The positionality that researchers bring to their work, and the personal experiences through which that positionality is shaped, may influence what researchers bring to research encounters, their choice of processes, and their interpretation of outcomes. (Foote & Bartell, 2011, p. 46)
As is the case with all researchers, my life experiences informed various aspects of this study. In this subsection I provided insights into the paradigm (worldview), hypotheses derived from my experiences with the concept of proof. All these aspects influenced the research process (for example, research questions, sample, methods, interpretations, and so on). For instance, though some township schools tend to achieve 100% pass rate in their Grade 12 examinations, the quality of these passes tend to be weaker than those in previously white schools with similar pass rate.
Hence I chose to compare the quality of learners’ functional understanding of proof and argumentation between fee-paying as well as no-fee high schools with a focus on mathematics and science (that is, Dinaledi schools).
Introduction to the study Researcher positionality
In this study, the pragmatic paradigm was suitable to serve as a framework with which to describe how this study unfolded. I adopted this paradigm informed by my belief in using “what works” to find answers to research questions. Pragmatism is defined as a paradigm that encompasses both quantitative and qualitative research methods (Johnson & Onwuegbuzie, 2004).
However, Creswell and Plano-Clark (2011) use the term “worldview” in this regard. Teddlie and Tashakkori (2010) define a paradigm as ‘a worldview together with the philosophical assumptions associated with that point of view’ (p. 84). Further, for Teddlie and Tashakkori (2010), pragmatism means typically a worldview associated with mixed methods research as it embraces features associated with both postpositivism and constructivism worldviews and rejects ‘the dogmatic either-or choice between constructivism and postpositivism and the search for practical answers to questions that intrigue the investigator’ (p. 86).
For the pragmatic paradigm, the view is that the research problem, rather than loyalty to any research paradigm, determine the data collection and analysis methods that are most likely to provide answers to the research questions. In a nutshell, pragmatism as a paradigmatic framework directed the research efforts in this project. In this respect, the use of mixed methods design was not merely a matter of combining qualitative and quantitative methodologies together, but arose from the need for pragmatic response to the research questions at hand. Put another way, my choice of research questions, data collection and analysis methods, and interpretation of findings reflected the underlying pragmatic view of the world.
Consistent with this paradigm, a mixed methods sequential explanatory research design was employed in this study. That is, a quantitative method, which took priority in this study, was followed by a qualitative method as a means to understand why Presh N tended to hold informed beliefs about the functions of proof. I agree with Clough and Nutbrown (2012) when they make this observation:
Since research is carried out by people, it is inevitable that the standpoint of the researcher is a fundamental platform on which enquiry is developed. All social science research is saturated (however disguised) with positionality. (p. 10)
Introduction to the study Researcher positionality
Given the philosophical differences in the structure and knowledge confirmation between quantitative and qualitative approaches (Foss & Ellefsen, 2002), in the next section I provide a personal background for the reader to understand how my experiences with the mathematics discipline might have influenced the results in the qualitative segment of this study. Put another way, it is necessary to disclose to my background to the reader to facilitate their evaluation of the findings of this study.
1.7.1 Early beginnings
I was born in Esilobela township, Carolina, in the Mpumalanga province, but spent my childhood in the Dundonald village situated a few kilometres from the Eswatini border. By township is meant a historically disadvantaged area characterised by, for example, poverty, high crime, antisocial behaviour, shortage of classroom resources that facilitate learning of mathematics, for example, dynamic geometry software (DGS)11, recreational facilities, and community libraries. After finishing high school in the mid-eighties I received a bursary to study for an integrated teachers’
degree with a pure mathematics major and was the first member of my family to attend university.
My parents (mother: self-employed; father has since passed on: underground mine worker) separated when I was very young.
Together with my stepbrother and mother who sold mostly second hand clothing items and worked the soil, we lived in a mud house. Compared to my privileged white counterparts, our house had no electricity nor flushing toilets. We relied on public transport as we had no car and attended under-resourced primary and high schools; reflecting realities of apartheid South Africa characterised by an inequitable social and political system. In general, I come from a less privileged background which informs my strong commitment to social justice.
11 The phrase “dynamic geometry” was originally invented and trademarked by publishers, Key Curriculum Press, to describe the Geometer’s Sketchpad (Jackiw, 2001).
