Three Dimensions : A Model of Goal and Theory Description in Mathematics Instruction - The Wiskobas Project

In Dutch "WISKOBAS" stands for a particular kind of mathematics in the elementary school (ages 6-12). In tum Wiskobas was one of the depart­ ments in the IOWO, the Institute for the Development of Mathematics Education. This institute was concerned with the development of material for mathematics education as well as the related research on the possibility of change from the then existing arithmetic instruction to the future mathematics education. The present publication Three Dimensions has three aims: to give a picture of the goals Wiskobas set for future mathematics education, at the same time to show how such goals can be described, and to show the theoretical framework of the Wiskobas curriculum. The problem at hand is not at all simple. What is more, Wiskobas' ideas about mathematics education cannot literally be translated into strings of words. So how can we face the accusation that our objectives are unattain­ able and the goal itself irrational? In order to avoid this vagueness as much as possible and for the sake of clarity, this book makes continuous use of illustrations of mathematics education. In these examples both the subject-matter and the methods of description of the goals are illustrated as explicitly as possible, while at the same time creating the opportunity to read between the lines. The reader is urged to follow carefully the mathe­ matical material at the start of each chapter. This advice applies both to the more general education oriented, and to the more mathematical! didactical reader.

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.