Advanced mathematical thinking:
Gespeichert in:
| Format: | Buch |
|---|---|
| Sprache: | English |
| Veröffentlicht: |
Dordrecht [u.a.]
Kluwer
1994
|
| Ausgabe: | [pbk] |
| Schriftenreihe: | Mathematics education library
11 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVII, 289 S. |
| ISBN: | 9780792328124 |
Internformat
MARC
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| 245 | 1 | 0 | |a Advanced mathematical thinking |c ed. by David Tall |
| 250 | |a [pbk] | ||
| 264 | 1 | |a Dordrecht [u.a.] |b Kluwer |c 1994 | |
| 300 | |a XVII, 289 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
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Datensatz im Suchindex
| _version_ | 1804148335744909312 |
|---|---|
| adam_text | TABLE
OF
CONTENTS
PREFACE
xiii
ACKNOWLEDGEMENTS
xvii
INTRODUCTION
CHAPTER
1 :
The Psychology of Advanced Mathematical Thinking
David Tall
3
1.
Cognitive considerations
4
1.1
Different kinds of mathematical mind
4
1.2
Meta-theoretical considerations
6
1.3
Concept image and concept definition
6
1.4
Cognitive development
7
1.5
Transition and mental reconstruction
9
1.6
Obstacles
9
1.7
Generalization and abstraction
11
1.8
Intuition and rigour
13
2.
The growth of mathematical knowledge
14
2.1
The full range of advanced mathematical thinking
14
2.2
Building and testing theories: synthesis and analysis
15
2.3
Mathematical proof
16
3.
Curriculum design in advanced mathematical learning
17
3.1
Sequencing the learning experience
17
3.2
Problem-solving
18
3.3
Proof
19
3.4
Differences between elementary and advanced
mathematical thinking
20
4.
Looking ahead
20
VI
TABLE OF CONTENTS
I
:
THE NATURE OF
ADVANCED MATHEMATICAL THINKING
CHAPTER
2 :
Advanced Mathematical Thinking Processes Tommy Dreyfus
25
1.
Advanced mathematical thinking as process
26
2.
Processes involved in representation
30
2.1
The process of representing
30
2.2
Switching representations and translating
32
2.3
Modelling
34
3.
Processes involved in abstraction
34
3.1
Generalizing
35
3.2
Synthesizing
35
3.3
Abstracting
36
4.
Relationships between representing and abstracting (in learning
processes)
38
5.
A wider vista of advanced mathematical processes
40
CHAPTER
3 :
Mathematical Creativity
Gontran Ervynck
42
1.
The stages of development of mathematical creativity
42
2.
The structure of a mathematical theory
46
3.
A tentative definition of mathematical creativity
46
4.
The ingredients of mathematical creativity
47
5.
The motive power of mathematical creativity
47
6.
The characteristics of mathematical creativity
49
7.
The results of mathematical creativity
50
8.
The fallibility of mathematical creativity
52
9.
Consequences in teaching advanced mathematical thinking
52
CHAPTER
4 :
Mathematical Proof
1.
Origins of the emphasis on formal proof
2.
More recent views of mathematics
3.
Factors in acceptance of a proof
4.
The social process
5.
Careful reasoning
6.
Teaching
Gila Hanna
54
55
55
58
59
60
60
TABLE
OF
CONTENTS
vii
II:
COGNITIVE
THEORY
OF ADVANCED MATHEMATICAL THINKING
CHAPTER
5 :
The Role of Definitions in the Teaching and Learning of
Mathematics Shlomo
Vinner
65
1.
Definitions in mathematics and common assumptions about
pedagogy
65
2.
The cognitive situation
67
3.
Concept image
68
4.
Concept formation
69
5.
Technical contexts
69
6.
Concept image and concept definition
-
desirable theory and
practice
69
7.
Three illustrations of common concept images
73
8.
Some implications for teaching
79
CHAPTER
6 :
The Role of Conceptual Entities and their symbols in building
Advanced Mathematical Concepts Guershon
Harel
&
James Kaput
82
1.
