Having now specified certain theoretical antecedents that have informed the orientation, structure and approach adopted in this study, focus in Part 2 of the study shifts to discussion of the empirical terrain that is the focus of this study – namely, the subject- matter domain of Mathematical Literacy. The structure of legitimate participation in the subject (and the structure of knowledge required to facilitate this legitimate
participation) is of particular concern in this impending discussion. This discussion is precluded by focus on general conceptions of mathematical literacy, numeracy and/or quantitative literacy and the structure of participation and behaviour prioritised in these different conceptions, together with clarification of the criteria according to which participation in the subject is legitimised. A framework to facilitate comparison of different conceptions of mathematically literate, numerate and/or quantitatively literate behaviour is also presented.
PART 1
THEORETICAL (AND OTHER) ANTECEDENTS
INTRODUCTION AND OVERVIEW
The intention of this part of the study is to outline certain theoretical considerations that have informed the development of this study. In particular, the key area of focus of this study: namely, the development of a ‘language of description’ of the structure of knowledge associated with a form of the knowledge domain of mathematical literacy (and associated practices and forms of participation in the South African high school subject-matter domain of Mathematical Literacy) that promotes the development of a life-preparedness orientation. The discussion in this regard moves from a macro to a micro level of analysis. Namely, from consideration of the positioning of the study as a whole within a particular paradigm and educational framework, to a more localised discussion on the contents of the study. This latter level of analysis is achieved through exploration of the components and theoretical grounding of the language of description developed in the study.
This part of the study is divided into four chapters (Chapter 1 to Chapter 4). In Chapter 1 I position the study within an interpretive paradigm in the field of mathematics education, and highlight the ontological, epistemological and methodological positions that stem from this paradigmatic orientation. In Chapter 2 I argue that the key intention of this study is the development of a ‘language of description’ (both internal and external) of the structure of knowledge associated with a particular form of the knowledge domain of mathematical literacy and explore more precisely what is meant by a language of description. In this chapter I also explore various facets requiring consideration with respect to the development of such a language, including identifying the claims and concepts of the language and the mode of theorising adopted in the language. In Chapter 3 I expand on the characterisation of the study as the development of a language of description of a knowledge structure by locating the study in the Field of Knowledge Production within Bernstein’s (2000) pedagogic device, and discuss the limiting perspective offered by the study as a result of this categorisation. In the final chapter (Chapter 4) I conceptualise and clarify the dominant orientation that characterises the language of description for the knowledge domain of mathematical literacy – namely, a
‘life-preparedness orientation’.
This chapter structure and the positioning of this study within the field of educational research are illustrated in Figure 2 below, with the hierarchy of the diagram intended to illustrate the movement from a macro to a micro level of discussion.
Figure 2: Overview and chapter structure of Part 1 of the study PARADIGM
Interpretative Paradigm
Mathematics Education THEORETICAL FIELD
THEORETICAL ORIENTATION
Sociology
RESEARCH FOCUS/INTENTION
Field of knowledge production Qualitative Study RESEARCH TYPE
Development of a language of description of the structure of knowledge characterised by a specific
relationship between mathematics and real-world contexts
RESEARCH LOCATION
CHAPTER 1
CHAPTER 2
CHAPTER 3
Language of description characterised by a
‘life-preparedness orientation’
CHAPTER 4
CHAPTER 1
RESEARCH METHODS, PARADIGMS AND MATHEMATICS EDUCATION THEORIES
In this chapter I position this study within an interpretive paradigmatic orientation and outline the ontological, epistemological and methodological positions that characterise this paradigmatic orientation. I argue further that this study is driven by a sociological impetus and is characterised by a qualitative research process that is located in the field of mathematics education.
1.1 Positioning this study in an interpretive paradigm:
considerations of ontological, epistemological and methodological orientations
A paradigm may be viewed as a set of basic beliefs (or metaphysics) that deals with ultimates or first principles. It represents a worldview that defines, for its holder, the nature of the "world," the individual's place in it, and the range of possible relationships to that world and its parts … (Guba & Lincoln, 1994, p. 107) (emphases in original) Drawing from the above, a paradigm is a particular way of looking at the world, informed by the values and assumptions of the researcher, which directs the focus and form of the intellectual structure on which the research process is based (Kuhn, 1962).
The particular paradigm in which a researcher is positioned, then, affects not only what is perceived to fall “within and outside the limits of legitimate inquiry” (Guba &
Lincoln, 1994, p. 108). Instead, what is also affected is how the researcher views the world and their own role or subjectivity in the research process, the kinds of questions that are asked with respect to a particular inquiry and how these questions are to be scrutinized, and the type of methodology that is chosen by the researcher to gather information relating to these questions.
