3.5.1 Conceptual framework in a theoretical framework
In this study, part of the critical engagement of previous literature on functions of proof and the concept of argumentation was meant to explore a theoretical framework (the two theories and various concepts) in order to develop a conceptual framework. In deciding to present both the theoretical and conceptual frameworks for this study, I was guided by Kumar’s (2005) view that a conceptual framework stems from a theoretical framework and concentrated, usually, on one aspect of that theoretical framework which formed the basis of a research problem. However, according to Miles and Huberman (1994), it seems that no consensus exists in literature on the difference between a theoretical framework and a conceptual framework (for example, Leshem &
Trafford, 2007; MacMillan & Schumacher, 2010; Maxwell, 2013; Miles & Huberman, 1994;
Sinclair, 2007).
The lack of consensus notwithstanding, Rocco and Plakhotnik (2009) makes a distinction between a theoretical framework and conceptual framework and argues that using these terms interchangeably in research causes confusion. Imenda (2014) and Miles and Huberman (1994) support this stance and venture to make a distinction between these two constructs. Imenda (2014) defines a theoretical framework as ‘the application of a theory, or a set of concepts drawn from one and the same theory, to offer an explanation of an event, or shed some light on a particular phenomenon or research problem’ (p. 189). Miles and Huberman define a conceptual framework as a visual or written product, one that ‘explains, either graphically or in narrative form, the main things to be studied—the key factors, concepts, or variables – and the presumed relationships among them’ (p. 18). Kitchel and Ball (2014) go further to point out that though a conceptual
The theoretical frameworks The conceptual framework for this study
framework or model indicates that a relationship exists, it lacks the rationale behind the relationship. In this study, whereas a theoretical framework is overarching and has its genesis in theories and constructs that have already been tested and thus generally accepted in literature, a conceptual framework emerges from a theoretical framework and is more specific in identifying relationships among concepts from the theories already identified in this study.
However, in view of the design of this study, there is a potential stumbling block suggested by Ngulube, Mathipa, and Gumbo (2014). These researchers point out that the use of research frameworks is not yet fully developed in mixed methods studies. In contrast, Grant and Osanloo (2014) believe that both theoretical and conceptual frameworks can be used in mixed methods designs. Taking the latter advice, I went ahead and constructed a conceptual framework on the basis of two theoretical frameworks. These frameworks are manifested to some degree in the methodology, arguments about what might happen, research questions, data collection and analysis, and synthesis of the findings (Bernard, 2013; Silverman, 2013; Royse, 2008).
3.5.2 Connecting theories, research problem and questions
Whereas a theoretical framework was used to ground the study in the van Hiele and Toulmin’s theories, the purpose of constructing a conceptual framework was to diagrammatically connect the relevant concepts guiding this study in the theories and constructs underpinning this study. It is worth noting that this conceptual framework which situates the study in relevant literature and illustrates the network of relationships among the concepts with a figure, was constructed rather than found ready-made in literature waiting to be utilised. I schematically present the multidimensional conceptual framework specific to this study in Figure 3—9. For example, the thicker arrows represent the notion that the collectivist culture is hypothesised as stronger than the textbook or teacher factors at accounting for the reasons why the learner held the beliefs she held about the functions of proof. In addition, the conceptual framework addressed both the qualitative and quantitative strands of this study and suggested interactions and relationships among the variables embedded in the problem investigated in this study.
The theoretical frameworks The conceptual framework for this study
Figure 3—9. The multidimensional framework of this study
According to Silverman (2013), concepts are essential in a research problem and need to be described as clearly specified ideas deriving from a particular theory. Thus, this section focuses on demonstrating the connection among the theories and constructs underpinning this study and their role in contributing to finding answers to the three problems highlighted in this study. The first problem relates to the contention that learners encounter difficulties with proof, yet very little insight on their functional understanding of proof has been provided despite acknowledgement in literature that this understanding makes the learning of proof a meaningful activity. The second problem is that very little is known about whether an association between learners’ argumentation quality and functional understanding of proof despite the recognition that argumentation is embedded in proof. Third and final is that while knowing learners’ functional understanding of proof is important, more important is knowing the factors that either promote or inhibit the acquisition of informed functional understanding of proof.
Guided by the theoretical and empirical literature previously reviewed and the presented multidimensional framework of the study, the following main research question has been
Teacher
Empirical arguments Semantic
contamination
Textbook Collectivi
st culture
Functional understanding
of proof Explanation Verification
Systematisation
Communication Discovery
Argumentation
Grounds
Rebuttals Claim
Empirical arguments
The theoretical frameworks The conceptual framework for this study
established to frame the design of the study, data collection, and data analyses: “How can meaningful learning of proof construction take place?” This question was broken into four sub- questions:
3.5.3 Quantitative phase questions
3.5.3.1 What functional understanding of proof do Grade 11 learners hold?
3.5.3.2 How is the relationship (if any) between learners’ quality of argumentation and their functional understanding of proof?
3.5.4 Qualitative phase questions
3.5.4.1 Why does Presh N hold informed beliefs about the functions of proof?
3.5.4.2 How is the interaction among the three constructs (that is, functional understanding of proof, argumentation ability, and factors influencing functional understanding?
For the purpose of this study, factors such as two-column proof, ability, proof-type, language, teaching methods, and DGS were not part of the investigation; some required examination of learners’ written work and others required observation of experimentation and conjecturing, aspects that were beyond the scope of this study. Thus, I would have been unreasonable to posit that current instructional practices were not advocating the aims in CAPS without empirical evidence collected from classroom observations. That said, data were collected and analysed and inconsistencies were explained in light of the theoretical framework.
This study was guided by the view that a theoretical framework dwells on established and tested theories that underpin the findings of numerous investigations on how variables in a phenomenon are interrelated while a conceptual framework is viewed a model that indicate or describe the relationships among specific variables identified in the study. One way of making sense of this differences is to consider the scale on which the frameworks differed in this study.
By way of example, the broader framework that provided direction is the van Hiele theory, learners at high schools were expected to understand the functions of proof in mathematics. In brief, theoretical perspectives of the theory informed the conceptual framework. Therefore, the
The theoretical frameworks Chapter summary
conceptual framework contains my idea on how the research problem identified in the theoretical framework was explored. This is what differentiates a conceptual framework from a theoretical framework.