REINVENTION OF EARLY ALGEBRA Developmental research on the transition from arithmetic to algebra The illustration on the cover is a mathematical riddle from a manuscript by Paolo Dagomari (ca.1281-1370). Source: Tropfke (1980), p. 577 CIP-data Koninklijke Bibliotheek, Den Haag Amerom, B.A. van Reinvention of early algebra. Developmental research on the transition from arith- metic to algebra. Utrecht: CD-ß Press, Center for Science and Mathematics Educa- tion. - (CD-ß wetenschappelijke bibliotheek ; 41) Thesis Utrecht University. - With ref. - With a summary in Dutch. ISBN 90-73346-48-7 Subject headings: mathematics education, early algebra, history of mathematics Cover: Zuidam Uithof Utrecht Press: Wilco, Amersfoort Copyright: Freudenthal Instituut, Utrecht 2002 REINVENTION OF EARLY ALGEBRA Developmental research on the transition from arithmetic to algebra Heruitvinden van aanvankelijke algebra Ontwikkelingsonderzoek rond de overgang van rekenen naar algebra (met een samenvatting in het Nederlands) Proefschrift ter verkrijging van de graad van doctor aan de Universiteit Utrecht op gezag van de Rector Magnificus, Prof.dr. W.H. Gispen, ingevolge het besluit van het College van Promoties in het openbaar te verdedigen op donderdag 16 mei 2002 des middags te 16.15 uur door Barbara Ann van Amerom geboren op 13 maart 1969 te Dordrecht Promotoren: Prof. dr. J. de Lange Faculteit Wiskunde en Informatica, Universiteit Utrecht Prof. dr. K.G.E. Gravemeijer Faculteit Sociale Wetenschappen, Universiteit Utrecht Faculteit Wiskunde en Informatica, Universiteit Utrecht Co-promotor: Dr. J.A. van Maanen Faculteit Wiskunde en Natuurwetenschappen, Rijksuniversiteit Groningen Preface To be honest, my appointment to the project ‘Reinvention of algebra’ was a rather unexpected turn of events. I never had the ambition to pursue an academic career. During the final month of my teacher training I still had every intention to become a secondary school mathematics teacher. However, when I was given the opportuni- ty to combine my two main interests in mathematics – history and didactics – the choice was easily made. Looking back, there have been no regrets. Soon after the project started I took a part-time teaching job on the side, and just as I had hoped, the combination of theory and practice has been inspiring and fulfilling. At this time I wish to commemorate Leen Streefland, who helped me get started on the project and who supervised my early work. It has been an honor and a privilege to have known him and to have worked with him. He made me feel competent as if I were his equal and not a rookie, but even now there is so much more that I could have learned from him. Of course I could have never completed this book without the help of others. First of all I want to thank my supervisors Jan van Maanen, Jan de Lange and Koeno Gravemeijer, for their honest and constructive criticism, advice and support. I am es- pecially grateful to them for letting me decide my own course, and on occasions that I seemed to go astray they helped me just in time to find my way back. A special thanks goes to Jan van Maanen, who has guided me since my university years and who brought this project to my attention. He has been more than a supervisor to me, he has also been my friend. It was never too much trouble for him to travel from Groningen early in the morning in order to work together on my project the whole day. Even in rough times he continued to believe in me, sometimes more than I did myself, reminding me not to be too hard on myself. I also extend my thanks to my colleagues at the Freudenthal Institute, for their ex- pertise, their interest and support – in meetings, during coffee and lunch breaks or while waiting for the xerox machine to finish. In particular I want to thank Hanneke Beemer for designing and testing the student unit The Fancy Fair, and Mieke Abels for testing it again with her students and giving suggestions for improvement after- wards. I am very grateful to Marjolein Kool en Julie Menne for their advice and en- couragements when the going got rough. Moreover, Marjolein contributed some re- ally good ideas and historical materials for the student activities. Meeting the dead- line for this book would not have been possible without the assistence of Betty Heijman, Nathalie Kuipers, Ank van der Heiden-Bergsteijn, Marianne Moonen- Harmsen, Anneleen Post and Ellen Hanepen. v Uiteraard wil ik ook de docenten bedanken die hebben deelgenomen aan de verschil- lende onderwijsexperimenten. Dit zijn de leerkrachten van groep 7 en/of 8 van RK Basisschool De Kameleon te Utrecht, Koningin Julianaschool te Woudenberg, RK Basisschool ‘t Zonnewiel te Almere, Basisschool Rijnsweerd te Utrecht, CBS Het Anker te Brielle, OBS Lorentz te Hilversum en Basisschool Klaverblad te Amster- dam, en de brugklasdocenten verbonden aan RK Scholengemeenschap College De Klop te