Three roles of conceptual entities
83
1.1
Working-memory load
84
1.2a Comprehension: the case of uniform and point-wise
operators
84
1.2b Comprehension: the case of object-valued operators
86
1.3
Conceptual entities as aids to focus
88
2.
Roles of mathematical notations
88
2.1
Notation and formation of cognitive
entities
89
2.2
Reflecting structure in elaborated notations
91
3.
Summary
93
CHAPTER
7 :
Reflective Abstraction in Advanced Mathematical Thinking
Ed Dubinsky
95
1.
Piaget
s
notion of reflective abstraction
97
1.1
The importance of reflective abstraction
97
1.2
The nature of reflective abstraction
99
1.3
Examples of reflective abstraction in children s thinking
100
1.4
Various kinds of construction in
reflective abstraction
101
2.
A theory of the development of concepts in advanced
mathematical thinking
102
viii TABLE OF
CONTENTS
2.1
Objects, processes and
schemas
102
2.2
Constructions in advanced mathematical concepts
103
2.3
The organization of
schemas
106
3.
Genetic decompositions of three
schemas
109
3.1
Mathematical induction
110
3.2
Predicate calculus
114
3.3
Function
116
4.
Implications for education
119
4.1
Inadequacy of traditional teaching practices
120
4.2
What can be done
123
III
:
RESEARCH INTO THE TEACHING AND LEARNING
OF ADVANCED MATHEMATICAL THINKING
CHAPTER
8 :
Research in Teaching and Learning Mathematics at an Advanced
Level Aline Robert
&
Rolph Schwarzenberger
127
1.
Do there exist features specific to the learning of advanced
mathematics?
128
1.1
Social factors
128
1.2
Mathematical content
128
1.3
Assessment of students work
130
1.4
Psychological and cognitive characteristics of students
131
1.5
Hypotheses on student acquisition of knowledge in
advanced mathematics
132
1.6
Conclusion
133
2.
Research on learning mathematics at the advanced level
133
2.1
Research into students acquisition of specific concepts
134
2.2
Research into the organization of mathematical content
at an advanced level
134
2.3
Research on the external environment for advanced
mathematical thinking
136
3.
Conclusion
139
CHAPTER
9 :
Functions and associated learning difficulties
Theodore
Eisenberg 140
1.
Historical background
140
2.
Deficiencies in learning theories
142
3.
Variables
144
4.
Functions, graphs and visualization
145
5.
Abstraction, notation, and anxiety
148
6.
Representational difficulties
151
TABLE
OF
CONTENTS
їх
7.Summary
CHAPTER
10 :
Limits
Bemard
Čomu
1.
Spontaneous conceptions and mental models
2.
Cognitive obstacles
3.
Epistemologicei
obstacles in historical development
4.
Epistemological obstacles in modern mathematics
5.
The didactical transmission of epistemological obstacles
6.
Towards pedagogical strategies
CHAPTER
11 :
Analysis
Michèle
Artigue
Historical background
1.1
Some concepts emerged early but were established late
1.2
Some concepts cause both enthusiasm and virulent
criticism
1.3
The differential/derivative conflict and its educational
repercussions
1.4
The non-standard analysis revival and its weak impact on
1.5
education
Current educational trends
2.
Student conceptions
2.1
A cross-sectional study of the understanding of
elementary calculus in adolescents and young adults
2.2
A study of student conceptions of the differential, and
of the processes of differentiation and integration
2.2.1
The meaning and usefulness of differentials and
differential procedures
2.2.2
Approximation and rigour in reasoning
2.2.3
The role of differential elements
2.3
The role of education
3.
Research in didactic engineering
3.1
Graphic calculus
3.2
Teaching integration through scientific debate
3.3
Didactic engineering in teaching differential equations
3.4
Summary
4.
Conclusion and future perspectives in education
CHAPTER
12 :
The Role of Students Intuitions of Infinity in Teaching the
Cantoria!
