Informal geometry as a basis for learning formal geometry

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THE PROBLEM-CENTRED PERSPECTIVE

The problem-centred approach currently being followed in many schools in South Africa utilizes the constructivist views on learning (Murray, Olivier and Human 1993:193). The following will be an account of the objectives of the problem-centred approach to learning, the role of the teacher in such learning, problem solving as a learning type and the role of social interaction in problem-centred learning. The problem-centred approach encompasses creativity and divergent thinking. Creativity and divergent thinking are inherent in problem solving. Engaging in activities of creating and divergent thinking in the teaching and learning of geometry facilitates teaching in context i.e. the correct teaching and learning environment. Creativity divergent thinking, problem solving and teaching in context are embedded in the problem-centred approach and this results in better learning. The empirical study in this research uses this approach and the following empirical format should be realized.

Instruction using the problem-centred approach

Bereiter (1992:337-340) makes a distinction between two types of knowledge viz: referent-based and problem-based knowledge. In the former, knowledge centres around attempting to understand what the text says i.e. acquiring knowledge for its own sake whereas in the latter, knowledge is constructed around the solution of problems, it is situation-based, the knowledge that one possesses is used to solve particular problems and is changed if not suitable (accommodation) and as a result new knowledge is constructed (constructivism). For knowledge to be functional in making sense of the world, it should be attached to persistent problems in the learner’s mind otherwise it will only be recalled when cues for such knowledge are given which is then referentbased. Bereiter advocates for problem-based rather than referent-based knowledge in instruction.

Problem solving as a learning type

Problem solving is a process and it entails the use of creative guidelines (heuristics) that help in the solution of problems. These guidelines are general suggestions and cannot guarantee success. Some of the guidelines/heuristics of problem solving involve: working backwards; making a drawing; creating your own problem and solving it and thinking of a similar problem that was solved successfully. Through the problem solving process, learners come to perceive mathematics as problem solving. In other words, children learn mathematics through problem solving, for example, in finding the nets of a cube, they represent a three-dimensional object in two dimensions.

The equilateral triangle

Five learners mentioned that in an equilateral triangle, three sides are equal. Of these learners, one also mentioned the fact that all the angles are equal and each is equal to 60E. The remaining seven learners could not state the properties of an equilateral triangle even though the majority of them had correctly identified the equilateral triangle in 1. This seems to indicate the importance of exposing learners to different ways of asking the same question.

Musical instruments

Learners made various types of musical instruments using cartridge paper. Parts of the musical instruments are made up of geometric shapes such as rectangles, squares and circles. A popular instrument is the guitar. Six learners constructed this type of instrument. All these learners made the portion with strings using a rectangle. The broad portions consisted of a square in two such instruments; a circle in two other instruments; a triangle in a bass guitar and a shape that is not a polygon as it has three straight sides with the fourth side a curve. The learners creativity is called to the fore as they came with these interesting designs. Some of the learners’ designs appear hereunder.

TABLE OF CONTENTS :

• CHAPTER 1 : BACKGROUND AND OVERVIEW OF THE STUDY
  • 1.1 Introduction
  • 1.2 Problem statement
  • 1.3 Aims and objective of the research project
    • 1.3.1 Aim of the study
    • 1.3.2 Objectives of the study
  • 1.4 Research design
    • 1.4.1 Literature study
    • 1.4.2 Empirical study
  • 1.5 Significance of the study
  • 1.6 Terminology
    • 1.6.1 Geometry education
    • 1.6.2 Creativity
    • 1.6.3 Divergent thinking
    • 1.6.4 Problem solving
    • 1.6.5 The problem-centred approach
  • 1.7 Progress of the investigation
• CHAPTER 2 : CREATIVITY AND DIVERGENT THINKING
  • 2.1 Introduction
  • 2.2 The concept of creativity
    • 2.2.1 Definitions of creativity
      • 2.2.1.1 The nature of the creative person
    • 2.2.2 The creative process
      • 2.2.2.1 The phases of creativity
• 2.3 Divergent thinking
  • 2.3.1 Orientation
  • 2.3.2 Teaching for thinking
  • 2.3.3 A taxonomy of critical thinking dispositions and abilities
  • 2.3.4 Developing reasoning skills
  • 2.3.5 Cognitive processes
  • 2.3.6 Metacognition
  • 2.3.7 Language and thought
• 2.4 Creativity, divergent thinkingand the problem-centred approach to teaching and learning
• 2.5 Conclusion
• CHAPTER 3 : PERSPECTIVES ON GEOMETRY EDUCATION
  • 3.1 Introduction
  • 3.2 Geometry education
    • 3.2.1 Orientation
    • 3.2.2 Spatial Perception / Visualization
      • 3.2.2.1 Visual education
      • 3.2.2.2 Spatial visualization in the mathematics curriculum
      • 3.2.2.3 Visualization in multicultural mathematics classroom
    • 3.2.3 Space and shape
    • 3.2.4 Spatial abilities
    • 3.2.5 Informal geometry as a basis for learning formal geometry
  • 3.3 Piaget
    • 3.3.1 Orientation
    • 3.3.2 Piaget’s stages of intellectual development
    • 3.3.3 Piaget and inhelder’s topological primacy theory
      • 3.3.3.1 Topological primacy
      • 3.3.3.2 Projective space
      • 3.3.3.3 Euclidean space
• 3.4 Vygotsky
• CHAPTER 4 :CREATIVITY AND DIVERGENT THINKING IN GEOMETRY TEACHING AND LEARNING IN A PROBLEM-CENTRED CONTEXT
  • 4.1 Introduction
  • 4.2 The problem-centred perspective
    • 4.2.1 Objectives
    • 4.2.2 Instruction using the problem-centred approach
      • 4.2.2.1 Problem solving as a learning type
      • 4.2.2.2 The role of social interaction
      • 4.2.2.3 The role of the teacher
      • 4.2.2.4 The role of learners
• 4.3 Creativity in Geometry teaching and learning
• CHAPTER 5 : METHOD OF RESEARCH
• CHAPTER 6 : DATA INTERPRETATION
• CHAPTER 7 : SUMMARY, CONCLUSIONS AND RECOMMENDATIONS

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CREATIVITY AND DIVERGENT THINKING IN GEOMETRY EDUCATION