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**Chapter 2** **Literature review and conceptual framework**

**Introduction**

This literature study is a critical and integrative synthesis of various researchers’ findings, justifying this research endeavour. It is imperative to remember that South Africa is the only country offering ML as a compulsory alternative to Mathematics in Grades 10 to 12. As the study concerns the ML teachers and the relationship between their knowledge and beliefs and their instructional practices, the literature review begins with a comparison of the international and national perspectives of mathematical literacy.Comparisons are made between the different conceptions of mathematical literacy; the contexts in which mathematical literacy can be applied; international studies measuring learners’ mathematical knowledge and literacy skills; meanings and definitions of mathematical literacy; and the role mathematical literacy plays in some school curricula. Following the review on mathematical literacy is a discussion of the meaning of teachers’ instructional practices and the value of various approaches to teaching. Moving to the core of the problem, literature regarding teachers’ knowledge and beliefs about the subject they teach are discussed. Attention is given to the different domains of teachers’ knowledge,teachers’ belief systems and the relationship between their knowledge and beliefs and their instructional practices. The literature review concludes with the conceptual framework which is based on concepts and theories from relevant work in the literature6.

**Mathematical literacy**

Mathematical literacy is not a clearly defined term and internationally there exists a range of different conceptions of mathematical literacy that are discussed in this section. As mathematical literacy (ML) is a school subject in South Africa, it is important to understand the motivation and purpose of ML in the South African curriculum and to compare it with the role mathematical literacy plays internationally.

**International perspectives on mathematical literacy**

In this section I mention the different terminology being used for mathematical literacy, compare different conceptions of mathematical literacy, discuss different contexts in which mathematics could be applied and refer to some international comparative studies that measure learners’ mathematical literacy skills in order to derive a general meaning or definition of mathematical literacy.There is an expanding body of literature that uses the terms “mathematical literacy” and “numeracy” as synonyms (Jablonka, 2003). The National Council on Education and the Disciplines however uses the term “quantitative literacy” to stress the importance of enquiring into the meaning of numeracy in a society that keeps increasing the use of numbers and quantitative information (Jablonka, 2003, p. 77). Jablonka prefers to use the term “mathematical literacy” to focus attention on its connection to mathematics and to being literate, in other words to a mathematically educated and well-informed individual (p. 77). In a comprehensive study by Jablonka (2003) in which different international perspectives on mathematical literacy were investigated, she found that the perspectives basically differ according to the stakeholders’ underlying principles and values. In her opinion there is a direct connection between a conception of mathematical literacy and a particular social practice. She acknowledges the difficulty of pointing out the distinct meaning of mathematical literacy as it varies according to the culture and context of the stakeholders who promote it (p. 76). The different conceptions of mathematical literacy relate to a number of relationships and factors. One of the relationships is between mathematics, the surrounding culture, and the curriculum (p. 80) while another is between school mathematics and out-of-school mathematics as mathematical literacy is about the individual’s ability to use the mathematics they are supposed to learn at school (p. 97). Varying with respect to the culture and the context four possible perspectives of mathematical literacy are:

• The ability to use basic computational and geometrical skills in everyday contexts.

• The knowledge and understanding of fundamental mathematical notions.

• The ability to develop sophisticated mathematical models.

• The capacity for understanding and evaluating another’s use of numbers and mathematical models (p. 76).

With the above-mentioned perspectives as background the different conceptions of mathematical literacy as found in the literature will subsequently be categorised.

