Three Dimensions

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.