Amongst Mathematicians (eBook)

About the author: Elena Nardi was born in 1968. She studied mathematics (BSc) in Thessaloniki, Greece and mathematics education at Cambridge (MPhil) and Oxford (DPhil) in the UK. Her research is in various areas of mathematics education, with a particular emphasis on the psychology of mathematical learning and the teaching and learning of mathematics at the undergraduate level. She is Senior Lecturer in Mathematics Education at the University of East Anglia in Norwich, UK. She is involved with graduate supervision and with teaching mathematics education and research methods, she co-ordinates UEA's group of researchers in mathematics education and she is Joint Editor in Chief (2008-2010) of Research in Mathematics Education the new international journal of the British Society for Research into the Learning of Mathematics. Her book Amongst Mathematicians: Teaching and Learning Mathematics at University Level is due in the spring/summer of 2007 by Springer. For more information on Elena's work: http://www.uea.ac.uk/~m011/

ACKNOWLEDGEMENTS 7
TABLE OF CONTENTS 8
PROLOGUE 13
CHAPTER 1 BACKGROUND AND CONTEXT 14
SUMMARY 14
1. TALUM: A GENERAL INTRODUCTION 15
2. A CERTAIN TYPE OF TALUM RESEARCH 17
3. THE TALUM STUDIES THE BOOK DRAWS ON 20
CHAPTER 2 METHOD, PROCESS AND PRESENTATION 26
SUMMARY 26
1. DATA SAMPLES AND M 27
2. THE DIALOGIC FORMAT 29
3. STYLE, FORMAT AND THEMATIC BREAKDOWN OF CHAPTERS 3 – 8 40
NOTE TO READER: RECOMMENDATIONS ON HOW TO READ CHAPTERS 3 – 8 49
CHAPTER 3 THE ENCOUNTER WITH FORMAL MATHEMATICAL REASONING: CONCEPTUALISING ITS SIGNIFICANCE AND ENACTING ITS TECHNIQUES 50
EPISODE 3.1 THE TENSION BETWEEN THE FAMILIAR (NUMERICAL, CONCRETE) AND THE UNFAMILIAR (RIGOROUS, ABSTRACT): RESORTING TO THE FAMILIARITY OF NUMBER 51
EPISODE 3.2 THE TENSION BETWEEN THE GENERAL AND THE PARTICULAR: CONSTRUCTING EXAMPLES AND APPLYING THEORETICAL KNOWLEDGE IN CONCRETE CONTEXTS 57
EPISODE 3.3 USING DEFINITIONS TOWARDS THE CONSTRUCTION OF MATHEMATICAL ARGUMENTS 66
EPISODE 3.4 LOGIC AS A BUILDING BLOCK OF MATHEMATICAL ARGUMENTS: RECONCILING WITH INCONCLUSIVENESS 73
EPISODE 3.5 PROOF BY CONTRADICTION: SPOTTING CONTRADICTION AND SYNDROME OF THE OBVIOUS 79
EPISODE 3.6 PROOF BY MATHEMATICAL INDUCTION: FROM N TO N+1 92
EPISODE 3.7 PROOF BY COUNTEREXAMPLE: THE VARIABLE EFFECT OF DIFFERENT TYPES OF COUNTEREXAMPLE 98
SPECIAL EPISODE SE3.1: ‘SCHOOL MATHEMATICS, UK 102
SPECIAL EPISODE SE3.2: ‘INEQUALITIES’ 111
SPECIAL EPISODE SE3.3: MATHEMATICAL REASONING IN THE CONTEXT OF GROUP THEORY 112
SPECIAL EPISODE SE3.4: ‘ALGEBRA / GEOMETRY’ 115
CHAPTER 4 MEDIATING MATHEMATICAL MEANING THROUGH SYMBOLISATION, VERBALISATION AND VISUALISATION 120
