I begin this section by touching on two prominent theories on argumentation. Perelman and Olbrechts-Tyteca (1982) and Toulmin (2003) are the most influential theorists on argumentation.
Perelman Olbrechts-Tyteca (1982) tries to find a description of techniques of argumentation used by people to obtain the approval of others for their opinions. Toulmin (2003), the other influential
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writer, developed his theory (starting in 1950’s) in order to explain how argumentation occurs in the natural process of an everyday argument. He calls his theory “the uses of argument”. Ribeiro (2012) points out that Toulmin’s model focuses precisely on studying the structure of arguments.
In contrast, Perelman and Olbrechts-Tyteca’s (1982) model does not seem to give rise to a structure that demonstrates how the components of an argument are related; in fact, they see argumentation as a process opposed to mathematical proof. Of course, as discussed earlier, I see proof and argumentation as inseparable.
As Aberdein (2005) points out, Toulmin’s The Uses of Argument is arguably the single most influential work in modern argumentation theory. Toulmin’s (2003) model explains how the six components of an argument link and also how the argument structure can be employed to analyse arguments. For these reasons, Toulmin’s scheme was useful in determining learners’
competence in generating arguments to support claims in this study. Further, Toulmin’s (2003) theory focuses on argumentation wherein the conclusion, belief or claim is produced by reasoning (justification) as the starting point for the construction of arguments. Hence, its account of argumentation has been found to be insightful on the basis that he focuses on the rhetoric of mathematical practice, arguments (Shapin, 2002). In addition, Toulmin’s (2003) layout is intended to encompass all forms of argument, mathematics included (Aberdein, 2005). Taking this brief analysis into account, a “Why?” question calls upon the interlocutor to justify their position which in turn transforms a mere statement into an argument.
Using Toulmin’ (2003) argument structure (TAP), Pedemonte (2007) not only describes a proof through argumentation, she also shows that argumentation is useful in the production of a conjecture. Hence, Lakatos (1991) views proof and conjecturing as inseparable. I could only concur with him for the simple reason that conjecturing is an activity undertaken to arrive at a mathematical proof whose validity would eventually be subjected to scrutiny through engaging in the process of argumentation. More specifically, following an analysis of mathematical proofs, Aberdein (2012) concludes that proofs consists of a number of argument structures rather than a single argument structure. In this study, I only focused on requiring participants to make single
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arguments which by definition, only comprised a claim, data, warrant and a rebuttal, for the purpose of gaining insight into learners’ ability to construct an argument.
According to Mariotti (2006) the learner must make sense of this difference between argumentation and proof, without rejecting one for the other. In her characterisation of proof and argumentation, like Pedemonte (2007), she argues that proof is a special case of argumentation, and I agree. She further points out that argumentation, the process of supporting the truth of a particular proposition, introduces learners to the practices of the mathematics discipline.
Some research studies showed that cognitive unity exists between the construction of a conjecture and the construction of a mathematical proof (Boero et al., 1996). In addition, recent researchers suggests that the major goal of teaching mathematics is to develop learners’ abilities to establish and defend their own positions while respecting the positions of others (Idris, 2006).
Understanding of the functions of proof were found to correlate with the ability to engage in argumentation or proof construction task (Clark & Sampson, 2008; Conner, 2007; Hanna, 2000).
For this study, argumentation reflected the communication function of proof in mathematics.
Although current research in mathematics education does not offer much insight into the relationship between proof and argumentation, both processes are characterised by being conducted when someone wants to convince (oneself or others) about the truth of a proposition (Pedemonte, 2007). Thus, I viewed proving as an activity that begins with the construction of an argument which is accomplished through argumentation. In this case, Toulmin’s model is seen as a powerful tool to characterise the two types of arguments discussed in this study: empirical (informal) and deductive (formal) arguments.
In mathematics education, Krummheuer (1995) started the trend of using Toulmin’s scheme of conclusion, data, warrant, and backing by analysing and documenting how learning progresses in a classroom. However, he employed a reduced version of the original scheme, omitting the use of the rebuttals and qualifiers and his study focused on primary mathematics (grade 2). Pedemonte (2007) investigated the structural differences between proof and argumentation. The study took place when 102 high school learners in France and Italy began to
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learn proof. The learners had prior experience of proof as a means to systematise and knew the theorems necessary to solve the proposed problems. She found that open problems which ask for a conjecture appear to be extremely effective for introducing the learning of proof and that argumentation activities seem to favour the construction of a proofs.
In a comparative study on proving processes in French and German lessons on the theorem of Pythagorean, carried out by Knipping (2003), it was found that proving discourses allow for reflection on underlying functions of proving in class. Conner (2008) study examined the argumentation in one preservice teacher’s high school geometry classes and suggested a possible relationship between the observed argumentation and the preservice teacher’s understanding of proof. She conducted two semi-structured to infer the teacher’s understanding of the concept of proof from her responses. Using Toulmin’s scheme, she found that there were difference in the order in which components of an argument were presented. She also found that the teacher’s understanding of the functions of proof—as, for example, a means to explain why a statement were true—influenced the support for argumentation in her classroom.