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**CHAPTER TWO ****LITERATURE REVIEW AND CONCEPTUAL FRAMEWORK**

The chapter is divided into two main sections, the literature review and the conceptual framework. The first section reviews studies that have been done on problem-solving,problem-solving skills, problem-solving models and challenges learners face in probability,as well as some common errors and misconceptions in probability as these factors could be relevant to the quintile ranking of schools. The second part of the chapter discusses the conceptual framework of the study. Bloom’s (1956) taxonomy and the CAPS (DBE, 2011) probability curriculum were used as frameworks to tease out the learners’ problem-solving skills in probability

**LITERATURE REVIEW**

This section reviews the literature related to the subjects of the study, namely: problemsolving; problem-solving skills;investigation into problem-solving skills; problem-solving models; evaluation of problem-solving skills; probability; probability in the South African school curriculum; studies on teaching and learning of probability; errors and misunderstandings of probability experienced by learners; learner achievement and the quintile ranking of schools in South Africa.

**Problem-solving**

For some years now, mathematical problem-solving has been seen as a vital aspect of the teaching of mathematics, the learning of mathematics and mathematics in general. It is one of the most important cognitive skills needed in many professions as well as in everyday life (Jonassen, 2000). This is because everyone encounters problems in their daily activities. Some of these problems may require simple solutions. Others may require a series of steps before one arrives at the desired solution. The act of solving problems and getting problems right demands some level of skill. Individuals who have these skills have greater opportunities in their everyday life and profession. For example, people’s capacity to solve complex problems enables them to adapt to changes in the community or the environment and to learn from their mistakes. Proficiency in problem-solving contributes to selfactualisation and leads to greater opportunities for employment as well as contributing to economic growth (Hanushek, Wößmann, Jamison & Jamison, 2008). Due to its importance,there have been calls for the teaching of problem-solving, as well as the teaching of mathematics, through problem-solving to be included in the mathematics curriculum (Liljedahl, Santos-Trogo Malaspina & Bruder, 2016; Zanzali & Lui, 2000). It is not surprising that the field has seen tremendous interest by researchers in mathematics education. As a result, the past decades have witnessed much research done on problem-solving in different disciplines. Studies on problem-solving have focused on different themes (Anderson, 1980; Jonassen, 2010; Mayer & Wittrock, 2006; Newell & Shaw, 1958). Among these themes is the emergence of a number of problem-solving models such as Polya’s (1957) problem-solving models, problem-solving assessment tools, problem-solving as a teaching method and the identification of students’ problem-solving skills. Researchers in the course of their studies have given different definitions of problem-solving. A few are captured here. Heppner and Krouskopf (1987) defined problem-solving as cognitive and effective behavioural processes for the purpose of adapting to internal or external demands or requests. Bingham (1988) defined problem-solving as a process that requires a series of efforts oriented towards eliminating the difficulties encountered in order to achieve a certain objective. According to Kashani-Vahid, Afrooz, Shokoohi-Yekta,Kharrazi, & Ghobari (2017, p. 176), “Problem solving is a self-directed cognitive-affectivebehavioural process” through which individuals or groups attempt to find effective solutions to problems they encounter in life. Krulik and Rudnick (1980, p. 3) defined a problem as “a situation, quantitative or otherwise, that confronts an individual or group of individuals, that requires resolution, and for which the individual sees no apparent or obvious means or path to obtaining solution”. Krulik and Rudnick (1980) opined that the problem-solving process required individuals to use previously acquired knowledge, skills and understanding to satisfy the demands of an unfamiliar situation. This implies that for one to be a good problem solver,one ought to have acquired certain skills that could engineer the easy solving of the problem from previous experience. The implication is that problems have some degree of difficulty that requires special skills to tackle. The key to becoming a good problem solver lies in the cognitive domain, since the process is a cognitive one. This study is therefore grounded in Krulik and Rudnick’s (1980) definition of problem-solving.Of all the various definitions of problem-solving, Mayer and Wittrock (2006, p 287)definition of problem solving, “a cognitive process directed at overcoming obstacles” is the most widely accepted by problem-solving advocates. According to the Meyer and Wittrock(2006, p 287) problem-solving is a means of “transforming a given situation into a desired situation when no obvious method of solution is available.” The various definitions presented all have something in common, namely overcoming an obstacle to reach the desired solution.

