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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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2022. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.