Utrecht, Almere College te Dronten en Mendelcollege te Haarlem. In het bij- zonder gaat mijn waardering uit naar Jaap Griffioen voor zijn op- en aanmerkingen op het lesmateriaal. Verder wil ik Monique Schipper en Natasja Capelle vermelden vanwege hun inzet bij de lesobservaties en -verslagen. Niet in de laatste plaats heb ik veel steun gehad van vrienden en familie. Mede dank zij hen is het me gelukt enkele tegenvallers te incasseren en sterker uit de strijd te komen. Met dit boek sluit ik een drukke, spannende en ook zeer dankbare periode in mijn leven af. De afgelopen maanden heb ik naar een dubbele bevalling toegeleefd – van een kind en van een boek – en nu is het tijd om daarvan te gaan genieten. vi Table of contents 1 Introduction 1.1 Reasons for this study 1 1.2 Aims of the study 1 1.3 Duration and partition of the study 2 1.4 Reading guide for this book 2 2 Learning and teaching of school algebra 2.1 Introduction 3 2.2 Traditional school algebra 3 2.3 Approaches to algebra 4 2.4 Typical learning difficulties 7 2.5 Symbolizing 10 2.5.1 Symbolizing and schematizing 11 2.5.2 Symbolizing equations 12 2.5.3 Models for equation solving 14 2.6 Arithmetic and algebra 17 2.6.1 A dual relationship 18 2.6.2 Accesses to algebra in the Dutch arithmetic curriculum 24 2.7 Pre-algebra: on the way to algebra 26 2.8 Algebra from a geometrical perspective 29 2.9 Early algebra in the Dutch curriculum 31 2.10 Conclusion 32 3 History in mathematics education 3.1 Introduction 35 3.2 Arguments for using history 35 3.3 History of algebra: a summary 37 3.4 Implementing historical elements of algebra 44 3.4.1 Reinvention of early algebra 45 3.4.2 Historical influences on the experimental learning strand 46 3.4.3 Expectations 47 4 Research plan and methodology 4.1 Introduction 49 4.2 Research questions 49 4.2.1 Main research questions and hypotheses 49 4.2.2 Sub-questions 50 4.3 Research method 51 4.3.1 Curriculum development and Realistic Mathematics Education 51 4.3.2 Developmental research 56 4.3.3 Explorative research 57 4.4 Research plan 57 4.4.1 Important changes of focus in the study 59 4.4.2 Audience 60 4.4.3 Educational design 60 vii 4.4.4 Teaching experiments 62 4.4.5 Analysis of student work 64 4.4.6 Peer review 67 4.5 Analysis of test tasks: a paradigmatic example 67 4.5.1 Test task Number Cards 67 4.5.2 Solution strategies for Number Cards 68 4.5.3 Progressive formalization 75 5 Design process 5.1 Introduction 83 5.2 The orientation phase 83 5.2.1 RME design heuristics 85 5.2.2 Mathematical content 88 5.2.3 Case studies 94 5.2.4 Evaluation and reflection 112 5.3 Additional try-outs for orientation 118 5.3.1 Mathematical starting level 118 5.3.2 Systems of equations 120 5.3.3 Pre-algebraic strategies from the past 123 5.4 Pilot experiment 124 5.4.1 Classroom experiences and results 126 5.4.2 A small group of students: experiences and results 145 5.5 Evaluation and reflection 158 5.6 Peer review 161 5.7 Conclusions 164 6 Final phase of the study 6.1 Introduction 171 6.2 State of affairs after the peer review 171 6.3 Research questions 173 6.4 Pre-algebra units revised 173 6.5 Field test 177 6.5.1 Experimental groups 177 6.5.2 Execution of the teaching experiment 179 6.6 Results of the Number Cards task 183 6.6.1 Purpose and expectations of Number Cards 183 6.6.2 The analysis: strategies observed 184 6.6.3 Notations 186 6.6.4 Summary of Number Cards results 188 6.6.5 Comparison with unit tasks 188 6.6.6 Theoretical reflections 194 6.6.7 Formulation of conjectures 196 6.7 Results of student work 199 6.7.1 Overview of results 200 6.7.2 Separate developments ofalgebraic reasoning and symbolizing 203 6.7.3 Symbolizing 238 viii Table of contents 6.7.4 Regression of strategy use 246 6.7.5 Understanding relations between quantities 253 6.7.6 Effect of history on the learning and teaching of early algebra 261 6.7.7 Role of the unknown 275 6.7.8 Recapitulation versus reinvention 277 6.8 Answers to the research questions 279 6.8.1 Conjectures 279 6.8.2 Sub-questions 280 6.8.3 Main research questions 283 6.9 Personal reflections 284 7 Discussion and recommendations 7.1 Reflections on the study’s turn-about 287 7.2 Discussion of the results 288 7.2.1 Symbolizing 288 7.2.2 Symbolizing versus reasoning 291 7.2.3 Regression of strategy use 292 7.2.4 Discontinuities between arithmetic and algebra 293 7.2.5 Indications of a suitable learning trajectory: crossing the gap 295 7.2.6 History as a didactical tool 297 7.2.7 Reinvention versus recapitulation 298 7.2.8 Further research 298 7.3 Recommendations for teachers and educational designers 299 References 303 Samenvatting 309 Appendix 323 Curriculum Vitae 339 ix x 1 Introduction 1.1 Reasons for this study educational argument In recent years, many research projects on mathematics education have focused on learning difficulties related to algebra, like the translation of word problems into simple algebraic equations.