Guba and Lincoln (1994, p. 108) argue that the basic beliefs that define the dominant view or perspective in a paradigm are encapsulated through the responses that would be given by proponents of a particular paradigm to three fundamental questions. Firstly, the ontological question – “What is the form and nature of reality and, therefore, what is there that can be known about it?” Secondly, the epistemological question – “What is the nature of the relationship between knower or would-be-knower and what can be known?”: namely, how does the researcher perceive knowledge is produced, acquired and communicated (Scotland, 2012, p. 9) and their influence over the knowledge being collected or uncovered, or do they perceive that they are external to it and/or do not influence it (University of South Hampton, 2011). And thirdly, the methodological question – “How can the enquirer (would-be-knower) go about finding out whatever he or she believes can be known?” (All quotations above: Guba & Lincoln, 1994, p. 108).
In short, the questions address the issue of what – in the eyes of the researcher - constitutes reality, how does the researcher position themselves with respect to this reality, and how will the researcher go about finding out something about the reality?
These three questions provide a “holistic view of how we view knowledge, how we see ourselves in relation to this knowledge and the methodological strategies we use to un/discover it.” (University of South Hampton, 2011) – in other words, these three questions provide a view or perspective of the researcher on “what is”, “what it means
to know”, and the “why, what, from where, when and how” of the data collection process (Scotland, 2012, p. 9). Collectively, then, these three questions provide not only a point of analysis for individual paradigms but also points of comparison between different paradigms.
The specific paradigm that is reflected in this study, and which represents my own world view on the topic under scrutiny in this study, is that of the Interpretive (or Social Constructivism) paradigm. From this paradigmatic orientation, the dominant ontological position is that of relativism (Scotland, 2012, p. 11): namely, ‘reality’ is viewed as a social construction such that different cultures are seen to make sense of the world differently as influenced by the specific social, economic, political and cultural environment and issues which define and affect participation in the culture at particular points in time. As such, from this ontological perspective there is no single or accurate reality. Rather, an experience of reality is dependent on the culture and context in which a person finds themselves at a particular point in time – reality is “socially and experientially based”
(Guba & Lincoln, 1994, p. 110); and this reality shifts over time, between cultures, and even within a culture between different groups (McKee, 2003, pp. 9-10). In other words, reality is subjective, is individually constructed, and there are multiple possible realities for differing individuals, groups, and cultures (Scotland, 2012, p. 11). This viewpoint is supported by constructivist and sociological perspectives that promote the notion of the social construction of knowledge and of an individual’s relation to that knowledge (McKee, 2003, pp. 9-10).
This ontological perspective of multiple realities gives rise to an epistemological position that views the potential for the existence of ‘multiple knowledges’ (Guba &
Lincoln, 1994, p. 113). Accordingly, the world is seen to not exist independently of our descriptions of it: “Meaning is not discovered; it is constructed through the interaction between consciousness and the world.” (Scotland, 2012, p. 11). Subjectivity is, thus, a key component of the interpretative epistemology, characterised by a research process involving constant interaction between the researcher and the object of investigation (Guba & Lincoln, 1994, p. 111). Furthermore, this position is accompanied with explicit recognition of the influence of the researcher’s own values and social, economic,
cultural and political orientations on choices and decisions made during the research process and, hence, on the intention, structure and outcome of this process (Scotland, 2012, p. 12). Given this adherence to relativism and a perspective on the potential for multiple knowledges, the primary aim of the interpretivist research process is
understanding of the ways in which different individuals or groups of people perceive their reality and their place in that reality (Guba & Lincoln, 1994, p. 113). Importantly, this intention for understanding can also be accompanied by intentions for “Advocacy and activism” (Guba & Lincoln, 1994, p. 113). In other words, interpretative research need not be exclusively descriptive and can, instead, be accompanied by critical components for change. This issue is particularly pertinent to the sociological impetus of this study (see page 15 below) in which a key intention in the study is the
modification of the existing structure of knowledge in the subject Mathematical
Literacy to overcome perceived structures of educational disadvantage afforded through participation in the subject.
Methodologically, Interpretivism promotes the use of research methods that facilitate investigation, analysis, interpretation and understanding of how different people and groups perceive reality and their place in that reality, together with social and cultural conditions that influence perceptions of reality (Scotland, 2012, p. 12). Interpretive methodology also promotes the use of methods that facilitate interpretation of varied and
multiple representations of reality from an individual’s perspective through direct interaction between researcher and the object of investigation (Guba & Lincoln, 1994, p.
