.

After they prepared their tree diagrams, Magda noted that she originally had 24combinations on her tree diagram for the four-topping pizza, but after crossing outduplicates, she was left with only one pizza that had exactly four toppings; this ledthe group to realize that they would have to cross out a lot of pizzas. Angela found15 different pizzas that can be made when there are four topping choices. In order toconfirm this answer, the group agreed that each member would work on a differentcase: 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 hadonly one pizza; for the other three cases, Angela and Sherly made lists and Michelleused a tree diagram. They concluded that there were 15 total combinations; fourwith one topping, six with two toppings, four with three toppings, and one with fourtoppings, confirming Angela’s earlier answer. When Researcher 2 stopped by to askabout their solution, Michelle pointed out that they had not counted the plain pizza;therefore, they now had 16 pizzas.

128 L.D. Tarlow

Next the group continued their work for the five-topping pizza question, withanchovies as the new topping. Again they distributed working on the different casesamong the group, with Magda adding the five-topping case to her original four-topping case, and the others keeping the same cases. Using this procedure, theyfound all 32 five-topping pizzas. Angela realized that the number of possible pizzasdoubled with the addition of one topping, and so she conjectured that for the three-topping case, there would be eight possible pizzas. She confirmed this by findingthose eight pizzas.

Researcher 2 asked the students to explain their work. Sherly and Magdaexplained that they had found combinations by substituting the new topping,anchovies, for each of the other toppings in their previous combinations. In thisway, they would not have to cross out answers.

Seeing the doubling pattern with the pizzas, Magda recalled that there was alsoa doubling pattern in the towers problem, which they had not been able to explainbefore. Angela recalled the formula for towers from the previous session (“we cameup with that whole like x to the n thing”). She did not observe that the problemswere structurally similar, though, noting that order seemed to make a difference withtowers (a tower with a red cube on top of a yellow cube is a different tower fromone with a yellow cube on top of a red cube), whereas a pepperoni and mushroompizza is the same pizza as a mushroom and pepperoni pizza.

Researcher 2 asked the group to continue to explore finding pizzas with othertopping 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, thenumber of possible pizzas would be 64.

Then Magda had what was described as a breakthrough: She noticed that thenumbers they were finding were also seen in Pascal’s Triangle. A discussion withResearcher 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’sTriangle (see Fig. 11.6).

Fig. 11.6 Magda and Sherlydiscuss Pascal’s Triangle


MAGDA: I am thinking. One, three, three, one. I don’t know, maybe it hassomething to do with this.

RESEARCHER: Why don’t you put it on a new piece of paper cause you areabout to give out -

MAGDA: I don’t know if that works though. Does it?RESEARCHER: Let’s see.MAGDA: No, it doesn’t work because we don’t have the four in here, or

something. You know that, like -SHERLY: Pascal’s.ANGELA: Pascal’s, yeah.MAGDA: Yeah But it doesn’t work, so scratch out my idea.RESEARCHER: No, wait just a minute. Explain what you see and what doesn’t

work.ANGELA: Wait, wait. How does it work?MAGDA: But it doesn’t work for this one [row 5].ANGELA: Six, four and four. Yes it does. What are you talking about?SHERLY: Yeah.MAGDA: Oh wow, it does. Maybe my idea works.ANGELA: 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 lengthydiscussion, Angela said that the 6 in row 4 (representing six pizzas with exactly twotoppings) would be generated from the two 3’s in row 3 (representing three pizzaswith one topping and three pizzas with two toppings): add the new toppings (pep-pers) 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 piz-zas that already had two toppings each (sausage/pepperoni, sausage/mushroom, andpepperoni/mushroom) would be unchanged and become part of the two-toppinggroup at the next level.

Fig. 11.7 Magda illustrateshow Pascal’s Triangle isrelated to the pizza problem

130 L.D. Tarlow

At Magda’s suggestion, the group tested this rule by explaining other additions onPascal’s Triangle using pizzas. When Magda was asked to explain the relationshipbetween pizzas and Pascal’s Triangle, she responded:

Okay, so we have the three 1-toppings ones, which was the sausage, pepperoni, and mush-room. So those are the three combinations for the 1-topping ones. Because we are movingto four toppings, we needed to add an extra topping. So we just added peppers because thisstands for two topping pizzas, so we just added the peppers.

The group noted that the other three represented pizzas that already had twotoppings, 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 andthe numbers in Pascal’s Triangle. They were able to explain the meaning of Pascal’sIdentity in terms of generating successive groups of pizzas, with more choices fortoppings. Finally, they noticed that the pizza and towers problems had the sameanswers.

11.4 Discussion

All of the students in this session solved the pizza problem and justified theirsolution using proof by cases. They connected their topping combinations to thenumbers on Pascal’s Triangle and used answers to the pizza problem in order toexplain the addition rule for Pascal’s Triangle. They also noted the doubling patternas 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 pizzaproblems in the earlier grades, explained their reasoning for the doubling rule usingboth pizzas and towers. Furthermore, they explained the addition rule for Pascal’sTriangle using towers as well as pizzas. Finally, the Table A group constructed athree-way isomorphism between the tower problem, the pizza problem, and thenumbers on Pascal’s Triangle. We also note that although the students at TableA made an attempt to use a partially remembered formula, they did not accept theformulaic answer but went on to use their own methods to verify the answer; inthe process, they abandoned that formula and instead generated their own correctformula.

These 11th-grade students exhibited advanced reasoning skills as they thoughtabout the problems, justified their solutions, and made connections. The studentswho had participated in the longitudinal study in the early grades demonstrated thatthey had benefited from exposure to thoughtful mathematical experiences over along period of time. They retrieved representations that had been built years beforeand then modified those representations in order to build and extend their mathe-matical knowledge and make connections to other mathematical ideas. The studentswho came to the study later on also demonstrated the ability to think deeply aboutthe problems and justify their solutions.


The students were presented with challenging problems and given the respon-sibility to solve them. They did not work in isolation; rather, they were activeparticipants in a learning environment where ideas were shared and discussed. Inthe course of making sense of their observations and of what their peers were say-ing and doing, they moved back and forth between their representations, which hadbecome less concrete and more abstract and symbolic. They developed, modified,refined, and extended representations, which enabled them to solve increasinglymore complex combinatorics problems. They built an understanding of fundamentalmathematical ideas and used those ideas to justify solutions to problems. Using theirpersonal notations, they extended their reasoning and made connections to otherideas in combinatorics. They were offered opportunities to work on rich, demandingproblems, to think carefully about their ideas, and to discuss their ideas with theirpeers 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 ofPascal’s Triangle based on personal experience with combinatorics problems andmaking use of personal notation. In the next chapter, we see how one group ofstudents made sense of standard mathematical notation by building on their personalnotation and knowledge.

Chapter 12Representations and Standard Notation

Elizabeth B. Uptegrove

Date and Grade: May 18, 1999; Grade 11Tasks: Towers, Pizzas, and Pascal’s TriangleParticipants: Ankur, Brian, Jeff, Mike, and RominaResearchers: Carolyn Maher and Regina Kiczek

12.1 Introduction

In the preceding chapters in this section, we considered how students made sense ofPascal’s Triangle and isomorphic combinatorics problems using their own increas-ingly sophisticated and abstract representations. In this chapter, we see how onegroup 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. Thisremarkable demonstration of how students can come to make sense of complexmathematical ideas was captured during the session that came to be referred to asthe “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 tow-ers 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 thesession described in this chapter, they made use of standard combinatorial nota-tion to communicate, clearly and concisely, the ideas about Pascal’s Triangle andPascal’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 interms of general versions of the pizza and towers problems.



134 E.B. Uptegrove

In the following sections we discuss the strategies used by the students to makesense of the standard notation. We note how the use of increasingly sophisticatednotation accompanied the students’ building of general notions about the meaningof Pascal’s Identity. We show how their organizational strategies proved key in theirmaking sense of the standard notation and of the relationships between the combina-torial problems, Pascal’s Triangle, and Pascal’s Identity. Further, we show that theyfound in the standard notation an essential tool for expressing their understandingof Pascal’s Identity in general form.

12.2 Summary of Earlier Student Work

As discussed in Chapter 10, during sophomore- and junior-year problem-solvingsessions, Ankur, Brian, Jeff, Mike, and Romina revisited and extended previouswork on two familiar combinatorial problems. In the first session of their sopho-more 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 provethat all solutions were present; they recognized that those problems were related toeach other, to the binomial coefficients, and to Pascal’s Triangle; they found generalsolutions to those problems; and they used those problems to form preliminary ideasabout the meaning of Pascal’s Identity.

The key organizational decision – to organize pizzas by number of toppings andtowers by number of cubes of a given color – not only helped the students show that

The coefficients of the binomial expansion:

The numbers also represent 5-tall towers with ...

The also represent pizzas with ...

1 5 10 10 5 1

1

0 red cubes

0 toppings

1 topping

2 toppings3 toppings

4 toppings

5 toppings

1 red cubes

2 red cubes 3 red cubes4 red cubes

5 red cubes

5 10 10 5 1

1 5 10 10 5 1

Fig. 12.1 Row 5 of Pascal’s Triangle

12 Representations and Standard Notation 135

they had all the solutions, but also helped them see the relationship between the twoproblems and Pascal’s Triangle. It enabled the students to realize, for example, thatthe fifth row of Pascal’s Triangle (1 5 10 10 5 1) contains not only the coefficients ofthe binomial expansion but also the solution to both the five-topping pizza problemand the five-tall towers problem. Refer to Fig. 12.1.

When the students were introduced to the use of standard notation to describe thebinomial coefficients, they were able to make use of the connection they had alreadyformed between the binomial coefficients and the towers and pizza problems. Theyapplied the standard notation to the pizza and towers problems; knowing how togenerate the answers to the pizza and towers problems in their own notation enabledthem to use the standard notation to describe the general pizza and towers answersand 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 investiga-tions of Pascal’s Triangle that they had begun in their sophomore year. By the endof this session, they had written Pascal’s Identity in standard notation and provideda sound explanation of its meaning. They did this by looking at general forms of thetowers and pizza problems, referring back to their previous explanations of specificinstances of Pascal’s Identity in terms of towers and pizzas, and making use of thebinary 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, andRomina) talked about that day’s class work, which had been to find the coefficientsof the expansion of (a+b) n. Jeff brought up what they called “choose” notation, thenotation to denote combinations, using the nCr function on their calculators.

In this episode, when Jeff was trying to explain how to find a particular coefficientof the expansion of (a+b)10, Romina spontaneously introduced the towers problemwith the words “ten high” and “two reds.” Jeff and Mike elaborated that this meantbuilding towers ten cubes tall, selecting from two colors, and counting how manythere are containing exactly two cubes of one of the colors (red).

JEFF: . . . If we were looking for a plus b to the tenth say, . . . it was1 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 howto figure it out. We knew it was the choose thing, whatever thatmeans. . . . What was it? Ten choose two? . . . Like, uh, was itN-C-R? [Jeff is referring to buttons on his calculator.] Two, isthat how you do it? [Jeff writes 10 nCr 2.] Right? . . . And thatequals 45, and that’s the answer. . . . We’re not really sure howall this works but it’s like . . .. If you have ten different. What isit? Ten different things . . .

ROMINA: Ten high. Ten high.JEFF: Ten high. How many.

136 E.B. Uptegrove

ROMINA: How many would have two reds, only two reds.JEFF: How many would have two, two reds.RESEARCHER: One more time.JEFF: If you have towers with ten high and two colors.MIKE: How many different places can you put two reds in there?JEFF: 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 beendifficult for anyone not familiar with their work to understand the references “tenhigh” and “how many would have two reds.” But the elaboration (although it stillassumed knowledge of the towers problem) shows that they knew that the coef-ficients of the binomial expansion to the tenth power were related to the ten-talltowers 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 theboard and explained that any row could be expressed in “choose” notation; for exam-ple, the row 1 3 3 1 could be called “3 choose 0” through “3 choose 3.” When theresearcher asked the students to talk about the addition rule for Pascal’s Triangle inthat notation, Romina suggested a new vehicle, the pizza problem, even though shehad just used the towers problem in the previous explanation. Mike used the pizzaproblem to explain a particular case of addition: think of the nth row of Pascal’sTriangle as representing all the possible pizzas that can be made when there are ntoppings to choose from, and think of generating new rows of Pascal’s Triangle asmaking new pizza toppings available. Then the pizzas represented by the first threein 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-toppingpizzas) or let them remain one-topping pizzas. If you add the new topping, you nowgroup them with the second three in that row (the pizzas that already have two top-pings), 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 thenew topping added to it), giving four pizzas with one topping. Figure 12.2 illustratesMike’s explanation. A portion of their discussion is given below.

Fig. 12.2 Examples of Pascal’s Identity


MIKE: 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.] . . .

JEFF: That would be a plus b to the third.MIKE: 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 categoryis everything with no toppings. [Mike points to the first 1 in row 3.] Andthat’s your number for that one. [Mike points to 000.] That’s like, likebinary numbers or something. Next would be- [Mike writes 001, 010,then 101.] There’s all the, the ones that have one topping.

JEFF: Right, you got to write that 0 at the end. You messed up. Last one shouldbe a hundred, not a hundred and one.

MIKE: I knew that. [Mike changes 101 to 100.] There’s all the ones that have onetopping. . . .. There’s your 3 choose 1 and there’s three different combina-tions you could put that. . . .. But, um, when you have a new – when youadd another place, another topping. [Mike draws dashes to the right of thefour numbers already there. Refer to Fig. 12.3.]

JEFF: That could be one or the other, one or the other, one or the other.MIKE: 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 threeswould 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 onlyhave one if they have the zero. [Mike points to the 4 in row 4.] So threeget 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.] withone topping won’t get one so, you know, you can put them in the samecategory as this one.

JEFF: That’s their four? Yeah.MIKE: 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. Referto Fig. 12.2.] And you could put them in together with the ones that didget something. That’s why you would add.

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

000 0

1

0010

1

0100

1

100 0

1

Fig. 12.3 Binary listing of 3 choose 0 and 3 choose 1

138 E.B. Uptegrove

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

RESEARCHER: Show me that 3 plus 3 is 6. Which ones would it be? . . .

MIKE: This one and that one. [Mike points from 3 choose 1 and 3choose 2 to 4 choose 2.] . . .

RESEARCHER: Okay, so you’re saying 3 choose 1 plus 3 choose 2 equals 4choose 2. Right? Okay. So what’s 4 choose 2 plus 4 choose 3?

JEFF: . . . 4 choose 2 plus 4 choose 3? That would be, that would be 5-ANKUR: 5 choose. . . .

MIKE: 5 choose 3.JEFF: Why is he 5 choose 3?ANKUR: Because it’s always the one on the right. [Ankur means that the

“choose” number of the sum is the same as the “choose” numberof the rightmost addend.]

Mike observed that the bottom number indicated the number of toppings actuallyused, so that when a topping was added, the bottom number changed and when atopping was not added, the bottom number did not change.

The researcher asked the group to continue by writing a general (nth) row ofPascal’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’ssuggestion to use lowercase n to indicate row number and r to indicate a numberin the middle of the row (following standard usage), Jeff used uppercase lettersN and X.

Brian arrived after the group had been working for almost an hour; first Jeffexplained to Brian how Fig. 12.5 related to that day’s work in their regular mathclass 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 whateverN equals.

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

N − 1

0 . . . N − 1

X . . .

N − 1

N − 1N

0…

N

X − 2

N

X −1

N

X

N

X+ 1

N

X +2…

N

N

⎛

⎝ ⎜

⎞

⎠ ⎟

⎛

⎝ ⎜

⎞

⎠ ⎟

⎛

⎝ ⎜

⎞

⎠ ⎟

⎛

⎝ ⎜

⎞

⎠ ⎟

⎛

⎝ ⎜

⎞

⎠ ⎟

⎛

⎝ ⎜

⎞

⎠ ⎟

⎛

⎝ ⎜

⎞

⎠ ⎟

⎛

⎝ ⎜

⎞

⎠ ⎟

⎛

⎝ ⎜

⎞

⎠ ⎟

⎛

⎝ ⎜

⎞

⎠ ⎟ Fig. 12.5 Rows N–1 and N

of Pascal’s Triangle


N+ += =X

NX+ 1

1X +N +

1nr

n −1r −1

n −1r

⎛ ⎝

⎞ ⎠

⎛ ⎝

⎞ ⎠

⎛ ⎝

⎞ ⎠

⎛ ⎝

⎞ ⎠

⎛ ⎝

⎞ ⎠

⎛ ⎝

⎞ ⎠

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

RESEARCHER: Can you write it as an equation? Just like you wrote three plusthree equals six.

ANKUR: N plus, just that plus that. [Ankur points to the entries N chooseX and N choose X + 1 in Fig. 12.5.]

MIKE: N choose X.JEFF: N choose X plus N choose X plus one. [Jeff writes on the board

as he speaks. Refer to Fig. 12.6.]MIKE: Equals that. . . .

JEFF: 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 Xand going into the X plus 1. [Jeff points to N choose X.]

MIKE: Yeah.JEFF: And this would be losing an X. [Jeff points to N choose X+1.]MIKE: No, no, not losing, not getting anything.ANKUR: Staying the same.MIKE: And the top numbers have changed because you have more.JEFF: Because you’re adding; you have more things [to choose from].RESEARCHER: Say it so Brian can follow it because he wasn’t here for the

earlier pizza discussion.JEFF: What, what we’re doing is the next line of the triangle.

