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

Bibliographic Details
Main Author: Wittmann, Erich Christian.
Format: eBook
Language:English
Published: Cham : Springer International Publishing AG, 2020.
Edition:1st ed.
Subjects:
Electronic books.
Online Access:Click to View
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).

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