Nardi Amongst Mathematicians

Teaching and Learning Mathematics at University Level
1. Auflage 2007
ISBN: 978-0-387-37143-6
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 341 Seiten

offers a unique perspective on the ways in which mathematicians perceive their students' learning, teach and reflect on their teaching practice; also on how they perceive the often fragile relationship between the communities of mathematics and mathematics education.

Elena Nardi employs two fictional, yet entirely data-grounded, characters to create a conversation on these important issues. While personas are created, the facts incorporated into their stories are based on large bodies of data including intense focused group interviews with mathematicians and extensive analyses of students' written work. This book demonstrates the pedagogical potential that lies in collaborative undergraduate mathematics education research that engages mathematicians, researchers and students. Nardi also addresses the need for action in undergraduate mathematics education and offers a discourse for reform through demonstrating the feasibility and potential of collaboration between mathematicians and mathematics education researchers.

is of interest to both the mathematics and mathematics education communities including university teachers, teacher educators, undergraduate and graduate students, and researchers.

Autoren/Hrsg.

Nardi, Elena

Inhaltsverzeichnis

1;ACKNOWLEDGEMENTS;7
2;TABLE OF CONTENTS;8
3;PROLOGUE;13
4;CHAPTER 1 BACKGROUND AND CONTEXT;14
4.1;SUMMARY;14
4.2;1. TALUM: A GENERAL INTRODUCTION;15
4.3;2. A CERTAIN TYPE OF TALUM RESEARCH;17
4.4;3. THE TALUM STUDIES THE BOOK DRAWS ON;20
5;CHAPTER 2 METHOD, PROCESS AND PRESENTATION;26
5.1;SUMMARY;26
5.2;1. DATA SAMPLES AND M;27
5.3;2. THE DIALOGIC FORMAT;29
5.4;3. STYLE, FORMAT AND THEMATIC BREAKDOWN OF CHAPTERS 3 – 8;40
5.5;NOTE TO READER: RECOMMENDATIONS ON HOW TO READ CHAPTERS 3 – 8;49
6;CHAPTER 3 THE ENCOUNTER WITH FORMAL MATHEMATICAL REASONING: CONCEPTUALISING ITS SIGNIFICANCE AND ENACTING ITS TECHNIQUES;50
6.1;EPISODE 3.1 THE TENSION BETWEEN THE FAMILIAR (NUMERICAL, CONCRETE) AND THE UNFAMILIAR (RIGOROUS, ABSTRACT): RESORTING TO THE FAMILIARITY OF NUMBER;51
6.2;EPISODE 3.2 THE TENSION BETWEEN THE GENERAL AND THE PARTICULAR: CONSTRUCTING EXAMPLES AND APPLYING THEORETICAL KNOWLEDGE IN CONCRETE CONTEXTS;57
6.3;EPISODE 3.3 USING DEFINITIONS TOWARDS THE CONSTRUCTION OF MATHEMATICAL ARGUMENTS;66
6.4;EPISODE 3.4 LOGIC AS A BUILDING BLOCK OF MATHEMATICAL ARGUMENTS: RECONCILING WITH INCONCLUSIVENESS;73
6.5;EPISODE 3.5 PROOF BY CONTRADICTION: SPOTTING CONTRADICTION AND SYNDROME OF THE OBVIOUS;79
6.6;EPISODE 3.6 PROOF BY MATHEMATICAL INDUCTION: FROM N TO N+1;92
6.7;EPISODE 3.7 PROOF BY COUNTEREXAMPLE: THE VARIABLE EFFECT OF DIFFERENT TYPES OF COUNTEREXAMPLE;98
6.8;SPECIAL EPISODE SE3.1: ‘SCHOOL MATHEMATICS, UK;102
6.9;SPECIAL EPISODE SE3.2: ‘INEQUALITIES’;111
6.10;SPECIAL EPISODE SE3.3: MATHEMATICAL REASONING IN THE CONTEXT OF GROUP THEORY;112