111). It is as a result of this positioning of the researcher as an active transactional participant in the construction and understanding of meaning in socially constructed representations of reality that lead Guba and Lincoln (1994, p. 111) to categorise Interpretivist methodology as “Hermeneutical and dialectical”. In other words, understanding of how others associate meaning to their own representations of reality is able to be deemed through derivation of expressed and/or hidden meaning in texts (i.e.
hermeneutics) or through dialogue between researcher and the objects of the research process.
Having outlined the general characteristics of the ontological, epistemological and methodological positions associated with the interpretative paradigm, it now becomes possible to reflect on the significance of these positions with respect to the commitment in this study to an interpretative paradigmatic orientation. Ontologically, the research process in this study is driven by an intention to interpret, analyse and understand the dominant perception of the ‘reality’ of the structure of knowledge and participation in the subject-matter domain of Mathematical Literacy. This commitment is further
characterised by an intention to understand how this reality is differently experienced by various role-players who participate in the subject (e.g. teachers vs. curriculum
developers vs. examiners of national exam papers vs. textbook authors vs. teacher trainers). Understanding of how this reality is different to the perceived reality in other subjects (such as Core [scientific] Mathematics) is also in focus. This ontological position recognises that these multiple representations of reality of knowledge and participation for the subject are directly influenced by various political, social,
economic and cultural factors, and, particularly, by pressure within the South African education system to prioritise mathematical knowledge as a preferenced form or structure of knowledge. However, in interpreting existing perceptions of reality for the subject Mathematical Literacy, there is also an impetus for ‘activism’ (Guba & Lincoln, 1994, p. 113). Namely, my own position is that the existing reality – which I claim is dominated by a prioritising of mathematical knowledge structures as the basis for legitimate participation in the subject – supports the promotion of Myths of Reference, Participation and Emancipation (Dowling, 1998)2. Consequently, a degree of
educational disadvantage with respect to future career, work and social advancement opportunity is sustained and reproduced through participation in the subject. As such, a key intention of this study is for the development of an alternative ‘reality’ of the structure of knowledge and participation for the subject-matter domain of Mathematical Literacy in which a life-preparedness orientation is prioritised over the development of mathematical knowledge. This alternative reality is theorised and discussed in detail in Part 4 of the study where I present an alternative ‘language of description’ for the knowledge domain of mathematical literacy.
Drawing from an epistemological position that views the potential for the existence of
‘multiple knowledges’, the research process in this study acknowledges varying possible interpretations and descriptions of the ways in which role-players in the subject Mathematical Literacy (as the object of study) perceive the structure of legitimate participation in the subject. The particular interpretation adopted in this study, as informed by my historical, social, economic, political and theoretical orientation, construes existing structures of participation as problematic in relation to issues of educational, social and economic positioning and access. As such, my interpretation leads
2 Refer to Part 3, Chapter 10 and sub-section 10.2.2 starting on page 156 below for a detailed discussion of these concepts.
me to question current pedagogic and assessment priorities in the subject and to present an alternative understanding of a structure of participation which aims to negate existing forms of positioning. It is this interpretation which has directly informed the structure of the developed alternative internal language of description for the knowledge domain of mathematical literacy (see Part 4 starting on page 181 below) and the promotion of a ‘life- preparedness’ (Venkat, 2010) orientation3 for the subject over the prioritising of mathematising processes in this alternative language of description. This is evidenced through the composition of the language of description of knowledge domains of practice that prioritise modelling processes (the ‘Modelling’ domain of practice) and understanding of patterns and structures of everyday practices (the ‘Everyday’ domain of practice) alongside mathematical structures (the ‘Mathematical Competency’ domain of practice). This interpretation of the structure of participation for the subject has also directly informed the structure of the external language of description developed to facilitate the means through which the internal language (and characteristics of the associated domains of practice) can be used as a lens for analysis of empirical resources relating to the subject (c.f. Part 5 starting on page 253). This external language of description focuses on identification and interpretation of the discursive resources (signifiers [words/vocabulary and visual mediators], routines and narratives) embedded and indexed in discursive practices in the subject, and is accompanied by recognition of the highly interpretive and subjective nature of this process of engaging with discursive resources and of any assumptions or conclusions made as a result of this process. My commitment to an interpretivist paradigmatic orientation has, thus, influenced the deliberately construed and highly interpretive description of the structure of knowledge and participation in the subject-matter domain of Mathematical Literacy adopted in this study, and also the “Transactional and subjectivist” (Guba & Lincoln, 1994, p. 111) role of me as the researcher in this process.