Remember how today in class the other triangle was one, two.BRIAN: Yeah.JEFF: Three, that whole row there. Well, that’s the increase in N and

then the X plus one. . . . Say we’re doing pizzas.BRIAN: All right.JEFF: If you add another topping onto it.ROMINA: You know how we get the triangle and how we go 1 2 1 and add

those two together?BRIAN: Yeah.JEFF: We were explaining why you add.BRIAN: All right, keep going.JEFF: 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 toppingis available, the second (“choose”) number in the expression isincreased 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?

BRIAN: Yes. It actually does.JEFF: So, so that would be the general addition rule in this case.

140 E.B. Uptegrove

Fig. 12.7 Two versions ofPascal’s Identity

The students’ version of Pascal’s Identity is equivalent to a standard textbookversion of this equation, with n equivalent to N+1 and r equivalent to X+1. (SeeFig. 12.6) Following the production of the equation in combinatorics notation, thestudents were asked to convert that notation to factorial notation. They did so; theirwork is shown in Fig. 12.7.

Earlier in this session, Mike had explained 3 + 3 = 6 (Fig. 12.3) by stating thatthe threes are from the three-toppings row of Pascal’s Triangle: the first three repre-sents the three one-topping pizzas that become two-topping pizzas, and the secondthree represents the three two-topping pizzas that remain two-topping pizzas; thesix represents the six two-topping pizzas that can be made when there is a fourthtopping available. Now the students generalized this rule using the standard nota-tion, which they called “choose” notation: N choose X gives the number of pizzasthat have exactly X toppings when there are N toppings to select from and N chooseX+1 gives the number of pizzas that have exactly X+1 toppings. Moving to the nextrow down in Pascal’s Triangle means that you increase the number of available top-pings 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 selectfrom.

The students had described instances of Pascal’s Identity in earlier sessions. Forexample, in their sophomore year, Ankur, Jeff, and Romina had provided a specificexplanation using towers: In order to create a six-tall tower with exactly three redcubes, add a red cube to the five-tall towers that have exactly two red cubes andadd a blue cube to the five-tall towers that have already have three red cubes. (Thiswas discussed in more detail in Chapter 10.) But this night session explanation wasthe first explanation of Pascal’s Identity using standard notation to state a generalresult.

12.4 Durability of Understanding

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


the formula, and all three were able to provide a cogent explanation of the additionrule. As an example, we discuss here the interview with Mike.

Over the years in which he worked on the combinatorial problems, Mike pro-gressed from drawings and codes through his personal (binary) representationsystem to the standard combinatorial notation. From the time he introduced his ideasabout binary notation to his fellow students to his most recent interview over 5 yearslater, 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 combinatoricstasks. Mike took the lead in devising representations, finding connections, and mak-ing sense of the tasks. He recognized structural similarities between problems; hemoved 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 torecall how the group used the pizza problem to think about Pascal’s Triangle. Mikespent a little time regenerating the meaning of specific entries in Pascal’s Triangleand then, in response to a question about a general rule, he reconstructed the for-mula that had been developed during the night session. A portion of their discussionfollows. Mike began with the two-topping pizza problem.

MIKE: Okay. If you had no toppings, that would be one pizza.RESEARCHER: Okay. So where is that on the triangle?MIKE: 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 possiblepizzas with that. And then if you have all the toppings, that’sone. Right. And then automatically I see that relates to thisrow. [Mike points to row 2 of Pascal’s Triangle (1 2 1).] AndI’m pretty sure it would go down, this is like a third toppingand a fourth topping. [Mike points to rows 3 and 4 of Pascal’sTriangle.] Now I think the way I thought about it is, like, the rowon the outside [leftmost entry in a given row] would be yourplain pizza. And there’s only one way to make a plain pizza.And . . . the next one over would be how many pizzas you couldmake using only one topping, and then so on until you get to thelast row [the rightmost entry in a given row] which is all yourtoppings. And, once again, you can only make one pizza out ofthat. . . .

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

RESEARCHER: 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. . . .

MIKE: Like a general equation?RESEARCHER: Well, that was what I was going for ultimately. . . .

MIKE: To give an amount for any spot in the row.

142 E.B. Uptegrove

RESEARCHER: Right. . . .

MIKE: 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 apause; then Mike writes the left part of the equation shown inFig. 12.7.] I’m just going to like work this out in my head andsee if it actually works. [A few seconds later, Mike adds theright part of the equation.]

This equation, shown below, is equivalent to the textbook version and to thenight 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.) Wecan see that Mike did not rely on symbol manipulation. He linked the numbers toa problem task that made sense to him, and then he expressed the relationships inthat task in symbolic form. We conclude that Mike was reconstructing the substancerather than merely remembering the form.

(rn

)+

(rn + 1

)=

(r − 1n + 1

)

12.5 Discussion

Exploring previously unexamined complexities of the towers and pizzas problemswas a mathematically challenging task of the sort recommended by Davis andMaher (1990) to foster students’ ability to engage in real mathematics – developingtheir own mathematical theories, for example. Conditions important for the devel-opment of new mathematical ideas were in place: these students had ample time forexploration of mathematical ideas and the opportunity to express their own ideas.The students’ existing representations were taxed by new questions about how torelate these problems to each other, to Pascal’s Triangle, and to the binomial coef-ficients 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 dealingwith these new ideas. We have shown that these students did make use of a newtool – standard mathematical notation – for dealing with their ideas about Pascal’sIdentity.

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 middleschool, they began instead to use symbolic notation. (For example, they used letterand number codes to stand for the objects they were investigating.) Besides con-tinuing the use of codes during high school, the students also found increasing usefor the standard notations of mathematical discourse. For example, the binary nota-tion that they began to use in high school was more powerful than the letter codesbecause it was easily extended (adding a cube to the tower or a topping choice to


a pizza corresponded to adding a binary digit) and it was applicable to both pizzaand towers problems, thus making it easier for the students to identify the similarstructures of the two problems. Using binary notation helped the students focus onthe isomorphic structural aspects of the combinatorial problems (the duality of thechoices) rather than the surface features (the different pizza toppings, for example).Binary notation was also an easily generalizable notation in three ways: (1) addingan extra digit corresponded to making a new pizza topping available and to increas-ing the tower’s height by one block; (2) adding a 1 corresponded to adding thatnewly 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 blockof the other color. This idea that it was not necessary to know the current number (ornames) of pizza toppings or the current height of the tower in order to describe whathappened next was important in the students’ production of the general equation forPascal’s Identity.

During the course of discussions over 4 months of their sophomore year, thesestudents first noted that the pizza and towers problems had the same answer in spe-cific cases. Then they linked specific answers to the pizza and towers problems tospecific entries in Pascal’s Triangle. Finally, they described the links among bino-mial coefficients, pizza toppings, and towers. (Blue block = a = topping on thepizza; white block = b = topping off the pizza.) During the night session, they builton their knowledge of these links in order to produce the general form of Pascal’sIdentity. We claim that their ability to map corresponding mathematical structuresamong these three representations is a strong indication of their mathematical com-petency and it indicates more competency than, for example, simply being able toreproduce or use a memorized formula.

The way these students organized their answers to the pizza and towers prob-lems was a key organizational element that helped them to form connections amongthose problems and Pascal’s Triangle. They also made extensive use of their per-sonal representations at the beginning of the process. But once those connectionswere formed, the students began to make general statements about Pascal’s Triangleand Pascal’s Identity, and they had less use for personal representations. Finally,although they were able to articulate general information about Pascal’s Triangleand Pascal’s Identity, they did not represent the generalizations symbolically untilthe night session.

After they had made the association between Pascal’s Triangle and the com-binatorial problems, the students demonstrated an ability to describe any selectedentries 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 throughsix red cubes, respectively. The fact that they could explain any instance suggestedthat they had an idea of the general rule; but without the standard notation, theycould express their general ideas most easily by referring to specific examples. Bythe time of the night session, these students seemed to know general rules aboutgenerating Pascal’s Triangle, but they lacked the notation to express these rules in aconcise way. They were at the point where they needed standard notation in orderto proceed further.

144 E.B. Uptegrove

We suggest that these findings point to one way that teachers can follow therecommendation by the NCTM (2000) to “use sound professional judgment whendeciding 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 formalnotation 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 thatthe students have developed on their own.

In this chapter, we have seen how this group of students learned about the rela-tionships among well-known combinatorics problems and Pascal’s Triangle. In thefollowing chapter, we observe the same students working on new problems in com-binatorics and using what they learned about the pizza and towers problems andtheir relation to Pascal’s Triangle in order to make sense of that unfamiliar problem– the Taxicab Problem.

Chapter 13So Let’s Prove It!

Arthur B. Powell

Date and Grade: May 5, 2000, Grade 12Tasks: The Taxicab ProblemParticipants: Brian, Jeff, Mike, and RominaResearchers: Carolyn A. Maher and Arthur B. Powell

13.1 Introduction

In previous chapters, we observed students throughout middle school and highschool working on and making sense of two isomorphic problems in combina-torics – the towers problems and the pizza problems. In this chapter, we see howstudents just finishing high-school work on another isomorphic problem, demon-strating the application of techniques and ways of thinking that they developedthroughout their previous years in the study. We further address the challenge thatDavis (1992a) proposes to mathematics education researchers to investigate theemergence among learners of what lies at the core of mathematics: mathematicalideas. 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 thepower 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

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


146 A.B. Powell

students collaborated on a culminating, performance-assessment task of the researchstrand on combinatorics – the Taxicab Problem:

A taxi driver is given a specific territory of a town, shown below. All trips originate at thetaxi stand. One very slow night, the driver is dispatched only three times; each time, shepicks up passengers at one of the intersections indicated on the map. To pass the time, sheconsiders all the possible routes she could have taken to each pick-up point and wonders ifshe 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? Ifnot, 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 claimsin their mathematical discourse require justification or proof. This is particularlytrue if the students deem the claim to be obvious or if the students are in themidst of group problem solving with intellectual peers. On May 5, 2000, in the lateafternoon, 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-shapedtable, on top of which are four black felt-tip markers, sheets of blank paper, anda problem statement. The statement is of a problem in which one is to determinein a given rectangular grid the number of different shortest paths between pairs ofspecified 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 whetherthey understand it. Jeff asks aloud whether one has to stay on the grid lines andwhether they represent streets. The researcher responds, “Exactly.” Each studenthas taken a marker. Among themselves, they observe that from the black endpoint

13 So Let’s Prove It! 147

or “taxi stand,” five and seven are respectively the number of blocks it takes to reachthe blue and red endpoints or “pick-up points.” Moreover, some assert that differ-ent routes to each point have the same length as long as one doesn’t go beyondthe particular pick-up point. Especially noteworthy from cognitive and pedagogi-cal viewpoints, Brian says to his colleagues, “So, let’s prove it!” After a few quietmoments, 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, andRomina decide that to determine the number of paths between three specialized pairsof endpoints they need to generalize the problem. This moment is a watershed eventin their mathematical work on the Taxicab Problem. Through their various heuris-tic actions, the students generate data that they consider reliable. They reflect onnumerical patterns in their array of data, observe that it resembles Pascal’s Triangle,and conjecture that Pascal’s arithmetic array underlies the mathematical structureof the problem. How do they justify this conjecture? They embark on building anisomorphism between the Towers Problem and the Taxicab Problem since fromprevious experience they know that Pascal’s Triangle underlies the mathematicalstructure of the Towers Problem. The students’ strategy can be interpreted as jus-tifying their conjecture by transitivity: (a) the mathematical structure of Pascal’sTriangle is equivalent to that of the Towers Problem and (b) the mathematical struc-ture of the Towers Problem is equivalent to that of the Taxicab Problem; implyingthat (c) the mathematical structure of Pascal’s Triangle is equivalent to that of theTaxicab Problem. The students knew that (a) is true and demonstrate (b) to justifyand conclude (c).

13.2.2 Reasoning and Justifying

In the following sections, we discuss students’ methods of reasoning and ways ofjustifying 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-to-student interaction. It centers on a question about a relation that she notices aboutwhich Romina invites her colleagues to comment.

ROMINA: Isn’t it like anyway you go-BRIAN: Pretty much, because look-ROMINA: As long as you don’t go like past it. [Facing Brian’s direction.]BRIAN: The first one- No.MIKE: Well what if you go to the last one-

148 A.B. Powell

BRIAN: You can go all the way down and go over and go down three and goover two. [Tracing the routes above the problem sheet with a blackmarker in his right hand.]

ROMINA: Isn’t it- Don’t they all come out to be the same amount of blocks?BRIAN: Five.JEFF: Five?ROMINA: Five? I got seven.JEFF: Uh, which one- Yeah, we were both looking at the red one.BRIAN: I’m looking at blue. [Mike is tapping his pen on the grid along

intersection points.]JEFF: Yeah.ROMINA: Oh, okay.JEFF: All right. I mean pretty much.ROMINA: As long as you don’t go like past it you’re fine. So it’s the same thing.BRIAN: 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 pastit”) paths or routes between the taxi stand and the red pick-up point. She observesthat as long as one does not go beyond the red pick-up point that the numbers ofblocks traversed or lengths of routes to red equal each other. Specialized to the redpick-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 shortestroute, and, her central observation, (c) efficient routes have the same length. Thesethree ideas are important and fundamental for progressing toward a resolution of theproblem task.

At first, Brian disagrees (“The first one- No, ’cause-”) and then, examining routesto the blue pick-up point, attempts to understand Romina’s remark (“You can goall 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 ofblocks 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 referringto the blue pick-up point (“I’m looking at blue.”). Taking up Romina’s observationfor 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 thestudents discuss how they know that Romina’s unchallenged assertion is true.

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

backwards a couple of times.


ROMINA: You can’t go, well-MIKE: I know that would be dumb.BRIAN: [inaudible] the shortest route only if you go forward.MIKE: But the only- You can’t go diagonal so you have to go up and down.

So if the thing is down this many andJEFF: Over that many, it’s the sameMIKE: It’s the same-ROMINA: It’s the same areaMIKE: No matter how you do it, no matter how you do it it’s- you have to-

you can’t get around doing that. [Pointing and gesturing around hisgrid]

ROMINA: All right.MIKE: You can’t get around going four down and right one ’cause -.JEFF: All right, yeah. All right.MIKE: You can’t go over there. You can’t get around doing that.JEFF: Yeah.ROMINA: What if I were to go like to the red when I go one, two, three, four-

[Pointing at her problem sheet.]MIKE: But they’re not asking for that.ROMINA: Five, six, seven.JEFF: Five, six, seven. It’s the same thing.ROMINA: Like how- how am I going to- like how would I-JEFF: It’s the same thing.MIKE: It’s the same.ROMINA: -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 whosevertices are taxi stand and the red pick-up point.]

BRIAN: Like plus and [Inaudible].JEFF: Well, it’s not area.MIKE: It’s not area. It’s just a-JEFF: It’s the perimeter. It’s like each one being one.MIKE: One, two, three, four, five, six, seven. [Pointing at Romina’s paper

and counting the length of a route to the red destination point.] [Jeffscratches his head.]

ROMINA: All right.MIKE: There’s no way you can get around going- [gesturing with his hands]JEFF: Going seven blocks.ROMINA: No, yeah, I understand.MIKE: Across that many and down that many because you can’t go diago-

nally. Can’t- [gesturing with his hands over his problem sheet acrossto the left and then down]

JEFF: Yeah.MIKE: Can’t get around it, so- [gesturing with his hands]JEFF: I mean, that’s the most sensible way I think to say that. Right? And

they want to know how many though.

150 A.B. Powell

Justifying Romina’s observation, reiterated by Jeff, or, equivalently, entertainingBrian’s proposal becomes a shared project of the participants. When Jeff poses hisquestion (“why is it the same every time?”) and the others understand his “it” tomean “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 silencethat Brian’s proposal received almost 2 min earlier. The ensuing discursive exchangehints that the issue of the why Romina’s observation was true in general remained aconcern 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 alwaysrequire 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 remindsthe others that only going forward will produce a shortest route. Mike generalizeshis awareness to all routes. Jeff signals that he is convinced, saying, “I mean, that’sthe 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 emer-gent cognition (Powell, 2006), providing discursive evidence to several ideas: (1)movement within the given portion of the taxicab plane goes left or right and upor down; (2) diagonal movements are not permissible; (3) the taxi stand and eachpick-up point together define a rectangle in which the pair of points are locatedat opposite ends of a diagonal, and the problem task involves moving along theperimeter but does not concern the extent of space that a rectangle occupies; (4) thenumber of units down plus the number of units across are objects related by additionto produce the length of a shortest path; (5) any route to the blue pick-up point willinvolve four blocks down and one block across; and (6) each horizontal and verticalline 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 reintroducedBrian’s proposal that they justify the idea that the length of efficient routes from thetaxi stand to a pick-up point are equivalent, expresses satisfaction with Brian andMike’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 theycan now turn their attention of the crux of their task: “And they want to know howmany though.”

The discursive exchanges in the three episodes quoted above are critical. Theypresent the major occasion in which the participants ferret out the nature of the prob-lem space and build fundamental ideas essential for investigating the problem task.The participants establish what are the basic objects of taxicab geometry (pointsand line segments or routes); basic awareness of the Taxicab Problem (there canbe more than one shortest route to an intersection point in the taxicab plane); andimplicitly note a distinguishing feature between Euclidean and taxicab geometries(how distance is measured). This distinction emerges when Mike observes that inthe context of the problem task, one cannot travel diagonally, he touches upon thefundamental distinction between the metric of Euclidean geometry and that of taxi-cab geometry. Moving forward with the ideas they have built that were illustratedin the three episodes, the students shift their focus to delve further into the problemtask and generate considerably more data.