**Chapter 1 ****Introduction and contextualisation**

**1.1 Introduction **

1.1.1 International perspective on mathematical literacy

1.1.2 National perspective on mathematical literacy

1.1.3 The experiences of ML teachers

1.1.4 Silence in the literature addressed in this study

**1.2 Rationale for the study **

**1.3 Statement of the problem **

**1.4 The purpose of the study**

**1.5 Research questions **

**1.6 Methodological considerations **

**1.7 Definition of terms**

**1.8 Possible contribution of the study **

**1.9 Limitations of the study **

**1.10 Summary**

**1.11 The structure of the thesis**

**Chapter 2 ** **Literature review and conceptual framework **

**2.1 Introduction **

**2.2 Mathematical literacy **

2.2.1 International perspectives on mathematical literacy

2.2.1.1 Different conceptions of mathematical literacy

2.2.1.2 Some contexts in which mathematical literacy can be applied

2.2.1.3 Studies measuring learners’ mathematical literacy skills

2.2.1.4 Defining mathematical literacy

2.2.1.5 The role of mathematical literacy in some international school curricula

2.2.1.6 Summary

2.2.2 An overview of ML

2.2.2.1 The history of ML

2.2.2.2 ML principles

2.2.2.3 Pedagogical approaches for teaching ML

2.2.2.4 The ML learner profile

2.2.2.5 Some general concerns about ML

2.2.2.6 Comparison between the national and international perspectives on mathematical literacy

2.2.2.7 An overview of ML and Mathematics

2.2.2.8 Summary

**2.3 Teachers’ instructional practices **

2.3.1 Tasks

2.3.2 Discourse

2.3.3 Learning environment

**2.4 Mathematics teachers’ knowledge and beliefs about mathematics and the teaching thereof **

2.4.1 Relationship between knowledge and beliefs

2.4.2 Overview of the different domains of teachers’ knowledge

2.4.2.1 Shulman’s (1986) categories of content knowledge

2.4.2.2 Grossman’s (1990) components of PCK

2.4.2.3 Borko and Putnam’s (1996) domains of knowledge

2.4.2.4 Ball, Thames and Phelps’ (2005) domains of knowledge for teaching

2.4.2.5 Hill, Ball and Schilling’s (2008) domain map for mathematical knowledge for teaching

2.4.2.6 Summary

2.4.3 An overview of mathematics teachers’ beliefs about mathematics and the teaching thereof

2.4.3.1 The nature of beliefs

2.4.3.2 Teachers’ belief systems

2.4.4 The influence of teachers’ knowledge and beliefs on their instructional practices

2.4.4.1 The influence of teachers’ knowledge and beliefs on the learners

2.4.4.2 The influence of teachers’ knowledge and beliefs on their teaching

2.4.5 Summary

**2.5 Conceptual framework **

2.5.1 General view on mathematics teachers’ knowledge and beliefs

2.5.1.1 Mathematics teachers’ MCK

2.5.1.2 Mathematics teachers’ PCK

2.5.1.3 Mathematics teachers’ beliefs

2.5.2 The three domains of PCK and beliefs

2.5.2.1 PCK and beliefs regarding content and learners

2.5.2.2 Knowledge and beliefs regarding content and teaching

2.5.2.3 Knowledge and beliefs regarding the curriculum

2.5.3 Teachers’ instructional practices

2.5.4 Summary

**2.6 Conclusion**

**Chapter 3 **

**Methodology **

**3.1 Introduction **

**3.2 Research paradigm and assumptions**

3.2.1 Research paradigm

3.2.2 Paradigmatic assumptions

**3.3 Research approach and design **

3.3.1 Research approach

3.3.2 Research design

**3.4 Research site and sampling **

**3.5 Data collection techniques **

3.5.1 Observations

3.5.2 Interviews

**3.6 Data analysis strategies **

**3.7 Quality assurance criteria **

3.7.1 Trustworthiness of the study

3.7.2 Validity and reliability of the study

3.7.2.1 The Hawthorne effect

3.7.2.2 The Halo effect

