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

Springer Netherlands, 1987. — 352 p. — (Mathematics Education Library 3). — ISBN 978-94-010-8160-3, 978-94-009-3707-9.In Dutch "WISKOBAS" stands for a particular kind of mathematics in the elementary school (ages 6-12). In tum Wiskobas was one of the departments 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 unattainable 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 mathematical material at the start of each chapter. This advice applies both to the more general education oriented, and to the more mathematical! didactical reader.Guide to the mathematical material
Mathematical material for chapter I: "Gulliver"From "New Math" to Wiskobas
Three trends
Wiskobas
The history of Wiskobas
The exploratory phase
The integration phase
Spin-off, further development and researchWiskobas between four trends
Wiskobas and the empirical trend
Wiskobas and the structural trend
Wiskobas and the arithmetical trend
Wiskobas and the current arithmetic educationInnovation according to Wiskobas
The innovation strategy
The innovation theme
The innovation
The problem
The problem of goal description
The question at issue
What is not dealt with?
Overview of what follows
Chapters
What is the function of the mathematical material?
Short summaryMathematical material for chapter II: "Counting problems"
Starting points
Mathematical activity
Flowers
Routes
Apples
To and fro
To and fro again
Didactical digression
Cards for the cube crawler
Routes on a highway network
Score progression
Families
Mathematising
Acting didactically
A mathematics lesson
Didactising
Starting points for mathematics education
Activity
Differentiation
Vertical planning
Structural character
Language aspect
Applicability
Dynamics
The specifically mathematical approachMathematical material for chapter III: "Grains on the chessboard"
One-dimensional goal description
Goal descriptions
General, intermediate and concrete goal descriptions
One-, two-, and three-dimensional goal descriptionsIntegral one-dimensional goals
Personal development
Socialisation
Preparation for further education
Social relevanceMathematical one-dimensional goals
Arithmetical aspect
Language aspect
Applicability
Practical use
Structural aspect
Methodological aspect
Dynamic aspect
Attitude aspect
Relationships between integral and mathematical goalsMathematical material for chapter IV: "The land of eight"
Two-dimensional goal description
b]Popham and Eisner: Two views on goal description[/b]
Popham's views on "instructional objectives"
Eisner's views on "expressive objectives"Variants of instructional objectives
Concrete product goals
Operationalised product goals
The goals approach
Concluding remarks
Variants of expressive objectives
PISA goals
Process goals
All-embracing process goals
Concluding remarks
Product and process goals in "The Land of Eight"
Product goals in "The Land of Eight"
Process goals in "The Land of Eight"The possibilities and limitations of two-dimensional goal descriptions
Possibilities and limitations of two-dimensional product goal descriptions
Possibilities and limitations of two-dimensional process goal descriptionsMathematical material for chapter V: "Freckleham"
Three-dimensional goal description
The history of "Freckleham" and the significance of its goals
Development as a process of making goals concrete
Development as a progressive structuring of activities
The objectives and history of "Freckleham" in Wiskobas
Conclusions
"Freckleham" in three dimensions
The people of "Freckleham"
Map of "Freckleham"
Greetings
Confusion
Thieves
The town meeting
New greeting suggestions
The Freckleham song in code
"Freckleham" in a 'wider' connection
2Basis of "Freckleham" in a 'deeper' connection
Holistic three-dimensional goal description
Different kinds of three-dimensional goal description
Characteristics of the holistic threedimensional goal description
Rough empirical basis of the holistic three-dimensional goal description
Functions of holistic three-dimensional goal descriptionMathematical material for chapter VI: "Algorithms"
VI Survey and justification
History
Overview
Justification
Conclusions
Mathematical material for chapter VII (Appendix): "The wiskobas curriculum"
Framework for instruction theory
Preamble
Starting points of a realistic instruction theory
One-dimensional description of the framework for instruction theory
Van Hiele's levels
Freudenthal's didactical phenomenology
Progressive mathematising guided by the five instruction principles
Schematic comparison of the four trends in arithmetic/mathematics instruction
Two-dimensional description of a framework for instruction theory
Progressive mathematisation in the Wiskobas programme
The five tenets of the framework for instruction theory revisited
Comparison of the four trends
Three-dimensional description of a framework for instruction theory
Progressive mathematising of long division
Two more examples: Number systems and fractions
The most conspicuous elements of the framework for instruction theory seen from the viewpoint of implementation of instructional ideas
The broader framework for instruction theory
Gagne, Dienes, Piaget, and Bruner
Recent investigations of subject matter in instruction theory
The almost complete absence of instruction theory ideas in general cognitive psychological research
Overview
Closure
Notes