13.2.2.2 Generalizing to Specialize

The students take a decisive turn in their investigation: they generalize the prob-lem. Instead of determining the number of shortest paths between each of the threespecialized pairs of endpoints, the work to uncover a pattern among the numericalvalues that represent the number of shortest paths between the taxi stand and anypoint on the grid. They first examine points in close proximity to the taxi stand. Thisdecision proves to be a watershed event in their mathematical work on the TaxicabProblem.

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 tojustify their solution transpires over many turns of speech, spanning from turn 159to turn 1,320. Space does not permit us to present a full illustration of the develop-ment of the ideas and reasoning that comprise the students’ work toward justifyingtheir solution. They have continual discursive interactions with the aim of build-ing an isomorphism between a rule for generating the entries of Pascal’s Triangleand 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 problemis discussed in Chapters 4, 5, 10, and 12.) Her public query catalyzes a negotiatoryinterlocution 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 dyadicchoice that one has at intersections of the taxicab grid. Furthermore, he wonderswhether one can find the number of shortest routes to a pick-up point by addingup the different choices one encounters in route to the point. Romina proposes thatsince the length of a shortest route to the red pick-up point is 10, then “ten couldbe like the number of blocks we have in the tower.” Romina’s query concerningthe application of towers to the present problem task prompts Mike’s engagementwith the idea, as well. As if advising his colleagues and himself, he reacts in part bysaying, “think of the possibilities of doing this and then doing that.” While utteringthese words, he points at an intersection; from that intersection gestures first down-ward (“doing this”), returns to the point, and then motions rightward (“doing that”).Similar to Jeff’s words and gestures, Mike’s actions also acknowledge cognitivelyand corporally the dyadic-choice aspect of the problem task. Through their negotia-tory interactions, Mike, Jeff, and Romina raised the prospect of as well as providedinsights for building an isomorphism between the Taxicab and Towers Problems.

The prospect and work of building such an isomorphism reemerges several moretimes in the participants’ interlocution and, each time, they further elaborate theirinsights and advance more isomorphic propositions. Eventually, the building of iso-morphisms dominates their conversational exchanges. Approximately 35 min afterRomina first broached the possibility of relating attributes of the Towers Problemto the problem at hand, the participants reengage with the idea. Romina speculatesthat between the two problems one can relate “like lines over” to “like the color” and

152 A.B. Powell

then “the lines down” to the “number of blocks.” What is essential here is Romina’sapparent awareness that each of the two different directions of travel in the TaxicabProblem 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 andher 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 hori-zontal distance with one Unifix cube of color A and one unit of vertical distancewith 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, oneof B color [pointing to the 2 in Pascal’s Triangle]. Here’s one- it’s either one- either wayyou go. It’s one of across and one down [pointing to a number on the transparency grid andmotions with her pen to go across and down]. And for three that means there’s two A colorand one B color [pointing to a 3 in Pascal’s Triangle], so here it’s two across, one down orthe other way [tracing across and down on the transparency grid] you can get three is twodown [pointing to the grid].

Furthering the building of their isomorphism, Mike offers another propositionalfoundation. Pointing at their data on the transparency grid and referring to its diag-onals as rows, he notes that each row of the data refers to the number of shortestroutes 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 inter-section point whose “shortest route is four.” Moreover, he remarks that one couldname 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’sobservation points to two different ideas (1) it relates diagonals of information intheir data to rows of numbers in Pascal’s Triangle and (2) it notes that intersectionpoints whose shortest routes have the same length can have different numbers ofshortest routes.

Later in responding to a researcher’s question, the participants develop a propo-sition that relates how they know that a particular intersection in the taxicab gridcorresponds to a number in Pascal’s Triangle. They focus their attention on theirinscriptions in Fig. 13.1, which shows empirical data of shortest routes between thetaxi stand and nearby intersection points. In array A, the green numbers (lightershade inside sqiares) show empirical data of shortest routes between the taxi standand 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 theparticipant perspective, to the left of Jeff’s numbers, Romina wrote in green (lightershade) 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 empir-ical data; the remaining two rows contain assumed data values based on the additionrule 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, Rominaassociates the length of its shortest route, which is seven, to a row of her Pascal’sTriangle by counting down seven rows and saying, “five of one thing and two ofanother thing.” Mike inquires about her meaning for “five and two.” Both Romina


Fig. 13.1 Participants’ data arrays A and B

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

13.3 Conclusion

The narrative of these four students working on the Taxicab Problem has threesections. The first concerns their recognition of the need to justify an observationthat they made immediately after reading the problem statement. The observa-tion maybe simple but their recognition of the significance of the observationand that it needed to be justified before progressing on with resolving the prob-lem is rather sophisticated. This sophistication in their mathematical work speaksto the sociomathematical norms (Yackel & Cobb, 1996) that they have devel-oped through their longitudinal experience working on open-ended problems. Thissociomathematical norm is subtle and akin to the way mathematicians work.

The second section of the narrative pertains to their decision to seek a generalsolution to the problem and that such a solution would be easier than trying to countthe 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, findinga general solution of a problem situation in order to answer more specific questionsof 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 recog-nition that claims needs to be justified but also with a particular proof strategy that

154 A.B. Powell

emerged in the students’ attempt to justify their resolution of a generalized form ofthe Taxicab Problem.

Important sociomathematical norms (Yackel & Cobb, 1996) are evident in thestudents’ mathematical interactions in the first and third threads: claims need to bejustified and a problem’s solution needs to be connected or linked to attributes ofthe problem. These norms emerge from the mathematical interactions of studentswho have a collective history of problem solving through occasional interactionsover their school years with researchers from Rutgers University who increasinglyover the years left the students to structure their own mathematical investigations inresponse to given tasks.

In this chapter and the previous chapters of this section, we have given theresearchers’ perspectives on the students’ mathematical work. In Chapter 14, weexamine this work from the point of view of the students.

Part IVExtending the Study, Conclusions,

and Implications

Chapter 14“Doing Mathematics” from the Learners’Perspectives

John M. Francisco

Date and Grade: 1999–2000; high schoolTasks: Clinical interviewsParticipants: Ankur, Brian, Jeff, Mike, and RominaResearcher/Instructor: John Francisco

14.1 Introduction

The previous chapters focused on aspects of the cognitive development of thestudents in the longitudinal study. The present chapter looks into the epistemologicalgrowth of the students. During the longitudinal study, individual clinical interviewswere conducted with the students with the goal of capturing the mathematical beliefsthat the students might have developed in connection with their experiences in thelongitudinal study. This chapter reports on the analysis of five such interviews.The results provide insights into the students’ views about mathematics and abouthow it should be learned and taught. The findings challenge the widespread viewthat students below college hold naïve epistemological views; support studies thatshow that students who experience constructivist learning environments tend todevelop sophisticated epistemological beliefs and highlight the important of pastmathematical experiences in framing individuals’ mathematical beliefs.

Research on students’ views about mathematics can be placed within the field ofpersonal epistemological beliefs. This is a field traditionally concerned with describ-ing individuals’ views about the nature of knowledge and knowing. A substantialamount of research has been conducted within the field since the pioneering workof Perry (1970) with Harvard college students. Even more research has been asso-ciated with this field since the epistemological construct was expanded to includeindividuals’ 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, USAe-mail: [email protected]


158 J.M. Francisco

Schommer (2002) and Schommer and Walker (1995) and some continental scholars(De Corte, Op’t Eynde, & Verschaffel, 2002; Leder et al., 2002). Even though theexpansion has not been free of controversy, it is recognized that this expansion hasbrought the research on the field closer to classrooms practice.

Students’ epistemological beliefs have been examined in relation to a variety ofconstructs. Students’ beliefs have been studied in relation to the students’ home andschool environments (Hammer & Elby, 2002) and their teachers’ epistemologicalbeliefs (Hofer, 1994; Lyons, 1990; Pirie & Kieren, 1992; Roth & Roychoudhury,1994). There has been also extensive research that has examined students’ beliefswithin 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 epistemologicalbeliefs 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 collegeand even fewer below high school. The review further points out that lack of suchresearch has resulted in students below college being assigned naïve epistemologi-cal beliefs only because research findings show that entering college students tendto hold such views. Another challenge is the lack of studies that have examined theepistemological beliefs of students who have experienced a constructivist learningenvironments. The few existing studies (e.g., Hofer, 1994) have been exploratory innature.

The present study grew out of a natural interest on the part of the researchers toexamine the epistemological growth that participating students in the longitudinalstudy might have experienced in connection with the particular conditions in whichthey were asked to do mathematics. The researchers were particularly interestedin the students’ beliefs about (1) success and failure in mathematics, (2) know-ing mathematics, (3) learning and teaching mathematics, and (4) how the practicesthat they assigned to doing or learning mathematics compared with those in otherdisciplines. However, the researchers also sought to make a contribution towarddeepening the research community’s understanding of the epistemological beliefsof students below college, particularly at the high school level, and of students whoexperienced constructivist mathematical environments. The researchers viewed thelongitudinal study as a “learning experiment,” rather than a “teaching experiment,”through which they tried to understand how students construct mathematical ideaswhile working on open-ended mathematical tasks in particular conditions. However,there were no preconceived ideas about what students were to learn or how theywere supposed to learn. Students’ constructed mathematical ideas and reasoningwere results, not preconceived goals, of the research. This was consistent with aconstructivist approach to learning in the sense that participants had plenty of oppor-tunities to construct and accordingly revise their ideas without any guidance fromthe 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.

14 “Doing Mathematics” from the Learners’ Perspectives 159

Their conversations provide insights on their mathematical beliefs and challenge thewidespread view that students below college level hold naïve views in contrast tostudies that show students who experience constructivist leaning environments tendto develop sophisticated epistemological beliefs. However, the results also highlightthe importance of past mathematical experiences in the development of individuals’mathematical beliefs.

This study used a phenomenological approach to the students’ experiences inthe longitudinal study. Researchers avoided imposing any interpretive frameworkon the students (Wilson, 1977) and sought to infer the students’ epistemologicalviews among the meanings that the students assigned to their experiences in thelongitudinal study (Creswell, 1998). Overall, the approach was similar to Perry’s(1970) idea of inferring individuals’ epistemological beliefs from their reflectionson educational experiences.

Data for the present study consisted of 1-h videotaped individual interviews withthe five participating students about their experiences in the longitudinal study. Thefour males and one female – Ankur, Brian, Jeff, Mike, and Romina – agreed tobe interviewed and videotaped. However, it was the students’ long experience inthe longitudinal study, starting in first grade, that constituted the major criterionfor selecting the students to take part in of the interviews. Their long-term partic-ipation 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 particu-lar practices and criteria of validity), but the interview proceeded by eliciting andbuilding on students’ reflections on their experiences in the longitudinal study toascertain the students’ epistemological views. Typically, the interviews started withthe open question, “What are your memories of the longitudinal study?” Then theresearchers 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 aboutpersonal success and failure in mathematics, mathematics as sense making, math-ematics as a discovery activity, mathematics as an activity involving discourse,and the relationship between mathematics and other disciplines. These themes aredescribed below, along with supporting statements from the students. (Emphasiswas 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

160 J.M. Francisco

Mike from describing himself as being better than a “normal kid.” There weredifferences, however, on how the students construed mathematical success. Mikeand Ankur emphasized personal interest and hard work as the ultimate sources ofsuccess. They argued that those who like mathematics can be successful becausethey are willing to work harder in mathematics than those who do not like it.The other students stressed the importance of previous mathematical experiencesand training. In particular, they singled out particular aspects of experiences in thelongitudinal study such as collaborative work and opportunities to come up withideas, as opposed to merely receiving them from teachers or experts, as having con-tributed to their success and confidence in mathematics. Romina further suggestedthat confidence and success is built over time:

In fourth grade, I didn’t know who you were. Now we’re comfortable with you. You’vebeen our teachers for 10 years. That’s what you’ve been to us, so now it’s easier, and weknow what’s expected of us, what we have to do. Before we would wait for you to give us alittle start or a little push and point us in the direction. Now you hand us a problem and youjust kind of leave, and we just do it ourselves. We just start experimenting and see what wecan give you.

She also suggested that lack of success can be a function of how success isdefined. 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-basedmathematics, 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 inmy own words, not in anyone else’s words. It’s not from some mathematician from a 1,000years 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 ortrait that people are born with. On the contrary, a closer analysis of the students’reflections suggests that the students converge on recognizing the importance ofpast mathematical experiences in promoting mathematical confidence and abilityeither directly or indirectly through motivation. Romina’s last statement also sug-gests that standardized testing has the potential to portray otherwise bright studentsas mathematically weak only because the students do not do mathematics in theways prescribed in textbooks or by experts.

14.2.2 Knowing Mathematics as Sense Making

The students’ reflections on their experiences in the longitudinal study emphasizedthe importance of understanding as opposed to memorization of concepts. For exam-ple, Mike reported gaining increased conceptual understanding in the longitudinalstudy:

It feels different now because I know a lot more than I did before. If I were to solve thesame problems, it would be easier. I understand a lot better too the whole concept behind


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

Romina emphasized the importance of building durable understanding. In par-ticular, she explained that it involved the ability to recall as well as reconstructpreviously learned mathematical concepts:

Because everything I do I understand, because it’s more than just the numbers to me. If youunderstand something from the beginning, you’re going to always understand it. You can’tforget something like that. And like an equation, I don’t really know any equations. It’s likethings, I don’t know any solid equations, but I could explain to you something and workfrom 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 todo it, that’s the important part, that’s what we learned. And that’s being able to do it, beingable 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, oranything like that.

Brian emphasized the importance of developing the ability to “look deeper thanjust the surface” and of always asking “why,” qualities which he asserts that hegained in the longitudinal study:

When I look back at things, I’m happy I got involved in this program. Because, I knowat times, I seem very frustrated with it. But if I think hard, I really have gained a lot ofknowledge, and I learned how to look into things deeper than just surface things like, “Whyis it like this?” Now, I start thinking like that. And it helps me compute things in my mindbetter. Like, I really don’t know how to put this, but it just helps me in doing things otherthan 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 clearabout 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 teacherwould teach, I’d understand, I’d participate, and it was just, I enjoyed it. And then we wentover to the Springfield [Regional High School], and I did not enjoy geometry class at all. Itwas one of the first times that we used the textbook. I don’t remember in Harding using amath textbook. And the teacher would just simply teach out of the book, and assign home-work, straight problems, and it wasn’t anything that I enjoyed. Then after Springfield, wecame here [to the local Kenilworth High School] and I had Mr. Pantozzi [his mathemat-ics teacher who was also involved with the longitudinal study as a researcher and graduatestudent] 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 orunderstanding mathematics has a personal dimension, Jeff’s statement suggests thatunderstanding has a social or interpersonal dimension to knowing mathematics.

162 J.M. Francisco

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 mathematicalenvironment where students [not teachers] came up with their mathematical ideasor knowledge as opposed to merely receiving them from teachers or textbooks. Theimplicit idea of learning as a discovery activity was present in the reflections ofall students, albeit articulated in slight different ways. Mike argued that discoverylearning helped the majority of students understand mathematics and emphasizedexploration 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. Likewe were learning about e [base of natural logarithms], and he told me that when he wasin school, the teacher told them, “e is this, 2.7, whatever.” The teacher told him what itis. In our class, all we did was just explore e. We took days at a time, and I have a goodunderstanding of it. I guess, in a normal class, only selected kids might understand it. But ina class where everybody’s working together, everybody’s a part of the teaching, everybodyor at least the majority of kids will understand it.

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


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 differentthings, but by the end of the year, you’ve forgotten them all, and you have nothing. If youwould, say, argue for a couple days or weeks or whatever on different topics, you cover tenthings, but when you walk away, you still know those ten things at the end. And that’s whyit’s important to do that, and not just get the answers.

Romina also singled out mathematical discussions during mathematical activ-ities. In particular, she added that disagreements during the discussions were animportant cognitive mechanism by which students built new knowledge and howshe, 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 getthrough everything. We just disagree. I’ve always had to argue to get somewhere, becausethey 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 helpedthem build durable understanding of mathematics:

164 J.M. Francisco

Fig. 14.3 Brian Interview

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

Ankur favored problem-solving activities involving interesting and challengingtasks 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 comesup with problems, and most of the problems are interesting problems, and a lot of themare challenging, and all the students participate. We enjoy working in groups. We help oneanother, 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 Jeffand Romina about mathematical arguments, suggest the idea of doing mathematicsas a discursive activity. The statements assign mathematical arguing the cognitiverole of fostering knowledge acquisition. Jeff’s statement about understanding asexplaining ideas also suggests the idea of arguments as a way of proving or estab-lishing the validity of mathematical claims. In the statement below, Jeff is even moreexplicit about the arguments as way of proving mathematical claims:


We didn’t know if we were right or wrong. You only knew so much, but I would have myidea about how to get to a certain point and you might have the same idea about how to getto it. But getting there was the hardest part. That is what we were arguing about, the rightway to get there, the right way to make sure we covered the basis, how to make sure, howto 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, hesuggests that because uncertainty is more common in probability than in any otherarea 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 youbelieve it’s right, and sometimes that influences your decision. That’s probably why weargue. I remember the problem with the World Series Problem [see Appendix A]. We hadtwo 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, somestudents might not be engaged and so might not learn. Above, Mike suggested thatdiscovery learning with group work can help the majority of students learn mathe-matics. Ankur’s statement below suggests a similar idea and tries to illustrate howit happens:

Usually, you think that only one person in the group is learning, but if the group fully par-ticipates and everyone is involved, everyone in the group learns. When the Rutgers groupcomes over here, we all learn. I don’t think there is a case when someone doesn’t under-stand. Because if one person doesn’t understand, they’ll say something, or even if they’requiet, someone else will suggest something, will ask them if they understand, or say “Couldyou 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 associatedwith learning or doing mathematics were specific to mathematics or applicable toother subjects or real life.