**3.8 Ethical considerations **

**3.9 Conclusion**

**Chapter 4 **

**Presentation and discussion of the findings **

**4.1 Introduction **

**4.2 The data collection process **

**4.3 Data analysis strategies**

4.3.1 Transcribing the data

4.3.2 Coding of the data

4.3.2.1 Theme 1: ML teachers’ instructional practices

4.3.2.2 Theme 2: ML teachers’ knowledge and beliefs

4.3.2.3 Inclusion criteria for coding the data

4.3.2.4 Exclusion criteria for coding the data

**4.4 Information regarding the four participants**

4.4.1 Monty

4.4.2 Alice

4.4.3 Denise

4.4.4 Elaine

**4.5 Theme 1: The ML teachers’ instructional practices**

4.5.1 Monty’s instructional practice

4.5.1.1 Tasks

4.5.1.2 Discourse

4.5.1.3 Learning environment

4.5.2 Alice’s instructional practice

4.5.2.1 Tasks

4.5.2.2 Discourse

4.5.2.3 Learning environment

4.5.3 Denise’s instructional practice

4.5.3.1 Tasks

4.5.3.2 Discourse

4.5.3.3 Learning environment

4.5.4 Elaine’s instructional practice

4.5.4.1 Tasks

4.5.4.2 Discourse

4.5.4.3 Learning environment

4.5.5 Summary of participants’ instructional practices

4.5.6 Discussion of Theme 1: ML teachers’ instructional practices

4.5.6.1 Tasks

4.5.6.2 Discourse

4.5.6.3 Learning environment

4.5.6.4 Summary of discussion on Theme 1

**4.6 Theme 2: ML teachers’ knowledge and beliefs **

4.6.1 Monty’s knowledge and beliefs

4.6.1.1 Mathematical content knowledge (MCK)

4.6.1.2 Knowledge and beliefs regarding ML learners

4.6.1.3 Knowledge and beliefs regarding ML teaching

4.6.1.4 Knowledge and beliefs regarding ML curriculum

4.6.2 Alice’s knowledge and beliefs

4.6.2.1 Mathematical content knowledge (MCK)

4.6.2.2 Knowledge and beliefs regarding ML learners

4.6.2.3 Knowledge and beliefs regarding ML teaching

4.6.2.4 Knowledge and beliefs regarding ML curriculum

4.6.3 Denise’s knowledge and beliefs

4.6.3.1 Mathematical content knowledge (MCK)

4.6.3.2 Knowledge and beliefs regarding ML learners

4.6.3.3 Knowledge and beliefs regarding ML teaching

4.6.3.4 Knowledge and beliefs regarding ML curriculum

4.6.4 Elaine’s knowledge and beliefs

4.6.4.1 Mathematical content knowledge (MCK)

4.6.4.2 Knowledge and beliefs regarding ML learners

4.6.4.3 Knowledge and beliefs regarding ML teaching

4.6.4.4 Knowledge and beliefs regarding ML curriculum

4.6.5 Summary of the participants’ knowledge and beliefs

4.6.6. Discussion of Theme 2: ML teachers’ knowledge and beliefs

4.6.6.1 ML teachers’ mathematical content knowledge (MCK)

4.6.6.2 ML teachers’ knowledge and beliefs regarding their learners

4.6.6.3 ML teachers’ knowledge and beliefs regarding the teaching of ML

4.6.6.4 ML teachers’ knowledge and beliefs regarding the ML curriculum

4.6.6.5 Summary of discussion on Theme 2

**4.7 Findings, trends and explanations **

**4.8 Conclusion**

**Chapter 5 **

**Conclusions and implications **

**5.1 Introduction **

**5.2 Chapter summary **

**5.3 Verification of research questions **

5.3.1 Question 1: How can ML teachers’ instructional practices be described?

5.3.2 Question 2: What is the nature of ML teachers’ knowledge and beliefs?

5.3.3 Question 3: How do ML teachers’ knowledge and beliefs relate to their instructional practices?

5.3.4 Question 4: What are the possible implications of the findings from Questions 1, 2 and 3 for teacher training?

5.3.5 Question 5: What is the value of the study’s findings for theory building in teaching and learning ML?

5.3.6 Summary of verification of research questions

**5.4 What would I have done differently? **

**5.5 Providing for errors in my conclusion**

**5.6 Conclusions **

**5.7 Recommendations for further research **

**5.8 Limitations of the study**

**5.9 Last reflections **

**References **

**Appendices **