Three Dimensions

Mathematical Material for Chapter I: “Gulliver”.- I Introduction.- 1. From “New Math” to Wiskobas.- 1.1 Three trends.- 1.2 Wiskobas.- 2. The history of Wiskobas.- 2.1 The exploratory phase.- 2.2 The integration phase.- 2.3 Spin-off, further development and research.- 2.4 Summary.- 3. Wiskobas between four trends.- 3.1 Wiskobas and the empirical trend.- 3.2 Wiskobas and the structural trend.- 3.3 Wiskobas and the arithmetical trend.- 3.4 Wiskobas and the current arithmetic education.- 3.5 Conclusion.- 4. Innovation according to Wiskobas.- 4.1 The innovation strategy.- 4.2 The innovation theme.- 4.3 The innovation.- 5. The problem.- 5.1 The problem of goal description.- 5.2 The question at issue.- 5.3 What is not dealt with?.- 6. Overview of what follows.- 6.1 Chapters.- 6.2 What is the function of the mathematical material?.- 6.3 Short summary.- 7. Conclusion.- Mathematical Material for Chapter II: “Counting Problems”.- II Starting Points.- 1. Mathematical activity.- 1.1 Flowers.- 1.2 Routes.- 1.3 Apples.- 1.4 To and fro.- 1.5 To and fro again.- 1.6 Didactical digression.- 1.7 Cards for the cube crawler.- 1.8 Routes on a highway network.- 1.9 Score progression.- 1.10 Families.- 1.11 Mathematising.- 2. Acting didactically.- 2.1 A mathematics lesson.- 2.2 Didactising.- 3. Starting points for mathematics education.- 3.1 Activity.- 3.2 Differentiation.- 3.3 Vertical planning.- 3.4 Structural character.- 3.5 Language aspect.- 3.6 Applicability.- 3.7 Dynamics.- 3.8 The specifically mathematical approach.- 4. Conclusion.- Mathematical Material for Chapter III: “Grains on the Chessboard”.- III One-Dimensional Goal Description.- 1. Goal descriptions.- 1.1 General, intermediate and concrete goal descriptions.- 1.2 One-, two-, and three-dimensional goal descriptions.- 1.3 Summary.- 2. Integral one-dimensional goals.- 2.1 Personal development.- 2.2 Socialisation.- 2.3 Preparation for further education.- 2.4 Social relevance.- 2.5 Summary.- 3. Mathematical one-dimensional goals.- 3.1 Arithmetical aspect.- 3.2 Language aspect.- 3.3 Applicability.- 3.4 Practical use.- 3.5 Structural aspect.- 3.6 Methodological aspect.- 3.7 Dynamic aspect.- 3.8 Attitude aspect.- 4. Relationships between integral and mathematical goals.- 5. Conclusion.- Mathematical Material for Chapter IV: “The Land of Eight”.- IV Two-Dimensional Goal Description.- 1. Popham and Eisner: Two views on goal description.- 1.1 Popham’s views on “instructional objectives”.- 1.2 Eisner’s views on “expressive objectives”.- 1.3 Summary.- 2. Variants of instructional objectives.- 2.1 Concrete product goals.- 2.2 Operationalised product goals.- 2.3 The goals approach.- 2.4 Concluding remarks.- 3. Variants of expressive objectives.- 3.1 PISA goals.- 3.2 Process goals.- 3.3 All-embracing process goals.- 3.4 Concluding remarks.- 4. Product and process goals in “The Land of Eight”.- 4.1 Product goals in “The Land of Eight”.- 4.2 Process goals in “The Land of Eight”.- 4.3 Summary.- 5. The possibilities and limitations of two-dimensional goal descriptions.- 5.1 Possibilities and limitations of two-dimensional product goal descriptions.- 5.2 Possibilities and limitations of two-dimensional process goal descriptions.- 5.3 Summary.- 6. Conclusion.- Mathematical Material for Chapter V: “Freckleham”.- V Three-Dimensional Goal Description.- 1. The history of “Freckleham” and the significance of its goals.- 1.1 Development as a process of making goals concrete.- 1.2 Development as a progressive structuring of activities.- 1.3 The objectives and history of “Freckleham” in Wiskobas.- 1.4 Conclusions.- 2. “Freckleham” in three dimensions.- 2.1 The people of “Freckleham”.- 2.2 Map of “Freckleham”.- 2.3 Greetings.- 2.4 Confusion.- 2.5 Thieves.- 2.6 The town meeting.- 2.7 New greeting suggestions.- 2.8 The Freckleham song in code.- 2.9 “Freckleham” in a ‘wider’ connection.- 2.10 Basis of “Freckleham” in a ‘deeper’ connection.- 3. Holistic three-dimensional goal description.- 3.1 Different kinds of three-dimensional goal description.- 3.2 Characteristics of the holistic three-dimensional goal description.- 3.3 Rough empirical basis of the holistic three-dimensional goal description.- 3.4 Functions of holistic three-dimensional goal description.- 4. Conclusion.- Mathematical Material for Chapter VI: “Algorithms”.- VI Survey and Justification.- 1. History.- 2. Overview.- 3. Justification.- 4. Conclusions.- Mathematical Material for Chapter VII (Appendix): “The Wiskobas Curriculum”.- VII Framework for Instruction Theory.- 1. Preamble.- 1.1 Starting points of a realistic instruction theory.- 2. One-dimensional description of the framework for instruction theory.- 2.1 Van Hiele’s levels.- 2.2. Freudenthal’s didactical phenomenology.- 2.3. Progressive mathematising guided by the five instruction principles.- 2.4. Schematic comparison of the four trends in arithmetic/mathematics instruction.- 3. Two-dimensional description of a framework for instruction theory.- 3.1 Progressive mathematisation in the Wiskobas programme.- 3.2. The five tenets of the framework for instruction theory revisited.- 3.3. Comparison of the four trends.- 4. Three-dimensional description of a framework for instruction theory.- 4.1 Progressive mathematising of long division.- 4.2. Two more examples: Number systems and fractions.- 4.3. The most conspicuous elements of the framework for instruction theory seen from the viewpoint of implementation of instructional ideas.- 5. The broader framework for instruction theory.- 5.1 Gagné, Dienes, Piaget, and Bruner.- 5.2 Recent investigations of subject matter in instruction theory.- 5.3 The almost complete absence of instruction theory ideas in general cognitive psychological research.- 5.4 Overview.- 5.5 Closure.- Notes.