Ankur suggested that he thought that teaching out of the textbook, as opposedto the discovery method, was more suited to other subjects such as history than tomathematics:

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 wouldsimply 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 teacherusually teaches differently. I don’t know how to explain it, but it just seems that way. Thisteacher taught straight out of the textbook, you wouldn’t learn anything more, just simplywhat the book stated.

Romina was the most categorical of all the students in her response. She claimedthat arguing about ideas was a learning practice specific to mathematics and couldnot be implemented in other subjects such as English and History:

166 J.M. Francisco

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 dothe same. In math, it’s like, well especially the way I’ve been taught, because I have neveractually 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 learningmathematical practices were also relevant to other disciplines. For example, Briansuggested that his history teacher also used a problem-solving approach as opposedto just telling students what to do:

Well, the closest thing to my math class would have to be my history class. My historyteacher is an incredible teacher. He always, like for instance, we’re doing the Cuban MissileCrisis 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 asmuch as he does.

Mike argued that his longitudinal study experiences were relevant to other sub-jects. “I think it’s relevant to a lot of other subjects, like science, history; I guess youcould apply it to, basically, all subjects” and claimed that he used his experiencesin the longitudinal study, which he called a “type of thinking” in real life and othersubjects:

I guess I use the type of thinking in, like other subjects in school; I don’t know how you canapply it to life. It’s not hard to recognize what style of thinking you’re thinking of. I can’tcompare 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.


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 othersubjects reflect differences in interpretation of the question asked. Jeff, Romina, andAnkur seem to have answered the question of whether the mathematical learningpractices were actually taking place in other subjects. Mike and Brian, however,seem to have understood the question as asking whether they believed that thosepractices were applicable to other subjects.

14.3 Conclusions

The analysis of the interviews with the five students who participated in this studysuggests that the students (1) are confident in their mathematical ability, (2) empha-size mathematical understanding over memorization, and view mathematics as(3) a discovery activity and (4) a discursive activity. The results also suggest anagreement that that (5) the practices were not being implemented in the regularschools, except in a few isolated cases (a history teacher and Mr. Pantozzi). Thefindings suggest a few insights.

The students’ emphasis on the importance of learning as a discovery activitysuggests that they view themselves as learners as active participants in the construc-tion and justification of their mathematical knowledge, and not as mere receiversof knowledge and truth from experts or textbooks written by experts. Within thedomain of personal epistemological beliefs, such a view is held by individualsholding sophisticated or powerful personal epistemological beliefs. As a result, thefindings of the present study challenge the aforementioned widespread belief thatstudents below college hold naïve epistemological beliefs based on research thatshow that to be the case among freshmen college students. Given that the conditionsof the longitudinal study were consistent with a constructivist approach to learning,

168 J.M. Francisco

the results also support findings from exploratory studies suggesting that studentswho experience constructivist learning environments tend to develop more sophis-ticated epistemological views than students who experience teaching approachesbased on showing and telling students what to do.

The students’ views about mathematics are also consistent with the non-traditional approach to mathematical learning and teaching, advocated by theresearch community and promoted through publications such as the 2000 Principlesand Standards for School Mathematics of standards of the National Council ofTeachers of mathematics (NCTM). This is evident in the students’ emphasis ondurable mathematical understanding as opposed to memorization of concepts orprocedures, discovery learning, convincing or explanatory arguments, and collab-orative work. The students articulate the merits of such practices in enhancinglearning for the majority of students and point out that these practices also motivatethem to learn. This suggests powerful beliefs within the particular field of personalepistemological beliefs and within the larger field of mathematics education.

Another dimension of the depth of the students’ mathematical views is reflectedin the nature of the students’ articulation of cognitive process involved in doing orlearning mathematics. The students provide different characterizations of mathemat-ical understanding concepts with qualifiers such as conceptual, operational, durable,personal, and interpersonal. Understanding is also defined not only as recall but alsoas the ability to reconstruct previously learned ideas. Arguing is associated withknowledge acquisition as well as justification or proof for mathematical claims.There are also rich descriptions of the conditions in which learning, particularlydiscovery learning, takes place: hands-on activities, explorations, problem solv-ing, interesting and challenging tasks, collaborative work as arguing or discussingideas, work on tasks over time, and so on. The students’ ability to articulate indetail different cognitive aspects involved in learning is another measure of depthof the students’ mathematical beliefs. In particular, Romina’s idea about know-ing or understanding mathematics as personal is particularly insightful. A greatdeal 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 pro-mote the conception between formally defined mathematics and students’ personalconceptualizations to promote understanding.

Finally, it is particularly interesting that the students’ mathematical views mirrorthe particular conditions within which they engaged in mathematical activities inthe longitudinal study. Under the idea that the longitudinal study was more of a“learning experiment” rather than “teaching experiment,” researchers encouragedthe students to work collaboratively with other students; justify their reasoningto classmates; be the arbiters of whether or not a solution was correct based onwhether 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 thelongitudinal study. This suggests that the importance of construing mathematical


beliefs within the particular experiences in which students engage in mathemat-ics. In particular, this highlights the importance of teachers paying closer attentionto the kind of beliefs that they might be promoting in their students through theirconscious or unconscious practices or beliefs in mathematics classroom. This isan area which remains largely unexplored, as few studies have examined the rela-tion between mathematical beliefs and particular settings, whether within or acrosscultural settings.

In summary, we provide here an existence proof that students below collegelevel are capable of building powerful mathematical beliefs and insights about thecognitive processes and conditions involved in doing mathematics. However, wealso emphasize the importance of examining mathematical beliefs within particularlearning conditions.

In this and preceding chapters, we followed students in the longitudinal studythrough elementary school, middle school, and high school, working on problemsin combinatorics. In the following chapter, we look at a group of college studentsworking on the towers and pizza problems, and we see how their work compares tothat of the younger students.

Chapter 15Adults Reasoning Combinatorially

Barbara Glass

Date and Grade: 1998–2000; College FreshmanTasks: Pizzas and TowersParticipants: 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 themfrom the elementary students of the same names discussedin other chapters.)

Researcher/Instructor: Barbara Glass

15.1 Introduction

In the preceding chapters of this book, we have provided considerable evidenceshowing elementary and secondary school students’ success in solving open-endedproblems, over time, under conditions that encouraged critical thinking. In thischapter, we address the question as to whether similar results can be achieved byliberal-arts college students within a well-implemented curriculum that includes astrand of connected problems to be solved over the course of the semester. From aperspective of conceptualizing reasoning in terms of solving open-ended problems,it was of interest to learn whether students in a liberal-arts college mathematicscourse could be successful in providing arguments to support their reasoning and inmaking connections in a problem-solving-based curriculum.

Students enrolled in college-level mathematics courses might be expected to havealready developed effective reasoning skills. Unfortunately this is too often not thecase. This may be explained, in part, by a history of mathematics instruction insettings that devalue thinking and focus on rote and procedural learning.

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


172 B. Glass

Often, in traditional mathematics classrooms, the answer key or the teacher isthe source of authority about the correctness of answers; unfortunately, quick, cor-rect answers are often valued more than the thinking that leads to the answer. Toooften, teachers ask students to explain their thinking only when answers are wrong,emphasizing the product rather than the process of problem solving. Sanchez andSacristan (2003) offer data to support this from studying students’ written work.They report that students are not accustomed to expressing mathematical ideas, andthey offer as an explanation that the emphasis in schools is mainly on producingcorrect solutions. One consequence is that students tend to develop the belief thatall problems can be solved in a short amount of time. Students often stop trying tobuild 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 areasonable 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 averagetime was 12 min. Further, students view school mathematics as a process of master-ing formal procedures. These rules are often removed from real-life experience andapplication. As a result, students can feel that answers need not make sense. It isnot surprising, then, that students accept and memorize what they are told withoutmaking any attempt to deal with meaning (Schoenfeld, 1987).

Since many of the students in this study were previously taught mathematicsin this fashion, it would not be entirely surprising if they were unable to applyknowledge from previous mathematics courses to novel situations. Moreover, sincea student’s willingness to think about a problem is influenced by notions about whatmathematics is and what should be expected of students, it is not surprising whenstudents do not display the level of reasoning of which they are capable.

In this chapter, we examine how a small group of community college studentsenrolled in a liberal-arts mathematics class solved open-ended non-routine prob-lems in which they had to build and justify a solution. The tasks were the towersand pizza problems and extensions of these tasks. Our questions were (1) How docollege students solve non-routine mathematical investigations? (2) How do col-lege students’ representations and level of reasoning contrast with those of youngerstudents from a longitudinal study engaged in the same investigations? (3) Whatconnections, if any, do the college students make to analogous problems and to therules learned in previous classes? (4) To what extent, if any, do the college studentsjustify and generalize their results?

15.2 The Study

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

15 Adults Reasoning Combinatorially 173

called “consumer math.” The two liberal-arts mathematics courses can be takenin either order, so some students in Mathematical Concepts had already takenContemporary Mathematics and others had not. Most of the students in theMathematical 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. Somehad taken college preparatory mathematics in high school, while others took onlygeneral mathematics courses. Some had already taken other college-level mathemat-ics courses, while for others this was their first college-level mathematics course.When asked on a questionnaire about their mathematics background, many stu-dents described themselves as being very poor mathematics students who dislikedand feared mathematics, while others stated that they liked mathematics and hadalways done well in mathematics classes. There was also a wide range of ages,some students having recently completed high school, with others not having takenany mathematics for many years.

The study took place in a relatively new community college of moderate sizein an area of New Jersey that ranges from rural to suburban with very little racialor cultural diversity. In the fall semester of 2000 there were 929 full-time and 1,357part-time students enrolled. As with other community colleges, some of the studentsattend because poor academic records prevent them from being accepted elsewhere.Others are excellent students who attend the college for a variety of reasons includ-ing lower costs and the convenience of being close to their homes and places ofemployment.

Nine classes ranging in size from 6 to 25 students were studied between 1998 and2000. Sections of the course met for 15 weeks for two 75-min classes each week orthree 50-min classes each week. The students spent approximately half of the classtime working on various non-routine problems in a small group setting. After theyworked together on these problems the students were encouraged to present theirsolutions to the class. In addition, a weekly problem-solving homework assignmentwas given. As a part of the assignment, students were required to give a writtenexplanation of their solution method and a justification of how they knew that theirsolution was correct. Students also submitted write-ups of the problems done inclass.

Two groups from each class were videotaped as they worked on the towers andpizza problems. In addition, task-based interviews with ten representative studentswere videotaped. Students were selected because they were willing to be video-taped while participating in problem-solving sessions and willing to participate invideotaped follow-up interviews.

15.3 Student Solutions

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

174 B. Glass

problem. They then were asked to consider the five-tall towers problem. Somegroups also worked with three-color towers problems.

The students worked on the pizza problem during the 13th or 14th week of eachsemester, first on the four-topping pizza problem and then on the five-topping pizzaproblem. After they solved the basic problems, some groups were asked to considerthe pizza with halves problem, in which a topping could be placed on either a wholepizza or a half pizza.

15.3.1 Towers Problems

Most of the college students used patterns or some other form of local organiza-tion immediately, and some immediately imposed a global organization scheme. Anorganization by cases according to the number of cubes of one color was the methodchosen by six students. One group, which started the problem by randomly gener-ating towers using a build and check method, switched to this organization by casesat the suggestion of Jeff C. He said,

Here, put the ones that have three yellows and a red all together. [Danielle rearranges thetowers.] Okay. So now we do three yellows and a red at the bottom, ’cause you don’t havethat. [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 andthe other.

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

RED YELLOW

Fig. 15.1 Dana’s family ofone red cube and four yellowcubes


The case with two cubes of one color and two cubes of the other color was moreproblematic. The students used a variety of methods to demonstrate that they hadfound 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 organizetheir towers and move toward a proof by cases. However, both groups still statedthat their justification for the claim that they had all towers with two of each colorwas that they could not find any more.

Three of the students, Lisa, Errol, and Wesley, tried to argue that the number oftowers is sixteen because four times four is sixteen. The instructor responded thatthey needed a reason why the answer should be four times four. Lisa then produceda proof by cases, although she had difficulty justifying the case of two cubes of eachcolor. During her interview 7 weeks later, Lisa found an organization that accountedfor all of the towers with two cubes of each color.

Wesley rearranged his towers, but he offered no explanation for why his arrange-ment produced all possible towers. About 7 weeks later, during the interview,Wesley produced a similar arrangement and used it to account for all possible com-binations with a proof by cases where his cases were (1) towers with four cubes ofthe same color together, (2) three cubes of the same color together, (3) two cubes ofthe 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 proofthat 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 wasthe correct answer. This arrangement grouped towers with a red on top together andtowers with a yellow on top together. When the instructor continued to question himas to why this showed that the answer should be four times four, Errol turned tosimpler cases in an attempt to verify his numerical argument. He then noticed thedoubling pattern and used that to develop an argument by induction, abandoning thefour-times-four argument.

Five of the college students did a proof by cases for the five-tall towers. Eachof these proofs referred to opposites (pairs of towers with opposite colors in thesame positions). They also all used a staircase pattern to account for the towers withone cube of one color and four cubes of the other color. They used a variety ofmethods to justify the cases with two cubes of one color and three cubes of the othercolor.

After Rob and his group had organized their towers by cases, they noticed that thenumber of five-tall towers was double the number of four-tall towers. They extendedthe doubling pattern to predict how many towers they would get if the towers werethree tall, two tall, and one tall. They then built the one-tall and two-tall towers inorder to test their theory. While justifying their answer to the five-tall towers prob-lem, 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 oftowers doubled. After a few minutes, Rob explained to Steve that the number dou-bled because you could add either a red cube or a yellow cube to the bottom of eachtower. He explained as follows, building from a generic original tower he called X.

176 B. Glass

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

Steve demonstrated that he understood Rob’s explanation by using Rob’s proce-dure 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 histowers of four. He then built the opposites of these towers to find all towers witha yellow cube on top. He justified that he had found all five-tall towers with aninductive argument, but he was unable to extend this reasoning to predict how manysix-tall towers there are. However, during an interview 7 weeks later, Wesley cor-rectly extended the doubling pattern beyond the case that went from four-tall tofive-tall and predicted that there would be 64 six-tall towers.

Jeff C. applied the fundamental counting principle to predict that there wouldbe 32 five-tall towers. After Jeff’s group had produced those 32 towers, he used aninductive argument to show that they had found all possible towers by pairing eachof 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 withfour-tall towers to predict that there would be 32 five-tall towers. Even though hedid not build the five-tall towers in class, he used an inductive method to producea list of all five-tall towers on his written assignment. Figure 15.2 shows Errol’smethod.

There are sixteen possibilities. To justify my answer we will start with the possibilities ifthe towers were two cubes high (R-red, W-white)

Now, if we want to go to towers three blocks high, we simply take the 4 towers we haveand add a white block to the top and do the same with the red block (8 towers)

Now, for 4-cube high towers, we do the same thing: add a white block to the top of alleight 3-cube high towers and add a red to each of the eight towers. (This would alsowork if you put them on the bottom instead). (16 towers)

W-W W-R

R-W-W

W-R-W-W W-W-W-W R-R-W-WW-R-W-RR-R-R-WR-R-R-R

R-W-W-WW-W-W-RR-W-R-WR-W-R-R

W-W-W-RW-W-R-WW-W-R-R

&W-R-W-RW-R-R-WW-R-R-R

W-W-WW-W-RW-R-WW-R-R

R-W-RR-R-WR-R-R

&

R-W R-RWe have 4 possibilities.

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


B

B

B

B

b

b B b B b B b B b B b B b B b

B b B b B b

Bb

b

b

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 prob-lem, did the problem at home. She invented a tree diagram strategy to produce aninductive argument for the four-tall and five-tall towers. Her written work is shownbelow. 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) mysecond 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 mypossibilities to 8, and for each of those eight combinations I can add a blue or a browncube which finally doubles my answer giving me 16 possibilities. (I am using big ’B’ forbrown and a little ‘b’ for blue.)

Tim used a binary coding system to justify his conclusion that he had found allpossible combinations. This is the same method used by tenth-grade Mike fromthe Rutgers longitudinal study to justify his assertion that there were 32 differentfive-topping pizzas.

Several groups also had time to work on the three-color towers problem. MikeC.’s group and Rob’s group worked on four-tall towers, while Jeff’s group workedon three-tall towers. Rob and his partners applied the inductive method that theyhad developed for towers with two colors to solve the problem quickly. Jeff C. usedthe fundamental counting principle to calculate the number of towers, but he did notuse an inductive method to build the towers. Mike’s group divided the problem intotwo 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 four-tall towers that contained those two colors. After eliminating the three duplicateone-color towers that they had produced using this method, they had a total of 45towers. Then they worked on the second case, towers that contained at least onecube of each color. They each chose one of the colors and built all 12 towers withtwo cubes of that color and one cube of each of the other colors. Their solution forthe 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

178 B. Glass

fixed and paired it with each of the other toppings. Then they moved to the nexttopping on the list. Some students paired each additional topping only with top-pings that were below it on the list, reasoning that they had already accounted forthe other combinations. Other students considered all pairs and then eliminated theones that they already had. For the three-topping pizzas, some of the students againsystematically went down the list of toppings, while others failed to exhaust bothmajor and minor items. For example, after some students combined pepperoni andgreen peppers with all other possibilities, they moved to the pepper and mushroomtoppings instead of exhausting all possibilities that contained pepperoni.