**DEDICATION**

**DECLARATION **

**ACKNOWLEDGEMENTS**

**ABSTRACT**

**TABLE OF CONTENTS**

**LIST OF TABLES**

**LIST OF FIGURES **

**LIST OF ABBREVIATIONS.**

**CHAPTER ONE BACKGROUND OF THE STUDY**

**1.1 CONTEXT**

**1.2 PROBLEM STATEMENT**

**1.3 AIM OF THE STUDY **

**1.4 OBJECTIVES**

**1.5 RESEARCH QUESTIONS**

**1.6 RATIONALE FOR THE STUDY**

**1.7 SIGNIFICANCE OF THE STUDY **

**1.8 DEFINITION OF TERMS**

**1.9 OUTLINE OF** **CHAPTERS**

**1.10 CONCLUSION**

**CHAPTER TWO LITERATURE REVIEW AND CONCEPTUAL FRAMEWORK**

**2.1 LITERATURE REVIEW**

2.1.1 Problem-solving

2.1.2 Problem-solving skills

2.1.3 Investigations into learners’ problem-solving skills

2.1.4 Problem-solving models

2.1.5 Evaluation of problem-solving skills (PSS)

2.1.6 Learners’ errors and misconceptions in probability

2.1.7 Learner achievement and quintile ranking of schools in South Africa

**2.2 CONCEPTUAL FRAMEWORK **

2.2.1 Bloom’s taxonomy

2.2.2 Probability content in the CAPS document

**2.3 SUMMARY**

**2.4 CONCLUSION**

**2.5 PROJECTION FOR THE NEXT CHAPTER**

**CHAPTER THREE RESEARCH METHODOLOGY**

**3.1 RESEARCH PARADIGM **

**3.2 METHODOLOGY **

**3.3 RESEARCH DESIGN **

**3.4 POPULATION OF THE STUDY**

**3.5 SAMPLING AND SAMPLING TECHNIQUE**

**3.6 DATA COLLECTION INSTRUMENT**

3.6.1 Development of the test

3.6.2 Data collection procedure

3.6.3 Marking memorandum

**3.7 VALIDITY AND RELIABILITY OF INSTRUMEN **

3.7.1 Validity of the instrument

3.7.2 Reliability of the instrument

**3.8 DATA ANALYSIS**

3.8.1 Quantitative data analysis

3.8.2 Qualitative data analysis

**3.9 ETHICAL ISSUES **

**3.10 SUMMARY**

**3.11 PROJECTION OF CHAPTER FOUR**

**CHAPTER FOUR FINDINGS**

**4.1 FINDINGS**

**4.2 FINDINGS FROM QUANTITATIVE DATA ANALYSIS**

**4.3 ANALYSIS OF QUALITATIVE DATA**

4.3.1 Content analysis of learners’ performance

**4.4 ERROR ANALYSIS IN THE DIFFERENT ASPECTS OF PROBABILITY**

4.4.1 Mutually exclusive events

4.4.2 Complementary events

4.4.3 Independent event

4.4.4 Dependent events

4.4.5 Use of Venn diagram

4.4.6 Use of the contingency table

4.4.7 Fundamental counting principle

**4.5 REPORT ON RESEARCH QUESTIONS **

4.5.1 Research question one

4.5.2 Research question two

4.5.3 Research question three

4.5.4 Research question four

**4.6 PROJECTION FOR THE NEXT CHAPTER**

**CHAPTER FIVE DISCUSSION OF FINDINGS **

**5.1 RESEARCH QUESTION ONE**

**5.2 RESEARCH QUESTION TWO**

5.2.1 Mutually exclusive events

5.2.2 Complementary event

5.2.3 Independent event

5.2.4 Dependent events

5.2.5 Venn diagrams

5.2.6 Contingency table

5.2.7 Fundamental counting principles

**5.3 RESEARCH QUESTION THREE **

**5.4 RESEARCH QUESTION FOUR **

5.4.1 MUTUALLY EXCLUSIVE EVENTS

5.4.2 Complementary events

5.4.3 Dependent events and independent events

5.4.4 Use of Venn diagrams as an aid to solving probability problems

5.4.5 Use of tree diagrams as an aid to solve probability problems

5.4.6 Use of contingency tables as an aid to solve probability problems

5.4.7 Fundamental counting principles

**5.5 SUMMARY OF CHAPTER**

**CHAPTER SIX SUMMARY, CONCLUSION AND RECOMMENDATIONS **

**6.1 SUMMARY OF THE STUDY **

**6.2 CONCLUSION**

**6.3 RECOMMENDATIONS**

**6.4 SUGGESTIONS FOR FUTURE RESEARCH**

**6.5 LIMITATIONS OF THE STUDY **

**6.7 FINAL THOUGHT**

**REFERENCES **

**APPENDICES **

**APPENDIX A**: COGNITIVE TEST ON PROBABILITY AND COUNTING PRINCIPLES

**APPENDIX B**: SOLUTION TO COGNITIVE TEST ON PROBABILITY AND COUNTING PRINCIPLE **APPENDIX C**: PARENT’S CONSENT FORM

**APPENDIX D**: VARIOUS GRAPHS SHOWING LEARNER PERFORMANCE IN DIFFERENT CATEGORIES

**APPENDIX E**: SPSS 23 OUTPUT OF RELIABILITY COEFFICIENT

**APPENDIX F**: LETTER OF PERMISSION TO CONDUCT RESEARCH

**APPENDIX G:** ETHICAL CLEARANCE CERTIFICATE FROM UNISA

**APPENDIX H**: GAMES-HOWELL POST-HOC MULTIPLE COMPARISON TEST