PROPOSED STRATEGIES FOR TEACHING AND LEARNING GEOMETRY

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CHAPTER THREE: RESEARCH METHODOLOGY

Introduction

The chapter presents, a detailed description of the research methodology and the designs which were employed to illustrate the devised strategies that were employed when conducting this research study. I also justify why I have to employ each of the selected methods in conducting my research study. The following sections are to be epistemologically justified: (a) the research methodology and (b) the research design which comprised:(i) the methods used to collect data, (ii) sample selection, (iii) sampling techniques, (iv) description and advantages of the instruments used in collecting data, (v) a detailed description of how the diagnostic and post- intervention tests were developed and validated to ensure that they are at the appropriate level and relevant standard for the target group, (vi) the analysis of data, (vii) the ethical issues and (viii) research validity.

Research methodology

In this section, the research methodology, which is the philosophical framework that addresses the research questions in relation to the entire research processes is presented and described in detail (Creswell & Plano Clark, 2007). This research study is informed by the mixed methods paradigm which is defined as the unification of quantitative and qualitative data analysis in a distinct research study from which the simultaneously collected facts are given priority. The paradigm involves the amalgamation of the facts in one or more phases in the procedure of investigation to ensure that no part is left without being examined (Creswell, 2003). Researchers argue that the mixed methods paradigm provides the most instructive, comprehensive, composed, and convenient study outcomes (Johnson et al., 2007). The philosophical and epistemological foundation for employing mixed methods in association with my research study was to obtain different but complementary data on the same topic to best understand and get solutions to the difficulties learners face in learning geometry.
The advantages of using the mixed methods approach to this research were the following:

• To augment research outcomes to ensure that one form of data did not consent to obtain a deeper understanding of one or more of the constructs under study (Brewer & Hunter, 1989; Tashakkori & Teddlie, 1998).
• To had an opportunity to simultaneously generalise results from a sample in order to gain a deeper understanding of how the use of polygon pieces assisted by mathematics dictionary influenced teaching and learning of geometry in an interesting way. The deeper understanding was gained by uniting numerical trends from quantitative data and specific details of the phenomenon under study from qualitative data (Hanson, Creswell, Plano Clark, Petska & Creswell, 2005).
• To have a prospect to experiment hypothetical models and to adapt them based on my research participants’ response which they gave after being engaged in the intervention programme (Hanson et al., 2005).
  The listed advantages imply that the mixed methods paradigm gave more room for a thorough data analysis. All aspects identified by the different research instruments of my study were to be analysed from different angles of focus so that a true reflection of how the use of physical manipulatives assisted by mathematics dictionary influenced teaching and learning of geometry was eventually brought to light.
  According to Denscombe (2008),the mixed methods paradigm offers quite a number of opportunities to the researcher, which are: (i) to advance the precision of the collected data (ii) to produce a more multi-faceted picture by merging information from a variety of kinds of sources that complement each other (iii) as a means of complementing for the specific strength or weakness which a particular method has (iv) as a way of developing the analysis and construct on initial findings using distinct kinds of data and (v) used as an utility for assessing the appropriateness for the research sample (Collins, Onwuegbuzie & Sutton, 2006). In my study, the following were reasons for using a mixed methods paradigm:
• To get an opportunity to scrutinise and understand the complexity of the phenomenon under study at a deeper level to ensure that there is strong correlation between the interpretation and usefulness of research findings (Collins et al., 2006). The understanding of how the use of physical manipulatives assisted by mathematics dictionary in teaching and learning geometry influenced learners’ conceptual understanding allowed me to develop a model that can help to improve the situation of teaching and learning geometry (Creswell, 2003).
• To had an opportunity to strategically position myself to explore, experiment and to have an in-depth understanding of how polygons pieces can be used in the teaching and learning of geometry.
• To assess the appropriateness and relevance of the chosen instruments which were scheduled for data collection (Collins et al., 2006). Terre Blanche and Durrheim (1999) argue that the in-depth understanding of the meaning of human inventions of ideas, words and experiences can only be established in relation to the context in which they happen. Hence, in view of Terre Blanche and Durrheim’s (1999) proposition and by employing the mixed methods paradigm, I was in a position to conduct an in-depth research in a context that was highly restricted and explicitly related to experiences of nine eighth-grade learners.
  When learners were engaged in various tasks of an intervention programme that utilised pieces of polygons assisted by mathematics dictionary to facilitate conceptual understanding of geometry it was possible to observe them working. For this reason an opportunity was created to collect relevant and rich data which was required in this study (Collins et al., 2006). Lastly, the mixed methods paradigm also opened a window of exploring learners’ mathematical proficiency as they were engaged in using polygon pieces as physical manipulatives assisted by mathematics dictionary for teaching and learning of geometry.