EPISODE 4.0 TO APPEAR AND TO BE: CONQUERING THE ‘GENRE’ SPEECH OF UNIVERSITY MATHEMATICS 121
EPISODE 4.1 STRINGS OF SYMBOLS AND ‘GIBBERISH’ SYMBOLISATION AND EFFICIENCY 129
EPISODE 4.2 PREMATURE COMPRESSION 143
EPISODE 4.3 VISUALISATION AND THE ROLE OF DIAGRAMS 148
EPISODE 4.4: UNDERVALUED OR ABSENT VERBALISATION AND THE INTEGRATION OF WORDS, SYMBOLS AND DIAGRAMS 160
SPECIAL EPISODE 4.1: THE GROUP TABLE 161
OUT-TAKE 4.1: TYPED UP 168
CHAPTER 5 THE ENCOUNTER WITH THE CONCEPT OF FUNCTION 170
EPISODE E5.1 CONCEPT IMAGES AND CONCEPT DEFINITION 171
EPISODE E5.2 RELATIONSHIP WITH GRAPHS: ATTRACTION, REPULSION, UNEASE AND UNCERTAINTY 177
EPISODE E5.3 THE TROUBLING / POWERFUL DUALITY AT THE HEART OF A CONCEPT: FUNCTION AS A PROCESS, FUNCTION AS AN OBJECT 181
SPECIAL EPISODE SE5.1 THE TREMENDOUS FUNCTION-LOOKALIKE THAT IS TANX 185
SPECIAL EPISODE SE5.2 POLYNOMIALS AND THE DECEPTIVE FAMILIARITY OF ESSENTIALLY UNKNOWN OBJECTS 186
OUT-TAKE OT5.1 HISTORY RELIVED 188
OUT-TAKE OT5.2 EVOCATIVE TERMS FOR 1-1 AND ONTO 189
OUT-TAKE OT5.3 RR: A GROTESQUE AND VULGAR SYMBOL? 189
CHAPTER 6 THE ENCOUNTER WITH THE CONCEPT OF LIMIT 190
EPISODE 6.1 BEGINNING TO UNDERSTAND THE NECESSITY FOR THE FORMAL DEFINITION OF CONVERGENCE 191
EPISODE 6.2 BEYOND THE ‘FORMALISTIC NONSENSE’: UNDERSTANDING THE DEFINITION OF CONVERGENCE THROUGH ITS VERBALISATION AND VISUALISATION – SYMBOLISATION AS A SAFER ROUTE? 194
EPISODE 6.3 THE MECHANICS OF IDENTIFYING AND PROVING A LIMIT 202
SPECIAL EPISODE SE6.1: IGNORING THE ‘HEAD’ OF A SEQUENCE 204
OUT-TAKE OT6.1= OR > N?
OUT-TAKE OT6.2 SERIES 209
OUT-TAKE OT6.3 CONTINUITY AND DIFFERENTIABILITY 209
CHAPTER 7 UNDERGRADUATE MATHEMATICS PEDAGOGY 214
EPISODE 7.1: INTERACTION / PARTICIPATION 215
EPISODE 7.2: INTRODUCING, CONTEXTUALISING THE IMPORTANCE OF NEW IDEAS 224
EPISODE 7.3: CONCEPT IMAGE CONSTRUCTION 226
EPISODE 7.4 ABSTRACTION AND RIGOUR VERSUS CONCRETISATION, INTUITION AND EXEMPLIFICATION 229
SPECIAL EPISODE SE7.1: TEACHING WITHOUT EXAMPLES 259
SPECIAL EPISODE SE7.2: DO NOT TEACH INDEFINITE INTEGRATION 260
SPECIAL EPISODE SE7.3: TEACHING OF FUNCTIONS, PROCESS – OBJECT, POLYNOMIALS 262
SPECIAL EPISODE SE7.4: RULES OF ATTRACTION 263
SPECIAL EPISODE SE7.5: CONTENT COVERAGE 264
OUT-TAKE OT7.1 DOES LEARNING HAPPEN ANYWAY? 264
CHAPTER 8 FRAGILE, YET CRUCIAL: THE RELATIONSHIP BETWEEN MATHEMATICIANS AND RESEARCHERS IN MATHEMATICS EDUCATION 266
EPISODE 8.1 BENEFITS 267
EPISODE 8.2 REFLECTION AND CRITIQUE OF THE PRACTICES OF RME 273
SPECIAL EPISODE 8.1: THE REVIEWS 294
EPILOGUE 302
POST-SCRIPT AMONGST MATHEMATICIANS: MAKING OF, COMING TO BE 305
BIBLIOGRAPHY 319
THEMATIC INDEX: MATHEMATICS 341
THEMATIC INDEX: LEARNING AND TEACHING 343
AUTHOR INDEX 345
MATHEMATICS TEACHER EDUCATION 349