European and Chinese Cognitive Styles and their Impact on Teaching Mathematics / Edition 1

by Filippo Spagnolo, Benedetto Di Paola

Overview

The book provides strong evidence that research on the cognitive processes from arithmetic thought to algebraic thought should take into consideration the socio-cultural context. It is an important contribution to the literature on linguistic structure in comparative studies related to Chinese student mathematics learning.

This book not only makes a great contribution to research in mathematics education, the findings of this study also addressed insightful approaches and thoughts of understanding the development of algebraic thinking in cultural contexts for classroom teachers. Using written Chinese language from different theoretical references provided wonderful approaches for understanding student algebra cognitive development in a different way and calls educators for to pay special attention to an epistemological and linguistic view of algebraic development. The findings inform classroom teachers that the cultural context plays an important role in student learning mathematics. A typical analysis of the cognitive dimension involved in some in the historical and cultural contexts is a great resource for classroom teachers.

I really enjoyed reading this book and learned a lot from its compelling analysis.

Shuhua An, Associate Professor and Director of Graduate Program in Mathematics Education, California State University, Long Beach

Product Details

ISBN-13:	9783642116797
Publisher:	Springer Berlin Heidelberg
Publication date:	04/21/2010
Series:	Studies in Computational Intelligence , #277
Edition description:	2010
Pages:	267
Product dimensions:	6.10(w) x 9.00(h) x 0.90(d)

Table of Contents

Introduction 1

1 A General Framework and Theoretical References 3

1.1 Principal Objective of the Book 3

1.2 The Experimental Epistemology of the Mathematics as Paradigma of Reference 3

1.3 The Differences and the Analogies of the Carrying Cultural Coordinates 4

1.4 What the References with the Mathematics, with to Deduce and to Conjecture (and to Conceptualize) 5

1.5 Experimental Epistemology of Mathematics, Didactics of Mathematics, Communication of Mathematics, Semiotics: Theoretical Possible Interpretation 6

1.5.1 Conclusions 10

2 The Chinese Written Language as Tool for a Possible Historical and Epistemological Reflections on the Mathematics and the Impact of Teaching/Learning of Mathematics 13

2.1 The Written Chinese Language as Possible Tool to Acquire Algebraic Competences in Mathematics 13

2.1.1 The Point of View of the Neurosciences 14

2.1.2 Origin and Development of the Chinese Writing: Some Fundamental Step 15

2.1.3 The Pinyin and the Chinese Language in the Chinese School 17

2.1.4 Observations Cognitive on the Chinese Language 19

2.1.5 Chinese Language and Mathematics 22

2.1.6 Rules of Composition of a Chinese Character 24

2.1.7 The Classification of the Chinese Characters 25

2.1.7.1 Constitutions of Ideograms and Algebraic Thought 27

2.1.8 The Research of the Characters on the Dictionary 30

2.1.9 The Parametric Structure of the Written Chinese Language 30

2.1.10 The Meta-rules of the Language and the Mathematics 32

2.2 Can the Natural Language Influence the Educational System of the Area Logical-Mathematics? Which Is the Impact in the Teaching of the Mathematics? 34

2.2.1 Contents and Objectives in the Historical Evolution of the Teaching of the Mathematics in the Chinese School: A Brief Panning 34