Theory
Dina Tirosh
152
153
154
158
159
162
163
165
167
168
168
168
169
172
173
174
176
180
180
182
184
186
186
187
191
193
195
196
199
1.
Theoretical conceptions of infinity
200
TABLE
OF
CONTENTS
4.
5.
Students
conceptions
of infinity
201
2.1
Students intuitive criteria for comparing infinite
quantities
203
First steps towards improving students intuitive understanding of
actual infinity
3.1
3.2
3.3
3.4
3.5
3.6
3.7
Changes in students understanding of actual infinity
Final comments
The finite and infinite sets learning unit
Raising students awareness of the inconsistencies in
their own thinking
Discussing the origins of students intuitions about
infinity
Progressing from finite to infinite sets
Stressing that it is legitimate to wonder about infinity
Emphasizing the relativity of mathematics
205
206
206
207
207
208
208
Strengthening students confidence in the new definitions
209
209
214
CHAPTER
13 :
Research on Mathematical Proof
Daniel
Alibert &
Michael Thomas
215
1.
Introduction
215
2.
Students understanding of proofs
216
3.
The structural method of proof exposition
219
3.1
A proof in linear style
221
3.2
A proof in structural style
222
4.
Conjectures and proofs
-
the scientific debate in a mathematical
course
224
4.1
Generating scientific debate
225
4.2
An example of scientific debate
226
4.3
The organization of proof debates
228
4.4
Evaluating the role of debate
229
5.
Conclusion
229
CHAPTER
14 :
Advanced Mathematical Thinking and the Computer
Ed Dubinsky and David Tall
231
1.
Introduction
231
2.
The computer in mathematical research
231
3.
The computer in mathematical education
-
generalities
234
4.
Symbolic manipulators
235
5.
Conceptual development using a computer
237
6.
The computer as an environment for exploration of fundamental
ideas
238
TABLE
OF
CONTENTS
xi
7.
Programming
241
8.
The future
243
Appendix to Chapter
14
ISETL
:
a computer language for advanced mathematical thinking
244
EPILOGUE
CHAPTER
15 :
Reflections
David Tall
251
BIBLIOGRAPHY
261
INDEX
275
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| institution | BVB |
| isbn | 9780792328124 |
| language | English |
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| series | Mathematics education library |
| series2 | Mathematics education library |
| spelling | Advanced mathematical thinking ed. by David Tall [pbk] Dordrecht [u.a.] Kluwer 1994 XVII, 289 S. txt rdacontent n rdamedia nc rdacarrier Mathematics education library 11 Mathematik Mathematics Mathematikunterricht (DE-588)4037949-8 gnd rswk-swf Mathematik (DE-588)4037944-9 gnd rswk-swf Methode (DE-588)4038971-6 gnd rswk-swf Mathematik (DE-588)4037944-9 s Methode (DE-588)4038971-6 s Mathematikunterricht (DE-588)4037949-8 s 1\p DE-604 Tall, David Sonstige oth Mathematics education library 11 (DE-604)BV000020735 11 Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024378090&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Advanced mathematical thinking Mathematics education library Mathematik Mathematics Mathematikunterricht (DE-588)4037949-8 gnd Mathematik (DE-588)4037944-9 gnd Methode (DE-588)4038971-6 gnd |
| subject_GND | (DE-588)4037949-8 (DE-588)4037944-9 (DE-588)4038971-6 |
| title | Advanced mathematical thinking |
| title_auth | Advanced mathematical thinking |
| title_exact_search | Advanced mathematical thinking |
| title_full | Advanced mathematical thinking ed. by David Tall |
| title_fullStr | Advanced mathematical thinking ed. by David Tall |
| title_full_unstemmed | Advanced mathematical thinking ed. by David Tall |
| title_short | Advanced mathematical thinking |
| title_sort | advanced mathematical thinking |
| topic | Mathematik Mathematics Mathematikunterricht (DE-588)4037949-8 gnd Mathematik (DE-588)4037944-9 gnd Methode (DE-588)4038971-6 gnd |
| topic_facet | Mathematik Mathematics Mathematikunterricht Methode |
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