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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
Several direct quotations are used in the literature review to avoid nuance chances of meaning to the matter under discussion.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
.The knowledge and understanding of fundamental mathematical
.The ability to develop sophisticated mathematical
.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.
Different conceptions of mathematical literacy
The literature revealed different conceptions of mathematical literacy but the resemblances between mathematical literacy and RME, mathematisation, mathematical modelling, as well as mathematics in action are most evident.
As there is opacity as to what each of these conceptions entail and how they differ a clarification of the concepts and notions will be provided.
Realistic Mathematics Education
Hope (2007) expressed the resemblance of mathematical literacy with the theory of RME. RME uses a theoretical framework that relies on real-world applications and modelling, a didactical belief propagated by Hans Freudenthal (Gates & Vistro-Yu, 2003, p. 67). According to Van den Heuvel-Panhuizen (1998), Freudenthal and his colleagues laid the foundations of RME in the early seventies to address the world- wide need to reform the teaching and learning of mathematics and to move away from mechanistic mathematics education. Freudenthal’s theory of RME rests upon the following five components:
.Using a real-world context as a starting point for
.Bridging the gap between abstract and applied mathematics by using visual
.Having students develop their own problem-solving strategies rather than memorise rules and
.Making mathematical communication, perhaps in the form of journaling or oral presentations, an integral part of the
.Making connections to other disciplines using meaningful real-world problems (Hope, 2007, 30).
Hope (2007) further believes mathematical literacy is a matter of the appropriate pedagogy that should be used in teaching mathematics. According to these fundamental pedagogical aspects of teaching mathematics, it is comprehensible that the traditional school mathematics instruction is too formal, less intuitive, more abstract, less contextual, more symbolic, and less concrete than the type of instruction that would expand student thinking and develop mathematical literacy (p. 30).
Mathematisation
Freudenthal believed that the focus should not be on mathematics as a closed system, but on the activity, on the process of mathematisation (Van den Heuvel-Panhuizen, 1998), and that mathematics should be seen as a human activity that is connected to reality and relevant to society. Treffers (1978) formulated the idea of two types of mathematisation, namely horizontal and vertical mathematisation. He stated that in horizontal mathematisation the students come up with mathematical tools which can help to organise and solve a problem located in a real-life situation whereas vertical mathematisation is the process of reorganization within the mathematical system itself, like, for instance, finding shortcuts and discovering connections between concepts and strategies and then applying these discoveries (Van den Heuvel-Panhuizen, 1998). Freudenthal (1991) explained horizontal mathematisation as going from the world of life into the world of symbols, while vertical mathematisation means moving within the world of symbols (p. 24). It would seem that vertical mathematisation refers to the more formal mathematics while horizontal mathematisation refers to the informal mathematical literacy part.
According to Hope (2007) mathematising is a term used by The Organisation of Economic Co- operation and Development (OECD) which involves five elements:
.Starting with a problem whose roots are situated in
.Organising the information and data according to mathematical
.Transforming a real-world, concrete application to an abstract problem whose roots are situated in
Solving the mathematical
.Reflecting back from the mathematical solution to the real-world situation to determine whether the answer makes sense (p. 29).
Mathematical modelling
ML bears a strong resemblance to mathematical modelling in that both require an application of Polya’s four basic steps in problem-solving namely: a) understanding the problem; b) designing a plan; c) carrying out the plan; and d) looking back on the problem. Mathematical modelling can further be described as a matter of constructing an idealised, abstract model which may then be compared for its degree of similarity with a real system. (Giere, 1999, p. 50).
Gellert et al. (2001) use mathematical literacy as a metaphor referring to well-educated and well- informed individuals. According to them different conceptions of mathematical literacy are based on the relationship between mathematics, reality and the society. Their concept of mathematical literacy involves gaining a level of mathematical understanding that goes beyond the minimal abilities of calculating, estimating, and gaining some number sense, and basic geometrical understanding … by seeing the power of mathematics in its potential of abstracting from concrete realities by generating concepts and structures for universal application (p. 59). They further believe these abilities can be developed by experiencing mathematical modes of thinking, such as searching for patterns, classifying, formalising and symbolising, seeking implications of premises, testing conjectures, arguing, and thinking propositionally (p. 59) which form the basis of mathematical modelling. Mathematical literacy requires the mathematical competence to understand the mathematical methods involved and the analytical competence to demystify the justifications for specific mathematical applications as well as to assess their consequences (p. 66).
