1.1 Background
1.1.1 Statement of the problem
In all educational research, proof has been found to be a notoriously difficult concept for learners to learn (de Villiers, 1998). Almost three decades ago, de Villiers (1990) suggested that learners’
lack of an appreciation of the functions of proof – considered as central in motivating learners to view proof as a meaningful activity – has long been identified as the primary source of their difficulty with proof. Similarly, more than a decade later, an investigation spanning over five
Introduction to the study Background
countries at different levels of schooling, Ball, Hoyles, Jahnke, and Movshovitz-Hadar (2002) found that learners’ difficulty with proof stemmed at least partly from a lack of more refined understanding of the functions of proof in mathematics. The common characteristic of both lamentations is that learning proof without regard to its functions is unfruitful.
Yet scant attention has been given to the extent to which learners appreciated the functions of proof, not even in Euclidean geometry; despite the fact that learning about the functions of proof not only motivates learners to do proof meaningfully, but also helps them to understand how mathematical knowledge develops. Functional understanding of proof is foregrounded by the Specific Aims in the CAPS perhaps on the realisation that other attempts to resolve the problem distorted learners’ understanding of the nature of mathematics. According to the curriculum as described in the CAPS document, learning ‘proofs without a good understanding of why they are important will leave learners ill-equipped to use their knowledge in later life’ (Department of Basic Education [DBE], 2011, p. 8). However, an examination of the CAPS seemed to suggest that little or no instructional time is devoted to functional understanding of proof in mathematics. In this regard, Segal (2000) makes an interesting observation, no less apt today than when it was written:
It is not clear that there ever is a golden age in which the majority of schoolchildren about to enter higher education understood the role of (especially deductive) proofs. (p. 196)
Personal experience gained from teaching and learning mathematics suggests that lack of appreciation of the functions that proof performs in mathematics invokes rote learning as learners see no value in doing proof. Learning proof this way seems to generate in learners negative attitudes towards mathematics. Aaron (2011) cautions us by pointing out that '[a]s long as students believe that mathematical proof is irrelevant they will not move from an empirical view of proof to a more advanced view of proof’ (p. 40). The potential for functional understanding of proof to improve the proving of propositions received a boost by the validation of the LFUP survey instrument designed to measure learners’ functional understanding of proof. An instrument is a tool designed for measuring, observing, or documenting quantitative data (Creswell, 2012).
Introduction to the study Background
This study is important because learning of mathematics cannot be separated from the need to hold informed functional understanding of its intrinsic means for validation, proof (Balacheff, 2010). For instance, learners, like mathematicians, need to gain insight into why a proposition is true. However, if learners are oblivious to the other functions of proof, it is in part not difficult to see why learner performance in Euclidean proof is poor. In addition, without empirical inquiry into learners’ functional understanding of proof, it is difficult to make meaningful recommendations to policymakers and curriculum implementation monitors.
Gaining insights into the character of learners’ functional understanding of proof and the factors contributing to learners’ persistent belief that empirical arguments (proof by cases) are mathematical proof could encourage further studies by mathematics education researchers. In addition, the insights gained may inform the judgements and decisions of policymakers and curriculum monitors interested in better understanding why learners’ performance in Euclidean geometry is poor. More specifically, understanding the impact of collectivist culture provided insights into how future studies may be undertaken to support and thus improve learners’
participation in mathematics generally and performance in Euclidean geometry specifically. My contention is that, to motivate learners to do proof meaningfully, it is necessary to capture their functional understanding of proof. The remedy for this problem lay, at least in part, with instructional practices; assessment of functions of proof to be given prominence as they portray the nature of mathematics and the how mathematical knowledge develops. Altogether, the case advanced in this study reflected a desire to make classroom practice akin to that of mathematicians.
As suggested by Driver, Newton, and Osborne (2000), argumentation theory provides a theoretical basis for developing tools to analyse and improve argumentative discourse, either in speech or in writing. Specifically, the CAPS curriculum emphasised the need for learners to be exposed to mathematical experiences that gave them many opportunities to develop their
Introduction to the study Background
mathematical reasoning8. One way in which learners could meet this need is through argumentation which allows them to externalise their thinking (Erduran, Simon, & Osborne, 2004). However, the potential of Toulmin’s (2003) argument pattern (TAP) as a tool to measure the quality of arguments in the mathematics classroom has been a neglected component of argumentation discourse analysis (Erduran et al., 2004). In addition, to date, I am not aware of empirical investigations involving Toulmin’s model with high school learners from a South African perspective. Apart from attempting to fill this gap, this study contributes to the building of empirical support for the model as it provides quantitative perspectives by measuring the quality of written argumentation in mathematics classrooms.