Introduction to the study Organisation of the thesis
1.7.2 Interest in mathematics
I was particularly fascinated by mathematics and enjoyed some admiration from friends who I assisted with homework and preparation for examinations. Hence I chose to major in mathematics for my undergraduate degree and particularly enjoyed teaching Euclidean geometry at numerous high schools in the early nineties.
I returned to university in 2010 and obtained an MSc in science education three years later and began teaching Physical Sciences in high schools. Currently, I am a mathematics education lecturer at a public university in South Africa and a proponent and advocate of assuring redress of the past imbalances in the allocation of resources that facilitate mathematics learning and teaching.
This fascination with proof may have influenced my thinking about the functions of proof in mathematics as well as the interpretation of the qualitative data with undue bias.
1.8 Organisation of the thesis
In the process of doing this study, I submitted three manuscripts, which are in part based on this thesis to peer-reviewed journals. This was done for two reasons. One was to ensure dissemination of empirical research results arising from this study. Two was to meet the requirement of the School of Humanities at the University of KwaZulu-Natal which stipulated that submission of this thesis must be accompanied by at least one published journal article. The next subsection provides a summary of the focus of the chapters which constitute this thesis.
1.8.1 Chapter 1: Introduction to the study
This introductory chapter broadly encapsulates the notions of functional understanding of proof and argumentation within the context of the South African high school geometry curriculum against the backdrop of reported poor performance of learners in relation to proof. In addition, the discrepancy between actual classroom practice and professed SA in CAPS is underscored. The significance of the study is described within these contexts. The chapter also provides the aims and the resulting research questions that underpin the study. The theoretical bases (van Hiele’s and
Introduction to the study Organisation of the thesis
Toulmin’s theories) of this study are then contextualised after which an overview of the research design is explicated. Finally, the delimitations of the study and my positionality as a researcher are described.
1.8.2 Chapter 2: The review of literature
This chapter critically examines literature on the concepts underlying this study and reports on the results of research pertinent to this study: proof functions in mathematics, argumentation, and the factors influencing informed beliefs about the functions proof. The purpose of critically examining and reporting on previous studies is to build the foundation for the present study and thus connect its problem, purpose, and discussions to previous studies. This chapter explores and discusses major concepts and ideas providing conceptual frameworks some of which are permeated by historical and philosophical analyses. The major terms that built the conceptual framework for data analysis purposes include; understanding what mathematics is; functions of proof; mathematical understanding; and, argumentation. The measures for assessing functional understanding of proof – based on previous literature on learner difficulties with proof and its functions, and my own classroom experiences – are discussed.
1.8.3 Chapter 3: The theoretical framework
This chapter presents a brief description of the historical development of the two theories underpinning this study through which data analyses and interpretation of results were undertaken.
One is the van Hiele (1986) theory of geometric thinking whose central idea is that learning geometry takes place in discrete levels of thinking and that progress to the next level is a function of instruction. The theory has played a major role in understanding learners’ difficulty with geometry. De Villiers’ (1990) model provides the concepts for investigating learners’ functional understanding of proof in mathematics. Two is Toulmin’s (2003) argument pattern (TAP) scheme which was developed for the purpose of explaining how argumentation takes place in the natural contexts of everyday life, especially in law. He suggests that arguments can be understood using six components comprising: claims, data, warrants, backings, qualifiers, and rebuttals. In talking
Introduction to the study Organisation of the thesis
about learning, I draw on the sociocultural theory on the basis that I view mathematics as a human activity in which all learners can participate. Having interrogated these two theories, I construct a conceptual framework to understand the relationships among the concepts.
1.8.4 Chapter 4: Research methodology
First, an overview of methodology and instruments utilised in previous studies on learners’
functional understanding of proof is undertaken. A methodological framework that graphically describes the research design is provided. Next, I argue why it is helpful to make a distinction between methods and methodology, terms often treated as synonyms. Then, I provide the rationale for using a mixed-methods sequential explanatory design. Next, instrumentation, data collection measures and procedures as well as analysis procedures are described and justified. Final, issues of rigour are discussed. For the quantitative phase, the data collection instruments, for example, the LFUP and the Argumentation Framework in Euclidean Geometry (AFEG) questionnaires employed to capture and characterise learners’ functional understanding of proof and their argumentation ability are also discussed.