Two students used a chart that was similar to the chart that fourth-grader Brandonhad created when he did the pizza problem (Maher & Martino, 1998). Instead of theones and zeroes that Brandon had used, these students made a check to indicate thata topping was on a pizza and left a blank space to indicate toppings that were not onthe pizza. Interestingly, this is the same method that Brandon’s partner Colin used.

Stephanie C., who was simultaneously enrolled in a statistics class, hypothe-sized that she could calculate the number of pizzas using combinations formulas.She and her partner Tracy systematically created a list of pizzas and compared thenumbers to those that Stephanie C. had conjectured using her formulas, confirmingStephanie’s prediction.

Several other students who had previously studied combinations tried to cal-culate the number of pizzas using combinations formulas. However, they did notunderstand combinations well enough to apply them to the problem correctly. Forexample, Melinda stated that the problem could be solved either by combinations orby permutations, but she could not remember which to use. Also, students who triedto use formulas attempted to do a single calculation instead of doing separate cal-culations for each number of toppings. In short, most students were not successfulat using combinations. The correct combinations formula for the number of pizzashaving exactly r toppings when there are n toppings to select from is

nCr =(

nr

)= n!

r! (n − r)!

To find the total number of four-topping pizzas, therefore, it is necessary to sum4C0 through 4C4 (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 betweenthe towers problem and the pizza problem. For example, Rob and Samanthaexplained as follows:

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


ROB: Right. Because it was convenient. So onion would be the secondblock, sausage would be the third block, and mushroom wouldbe the bottom block.

INSTRUCTOR: Okay. What would your colors be?SAMANTHA: Orange and yellow.ROB: Yeah.INSTRUCTOR: What would orange be?ROB: Orange means it’s on the pizza.INSTRUCTOR: And yellow?ROB: It’s off.SAMANTHA: Off the pizza.

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

JEFF C.: What you could do to relate that to the topping problem. Is that,you could say, you could designate each spot for a topping. Thisis the pepperoni. This is the green pepper spot, and this is thesausage spot. [Jeff C. writes toppings by the drawing of the firsttower.]

INSTRUCTOR: Okay.JEFF C.: Or onion, or whatever you want to have it.INSTRUCTOR: Whatever.JEFF C.: So with nothing on it, with no toppings, there’s only one pos-

sibility. [Jeff C. points to the first tower he drew.] Now forone topping, there’s three possibilities for a one topping pizza.There’s pepperoni. [Jeff C. marks the first block of the secondtower.] There’s green pepper [Jeff C. marks the second block ofthe third tower], and there’s sausage. [Jeff C. marks the bottomblock of the fourth tower.]

INSTRUCTOR: Okay.JEFF C.: Then for a two topping pizza [Jeff C. draws three more tow-

ers.], there’re three possibilities. [Jeff C. moves the actual towerswith one white cube.] Pepperoni and green pepper [Jeff C. marksthe first and second blocks on the first tower.], pepperoni andsausage [Jeff C. marks the first and third blocks.], and greenpepper 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 otherwords, 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’dhave two pizzas with pepperoni and green pepper.

INSTRUCTOR: Okay.

180 B. Glass

JEFF C.: So those are the three possibilities. The ones with two toppings.And then for three toppings, there’s only one possibility – pep-peroni, green pepper, and sausage [Jeff C. draws another toweron paper and marks all the blocks.], or blue, blue, and blue. [JeffC. 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. Thisis the same explanation that the third-grader Meredith had originally given for therelationship between the two problems. Meredith later revised her explanation tonote correctly that the blocks in the towers corresponded to the toppings and the col-ors 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 taketheir exploration any further and so did not revise their explanation.

Melinda, Stephanie C., Wesley, and Lisa mentioned the relationship between thetwo problems during their interviews, approximately 1 week after the class ses-sion on the pizza problem. At this time, each student displayed an understanding ofthe isomorphism between the two problems. During her interview, Lisa first builtthe four-tall towers and then she produced a pizza problem chart similar to theone that she had made in class. As she completed the chart, she discovered thatthe rows of the chart looked like the towers. As a consequence, she was able tomatch the towers with the rows in her chart. A portion of Lisa’s chart is shown inFig. 15.4.

Pep. Onion

0 0

0

0

0

0

0

0

0

0

0 0

0

0

0

0

0

0

0

0

Mush. Bk.

Fig. 15.4 Lisa’s chart for thepizza problem


Lisa remarked that the entries in her chart were like the binary system that theclass 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 patternand verified that it continued to hold for smaller numbers of available toppings. AsMike C. thought about why the number of pizzas should double, he discovered thatthe numbers from the pizza problem matched the rows of Pascal’s triangle. He wasunable to explain, however, why the addition rule of Pascal’s triangle applied topizzas.

After Rob and Steve’s group completed the four-topping pizza problem, Steveremarked that they should look for a pattern before moving on to the five-toppingproblem. Rob noticed that the pizzas from the four-topping problem formed a rowof Pascal’s triangle. After figuring out how the addition rule of Pascal’s triangleworked with pizzas, Rob used an inductive method and Pascal’s triangle to create alist of pizzas for the five-topping pizza problem. Rob’s explanation of the additionrule for the Pizza Problem was similar to that used by Mike and the other studentsin the Night Session, discussed in Chapter 12. Rob wrote,

ZerotoppingsZero row

Onetopping1st row

Twotoppings

Threetoppings

Fourtoppings

Fivetoppings

2nd row

3rd row

4th row

5th row 1 5 10 10 5 1

PGMOS

PGMO

PMOSGMOSPGOSPGMSPGMO

0

1

1

0

0

1

0

1

0

1

1

1

1

14

0

P

PG

PGM

GMOPMO

PGOPGM

PMGMPG

POGOMOPMGMPG

PS, GO POS, PGOGOS, PMOMOS, GMOPMS, GMSPGS, PFM

GS, MOMS, PMOS, GMPO, PG

6

3

P

PGM3

4

PGMO

PGMOS

G2

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

182 B. Glass

The reason this [Pascal’s triangle] works is because every time we add another topping weare increasing the possibility of choice, without losing the old ones. In other words, all thetwo topping pies in the two topping total still apply when there are three total toppings. Alsoall 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 possiblecombinations – 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 newpossibilities 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 thesame ways as the Kenilworth students had done in elementary school and highschool. However, the college students solved the problems more quickly than thethird and fourth grade children and, unlike the elementary school children, the col-lege students did not generally rely on random checking as a primary strategy. Allcollege students were able to solve the four-tall towers problem and start the five-tall towers problem within a 50-min or a 75-min period. Most also finished thefive-tall towers problem within the same period. Several students also had time towork on extensions of the problem. However, the college students also showed lessinclination than the Kenilworth students to think about problems for extended peri-ods of time. Many stopped thinking about the problem after they had arrived at ananswer, even when the instructor asked them to spend some more time consideringthe 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. Thisis similar to what Milin had done in the fourth grade. (Refer to Chapter 5 fordetails.) Mike C. also noticed that the number of towers and pizzas was doubling, butMike C. was unable to provide a reason for the doubling pattern. It is interesting tonote that in fourth grade both Stephanie and Milin had noticed the doubling patternin the towers problem. Stephanie’s progression from pattern recognition to devel-opment of an inductive argument took about 8 months while Milin’s understandingdeveloped more quickly (Alston & Maher, 2003; Maher & Martino, 2000). PerhapsMike C. would also have recognized the reason for the doubling pattern if, likeStephanie, 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 theproblem.

The college students made very few connections with previous mathematicalknowledge, and most of these connections were trivial connections. Several stu-dents recognized that the problems were related to permutations or combinations,but most did not understand these concepts well enough to correctly apply themto the solutions of novel problems. This would suggest that learning about math-ematics in an atmosphere in which students are told what to do does not enable


them to develop genuine understanding. In contrast, the students in this study diddemonstrate a high level of reasoning as they thought about the problems, justifiedtheir solutions, and made connections between problems and the mathematics theylearned within the course.

Some of the conditions of the Rutgers University longitudinal study, such asextended classroom sessions and revisiting the same problem several times withinan extended time frame, could not be replicated because of the time constraints ofa college course. However, many of the conditions that enabled the elementary andsecondary students to become thoughtful problem solvers were duplicated. Bothgroups were given rich mathematical tasks and were encouraged to explain theirreasoning and methods of solution and to justify their solutions to the problems.Both groups were engaged in thoughtful mathematics. They found patterns, devel-oped methods of justifications, and provided justifications that their patterns werereasonable. It cannot be disputed that the students in the Rutgers longitudinal studybenefited from exposure to rich mathematical experiences over an extended periodof time. It is also significant that the students in this study, who had previously expe-rienced a variety of generally traditional mathematics instruction, demonstrated thatit is not too late to introduce rich mathematical experiences in a collegiate levelmathematics class. The level of reasoning that these students demonstrated providesevidence that it is possible to experience thoughtful mathematics within a traditional15-week college semester.

In this chapter, we have shown how college students worked on the towers andpizza problems. In the following chapter, we will follow up on the work on thesecollege students and other college students, comparing the work of these collegeundergraduates to the work of the longitudinal study high-school students (seeChapter 8) on the extension of the towers problem called Ankur’s Challenge (findingthe number of four-tall towers, built when choosing from three cube colors, havingat least one cube of each color).

Chapter 16Comparing the Problem Solving of CollegeStudents with Longitudinal Study Students

Barbara Glass

16.1 Introduction

In this chapter we consider a variety of solutions to Ankur’s Challenge from studentsranging from high school to graduate level study of mathematics. The problem wascreated by Ankur and first posed to tenth-grade classmates in an after-school sessionof the Rutgers longitudinal study (see Chapter 8).

Find all possible towers that are four cubes tall, selecting from cubes available in threedifferent 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 class-mates, we have given his problem to several cohorts of students enrolled in liberalarts mathematics classes and in graduate mathematics-education courses. Studentspresented their written work and gave further verbal explanations and clarificationsof their solutions. Researcher notes provided the data for the oral explanations. Weclassified their forms of reasoning into four categories: (1) justification by cases,(2) inductive argument, (3) elimination argument, and (4) analytic method (use offormulas). We present here some representative solutions from the high school (H),undergraduate (U), and graduate (G) students, listed according to the arguments thatwere used.

16.2 Justifications by Cases

We list below nine ways that students used justification by cases; one from highschool students, five from undergraduates, and two from graduate students. All thosewho used justification by cases began with the observation that one color would

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


186 B. Glass

appear twice and each of the other colors would appear once. Many then proceededby selecting one color and listing all possibilities for that color in some organizedfashion 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 generalcodes 1, 0, and X to indicate the three colors, and they focused on the placementof the duplicate color, using 1 to indicate the color that was duplicated and X andO to indicate the other two colors. As shown in Fig. 16.1, they found six possibleways to place the two 1’s in a four-tall tower. They observed that each tower pic-tured in Fig. 16.1 represents two towers, one with the X and O as shown on thetop of the diagram and the other with the X and O in the bottom position, giving12 towers. Finally, arguing from symmetry, they noted that X and O could also bethe duplicate color, and so there are 12 towers for the duplicate X case and 12 forthe duplicate O case. They concluded that there are therefore 36 towers that fulfillAnkur’s condition.

1

1 1

1 1

11

1 1

1 1

1 Ox

Ox

Ox

Ox

Ox

Ox

Ox

Ox

Ox

Ox

Ox

Ox

Fig. 16.1 Romina, Jeff, andBrian’s six prototype towers

16.2.2 Joanne and Donna’s Solution (U)

Joanne and Donna used red, blue, and green cubes. They found six ways to placetwo blocks of the same color in a tower containing four blocks, counting from thetop: positions 1 and 4, 1 and 3, 1 and 2, 2 and 4, 2 and 3, and 3 and 4. Each

16 Comparing the Problem Solving of Students 187

Fig. 16.2 Joanne and Donna’s listing of towers

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

16.2.3 Rob and Jessica’s Solution (U)

Rob and Jessica used yellow, red, and blue cubes. For their first case (which wedesignate as Case I), they chose yellow as the duplicated color. For Subcase A, theyfixed the blue cube in the top position and then they moved the red cube into thesecond, 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 cubeinto 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 thesecond yellow cube in the second, third, and fourth positions in turn. They notedthat each position of the second yellow cube produces two towers, because the redand blue cubes can be reversed. Therefore, there are six towers for Subcase C. Thetotal 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, givinga total of 36 towers. Refer to Fig. 16.3 for Rob and Jessica’s list of 12 towers inCase I.

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

188 B. Glass

16.2.4 Marie’s Solution (U)

Marie used blue, red, and yellow cubes. She started by assuming that the blue cubewas the color that appears twice. For the three cases, under this assumption, Mariecontrolled for variables. For the first case (Case 1) under this assumption, she fixedthe position of the first blue cube on the top of the tower and then she moved thesecond blue cube to the second, third, and fourth positions. Reasoning from symme-try, Marie noted that there are two towers for each of those positions because the redand 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 cubeinto the two remaining possible positions. As noted above, reasoning from symme-try, Marie knew that there are two towers for each position; therefore, there are fourtowers for Case 2. For Case 3, Marie placed the two blue cubes in the third andfourth position; that gives two towers for this case. Refer to Fig. 16.4 for a diagramof the 12 towers that Marie found under the assumption that the duplicated coloris blue. Again reasoning from symmetry Marie noted that this process could berepeated 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 thatappears twice, while the other two colors appear once. If the blue cube appearstwice, keep the two blue cubes together and move to all possible positions. Thereare two towers for each position because the other two colors can be reversed. Nextseparate the two blue cubes by one and move into all possible positions. Again eachposition will give two towers. Finally place the two blue cubes in the first and fourthposition, separated by two cubes, to give two more towers. This process can berepeated 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 consideringthe case where blue is the top cube in the tower. April wrote, “Start with blue onthe top. If there is also a blue in the second position, the third and fourth position


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 secondcube is purple, the other two cubes must have at least one white cube. They can beWW, BW, WB, WP, or PW. If the second cube is white, the other two cubes musthave at least one purple cube. They can be PP, PB, BP, PW, or WP. This gives 12combinations with blue on top. There are also 12 combinations with white on topand 12 combinations with purple on top for a total of 36 towers.” Figure 16.6 is adiagram 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 sim-ilar to April’s, except she worked from the bottom up; her three cases were: bluecube on the bottom of the tower, white cube on the bottom, and purple cube on thebottom. Figure 16.7 shows Bernadette’s three cases.

190 B. Glass

Fig. 16.7 Bernadette’s threecases: blue, white, and purpleon the bottom

Bernadette made an exhaustive list of all possibilities for towers with a blue cubeon the bottom and the repeated the procedure for the other two colors. She wrote ofher 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 finallycame 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 waspositioned 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 didthe same with the white and purple blocks, replacing the blue, getting only 36 possibilitiesof towers.

16.2.8 Tim’s Solution (G)

Tim used the colors red, yellow, and green (R, Y, and G); he described three cases oftowers with the required conditions and reasoned from symmetry that all three caseswould have the same number of towers. He built the required towers by startingwith 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 cubefixed 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

R

R

R

R R R

R R

R R

R R G

G

G

G G G

G G

G G

G G

Fig. 16.8 Tim’s diagram forthe four-tall towers with twored and two green


exchanged each of the red cubes for a yellow cube in order to fulfill the conditionsof the problem; since this can be done in two ways (first exchange one red cube andthen 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 theother two cases. He described his reasoning as follows: “Given three colors RedYellow and Green, towers 4 tall containing at least one cube of every color willyield towers with 1R, 1Y, 2G; 1 R, 2Y, 1G; and 2R, 1Y, 1G. All these cases will beequal in number. Consider 2R and 2G. There are six towers that are four tall with2R and 2G. Now exchange a Y for one of the two R’s in each tower. There are twoways to do this for each tower. Therefore there are 2 × 6 = 12 towers of [each ofthe 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 conditionwith color A on the bottom (Case 1). For her first subcase (AB), she placed a cubeof color A on the bottom and a cube of color B next. Within this subcase, Traci firstplaced C in the third position; the top position can thus be any of the colors, givingthree towers ABCA, ABCB, and ABCC. Continuing the AB subcase, Traci kept ABin place and determined that there were two more towers that started with AB andthat did not have C in the third place; these are ABAC and ABBC. Thus for theAB subcase, she had a total of five towers. The second subcase was AC. Within thiscase, Traci first assigned B to the third position. As with subcase AB, the top positioncan 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 additionaltowers, ACAB and ACCB. Thus for subcase AC, there are again five towers. Forthe last subcase (AA), Traci assigned AA to the bottom two cubes; the two towersthat can be made with AA are AABC and AACB. These three subcases cover allpossibilities for the case in which color A is on the bottom; they give a total of 12towers with A. Arguing from similarity, Traci found that there are also 12 towerswith B on the bottom and 12 with C on the bottom, for a total of 36 towers. Traci’swritten work is shown in Fig. 16.9.