Mathematics in action
Although some of the above-mentioned conceptions are formal, involving higher-order mathematical skills, there are other researchers who regard mathematical literacy as a fundamental requirement for all people, recognising its essential value to learners in contexts forming part of their everyday living (McCrone & Dossey, 2007; Powell & Anderson, 2007; Skovsmose, 2007). McCrone and Dossey (2007) believe mathematical literacy is not about studying higher levels of formal mathematics, but about making mathematics relevant and empowering for everyone (p. 32). They further call for mathematics to play an even
greater part in non-mathematics classes where teachers promote the mathematics embedded in their subjects.
Skovsmose (2007) refers to mathematical literacy as mathematics in action and considers the role of mathematical literacy in both mathematicians’ and non-mathematicians’ lives. He based his study on two types of literacy being either functional or critical, terms introduced by Apple in 1992. Functional literacy is defined by competencies a person possesses to fulfil a particular job function (p. 4) whereas critical literacy addresses themes such as working conditions and political issues. Skovsmose prefers to talk about reflective knowledge with respect to mathematics instead of critical literacy. Reflective knowledge refers to a competence in evaluating how mathematics is used or could be used (p. 4). Critical literacy is associated with the skill to create or design models using mathematics whereas functional literacy is the skill to use and apply those models. To make unambiguous distinctions between these two types is not that simple and it could have very different interpretations depending on the context of the learner (p. 4).
Clarification of basic concepts and notions
From the discussion above it is clear that the lines between the different concepts such as mathematisation and mathematical modelling are blurred. Blum and Niss (2010) provided a clarification of the different concepts and notions when they described the process of applied problem solving. The process of applied problem solving commences with a real problem situation and through a process of simplification, idealisation and structuring of the situation, the process ends with a real model of the original situation. Through the process of mathematisation, the real model is translated into mathematics. Mathematisation is therefore the process of converting the data, concepts, relations, conditions and assumptions (p. 208) of the real model into a mathematical model of the original situation. Mathematical modelling is the entire process leading from the real problem situation to the mathematical model. Then the mathematical model must be processed to obtain certain mathematical results. This includes mathematical activities such as drawing conclusions, calculating and checking concrete examples, applying known mathematical methods and results as well as developing new ones etc. (p. 208). The next process is to retranslate the results into the real world, i.e. to be interpreted in relation to the original situation (p. 208). The model is then validated and if discrepancies of any kind occur, they may lead to the modification of the model or replacement of the model by going through the process cycle more than once.
Summary
In the light of the above discussion of the different conceptions from the literature, mathematical literacy cannot adequately be described in terms of skills only, as it involves mathematical problems in contexts that require attributes such as conceptual understanding of formal mathematical knowledge and problem-solving skills (Gellert et al., 2001). Gellert et al. also believe that the differences between various conceptions of mathematical literacy consist of the problems to which mathematics is applied (p. 61). Hope (2007) on the other hand believes mathematical literacy is a matter of the appropriate pedagogy that should be used in teaching mathematics. All mathematics learners should therefore be provided with the opportunity to apply their knowledge and logic to real-world situations that form part of their daily lives. Mathematical literacy implies bridging the gap between abstract and applied mathematics where the contexts and degree of complexity differ. Jablonka’s (2003) question of: mathematical literacy for what?, further calls attention to the need for discussing the different contexts in which mathematics could be applied to further explicate the purpose of mathematical literacy.
Some contexts in which mathematical literacy can be applied Prescribing the different contexts in which mathematical literacy can be applied is as complicated as conceptualising mathematical In this section the different categories of contexts are discussed as well as the role technology plays in determining these contexts.
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
