Connecting Mathematics and Mathematics Education : : Collected Papers on Mathematics Education As a Design Science.

• Author: Wittmann, Erich Christian.
• Place / Publishing House: Cham : : Springer International Publishing AG,, 2020.; ©2021.
• Year of Publication: 2020
• Edition: 1st ed.
• Language: English
• Physical Description: 1 online resource (332 pages)

Table of Contents

• Intro
• Foreword
• Preface
• Contents
• About the Author
• 1 Unfolding the Educational and Practical Resources Inherent in Mathematics for Teaching Mathematics
• 1 From ``Instruction and Receptivity'' to ``Organization and Activity'' in Teaching
• 2 The Learning Environment ``Calculating with Remainders''
• 3 Mathematics for Specialists and Mathematics for Teachers
• 4 From ``Instruction and Receptivity'' to ``Organization and Activity'' in Teacher Education
• References
• 2 Teaching Units as the Integrating Core of Mathematics Education
• 1 Discussion of the Status and Role of Mathematics Education
• 2 Problems of Integration
• 3 Some Views on Mathematics Teaching
• 4 Teaching Units as the Integrating Core of Mathematics Education
• 4.1 Some Teaching Units
• 4.2 Teaching Units in Teacher Training
• 4.3 Teaching Units in Didactical Research
• 5 Conclusion
• References
• 3 Clinical Interviews Embedded in the ``Philosophy of Teaching Units''-A Means of Developing Teachers' Attitudes and Skills
• 1 Cooperation Between Theory and Practice Through ``Intermediate Practice''
• 2 Clinical Interviews as a Special Kind of Intermediate Practice
• 3 Concluding Remarks
• References
• 4 The Mathematical Training of Teachers from the Point of View of Education
• 1 The Problem of Integrating Mathematical and Educational aspects in Mathematics Education and Teacher Training
• 2 The Educational Substance of Subject Matter
• 3 Elementary Mathematics in Teacher Training
• 4 The Elementary Mathematics Research Program of Mathematics Education
• References
• 5 When Is a Proof a Proof?
• 1 Proofs and ``Proofs''
• 2 Formalism as a Fiction: The Indispensability of Intuition ...
• 3 The Elementary-Mathematics-Research-Program of Mathematics Education
• References
• 6 Mathematics Education as a `Design Science'.
• 1 The `Core' and the `Related Areas' of Mathematics Education
• 2 A Basic Problem in the Present Development of Mathematics Education: The Neglect of the Core
• 3 Mathematics Education as a Systemic-Evolutionary `Design Science'
• 4 The Design of Teaching Units and Empirical Research
• 5 And the Future of Mathematics Education?
• References
• 7 Designing Teaching: The Pythagorean Theorem
• 1 Introduction
• 2 Thinking About the Pythagorean Theorem within the School Context
• 3 Understanding the Structure of the Pythagorean Theorem
• 3.1 Different Proofs of the Pythagorean Theorem
• 3.2 Heuristic Approaches to the Pythagorean Theorem
• 3.3 Exploring Students' Understanding of Area and Similarity
• 4 Designing Teaching Units on the Pythagorean Theorem
• 4.1 Approaching the Pythagorean Theorem via the Diagonal of a Rectangle
• 4.2 Japanese Approach to the Pythagorean Theorem
• 5 Reflecting on the Units: Some Key Generalizable Concepts
• 5.1 Informal Proofs
• 5.2 ``Specializing''-A Fundamental Heuristic Strategy
• 5.3 The Operative Principle
• 6 Appendix: Solutions to the Problems in Exploration 3
• References
• 8 Standard Number Representations in the Teaching of Arithmetic
• 1 Principles of Learning and Teaching
• 2 The Epistemological Nature of Number Representations
• 2.1 Notes on the History of Number Representations: From Tools of Teaching to Tools of Learning