16.3 Inductive Arguments

Four students (three undergraduates and one graduate student) used inductivearguments, 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

192 B. Glass

Find all permutations with A on bottom

A B C

C C C

B B B

A A A

C C

A B

B B

A A

A B C

B B B

C C C

A A A

B B

A C

C C

A A

C B

B C

A A

A A

Use all three colors

once, then add extra

A bottom, B

next remaining

A bottom C next all colors

once, then alt.

A bottom, C

next remaining

A bottom, C

next remaining

There will be 12 permutations w/B on the bottom, and another 12 permutations w/C on the bottom.

12 X 3 = 36

12 perm, w/A

on the bottom

Fig. 16.9 Traci’s justification by cases

subcases (second cube white, second cube blue, and second cube red). If the secondcube is white, then Errol said that the third and fourth cubes would be one of thefour combinations blue–white, white–blue, blue–red, or red–blue; Errol missed thefifth 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 thirdand fourth cubes would have the other two colors (white–blue or blue–white), for atotal of two combinations. He thus found 10 of the 12 possible towers, and so whenhe multiplied this case by three (“since there are three colors involved, we can findout how many possibilities there are with one specific color on top, and multiply theanswer by 3”), his answer of 30 was off by 6. Figure 16.10 shows Errol’s writtenwork.

16.3.2 Christina’s Solution (U)

Using colors A, B, and C, Christina built up the four-tall towers by starting withall 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-talltowers.) 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-talltowers produced three four-tall towers when she added a fourth cube of each colorto the bottom, for a total of 27 towers. After Christina crossed out the 15 towers thatdid not fulfill Ankur’s condition (see Fig. 16.13), she was left with 12 towers. She


Fig. 16.10 Errol’s written justification

Fig. 16.11 Christina’s set oftwo-tall towers

Fig. 16.12 Christina’s set ofthree-tall towers with an A ontop

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

194 B. Glass

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

16.3.3 Bob’s Inductive Solution (U)

Working with colors red, blue, and yellow, Bob started by making the six three-talltowers that have all three colors, as shown in Fig. 16.14. He demonstrated that allsuch towers were accounted for by controlling for variables: he held the top colorfixed and moved the other two cubes in both possible positions.

B B

B

B

B

B

R

R

R R

R

R

Y

Y

Y

Y

Y YFig. 16.14 Bob’s starting setof towers

Then Bob built four-tall towers by adding a red, yellow, or blue cube to the bot-tom of each of the six three-tall towers. He noted that this would give all towers withtwo 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 allpossible positions. But with this procedure, Bob missed the towers with the dupli-cated color in the middle. When the instructor asked him to justify his solution, Bobabandoned this method and returned to a proof by cases. Bob’s solution by cases isdiscussed 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 wasto start with color B. Describing Case I on the tree diagram, she said, “(1) I startedon 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 isY, then the fourth must be B. If the third is B, the fourth must be Y [2 towers]. (3) Ifthe 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 totalof 12].”

Frances made the same inductive arguments for Cases II and III, for a total of 36towers.


R

R

R

R

B B B

B B

B

R R

R

R

R

R

R R R R R

RR

R R

Y

R Y

Y

Y

Y

Y

Y

Y

YY

Y Y Y Y

B

B

BB

B

12

12

12

36

BB

BB

R

Y

Y Y Y Y Y

YB

B B B B

B

B

B

B

YY

YR

R

R

R

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 eliminationarguments. All started with the fact that 81 (34) towers can be made when buildingfour-tall towers and selecting from three colors. Then they eliminated the towersthat did not meet Ankur’s criteria. Penny listed all the towers and crossed off theones that did not meet the criteria. Robert, Liz, and Mary calculated the number oftowers 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 towerswhen selecting from three colors is 3n.

16.4.1 Penny’s Solution (U)

Penny’s argument was a combination of inductive reasoning and elimination. Sheused a tree diagram to list all 81 four-tall towers with three colors, and then shecrossed off the towers that did not meet Ankur’s criteria.

196 B. Glass

16.4.2 Robert’s Solution (U)

Robert used colors red, blue, and green. Robert started with the 81 four-tall towersthat can be built when selecting from three colors; then he subtracted the towers thatdid not meet Ankur’s criteria; these are the towers with exactly one color and thetowers with exactly two colors. Robert gave the number of towers with exactly onecolor as 3×14, or three (one all red, one all blue, and one all green). He found thenumber of four-tall towers with exactly two of the colors by calculating the numberof all possible four-tall towers (which is 24) and subtracting the towers with onlyone of the colors (which is 2×14). Since there are three combinations of two colors(red/blue, red/green, and blue/green), he multiplied the two-color number by three.His complete calculation is: 34 − 3

(24 − 2

(14

)) − 3(14

) = 81 − 45 = 36Refer to Fig. 16.16 for Robert’s written work.

Problem: Want to build a tower 4 high with 3 uniquer colors (I chosered, green and blue) with the restriction, that at least 1 of each color isused.

I started by first calculating the total number of towers that are 4 tallw/3 colors and restrictions. So I drew a tower and mapped outpossiilities

Red, Blue or Green

3

R R R R

34 –3(24 –2(14)) – 3(14) = 34 – 45 = 36

B B B B G G G G

3 3 3

Next, I decided to subtract the towers that did not meet the clause of“at least one of each color” - I started with the easiest ones–oneswhich were 4 tall, but used only one color.

Which I found to be 3, but decided to write as 3 (14). Which helpedme see something better. Then I decided to subtract the ones that are4 tall, 2 colors, at least one of each which 24.

But I had to subtract 2 (or2(14)) because that left possibility ofgetting all one color or another. Since there are three combinations oftwo colors (red/blue, red/green, blue/green) I multiplied by 3 givingtotal number of 2 of 3 colors, 3 high, at least one of each color to3(24–2(14)). That gave me the total number of 4 tall, 3 colors, atleast 1 of each color to

Each “segment” of the tower had threepossible choices for color so total amountwas 3×3×3×3 = 34

Fig. 16.16 Robert’s written justification


16.4.3 Liz’s Solution (G)

Liz’s strategy was similar to Robert’s; she started with the 81 four-tall towers thatcan be built selecting from three colors, and then she subtracted the towers thatdo not meet Ankur’s criteria, which are the towers with at most two colors. Sheobserved that there are 16 towers with just red and green cubes (including the allred 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 greencubes (including the all blue tower and the all green tower). Since the three towersof all one color each appeared twice in the three groups of 16, Liz concluded thatthere are 3×16 – 3 or 45 towers that do not fit Ankur’s criteria. Subtracting 45 from81 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 ofthem have at least one of each color. Subtract out the ones that don’t have at leastthree 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 hereThere are 3 × 16 – 3 = 45 with at most two colors. 81 – 45 = 36 with at least oneof each color.

16.4.4 Mary’s Solution (G)

Like Liz, Mary started with the fact that 81 four-tall towers can be built whenselecting from three colors, and she also calculated that there are three sets of16 four-tall towers that can be built when selecting from two colors. She sub-tracted 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 tow-ers), and she added back the three, giving 36. Figure 16.17 shows Mary’s writtenexplanation.

16.5 Analytic Method

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

198 B. Glass

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 numberof 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 againwhen C is the repeated color, giving 36 towers fulfilling Ankur’s criteria. Figure16.18 shows Leana’s written analysis.

16.6 Discussion

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


4tall 3 colors at least 1 of each color

Colors A B C

CASE II CASE IIICASE I

A A B C B B A C

Tower

How many differentways can you arrange

AABC?

4!

Same Same

12

12 + 12 + 12 = 36

Another way: A is the color being repeated

A1A2BC 12×3 = 36

Works for each color4.3.2.1

2 2= = 12

Gets rid of repeats

24

12

2!

eliminates repeats

4.3.2.12

= 12


BBAC?


CCAB?

C C A B

A is the color repeated B is the color repeated C is the color repeated

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 calcu-late the total number of towers. That one student (Penny) used an inductive methodto generate her list of all 81 towers. All but two of the students who chose to doa 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 fixedcubed and rotated it exhaustively for particular cases. The approaches to arguingby cases varied. Students chose different cases into which to separate the towersand different variables for which to control as they built their justification. Therewere also variations within the other approaches. For example, students started theirinductive argument at different tower heights. Errol and Francis started at height oneand Christina started at height two. Bob started at height three, but he missed someof the towers as a result. He eventually resolved the discrepancy when he used a

200 B. Glass

solution by cases. Bob was the only student who used two different methods, andthat 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 theinterchange 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 thework and thus a greater confidence in the reasoning offered. For example, it was onlyafter sharing her justification with the instructor that April became confident thatshe had indeed found all possible towers. Also, Mary found her error (subtractingout some towers twice) when reviewing her work for presenting to others. In somecases, the discussion revealed to students flaws in their reasoning, resulting in a re-examination of the solution. As an illustration, the process of justifying that he hadthe correct number of towers enabled Bob to realize that his inductive method hadcaused him to miss several combinations. In this context, we can observe how theprocess of justifying their answers can enable students to reflect upon what theyhave done and on whether their answer is reasonable.

While the forms of reasoning generally fell into the four categories, the distri-bution of correct solutions was not uniform according to category. Few students(mostly graduate students) used formulas, and most of those students also used anelimination argument. Also, only graduate students and one senior undergraduatestudent used formulas correctly. Undergraduates successfully used arguments bycases 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 oflevels. Ankur’s challenge, a problem initially proposed to a group of high school stu-dents, has turned out to be of interest to students at many levels, and it has resultedin multiple kinds of thoughtful arguments. An important feature of this problemwas that students were required to account for all of the towers and then to buildarguments that were convincing to themselves and to others. It may be that prob-lems 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 explainand justify their ideas in writing and in the verbal sharing of results be included inmathematics courses at all levels.

Chapter 17Closing Observations

Arthur B. Powell

In the previous 16 chapters, we have witnessed ordinary students develop extraordi-nary mathematical ideas, forms of reasoning, and heuristics. Extraordinary are thesestudents’ accomplishments since their mathematical behaviors emerged not fromquickly parroting rules and formulae but rather from deliberately engaging their owndiscursive efforts. As Speiser (Chapter 7, this volume) notes, these students builtfundamental mathematical understanding, over time, through extended task-basedexplorations. They created models, invented notation, and justified, reorganized, andextended previous ideas and understandings to address new challenges. That is, theyperformed mathematics: created mathematical ideas and reasoned mathematically.These behaviors – ideating and reasoning – are fundamental human activities andhow they occur in the realm of mathematics, specifically elementary combinatorics,is what this book contributes.

Internationally, a community of mathematics education researchers has recog-nized this how question as substantially important. In January 1983, David H.Wheeler (1925–2000), the founding editor of the international journal, For theLearning of Mathematics, sent a letter to 60 or so mathematics educators invit-ing them to engage a daunting task: “suggest research problems whose solutionwould make a substantial contribution to mathematics education” (Wheeler, 1984,p. 40). The varied and thought-provoking responses of more than 15 educators werepublished, some in each of the three issues of the fourth volume of the journal. OnWheeler’s mind was the famous example of the 23 problems from various branchesof mathematics that David Hilbert (1862–1943) announced in an address deliveredto 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, USAe-mail: [email protected]


202 A.B. Powell

be expected” (Hilbert, 1900, p. 5).1 Of all the published responses to Wheeler’schallenge, Tall (1984) offered the briefest list of what he considered to be “the cen-tral questions”: (1) how do we do mathematics? and (2) how do we develop newmathematical ideas? (p. 25, emphasis added).

Hilbert’s 23 problems contributed to more than a century of vigorous, fruitfulresearch activity in physics and mathematics.2 Similarly, considered responses toTall’s two questions require substantial research efforts in different environmentsover extended periods of time. For researchers in mathematics education to enter-tain these questions, we must find ways to observe what learners do as they domathematics as well as to describe and analyze how they develop their mathematicalideas.

It bears noting that 8 years after Tall issued his central questions, Davis et al.(1992) similarly challenged mathematics education researchers to study the emer-gence 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 math-ematics education has focused on the actual ideas in students’ minds or on howwell teachers are able to identify these ideas, interact with them, and help studentsimprove on them” (p. 732). The chapters of this book have presented rich descrip-tions and analyses of actual ideas that students built in the realm of combinatorialreasoning. This work has implications for teaching both in the design and sequenceof effective tasks and in demonstrating how teachers could productively interactwith student ideas.

The global picture depicted in the chapters of this book underscores the needfor time to think deeply and discursively. A special issue of Educational Studies inMathematics collected several analyses concerning discourse in mathematics class-rooms. Commenting on these studies, Seeger (2002) wonders about the possibilityof a grand, panoramic theory of learning. In arguing for a comprehensive theoryof mathematics education, he suggests that such a theory needs to embrace fourmetaphors 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, Seegerfurther suggests that “theoretical work has to be balanced by the systematic develop-ment of focal problems for practice, theory, and research in mathematics education”

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

17 Closing Observations 203

(p. 289). He proposes two focal problems for mathematics education, one concern-ing ecological validity and representation and the other referring to the question oftime and change.

Besides epistemological concerns, the question of time and change also concernsmethodological issues. Building ideas and understanding are certainly temporal andunbounded. Consequently, there are complex judgments an investigator has to makewhen inquiring into what learners build, understand, or acquire from a discussion orlesson 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 thediscussion or lesson, in some other, more distinct time, or in some combination ofthese times? Ball and Lampert (1999) raise somewhat similar questions in a studyof teaching practice.

Epistemologically and methodologically, this book contributes to understandingthe relation between time and development. As outcomes of individual and collec-tive constructive actions over the course of the longitudinal study, the participantsbuild ideas, reason, and employ heuristics to resolve various tasks. They revealand make salient the important relationship between time and development. Theprocesses by which the participants build their ideas evidence an epistemologicalreality: knowledge construction is often a slow process. Mathematical ideas do notdevelop instantaneously and robustly but rather emerge slowly and in their nascentstate are rather fragile. Ideas dawn and mature over time. To loose fragility, amongother things, ideas need to be reflected on deeply, presented publicly, submitted tochallenge, available for negotiation, and subject to modification. That is, the essenceof developing and understanding mathematical ideas is often a protracted, iterative,and recursive phenomenon, occurring over more time than is usually appreciatedor acknowledged in practice in classrooms and in reports in the literature (Pirie &Kieren, 1994; Seeger, 2002). If learners are to develop deep understandings that areless fragile and more durable than is often witnessed by teachers in schools, theyneed to be offered extended periods of time to wrestle with a problem as well asto debate and negotiate heuristics, to articulate and justify their results, and to havetheir ideas challenged and then defend or modify their ideas.

If we agree that students must be actively and purposely engaged in their learningso that they can take ownership and be proud of their accomplishments, we need tocreate opportunities for this to occur. For example, strands of investigations can beintegrated into the regular curriculum as enrichment. When we eliminate the pres-sure of testing and grades, students can invest in thinking and reasoning for its ownsake and for the intrinsic rewards that knowing deeply entails. Perhaps every severalweeks, within particular strands, investigations can be revisited, and students canbring their more recent, accumulated knowledge to a more sophisticated examina-tion of earlier solutions, and thereby extend their knowing. A focus on justificationas a strand of school mathematics has great potential for building a solid foundationfor the later study in many fields and certainly of mathematics. A focus on rea-soning and sense making is an important requirement for a productive, responsiblecitizenry. Questioning, challenging, analyzing, revisiting all lead to better ways ofknowing. Can we as educators meet the challenge of educating thoughtful students

204 A.B. Powell

who are motivated by sense making and the critical review of ideas? The challengeawaits us.

The Video Mosaic Collaborative at Rutgers University provides a mechanismfor the ongoing building and sharing of knowledge. We invite readers to visitour website at http://www.video-mosaic.org/, view the videos and accompanyingobjects that provided the data for this book and join our expanding community ofresearchers by providing additional study and analysis.

Appendix ACombinatorics Problems

Listed here are the major combinatorics problems the students encountered fromelementary school through high school, along with brief discussions of solutions.

1. Shirts and Jeans (May 1990, Grade 2; October, 1990, Grade 3) – Stephen hasa white shirt, a blue shirt, and a yellow shirt. He has a pair of blue jeans and apair of white jeans. How many different outfits can he make?

He can make six different outfits; each of the two pairs of jeans can bematched 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, whitejeans/blue shirt, and white jeans/yellow shirt.

2. Shirts and Jeans Extended (October, 1990, Grade 3) – Suppose Stephen hadanother 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) – Yourgroup has two colors of Unifix cubes. Work together and make as many differ-ent towers four cubes tall as is possible when selecting from two colors. See ifyou 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 are2×2×2×2=16 possible towers that are four cubes tall. This can be generalizedto 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 mcolors to choose from. In the following discussions, we will call the first gener-alization (the n-tall tower with two colors) the towers problem, and we will callthe second generalization (the n-tall tower with m colors) the generalized towerproblem.

4. Cups, Bowls, and Plates (April 1991, Grade 3) – Pretend that there is a birthdayparty in your class today. It’s your job to set the places with cups, bowls, and

205C.A. Maher et al. (eds.), Combinatorics and Reasoning, MathematicsEducation Library 47, DOI 10.1007/978-0-387-98132-1,C© Springer Science+Business Media, LLC 2010

206 Appendix A: Combinatorics Problems

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 differentcombination of cup, bowl, and plate? Show how you figured out the answer tothis 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 platechoices. Therefore, there are 2×2×3 = 12 possibilities. Therefore, yes, it ispossible for ten children at the party each to have a different combination ofcup, bowl, and plate.

