Mathematical reasoning and its importance

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Learners’ verbal images of translation

Translation is formally defined as “a mapping that changes position of a point, line or a polygon by sliding it in a specific direction through a specific distance” (Tapson, 2006). This description has two aspects, namely that points move the same distance, and movement is in a common direction. Both of these aspects need to be stated for the definition to be considered complete. The process of evaluating learner definitions was complicated due to the varied language used, and the sometimes imprecise descriptions, hence careful scrutiny was necessary. The same parameters were used to assess the answers as in section 5.1.1.1: (a) A correct definition of translation, which refers to both aspects of translation namely (i) displacement (distance) and (ii) specific direction (or just ‘in a straight line’ displacement). (b) A partly correct definition which refers to only one of the two aspects of translation mentioned in (a). (c) An incorrect definition which is either so broad that it applies to transformations in general and not translation specifically, or is not related to translation at all. (d) Did not attempt to answer, where the learner left blank the space provided for the answer.

Learner symbolical images for a two-way translation

The question read as follows: Write down the formula for a translation of 4 units to the right and 3 units upward. The question required learners to write the formula as . Only four learners (4%) were able to give the correct formula. Twenty-two learners (23%) gave the equation as thereby showing a misconception of the horizontal translation direction. They associated the ‘plus’ sign with translation upwards and to the right. A total of 69 learners (72%) had misconceptions. Many misconceptions related to misunderstanding the resultant direction of the translation and the incorrect use of brackets. Twenty-three learners (24%) left the answer space blank. Some examples of learner responses for formulae are shown in Vignettes 11 and 12 below.

THEORETICAL FRAMEWORK

Learning has been explained from different theoretical perspectives, such as behaviourist, cognitivist or constructivist. A learning theory can function as a lens through which facts about how learning takes place are viewed and it normally influences what is seen and not seen about the facts. Learning theories help us to interpret facts, for example, good learning processes are likely to result in appropriate concept image formation within the learner’s mind. Concept images can be interpreted using learning theories. Contemporary psychology of mathematics education is centred on the constructivist and cognitivist philosophies. The formation of concept images by learners can be explained through these philosophies. Constructivist theories of thinking and reasoning can be traced back as far as Giambattista Vico in the 1700s (Glasersfeld, 1984), but Piaget and Vygotsky, writing in the 1970s, are considered the first true constructivist scholars with regards to education. Constructivism is now considered to have two major streams: personal constructivism, of which the major proponents are Piaget and Von Glasersfeld (Glasersfeld’s views of learner independence in learning being more radical than Piaget’s).

Radical Constructivism and Concept Images

Von Glasersfeld refers to concept images as conceptual structures. His version of constructivism is regarded as being more radical than Piaget’s because he maintains that knowledge is individually created and adjudicated and that experience is what brings forth knowledge claims. According to Von Glasersfeld (1984), knowledge consists of conceptual structures (concept images) that act as epistemic (knowledge) agents and it is actively built by the thinking individual through the senses or any other communication forms experienced within the individual learner’s minds. The emphasis is on active involvement of learners in the process of learning. Von Glasersfeld did not rate social interactions among learners as being important to knowledge building.

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Social-cultural constructivism and concept images

Vygotsky refers to concept images using the term knowledge structure. His primary notion of social–cultural constructivism insists that knowledge construction does not happen in the mind of a learner. It stresses that knowledge acquisition and construction happen as a learner interacts with his or her surroundings (1986). This points to the importance of interaction with other people in the school context for learning concepts, both dialogue with other learners and assistance from teachers and fellow learners. Dialogue aids understanding of concepts, and assistance from others strengthens the learning process 58 within a learner’s zone of proximal development (ZPD) 17 . Readiness-to-learn and scaffolding18 are two of the factors that influence learning within the ZPD (Vygotsky, 1978). Scaffolding is built through the learner-support materials or tools that are used. These could be in the form of hints or advice that prompts reflection, coaching, articulation of different ideas, or making links between every day and formal concepts. All these pathways facilitate concept image formation.

Table of Contents :

CHAPTER ONE Introduction
- 1.1 THE CONTEXT OF THE STUDY
  - 1.1.1 GET – NCS – CAPS Syllabi Relative Objectives
  - 1.1.2 FET – NCS – CAPS Syllabi Related Objectives
  - 1.1.3 Aggregating the NCS – CAPS Syllabi Objectives for Transformations of Functions and the Aims of the Mathematics in the FET Phase
- 1.2 THE BACKGROUND OF THE STUDY
  - 1.2.1 Weighting of transformation of functions
- 1.3 THE PURPOSE STATEMENT
- 1.4 THE PROBLEM STATEMENT
- 1.5 THE RESEARCH QUESTIONS
- 1.6 THE AIMS OF THE STUDY
- 1.7 THE SIGNIFICANCE OF THE STUDY
- 1.8 ASSUMPTIONS OF THE STUDY
- 1.9 DESCRIPTIONS OF KEY TERMS AND CONCEPTS
  - 1.9.1 Concept Image
  - 1.9.2 Mathematical Reasoning
  - 1.9.3 Coherence of Concept Image
  - 1.9.4 A Function
  - 1.9.5 A Functional Representation
  - 1.9.6 A transformation
  - 1.9.7 Transformation of a Function
- 1.10 STRUCTURE OF THE THESIS
CHAPTER TWO Conceptualising Concept Image
- 2.1 WHAT CONCEPT IMAGES ARE AND WHAT THEY CONSTITUTE
- 2.2 FORMATION OF CONCEPT IMAGES AND MATHEMATICAL REASONING
- 2.3 HOW THE APOS MODEL EXPLAINS LEARNING CONCEPTS THROUGH MATHEMATICAL THINKING AND REASONING
- 2.4 CONCEPTUAL UNDERSTANDING
- 2.5 MATHEMATICAL REASONING AND ITS IMPORTANCE
- 2.6 AQUISITION OF CONCEPT IMAGES AND MATHEMATICAL REASONING
- 2.7 REVISITING THE FACTS ABOUT CONCEPT IMAGE AND ITS COHERENCE
- 2.8 VINNER’S MODEL FOR CONCEPT DEFINITIONS AND CONCEPT IMAGES FRAMEWORK
- 2.9 DUBINSKY’S A.P.O.S. MODEL OF CONCEPTUAL FORMATION
  - 2.9.1 How the APOS model explains learning and understanding concepts in mathematics in relation to reflection, translation and stretch of functions
- 2.10 SFARD’S MODEL OF CONCEPT FORMATION
- 2.11 A NEW MODEL
- 2.12 CONCLUDING REMARK
CHAPTER THREE Theoretical Framework and Literature Review
- 3.1 THEORETICAL FRAMEWORK
  - 3.1.1 Cognitive constructivism and concept images
  - 3.1.2 Radical Constructivism and Concept Images
  - 3.1.3 Social-cultural constructivism and concept images
  - 3.1.4 Social-cultural Constructivism as viewed by Ernest
  - 3.1.5 Models of human memory structures
- 3.2 LITERATURE REVIEW
CHAPTER FOUR Methodology
- 4.1 THE RESEARCH DESIGN
- 4.2 SAMPLING PROCEDURE
- 4.3 THE PARTICIPANTS
- 4.4 DATA COLLECTION INSTRUMENTS
  - 4.4.1. The diagnostic test
  - 4.4.2 The follow-up clinical interview
- 4.5 VALIDITY AND RELIABILITY CHECKS
CHAPTER FIVE The Data and its Analysis
CHAPTER SIX Relating Research Findings to Research Questions
CHAPTER SEVEN Summary of the Study, Conclusions and Recommendations