6.11;SPECIAL EPISODE SE3.4: ‘ALGEBRA / GEOMETRY’;115
7;CHAPTER 4 MEDIATING MATHEMATICAL MEANING THROUGH SYMBOLISATION, VERBALISATION AND VISUALISATION;120
7.1;EPISODE 4.0 TO APPEAR AND TO BE: CONQUERING THE ‘GENRE’ SPEECH OF UNIVERSITY MATHEMATICS;121
7.2;EPISODE 4.1 STRINGS OF SYMBOLS AND ‘GIBBERISH’ SYMBOLISATION AND EFFICIENCY;129
7.3;EPISODE 4.2 PREMATURE COMPRESSION;143
7.4;EPISODE 4.3 VISUALISATION AND THE ROLE OF DIAGRAMS;148
7.5;EPISODE 4.4: UNDERVALUED OR ABSENT VERBALISATION AND THE INTEGRATION OF WORDS, SYMBOLS AND DIAGRAMS;160
7.6;SPECIAL EPISODE 4.1: THE GROUP TABLE;161
7.7;OUT-TAKE 4.1: TYPED UP;168
8;CHAPTER 5 THE ENCOUNTER WITH THE CONCEPT OF FUNCTION;170
8.1;EPISODE E5.1 CONCEPT IMAGES AND CONCEPT DEFINITION;171
8.2;EPISODE E5.2 RELATIONSHIP WITH GRAPHS: ATTRACTION, REPULSION, UNEASE AND UNCERTAINTY;177
8.3;EPISODE E5.3 THE TROUBLING / POWERFUL DUALITY AT THE HEART OF A CONCEPT: FUNCTION AS A PROCESS, FUNCTION AS AN OBJECT;181
8.4;SPECIAL EPISODE SE5.1 THE TREMENDOUS FUNCTION-LOOKALIKE THAT IS TANX;185
8.5;SPECIAL EPISODE SE5.2 POLYNOMIALS AND THE DECEPTIVE FAMILIARITY OF ESSENTIALLY UNKNOWN OBJECTS;186
8.6;OUT-TAKE OT5.1 HISTORY RELIVED;188
8.7;OUT-TAKE OT5.2 EVOCATIVE TERMS FOR 1-1 AND ONTO;189
8.8;OUT-TAKE OT5.3 RR: A GROTESQUE AND VULGAR SYMBOL?;189
9;CHAPTER 6 THE ENCOUNTER WITH THE CONCEPT OF LIMIT;190
9.1;EPISODE 6.1 BEGINNING TO UNDERSTAND THE NECESSITY FOR THE FORMAL DEFINITION OF CONVERGENCE;191
9.2;EPISODE 6.2 BEYOND THE ‘FORMALISTIC NONSENSE’: UNDERSTANDING THE DEFINITION OF CONVERGENCE THROUGH ITS VERBALISATION AND VISUALISATION – SYMBOLISATION AS A SAFER ROUTE?;194
9.3;EPISODE 6.3 THE MECHANICS OF IDENTIFYING AND PROVING A LIMIT;202
9.4;SPECIAL EPISODE SE6.1: IGNORING THE ‘HEAD’ OF A SEQUENCE;204
9.5;OUT-TAKE OT6.1= OR > N?;208
9.6;OUT-TAKE OT6.2 SERIES;209
9.7;OUT-TAKE OT6.3 CONTINUITY AND DIFFERENTIABILITY;209
10;CHAPTER 7 UNDERGRADUATE MATHEMATICS PEDAGOGY;214
10.1;EPISODE 7.1: INTERACTION / PARTICIPATION;215
10.2;EPISODE 7.2: INTRODUCING, CONTEXTUALISING THE IMPORTANCE OF NEW IDEAS;224
10.3;EPISODE 7.3: CONCEPT IMAGE CONSTRUCTION;226
10.4;EPISODE 7.4 ABSTRACTION AND RIGOUR VERSUS CONCRETISATION, INTUITION AND EXEMPLIFICATION;229
10.5;SPECIAL EPISODE SE7.1: TEACHING WITHOUT EXAMPLES;259
10.6;SPECIAL EPISODE SE7.2: DO NOT TEACH INDEFINITE INTEGRATION;260
10.7;SPECIAL EPISODE SE7.3: TEACHING OF FUNCTIONS, PROCESS – OBJECT, POLYNOMIALS;262
10.8;SPECIAL EPISODE SE7.4: RULES OF ATTRACTION;263
10.9;SPECIAL EPISODE SE7.5: CONTENT COVERAGE;264
10.10;OUT-TAKE OT7.1 DOES LEARNING HAPPEN ANYWAY?;264
11;CHAPTER 8 FRAGILE, YET CRUCIAL: THE RELATIONSHIP BETWEEN MATHEMATICIANS AND RESEARCHERS IN MATHEMATICS EDUCATION;266
11.1;EPISODE 8.1 BENEFITS;267
11.2;EPISODE 8.2 REFLECTION AND CRITIQUE OF THE PRACTICES OF RME;273
11.3;SPECIAL EPISODE 8.1: THE REVIEWS;294
12;EPILOGUE;302
13;POST-SCRIPT AMONGST MATHEMATICIANS: MAKING OF, COMING TO BE;305
14;BIBLIOGRAPHY;319
15;THEMATIC INDEX: MATHEMATICS;341
16;THEMATIC INDEX: LEARNING AND TEACHING;343
17;AUTHOR INDEX;345
18;MATHEMATICS TEACHER EDUCATION;349