Methodologically, the ontological and epistemological orientations adopted in this study necessitate a specific methodology. In particular, this methodology must facilitate interpretation and analysis of how the practices of participants in the subject either reflect coherence or divergence with the alternative conception of knowledge and participation developed for the subject-matter domain of Mathematical Literacy
(presented through the internal language of description and operationalised through the external language of description). This methodological intention is accompanied by a descriptive rather than normative approach in this sense that there is recognition that the structure of knowledge and participation as presented in the internal language of
description for the domain of mathematical literacy presented a particular, a limited, and only one of many possible ‘world-views’ of the structure of knowledge and
participation for the subject. The specific site of empirical analysis involves textual resources (i.e. a textbook section, a section of the curriculum, Grade 12 national examinations, and course notes for a teacher education course) that reflect different aspects of knowledge and participation in relation to the subject-matter domain of Mathematical Literacy. As such, the methodology selected and employed to facilitate analysis of these empirical resources in a way that is consistent with interpretivist ontological and epistemological orientations is that of textual analysis, as a sub-set of a wider methodology of discourse analysis (c.f. Part 6 starting on page 360). This method is drawn from the realm of hermeneutics (which is identified by both Guba and Lincoln (1994, p. 111) and Scotland (2012, p. 12) as reflecting interpretivist intentions).
Hermeneutics involves analysis of the discursive resources (signifiers
[words/vocabulary and visual mediators], routines and endorsed narratives) embedded
3 Refer to Chapter 4 on page 26 below for an elaborated discussion of the notion and components of a life-preparedness orientation.
in (and associated discourse communicated through) textual resources for the subject.
This process also involves comparison of these discursive resources to the
characteristics of the discursive resources of the domains of practice that constitute the internal language of description for the knowledge domain of mathematical literacy.
This methodology provides the means for identification and interpretation of possible
‘world-views’ of the authors of these texts and, specifically, of how these authors perceive the structure of legitimate knowledge and the criteria for legitimate
participation in practices involving the use of mathematics in contextualised problem situations. And since this methodology recognises the interpretative, subjective and transactional nature of the investigative research process, the methodology is entirely consistent with interpretivist ontological and epistemological orientations.
Ontologically, epistemologically and methodologically, then, this study is driven by an interpretivist paradigmatic orientation, albeit with an impetus for activism directed towards understanding how knowledge and practices associated with the subject-matter domain of Mathematical Literacy can be changed to facilitate that participation in the subject does not reinforce social and/or educational disadvantage and positioning.
1.2 Positioning this study as a form of qualitative research
Despite many proposed differences between quantitative and qualitative epistemologies, ultimately, the heart of the quantitative-qualitative “debate” is philosophical, not methodological. (Krauss, 2005, p. 759)
As suggested by the above quotation, although quantitative and qualitative studies are often differentiated according to the type of methodology employed in gathering and analysing information about an aspect of reality, these differences in methodology are the result of a deeper and, ultimately, epistemological difference between the nature of quantitative and qualitative research.
For Krauss (2005, p. 759), “Epistemology poses the following questions: What is the relationship between the knower and what is known? How do we know what we know?
What counts as knowledge?” The difference between quantitative and qualitative research is then defined in large part by the differential answers to each of these questions for each type of research. For quantitative research, the object of study is viewed as independent of the researcher and, so, it is possible to use a quantitative analysis of the object, deemed through observation rather than direct interaction, and to make connections across multiple different realities. Qualitative research, on the other hand, deems that knowledge is directly attached to specific instances or phenomenon and that it is not possible to separate meaning from this instance. As such, it is only through direct interaction with the phenomenon and the object(s) or subject(s) under investigation that a more complete view of the phenomenon is gained. From this perspective, an exclusively numerically based analysis of a situation that does not taken into account the specifics of the individual object(s) or subject(s) in the phenomenon is incapable of providing adequate interpretation or description of the situation. Furthermore, given the distinctive nature of each phenomenon and the need for direct intervention in the phenomenon, it is not possible for a single tool to facilitate connections across multiple realities since every reality is unique and different (Krauss, 2005, pp. 759-760).
In the previous section it has already been established that this study falls within an interpretivist paradigm, hereby acknowledging the role of the researcher in shaping the