5. Relay Race (October 1991, Grade 4) – This Saturday there will be a 500-mrelay race at the high school. Each team that participates in the race must have adifferent uniform (a uniform consists of a solid colored shirt and a solid coloredpair 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 differentrelay 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) – Yourgroup has two colors of Unifix cubes. Work together and make as many differ-ent towers five cubes tall as is possible when selecting from two colors. See ifyou 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 hasthree colors of Unifix cubes. Work together and make as many different towersfour cubes tall as is possible when selecting from three colors. See if you andyour 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 = 81possible 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 com-binations 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 par-ticipate in a TV Quiz Show and the opportunity to win a vacation to DisneyWorld. The game is played by choosing one of four possibilities for winningand then picking a tower out of a covered box. If the tower you pick matchesyour choice, you win. You are told that the box contains all possible towers thatare 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:

Appendix A: Combinatorics Problems 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 anyof the others?

In order to decide which is the best choice, we need to find the probability ofeach 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 allyellow). The probability is 2÷8 = 0.25.

There is only one red cube: There are three ways; the red cube can be on thetop, 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 isyellow. 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 exactlytwo cubes are yellow or exactly three cubes are yellow. As discussed above,the probability that exactly two cubes are yellow (the same as the probabilitythat exactly two cubes are red) is 0.375. Since there is one way for exactlythree cubes to be yellow, that probability is 1÷8 = 0.125. The probability ofeither event is therefore 0.375 + 0.125 = 0.5. (We can add because the twoevents 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 youcan take a friend to Disney World. But now your box has all possible towersthat are four tall (built by selecting from the two colors yellow and red). Youare to select from the same four possibilities for a winning tower. Which choicewould you make this time and why would this choice be better than any of theothers?

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 allyellow). The probability is 2÷16 = 0.125.

There is only one red cube: There are four ways; the red cube can be on thetop, second from the top, second from the bottom, or on the bottom. Theprobability 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.


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 discussedabove, the probability that exactly two cubes are yellow (the same as theprobability that exactly two cubes are red) is 6÷16 = 0.375. The probabilitythat exactly three cubes are yellow is the same as the probability that onecube is red: 4÷16 = 0.25. Since there is one way for exactly four cubes tobe yellow, that probability is 1÷16 = 0.0625. The probability of any one ofthe 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 shophas 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 twotoppings could be added to either half of the plain pizza or the whole pie. Howmany choices do customers have if they could choose from two different top-pings (sausage and pepperoni) that could be placed on either the whole pizzaor half of a cheese pizza? List all possibilities. Show your plan for determiningthese choices. Convince us that you have accounted for all possibilities and thatthere 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 fourchoices are: plain (sausage off, pepperoni off), sausage (sausage on, pepperonioff), 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, thefour 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 wealready accounted for the plain-sausage pizza in Case 1.)

Case 3: Pepperoni. There are two possibilities remaining for the other half ofthe pizza: pepperoni and sausage/pepperoni. (Plain and sausage are alreadyaccounted for.)

Case 4: Sausage/pepperoni. There is only one possibility left for the otherhalf 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 shophas asked us to help design a form to keep track of certain pizza choices. Theyoffer a cheese pizza with tomato sauce. A customer can then select from thefollowing toppings: peppers, sausage, mushrooms, and pepperoni. How manydifferent 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 possiblechoices.

There are 2×2×2×2 = 16 possible pizzas.

12. Another Pizza Problem (April 1993, Grade 5) – The pizza shop was so pleasedwith your help on the first problem that they have asked us to continue ourwork. Remember that they offer a cheese pizza with tomato sauce. A customercan then select from the following toppings: peppers, sausage, mushrooms, andpepperoni. 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 allthe possible choices. Find a way to convince each other that you have accountedfor all possible choices.

Each of the 16 four-topping pizzas has two choices of crust, so there are 32pizzas.

13. A Final Pizza Problem (April 1993, Grade 5) – At customer request, the pizzashop 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 canthen select from the following toppings: peppers, sausage, mushroom, and pep-peroni. There is a choice of crusts: regular (thin) and Sicilian (thick). How manydifferent 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 possiblechoices.

The first half of the pizza can have 24 = 16 possible topping configurations,as described above. Consider each of those configurations in turn. Followingthe procedure described above for the two-topping half-pizza problem, we findthat there are 16+15+14+. . . +3+2+1 possible pizzas; this sum is given by16×17÷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 two-digit numbers can be made from the digits 1, 2, 3, and 4? Each of four cards islabeled with a different numeral: 1, 2, 3, and 4. How many different two-digitnumbers can be made by choosing any two of them?

Counting I: Assuming that you are not permitted to reuse digits, there are fourchoices for the first digit and three for the second digit, giving 12 two-digitnumbers. (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 thesecond 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 thebinomial coefficients.


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

(a + b)n =n∑

r=0

(nr

)an−rbr

The coefficient of an–rbr is given by:

(nr

)= n!

r!(n − r)!This number is the rth entry in the nth row of Pascal’s Triangle, and it givesthe number of towers with exactly r cubes of one color, when building towersthat are n-tall and there are two colors to choose from. Hence, the binomialexpansion and the towers problem are isomorphic, with the number of instancesof 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) – Youhave two colors of Unifix cubes (red and yellow) to choose from. How manyfive-tall towers can you build that contain exactly two red cubes?

You are selecting two items (the positions of the two red cubes) from fivechoices (the number of cubes in the tower); there are ten ways to do this:

(52

)= 5!

2!(5 − 2)! = 10

17. Ankur’s Challenge (January 1998, Grade 10) – Find all possible towers that arefour cubes tall, selecting from cubes available in three different colors, so thatthe resulting towers contain at least one of each color. Convince us that youhave found them all.

Suppose the colors are red, blue, and green. We are counting the towers in threecases: (1) those with two red cubes, one blue cube and one green cube, (2) thosewith one red cube, two blue cubes, and one green cube, and (3) those with onered cube, one blue cube, and one green cube. The following equation gives thenumber of ways of selecting m groups of objects of size r1 through rm:

(nr1, r2, ..., rm

)= n!

r1! · r2! · ... · rm! , where∑

ri = n

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

(42, 1, 1

)= 4!

2! · 1! · 1! = 12


Similarly for the other two cases:

(41, 2, 1

)=

(41, 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, twoteams play each other in at least four and at most seven games. The first team towin four games is the winner of the World Series. Assuming that the teams areequally matched, what is the probability that a World Series will be won: (a) infour 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 isthe number of ways it can win three times in n–1 games (and then win thelast game). This is given by C(n–1,3). The probability of any given set of out-comes for n games is 1÷2n (since there are two equally likely outcomes foreach game). So the probability that one team wins the series in n games isgiven by C(n–1,3) ÷2n, and the probability of a win for either team is doublethat: 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 aresitting in a café in Paris and decide to play a game of flipping a coin. If thecoin comes up heads, Fermat gets a point. If it comes up tails, Pascal gets apoint. The first to get ten points wins. They each ante up 50 francs, making thetotal pot worth 100 francs. They are, of course, playing “winner takes all.” Butthen a strange thing happens. Fermat is winning, eight points to seven, when hereceives an urgent message that his child is sick and he must rush to his homein Toulouse. They carriage man who delivered the message offers to take him,but only if they leave immediately. Of course, Pascal understands, but later, incorrespondence, the problem arises: how should the 100 francs be divided?

We can list all the circumstances where Fermat gets two points before Pascalgets three points. He can do this in two flips, three flips, or four flips. (The gamecannot proceed past four flips. As soon as both players get to nine points, thenext 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.


(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. ThereforeFermat should get 100 × 11÷16 Francs ≈ 69 Francs and Pascal should get 31Francs.

20. The Taxicab Problem (May 2002, Grade 12) – A taxi driver is given a specificterritory of a town, shown below. All trips originate at the taxi stand. One veryslow night, the driver is dispatched only three times; each time, she picks uppassengers 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 pointand wonders if she could have chosen a shorter route. What is the shortest routefrom a taxi stand to each of three different destination points? How do youknow it is the shortest? Is there more than one shortest route to each point? Ifnot, 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 theright. 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 andto the right). Finding the number of shortest paths from the taxi stand (0,0) toany point (n,r) involves the number of ways to select r segments of one kind ofmovement in a path that includes two kinds of movements; i.e., the number ofshortest paths to (n,r) is C(n,r). For the specific cases given above, the shortestpaths are:

Blue: C(5,1) = 5.

Red: C(7,4) = 35.

Green: C(10,6) =210.

Appendix BCounting and Combinatorics Dissertationsfrom the Longitudinal Study

Franciso, J. M. (2004). Students’ reflections on their learning experiences: Lessonsfrom a longitudinal study on the development of mathematical ideas andreasoning. Unpublished doctoral dissertation, Rutgers University, Newark, NJ

Glass, B. H. (2001). Mathematical problem solving and justification with commu-nity college students. Unpublished doctoral dissertation, Rutgers University, NewJersey.

Kiczek, R. D. (2001). Tracing the development of probabilistic thinking: Profilesfrom a longitudinal study. Unpublished doctoral dissertation, Rutgers University,New Jersey.

Martino, A. M. (1992). Elementary students’ construction of mathematical knowl-edge: 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 mathematicalideas and reasoning in the discourse and inscriptions of learners engaged in acombinatorial task. Unpublished doctoral dissertation, Rutgers University, NewJersey.

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 casestudy. Unpublished doctoral dissertation, Rutgers University, Newark, NJ.

Steffero, M. (2010). Tracing beliefs and behaviors of a participant in a longitudinalstudy 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 andproof. Unpublished doctoral dissertation, Rutgers University, New Jersey.

Uptegrove, E. B. (2005). To symbols from meaning: Students’ investigations incounting. Unpublished doctoral dissertation, Rutgers University, New Jersey.

213

References

Alston, A. S., & Maher, C. A. (2003). Modeling outcomes from probability tasks: Sixth gradersreasoning together. In N. A. Pateman, B. J. Dougherty, & J. T. Zilliox (Eds.), Proceedingsof the 27th annual conference of the International Group for the Psychology of MathematicsEducation (vol. 2, pp. 25–32). Honolulu, HI: CRDG, College of Education, University ofHawaii.

Ball, D. L., & Lampert, M. (1999). Multiples of evidence, time, and perspective. In E. C.Lagemann & L. S. Schulman (Eds.), Issues in education research: Problems and possibilities(pp. 371–398). San Francisco, CA: Jossey-Bass.

Bellisio, C. W. (1999). A study of elementary school students’ ability to work with algebraic nota-tion and variables. Unpublished doctoral dissertation, Rutgers, The State University of NewJersey, New Brunswick, NJ.

Bruner, J. (1960). The process of education. Cambridge, MA: Harvard University Press.Carey, S., & Smith, C. (1993). On understanding the nature of scientific knowledge. Educational

Psychologist, 28(3), 235–251.Case, R. (1992). Neo-Piagetian theories of child development. In R. J. Sternberg & C. A. Berg

(Eds.), Intellectual development. New York: Cambridge University Press.Cathcart, W. G., Pothier, Y. M., Vance, J. H., & Bezuk, N. S. (2006). Learning mathematics in

elementary and middle schools: A learner-centered approach. Upper Saddle River, NJ: PearsonPrentice Hall.

Ceci, S. J. (1989). On domain specificity. . .More or less general and specific constraints oncognitive development. Merril-Palmer Quarterly, 35(1), 131–142.

Cobb, P., Wood, T., Yackel, E., & McNeal, B. (1992). Characteristics of classroom mathematicaltraditions: An interactional analysis. American Educational Research Journal, 29, 573–604.

Creswell, J. W. (1998). Qualitative inquiry and research design: Choosing among five traditions.Thousand Oaks, CA: Sage.

Davis, R. B. (1984). Learning mathematics: The cognitive science approach to mathematicseducation. Hillsdale, NJ: Lawrence Erlbaum Associates.

Davis, R. B. (1992a). Reflections on where mathematics education now stands and on where it maybe going. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning(pp. 724–734). New York: Macmillan.

Davis, R. B. (1992b). Understanding “Understanding”. Journal for Research in MathematicsEducation, 11, 225–241.

Davis, R. B., & Maher, C. A. (1990). The nature of mathematics: What do we do when we “domathematics”? [Monograph]. Journal for Research in Mathematics Education, 4, 65–78.

Davis, R. B., & Maher, C. A. (Eds.). (1993). Schools, mathematics, and the world of reality.Needham, MA: Allyn & Bacon.

Davis, R. B., & Maher, C. A. (1997). How students think: The role of representations. In L. D.English (Ed.), Mathematical reasoning: Analogies, metaphors, and images (pp. 93–115).Hillsdale, NJ: Lawrence Erlbaum Associates.

215

216 References

Davis, R. B., Maher, C. A., & Martino, A. M. (1992). Using videotapes to study the construc-tion of mathematical knowledge by individual children working in groups. Journal of Science,Education and Technology, 1(3), 177–189.

De Corte, E., Op’t Eynde, P., & Verschaffel, L. (2002). “Knowing what to believe": The relevanceof students’ mathematical beliefs for mathematics education. In B. K. Hofer & P. R. Pintrich(Eds.), Personal epistemology: The psychology of beliefs about knowledge and knowing.Mahwah, NJ: Lawrence Erlbaum Associates.

diSessa, A. A., & Sherin, B. L. (2000). Meta representation: An introduction. Journal ofMathematical Behavior, 19(4), 385–398.

Franciso, J. M. (2004). Students’ reflections on their learning experiences: Lessons from a longi-tudinal study on the development of mathematical ideas and reasoning. Unpublished doctoraldissertation, Rutgers University, Newark, NJ.

Francisco, J. M., & Maher, C. A. (2005). Conditions for promoting reasoning in problem solving:Insights from a longitudinal study. Journal of Mathematical Behavior, 24(2/3), 361–372.

Freudenthal, H. (1991). Revisiting mathematics education: China lectures. Dordrecht, TheNetherlands: Kluwer Academic Publishing.

Grattan-Guinness, I. (2000). A sideways look at Hilbert’s twenty-three problems of 1900. Noticesof the American Mathematical Society, 47(7), 752–757.

Gray, J. (2000). The Hilbert problems 1900–2000. Newsletter 36 of the EuropeanMathematical Society. Retrieved February 10, 2003, from http://www.mathematik.uni-bielefeld.de/∼kersten/hilbert/gray.html

Hammer, D., & Elby, A. (2002). On the form of a personal epistemology. In B. K. Hofer &P. R. Pintrich (Eds.), Personal epistemology: The psychology of beliefs about knowledge andknowing. Mahwah, NJ: Lawrence Erlbaum Associates.

Harvard-Smithsonian Center for Astrophysics. (2000). Private universe project in mathematics.Retrieved July 4, 2008, from http://www.learner.org/channel/workshops/pupmath

Hilbert, D. (1900). Mathematical problems. Retrieved February 7, 2003, from http://babbage.clarku.edu/∼djoyce/hilbert/problems.html

Hofer, B. K. (1994). Epistemological beliefs and first-year college students: Motivation and cogni-tion in different instructional contexts. Paper presented at the annual meeting of the AmericanPsychological Association, Los Angeles, CA.

Hofer, B. K. (2002). Personal epistemology as a psychological and educational construct: An intro-duction. In B. K. Hofer & P. R. Pintrich (Eds.), Personal epistemology: The psychology ofbeliefs about knowledge and knowing. Mahwah, NJ: Lawrence Erlbaum Associates.

Konold, C., Pollatsek, A., Well, A., Lohmeier, J., & Lipson, A. (1993). Inconsistencies in students’reasoning about probability. Journal for Research in Mathematics Education, 24(5), 392–414.

Lampert, M. (1990). When the problem is not the question and the solution is not the answer:Mathematical knowing and teaching. American Educational Research Journal, 27, 29–63.

Landis, J. H. (1990). Teachers’ prediction and identification of children’s mathematical behaviors:Two case studies. Unpublished doctoral dissertation, Rutgers, The State University of NewJersey, New Brunswick, NJ.

Landis, J. H., & Maher, C. A. (1989). Observations of Carrie, a fourth grade student, doingmathematics. Journal of Mathematical Behavior, 8(1), 3–12.

Leder, G., Pehkonen E., & Törner, G, (Eds.) (2002). Beliefs: A hidden variable in mathematicseducation? Dordrecht: Kluwer.

Lyons, N. (1990). Dilemmas of knowing: Ethical and epistemological dimensions of teachers’work and development. Harvard Educational Review, 60(2), 159–180.

Maher, C. A. (1988). The teacher as designer, implementer, and evaluator of children’s mathemat-ical learning environments. The Journal of Mathematical Behavior, 6, 295–303.

Maher, C. A. (1998). Constructivism and constructivist teaching: Can they coexist? InO. Bjorkqvist (Ed.), Mathematics teaching from a constructivist point of view (pp. 29–42).Finland: Abo Akeademi, Pedagogiska fakulteten.

References 217

Maher, C. A. (2002). How students structure heir own investigations and educate us: What we havelearned from a fourteen year study. In A. D. Cockburn & E. Nardi (Eds.), Proceedings of thetwenty-sixth annual meeting of the International Group for the Psychology of MathematicsEducation (PME26) (Vol. 1, pp. 31–46). Norwich, England: School of Education andProfessional Development, University of East Anglia.

Maher, C. A. (2005). How students structure their investigations and learn mathematics: Insightsfrom a long-term study. Journal of Mathematical Behavior, 24(1), 1–14.

Maher, C. A. (2008). The development of mathematical reasoning: A 16-year study (Invited SeniorLecture for the 10th International Congress on Mathematics Education, published in book withelectronic CD). In M. Niss (Ed.), Proceedings of ICME 10 2004. Roskilde, DK: RoskildeUniversity, IMFUFA, Department of Science, Systems and Models.