2.2.2 The Teaching of the Mathematics in the Today's Didactic Practice 39

2.3 A Resume Chart on the Relationship Natural Language Mathematics 44

3 The Meta-rules between Natural Language and History of Mathematics 47

3.1 The Algebra between the History and the Didactics, a Variety of Perspectives 47

3.1.1 The Algebraic Tradition in Antiquity: An Overview 52

3.1.2 The Elements of Euclid as a Canon of Reference for the Western Mathematical Tradition 57

3.1.3 An Initial Comparison between the "Elements" of Euclid and the "Jiuzhang Suanshu" 58

3.1.4 One Possible Comparison between Diofanto and Babylonians 59

3.1.5 The Algebraic Tradition among the Arabs and the Indians 60

3.1.6 Significant Developments for the Symbolic Algebra 69

3.2 A Look at the Ancient Chinese Mathematics 73

3.2.1 The Historical-Cultural 73

3.2.2 The Jiuzhang Suanshu 75

3.2.3 The Rule of Fangcheng as Meta-rule for the Chinese Algebra 80

3.2.4 Argue, and Demonstrate in Conjecturing Jiuzhang Suanshu an Example through the Algebra 84

3.3 Conclusion 87

3.4 How to Summarize the Role of History? 89

4 Common Sense and Fuzzy Logic 91

4.1 Fuzzy Logic, Fuzzy Thinking and Linguistic Approach 91

4.2 Fuzzy Sets and Their Representations 92

4.3 The Representative Point of View of Kosko 95

4.4 Some Epistemological Reflections on the Approaches to the Fuzzy Logic 98

4.5 Some Experimental Observations on Common Sense and Fuzzy Logic 99

4.5.1 Common Sense and Fuzzy Logic M. Ajello F. Spagnolo 99

4.5.2 Fuzzy Logic and Complexity 100

4.5.3 The Hypothesis 101

4.5.4 The Assignment a Computer Science Activity 102

4.5.5 A Priori Analysis 103

4.5.6 Implicative Analysis using the Chic Program 104

4.5.7 From the Pupil Records 105

4.5.8 A Possible Interpretation Key 106

4.5.9 Conclusions 107

4.5.10 Open Problems 108

Conclusions 117

5 The Experimental Epistemology as a Tool to Observe and Preview Teaching/Learning Phenomena 119

5.1 The Experimental Context 119

5.1.1 Choice of the Problematic Situations in Accordance with the Theoretical Framework and the Hypotheses of Research 119

5.1.2 The First Experimental Investigation: "Sudoku Magic Box" 127

5.1.2.1 A Priori Analysis 130

5.1.2.2 Quantitative and Qualitative Analyses of the Data 131

5.1.2.3 Second Experimental Investigation: "The Problem of Fermat", "Varying Questionnaire and Parameter in Different Semiotic Context" 136

5.1.2.4 A Priori Analysis: "Fermat Problem" 145

5.1.2.5 Quantitative and Quantitative Analysis: "Fermat Problem" 146

5.1.2.6 A Priori Analysis: "Questionnaire Variable and Parameter in Different Semiotic Context" 151

5.1.2.7 Quantitative and Qualitative Analysis: "Questionnaire Variable and Parameter in Different Semiotic Context" 154

5.1.2.8 Third Experimental Investigation: "The Sequence", "The Grid of Numbers" 165

5.1.2.9 A Priori Analysis "The Sequence" 170

5.1.2.10 Quantitative and Qualitative Analysis of "The Sequence" 171

5.1.2.11 A Priori Analysis of "The Grid of Numbers" 174

5.1.2.12 Qualitative Analysis of "The Grid of Numbers" 174

5.2 From Natural Languages to Logical Linguistic Aspects: Cognitive Styles of European and Chinese Students M. Ajello F. Spagnolo 185

5.2.1 Some Reflections on "Arguing, Conjecturing and Demonstrating" in Chinese Culture with Relation to Occidental Culture 186

5.2.2 The Algorithm as Fundamental Element of Arguing and Demonstrating? 186

5.2.3 What are the Stable Reasoning Patterns in Chinese Culture? 187

5.2.4 How were the Situations/Problems Chosen? 189

5.2.4.1 The a Priori Analysis of the Situations. We Report the a Priori Analysis of the Hypothesized Behaviours 190

5.2.4.2 Presentation of the Experimental Work in the Italian Classes 191

5.2.4.3 The Quantitative Analysis: Implicative and Factorial 191

5.2.4.4 Implicative and Similarities Analysis 192

5.2.4.5 General Considerations on Factorial Analysis, Implicative Analysis and Similarities 193

5.2.5 Interviews with Two Chinese: Qualitative Considerations 194

5.2.6 The Experience in the Chinese Classes 195

5.2.6.1 Factorial Analysis 196

5.2.6.2 Supplementary Variables 196

5.2.6.3 Considerations on the Analysis of the Data Relative to the Chinese Sample 197

5.2.7 General Conclusions and Future Perspectives 197

6 Strategy and Tactics in the Chinese and European Culture: Chess and Weich'I G. D'Eredità F. Spagnolo 201

6.1 Introduction 201

6.2 Strategy and Tactics 203

6.3 Notes on the Conceptions of Strategy and Tactics in the Orient and the Occident 205

6.4 Historical Games of Strategy in the Orient and Occident: Chess, Wei-ch'i and the Different Conceptions of Strategy and Tactics 206

6.5 Connection to Didactics and Open Problems 211

6.6 Conclusions 212

References 212

7 Rhythm and Natural Language in the Chinese and European Culture D. Galante F. Spagnolo 219

7.1 Rhythm and Natural Language in the West 219

7.2 Rhythm and Natural Language in China 222

7.3 Conclusions 227

References 227

8 Conclusions 229

8.1 The Conclusions of the Experimental Work 229

8.1.1 Topics such as Research Agreements? 230

8.2 The Experimental Epistemology of Mathematics: Concluding Observations on the Experimental Investigations Discussed 231

8.1.2 The First Experimental Investigation 232

8.3 The Second Experimental Investigation 234

8.4 Conclusions and Open Problems Related to "Strategy and Tactics" 238

8.4.1 Links with Education 239

8.5 Conclusions and Open Problems Related to Music 240

8.5.1 The References to Mathematics, the Reasoning and the Conceptualisation 240

References 243