• 2.2 Representations in Mathematics
• 3 Selection of Standard Number Representations
• 3.1 Criteria for Selecting and Designing Standard Representations
• 3.2 Fundamental Ideas of Arithmetic
• 3.3 Standard Number Representations
• 4 Some Teaching Units
• 4.1 The Twenty Frame and the Addition Table (Grade 1)
• 4.2 Multiplication Chart (Grade 2)
• 4.3 An Introduction into the Thousand Book (Grade 3)
• 4.4 ``Always 22'' (Grade 3).
• 4.5 Place Value Chart (Grade 4)
• 5 Conclusion
• References
• 9 Developing Mathematics Education in a Systemic Process
• 1 Bridging the Gap Between Theory and Practice: …
• 2 (Burst) Dreams
• 2.1 Descartes' Dream
• 2.2 Hilbert's Dream
• 2.3 Comenius' Dream
• 2.4 The `Systemic-Evolutionary'' Versus the ``Mechanistic-Technomorph'' Approach to the Management of Complexity
• 3 Consequences for Mathematics Education
• 4 Substantial Learning Environments for Practising Skills
• 5 Substantial Learning Environments in Teacher Education
• 5.1 Didactics Courses
• 5.2 Mathematics Courses
• 6 Conclusion
• References
• 10 The Alpha and Omega of Teacher Education: Organizing Mathematical Activities
• 1 Introduction
• 2 Mathematics in Contexts
• 3 The Context of Teacher Education
• 4 The O-Script/A-Script Method
• 5 Operative Proofs
• 6 Experiences with the Course
• References
• 11 Operative Proofs in School Mathematics and Elementary Mathematics
• 1 Some Learning Environments with Embedded Operative Proofs
• 1.1 Even and Odd Numbers
• 1.2 Multiplicative Arrow Strings
• 1.3 Egyptian Fractions
• 1.4 Fitting Polygons
• 2 The Concept of Operative Proof
• 3 The Theoretical Background of Operative Proofs
• 3.1 Mathematics as the Science of Patterns
• 3.2 The Quasi-empirical Nature of Mathematics
• 3.3 The Operative Principle
• 3.4 Practicing Skills in a Productive Way
• 4 Concluding Remarks
• References
• 12 Collective Teaching Experiments: Organizing a Systemic Cooperation Between Reflective Researchers and Reflective Teachers in Mathematics Education
• 1 Mathematics Education as a ``Systemic-Evolutionary'' Design Science
• 2 Taking Systemic Complexity Systematically into Account: Lessons …
• 3 Empowering Teachers to Cope with Systemic Complexity as Reflective Practitioners.
• 4 Collective Teaching Experiments: A Joint Venture of Reflective Teachers …
• 5 Closing Remarks: The Role of Mathematics in Mathematics Education
• References
• 13 Structure-Genetic Didactical Analyses-Empirical Research ``of the First Kind''
• 1 Introduction of the Multiplication Table in Grade 2
• 2 Designing a Substantial Learning Environment for Practicing Long Addition
• 3 Nets of a Cube
• 4 Structure-Genetic Didactical Analyses
• 5 Conclusion
• References
• 14 Understanding and Organizing Mathematics Education as a Design Science-Origins and New Developments
• 1 Origins
• 1.1 The Rise of the Sciences of the Artificial
• 1.2 Developments in Management Theory
• 1.3 Prototypes of Design in Mathematics Education
• 1.4 The Map of Mathematics Education as a Design Science
• 2 Conceptual Developments
• 2.1 The Natural Theory of Teaching: ``Well-Understood Mathematics''
• 2.2 Structure-Genetic Didactical Analyses
• 2.3 A Differentiated Conception of Practicing Skills
• 2.4 Awareness of Systemic Constraints
• 3 Practical Consequences
• 3.1 Integrating ``Well-Understood Mathematics''
• 3.2 Designing a Consistent and Coherent Curriculum
• 3.3 Including Operative Proofs
• 3.4 Addressing Teachers as ``Reflective Practitioners''
• 4 Final Remarks
• References
• Appendix Excerpts from The Book of Numbers (BN).