Maher, C. A., & Davis, R. B. (1990). Building representations of children’s meanings. In R. B.Davis, C. A. Maher, & N. Noddings (Eds.), Constructivist views on the teaching and learning ofmathematics: Journal for Research in Mathematics Education monograph (vol. 4, pp. 79–90).Reston, VA: National Council of Teachers of Mathematics.

Maher, C. A., & Davis, R. B. (1995). Children’s explorations leading to proof. In C. Hoyles &L. Healy (Eds.), Justifying and proving in school mathematics (pp. 87–105). London:Mathematical Sciences Group, Institute of Education, University of London.

Maher, C. A., & Martino, A. (1991). The construction of mathematical knowledge by individ-ual children working in groups. In P. Boero (Ed.), Proceedings of the 15th conference of theInternational Group for the Psychology of Mathematics Education (vol. 2, pp. 365–372). Assisi,Italy.

Maher, C. A., & Martino, A. M. (1992a). Teachers building on students’ thinking. The ArithmeticTeacher, 39, 32–37.

Maher, C. A., & Martino, A. M. (1992b). Individual thinking and the integration of the ideas ofothers in problem solving situations. In W. Geeslin, J. Ferrini-Mundy, & K. Graham (Eds.),Proceedings of the sixteenth annual conference of the International Group for the Psychologyof Mathematics Education (pp. 72–79). Durham, NH: University of New Hampshire.

Maher, C. A., & Martino, A. M. (1996a). The development of the idea of mathematical proof:A 5-year case study. Journal for Research in Mathematics Education, 27(2), 194–214.

Maher, C. A., & Martino, A. M. (1996b). Young children inventing methods of proof: The gang offour. In L. Steffe, P. Nesher, P. Cobb, G. Goldin, & B. Greer (Eds.), Theories of mathematicallearning. Hillsdale, NJ: Erlbaum.

Maher, C. A., & Martino, A. M. (1997). Conditions for conceptual change: From pattern recogni-tion to theory posing. In H. Mansfield (Ed.), Young children and mathematics: Concepts andtheir representations. Durham, NH: Australian Association of Mathematics Teachers.

Maher, C. A., & Martino, A. M. (1998). Brandon’s proof and isomorphism can teachers helpstudents make convincing arguments? (pp. 77–101). Rio de Janeiro: Universidade Santa Ursala.

Maher, C. A., & Martino, A. M. (1999). Teacher questioning to promote justification and gen-eralization in mathematics: What research practice has taught us. Journal of MathematicalBehavior, 18(1), 53–78.

Maher, C. A., & Martino, A. M. (2000). From patterns to theories: Conditions for conceptualchange. Journal of Mathematical Behavior, 19, 247–271.

Maher, C. A., Martino, A. M., & Alston, A. S. (1993). Children’s construction of mathematicalideas. In B. Atweh, C. Kanes, M. Carss, & G. Booker (Eds.), Contexts in mathematics educa-tion: Proceedings of the 16th annual conference of the Mathematics Education Research Groupof Australia (MERGA), pp. 13–39. Brisbane, Australia.

Maher, C. A., & Speiser, R. (1997a). How far can you go with block towers? Proceedings of theInternational Group for the Psychology of Mathematics Education (PME 21), Lahti, Finland,4, 174–181.

Maher, C. A., & Speiser, R. (1997b). How far can you go with block towers? Stephanie’sintellectual development. Journal of Mathematical Behavior, 16(2), 125–132.

218 References

Martino, A. M. (1992). Elementary students’ construction of mathematical knowledge: Analysisby profile. Unpublished doctoral dissertation, Rutgers, the State University of New Jersey, NewBrunswick, NJ.

Martino, A. M., & Maher, C. A. (1999). Teacher questioning to promote justification and gen-eralization in mathematics: What research practice has taught us. Journal of MathematicalBehavior, 18(1), 53–78.

Miles, M. B., & Huberman, M. A. (1994). Qualitative data analysis: An expanded sourcebook(2nd ed.). Thousand Oaks, CA: Sage.

Muter, E. M. (1999). The development of student ideas in combinatorics and proof: A six yearstudy. Unpublished doctoral dissertation, Rutgers University, Newark, NJ.

National Council of Teachers of Mathematics. (2000). Principles and standards for schoolmathematics. Reston, VA: National Council of Teachers of Mathematics.

O’Brien, M. (1994). Changing a school mathematics program: A ten-year study. Unpublisheddoctoral dissertation, Rutgers, the State University of New Jersey, New Brunswick, NJ.

Papert, S. (1980). Mindstorms: Children, computers and powerful ideas. New York: Basic Books.Perry, W. G. (1970). Forms of intellectual and ethical development in the college years: A scheme.

New York: Holt, Rinehart and Winston.Pintrich, P. R., & Hofer, B. K. (1997). The development of epistemological theories: Beliefs about

knowledge and knowing and their relation to learning. Review of Educational Research, 67(1),88–140.

Pirie, S. E. B., & Kieren, T. (1992). Creating constructivist environments and constructing creativemathematics. Educational Studies in Mathematics, 23, 505–528.

Pirie, S. E. B., & Kieren, T. (1994). Growth in mathematical understanding: How can wecharacterise it and how can we represent it? Educational Studies in Mathematics, 26(2–3),165–190.

Powell, A. B. (2006). Socially emergent cognition: Particular outcome of student-to-studentdiscursive interaction during mathematical problem solving. Horizontes, 24(1), 33–42.

Powell, A. B., Francisco, J. M., & Maher, C. A. (2003). An evolving analytical model forunderstanding the development of mathematical thinking using videotape data. Journal ofMathematical Behavior, 22(4), 405–435.

Roth, W.-M., & Roychoudhury, A. (1994). Physics students’ epistemologies and views aboutknowing and learning. Journal of Research in Science Education, 31(1), 5–30.

Sanchez, E., & Sacristan, A. S. (2003). Influential aspects of dynamic geometry activities in theconstruction of proofs. Proceedings of the 27th annual conference of the International Group forthe Psychology of Mathematics Education and the 25th annual meeting of the North AmericanChapter for the Psychology of Mathematics Education (vol. 4, pp. 119–126). Honolulu, HI.

Schoenfeld, A. H. (1987). What’s all the fuss about metacognition. In A. H. Schoenfeld (Ed.),Cognitive science and mathematics education (pp. 189–215). Newark, NJ: Lawrence EarlbaumAssociates, Inc. Hillsdale.

Schoenfeld, A. H. (1989). Explorations of students’ mathematical beliefs and behavior. Journal ofResearch in Mathematics Education, 20(4), 338–355.

Schommer, M. (2002). An evolving theoretical framework for an epistemological belief system.In B. K. Hofer & P. R. Pintrich (Eds.), Personal epistemology: The psychology of beliefs aboutknowledge and knowing. Mahwah, NJ: Lawrence Erlbaum Associates.

Schommer, M., & Walker, K. (1995). Are epistemological beliefs similar across domains? Journalof Educational Psychology, 87(3), 424–432.

Seeger, F. (2002). Research on discourse in the mathematics classroom: A commentary.Educational Studies in Mathematics, 49(1–3), 287–297.

Sfard, A. (1998). Two metaphors for learning and the dangers of choosing just one. EducationalResearcher, 27(2), 4–13.

Sfard, A. (2001). Learning mathematics as developing a discourse. In R. Speiser, C. Maher, &C. Walter (Eds.), Proceedings of 21st conference of PME-NA (pp. 23–44). Columbus, OH:Clearing House for Science, Mathematics, and Environmental Education.

References 219

Speiser, R. (1997). Block towers and binomials. Journal of Mathematical Behavior, 16(2),113–124.

Sran, M. K. (2010). Tracing Milin’s development of inductive reasoning: A case study.Unpublished doctoral dissertation, Rutgers University, Newark, NJ.

Sternberg, R. (1989). Domain-generality versus domain-specificity: The life and impending deathof a false dichotomy. Merrill-Palmer Quarterly, 35(1), 115–130.

Steffero, M. (2010). Tracing beliefs and behaviors of a participant in a longitudinal study forthe development of mathematical ideas and reasoning: A case study. Unpublished doctoraldissertation, Rutgers University, Newark, NJ.

Tall, D. (1984). Communications: Research problems in mathematics education—III. For theLearning of Mathematics, 4(3), 22–29.

Tarlow, L. D. (2004). Tracing students’ development of ideas in combinatorics and proof.Unpublished doctoral dissertation, Rutgers, The State University of New Jersey, NewBrunswick, NJ.

Torkildsen, O. (2006). Mathematical archaeology on pupils’ mathematical texts. Un-earthing ofmathematical structures. Unpublished doctoral dissertation, Oslo University, Oslo.

Uptegrove, E. B. (2004). To symbols from meaning: Students’ investigations in counting.Unpublished doctoral dissertation, Rutgers, The State University of New Jersey, NewBrunswick, NJ.

Weil, A. (1984). Number theory, an approach through history. Boston, MA: Birkhauser.Wheeler, D. (1984). Communications: Research problems in mathematics education—I (The

letter). For the Learning of Mathematics, 4(1), 40–47.Wilson, S. (1977). The use of ethnographic techniques in educational research. Review of

Educational Research, 47(1), 245–265.Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in

mathematics. Journal for Research in Mathematics Education, 27(4), 458–477.

Index

AAbstraction, 10, 17, 24, 77, 85Addition rule, 13, 73, 77, 82, 111–114,

124–125, 129–130, 133, 136–138, 141,152, 181

Addition rule of Pascal’s Triangle, 181Adults, 171–183Ali, 99–100Alston, Alice S., 5–6, 35, 45, 67, 97,

121, 182Amy-Lynn, 59, 62–63, 66–67, 121–122,

124–126Angela, 97–98, 103, 121, 127–129Angela’s Law of Towers, 98Ankur, 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, 209Argument, 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

BBeliefs, 157–159, 168–169Binary notation, 106, 108, 116, 118–120, 135,

141, 143Binary numbers, 106, 137Binomial, 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, 62Branches, 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

CCases, 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, 140Claims, 145–153, 164, 168, 202Coefficient, 77, 106, 135Collegiate, 183Combinations, 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, 202Combinatorics, 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, 147Connections, 3–4, 9, 18, 72, 74, 82, 105–120,

130–131, 141, 143, 171–172,178–182

Consumer math, 173

221

222 Index

Contemporary Mathematics, 172–173Contradiction, 14, 31, 40, 66–67, 97, 99Controlling for variables, 14, 35, 43,

97, 99Convincing arguments, 4, 104Counting I, 209Counting II, 209Counting methods, 172–173Cousin, 31–33, 36, 42, 46Critical events, 9Cups, Bowls, and Plates, 205–206

DDana, 11, 17, 19, 21–25, 27–32, 43, 60Danielle, 171, 174Discourse, 4, 74, 142, 146, 150, 159,

164–165, 202Discursive, 150–151, 164, 167, 201Donna, 171, 186–187Doubling rule, 39–40, 42, 49, 56,

106, 130Duplicates, 19, 29–30, 33, 36, 56, 61, 72,

78–82, 100, 122, 127, 197Durability of ideas, 141Dyadic choice, 151

EEducational Studies in Mathematics, 202Elevator, 31, 34, 37Empirical, 82, 152Epistemological beliefs, 157–159, 168Errol, 171, 175–176, 192, 199Euclidean geometry, 150Explanation, 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

FFactorials, 122Family tree, 52–53Fermat, 77Fermat’s Recursion, 77–78Final Pizza Problem, 70, 209Five-tall towers with exactly two red cubes, 33,

109, 119Five-tall towers selecting from two colors, 31Five-topping pizza problem, 13, 106, 112, 135,

174, 181Forms of reasoning, 6, 11, 14, 39, 72–73, 185,

200–201Formula, 77–78, 82, 103–104, 108–109, 122,

125, 128, 130, 141, 143, 178, 199For the Learning of Mathematics, 201

Four-tall towers selecting from 2 colors, 28,196–197

Four-tall towers selecting from 3 colors, 195Four-topping pizza problem, 68–70, 123, 127,

174, 181, 208Francisco, John M., 157–169

GGang of Four, 38, 42–43, 51, 75Generalization, 3, 17, 91, 94, 98, 106, 126Generalize, 3, 10, 18, 41, 90, 95, 104–105,

133, 147, 151, 172General rule, 97–98, 101, 124, 135, 141, 143Geometric, 108–109Glass, Barbara, 171–183, 185–200Grade 1, 7Grade 2, 20–22, 24, 73Grade 3, 3, 7, 22–23, 28, 74Grade 4, 5–6, 31, 33–42, 77Grade 5, 45, 59, 77Grade 6, 7Grade 7, 75Grade 8, 33, 73, 75, 85Group work, 25, 31, 36, 70, 103, 121, 162,

165, 177Guess My Tower, 42, 45, 50

HHarding School, 6Heuristic, 11–13, 25, 27, 43, 59, 67, 147,

201, 203High 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

IInductive 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, 151Isomorphic relationship, 108, 118, 120Isomorphism, 4, 14, 77, 109, 126, 130, 133,

147, 151–153, 179–180

JJaime, 17, 23Jeff C., 171, 174, 176–177, 179–180Junior year, 134Justification, 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 223

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

KKenilworth, 4–7, 12, 59, 69–70, 161, 182Kiczek, Regina, 121, 133Knowing deeply, 203

LLearning environment, 131Lecture classes, 118–120Letter codes, 106, 122, 142Liberal-arts, 171–173, 185Linda, 171Lisa, 171, 175, 180Logic, 39, 172Logical reasoning, 10, 17Longitudinal 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

MMagda, 97–99, 103, 121, 127–130Maher, Carolyn A., 4, 45–57, 59–72, 89–95,

105, 121, 133Making sense, 131, 134, 141, 145Martino, Amy M., 4, 9, 17–21, 27–28, 33, 35,

39, 41, 47, 49, 74–75, 178, 180, 182Mary, 171, 175, 195, 197–198, 200Mathematical beliefs, 157, 159, 168–169Mathematical concepts, 6, 119, 161, 172–173Mathematical 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, 147Melinda, 171, 178, 180Metaphor, 13, 85Michael, 17, 19–21, 23, 35Michelle, 27, 38, 41, 45, 50–54, 56, 59–60, 62,

64, 97, 99–103, 121–122, 127Michelle I., 27, 45, 50–54, 56, 59, 64–65Michelle R., 45, 50, 52–53, 56, 59, 62Mike, 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

NNational Council of Teachers of Mathematics

(NCTM), 5, 17, 144, 168National Science Foundation, 5Night session, 112, 116, 118, 133,

135–143, 181Notation, 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, 27N-tall towers selecting from r-colors,

42, 77

OOpposite, 29, 31, 35–36, 42, 46, 54, 67,

99–100, 150, 175Order, 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, 43Over time, 73, 75, 160, 162, 168, 171,

201, 203

PPantozzi, Ralph, 121, 161–162, 166–167Papert, 74–75Partial cases, 42Pascal, 211–212Pascal’s Identity, 13–14, 105, 111–115, 118,

120–121, 124–125, 127–130Pascal’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, 199Permutations, 172, 178, 182, 192Pirie, Susan, 97, 158, 203Pizza

with halves problem, 12–13, 67–69, 174plain, 12, 59, 106–107, 113, 117–118,

122–123, 125, 127, 141, 208problem, 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, 145–154Power of 2, 122, 124Probability, 165, 172

224 Index

Problem of Points, 211Proof

by cases, 41, 72, 119, 130, 175, 191, 194by contradiction, 99

RReasoning, 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, 97Recursive argument, 72, 199Relay Race, 206Representation, 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, 4–5Rob, 171, 175, 177–178, 181–182, 187Robert, 121–122, 124–127

SSamantha, 171, 179Scheme, 21, 24, 94, 105–106, 109, 174Second International Congress of

Mathematicians, 201Senior year, 145, 200Sense making, 6, 11–12, 24, 159–162, 200,

203–204Shelly, 121–127Sherly, 99–100, 121, 127–129Shirts and jeans, 11, 17–24, 27, 29, 59–60,

64, 73Shirts and jeans extended, 205Socially emergent cognition, 150Sociomathematical norms, 153Sophomore year, 119, 134–135, 140, 143Specialize, 147–148, 151Speiser, Robert, 33, 73–86, 105–112, 201Sran, Manjit K., 27–43, 45–57, 59–72

Staircase, 33, 40, 42, 174–175Standard notation, 133–144Statistics, 172, 178Stephanie, 11, 17, 19–25, 28–35, 40–42,

45–62, 71, 73–85, 121–127, 171, 178,180, 182

Stephanie C., 171, 178, 180Steve, 171, 175, 181Strategy, 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

TTarlow, Lynn D., 59, 97–104, 121–131Taxicab geometry, 150, 202Taxicab problem, 14, 144–147, 150–153, 212Theory, 51, 74, 102–103, 175, 202Tim, 171, 177, 190Tower

binomial relationship, 209families, 31–32, 34

Tracy, 171, 178Tree diagrams, 122, 127Trial and error, 29, 31, 35–42, 51

UUnderstanding, 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, 126Uptegrove, Elizabeth B., 9–14, 97–104,

105–120, 133–144

WWesley, 171, 175–176, 180

YYankelewitz, Dina, 17–25, 27–43, 45–57,

59–72

Combinatorics and Reasoning: Representing, Justifying and Building Isomorphisms

Documents