Research design

A detailed description of the contextual issues in relation to this study is provided. These contextual issues consist of the geographical background of: (i) the site where my research study was conducted, (ii) a clear description of the South African senior phase mathematical content in relation to van Hiele’s (1999) stages of geometric intellectual and mathematical background of the chosen sample, (iii) the methods used to collect data, (iv) learners’ sample selection, (v) learners’ sampling techniques, (vi) description and advantages of the instruments used in collecting data, (vii) a detailed description of how the diagnostic and post-intervention tests were developed and validated to ensure that they were at an appropriate level and relevant standard for the target group, (viii) the analysis of data, (ix) the ethical issues and (x) research validity.

Geographical background

My research site was one of the section 21 secondary schools in the Eastern Cape Province of South Africa in the Queenstown district. Section 21 secondary schools are semi-urban secondary schools in a low-income group residential area.This was one of the schools within my reach, which means that I could easily obtain access and informed consent.

1.1 INTRODUCTION OF THE CHAPTER 
1.2 BACKGROUND OF THIS RESEARCH STUDY
1.3 THE RESEARCH PROBLEM
1.4 RESEARCH QUESTIONS
1.5 UNDERLYING ASSUMPTIONS THAT INFLUENCED THE INTERVENTION
1.6 RATIONALE
1.7 CONCEPTUAL FRAMEWORK
1.8 RESEARCH METHODOLOGY
1.9 RESEARCH DESIGN
1.11 LIMITATIONS
1.12 AN OVERVIEW OF THE RESEARCH METHODOLOGY AND ITS DESIGN
1.13 OUTLINE OF MY THESIS
2.1 INTRODUCTION 
2.2 THE BACKGROUND OF GEOMETRY
2.3 PROPOSED STRATEGIES FOR TEACHING AND LEARNING GEOMETRY
2.4 DEFINITION OF PHYSICAL MANIPULATIVES
2.5 THE HISTORY OF PHYSICAL MANIPULATIVES USE
2.7 THE USE OF PHYSICAL MANIPULATIVES ASSISTED BY MATHEMATICS DICTIONARY IN THE TEACHING OF MATHEMATICS
2.8 SUGGESTIONS ON USEFUL WAYS OF USING PHYSICAL MANIPULATIVES
2.9 THEORETICAL FRAMEWORK
2.10 RESEARCH INTO THE VAN HIELE LEVELS OF GEOMETRIC THINKING
2.11 PHYSICAL MANIPULATIVES FOR VISUALISATION
2.12 PHYSICAL MANIPULATIVES FOR THE ANALYSIS OF GEOMETRIC CONCEPTS
2.13 PHYSICAL MANIPULATIVES FOR ABSTRACTION
2.14 CONCLUSION
3.1 INTRODUCTION 
3.2 RESEARCH METHODOLOGY
3.3. RESEARCH DESIGN
3.4 CONCLUSION
4.1 INTRODUCTION 
4.2 RESULTS
4.3 DISTRIBUTION OF DIAGNOSTIC AND POST-INTERVENTION TESTS MARKS .
4.5 WHY DID THE MODEL INFLUENCE MATHEMATICAL DEVELOPMENT?
4.6 LESSONS LEARNT FROM THESE RESULTS
4.7 THE ACTUAL MODEL OF TEACHING AND LEARNING GEOMETRY EMERGED DURING MY RESEARCH
4. 8 CHIPHAMBO’S REFLECTIVE MODEL FOR TEACHING AND LEARNING GEOMETRY CONTRIBUTIONS
5.1 INTRODUCTION TO THE CHAPTER 
5.2 FINDINGS AND CRITIQUE OF THE RESEARCH
5.3 KEY FINDINGS
5.4 UNEXPECTED OUTCOMES
5.5 REFERENCE TO PREVIOUS RESEARCH
5.6 THE DETAILED EXPLANATION OF MY RESEARCH RESULTS
5.7 ADVICE TO THE RESEARCHERS AND EDUCATORS IN INTERPRETATION OF MY RESEARCH FINDINGS
5.8 SUGGESTIONS FROM CHIPHAMBO’S REFLECTIVE MODEL FOR TEACHING AND LEARNING GEOMETRY
5.9 PRESENTATION OF IMPLICATIONS OF THE RESEARCH
5.10 COMMENTING ON FINDINGS
5.11 LIMITATIONS OF MY RESEARCH STUDY
5.12 RECOMMENDATION FOR FUTURE RESEARCH WORK
5.12 CONCLUSION
APPENDIX
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A CASE STUDY: INVESTIGATING A MODEL THAT INTEGRATES DICTIONARY AND POLYGON PIECES IN TEACHING AND LEARNING OF GEOMETRY TO GRADE 8 LEARNERS.