Given that the LFUP instrument was already established, its reliability and validity evidence is stipulated. For the qualitative phase, sample task-based questions from the Interview Schedule which were meant to elicit Presh N’s beliefs influencing her understanding of the functions of proof in mathematics are provided. The results (analysis, interpretation, and discussed) of this study are presented in independent chapters (5, 6, and 7) using the research questions as an organizing framework. The areas in which the methods were combined are identified and justified. Then, rigour and limitations of the design is discussed.
1.8.5 Chapter 5: Functional understanding of proof in mathematics
The LFUP questionnaire results are analysed (presented and interpreted) to answer the first research question, What functional understanding of proof do Grade 11 learners hold? This question sought to understand whether learners held naïve (empiricist), hybrid or informed
Introduction to the study Organisation of the thesis
understanding of the functions of proof in mathematics (verification, explanation, communication, discovery, and systematisation). The SPSS v.24 (2017) software is used to analyse the data. I use descriptive statistics to report on patterns in participants’ responses and multivariate techniques to identify factors accounting for variability in LFUP scores and how these vary across participating schools. The results show that learners’ functional understanding of proof are of a hybrid nature and inconsistent with those espoused in CAPS and held by contemporary mathematicians.
1.8.6 Chapter 6: The relationship between functional understanding of proof and argumentation ability
This chapter describes the results of the nature of the relationship between learners’ functional understanding of proof and their ability to argue, to answer the second quantitative research question; How is the relationship between learners’ functional understanding and argumentation ability? To this end, statistical techniques are used to describe this relationship. Specifically, SPSS v.24 (2017) is used to analyse the data. The results show that a weak, positive and significant correlation exists between the two constructs. Using Toulmin’s theory, the results show that learners’ argumentation ability was poor. In addition, multiple regression analysis indicates that the verification function accounts for the largest variability in learners’ functional understanding of proof.
1.8.7 Chapter 7: Beliefs about the functions of proof: The case of Presh N
The purpose of this chapter is to answer the third and third research question, Why does Presh N hold informed beliefs about the functions of proof? The participant (Presh N) is purposively sampled which means that she is selected on the basis that she is an information-rich individual for the most effective use of resources (Patton, 2002). The van Hiele theory is used to examine the findings after the results were analysed with the aid of ATLAS.ti and STATA, a framework for understanding the factors (deductive arguments, semantic contamination, collectivist culture, empirical arguments, teacher, and textbook) influencing understanding of the functions of proof is suggested.
Introduction to the study Organisation of the thesis
1.8.8 Chapter 8: Exploring the interaction among the three constructs
The purpose of this chapter is to answer the fourth and final research question, “What is the nature of the interaction among the three constructs (that is, functional understanding of proof, argumentation ability, and factors influencing functional understanding?” This chapter integrates the results of the quantitative and qualitative phases to discuss the outcomes of the entire study.
As indicated at the beginning of this study, both quantitative and qualitative research questions were posed to better understand Grade 11 learners’ functional understanding of proof, their argumentation ability, and the factors affecting functional understanding of proof in mathematics.
This chapter combines the results from both phases of the study to develop a more robust and meaningful picture of the research problem.
1.8.9 Conclusions
This final chapter concludes the study. In the discussion section, both quantitative and qualitative results were described simultaneously, having kept them independent in previous sections. The two phases are independent in that quantitative and qualitative research questions, data collection, and data analysis are separated for each phase. I take into account results of past empirical investigations in literature concerning learners' functional understanding of proof, argumentation ability, and reasons behind Presh N’s informed beliefs about the functions of proof. In the conclusions sections, I consider the overall investigation including the three unique contributions this study makes. One is that this study used a mixed methods design in which participating schools were randomly selected to improve the trustworthiness of the results. Two is that I validated a new measurement scale (LFUP) that allows teachers to gain insights into their learners’ understanding of the functions of proof and thus tailor instruction on the meaningful construction of proof. Three is that factors influencing beliefs about the functions of proof were investigated culminating in a suggested model for describing learners’ understanding of the functions of proof. In the conclusions section, I provide an overview of the study in relation to the research questions, describe findings and their implications, make recommendations and suggestions that other
Introduction to the study Chapter summary
researchers can consider, acknowledge several limitations, and reflect on the research project as a whole.