The review of literature The notion of argumentation
include a consideration of proof in each of these five functions it performs in mathematics (Knuth, 2002).
This probably prompted Balacheff (1991) to suggest that there is a long distance between these functions of proof and their manifestation in school mathematical practices. In particular, most challenging is finding more effective ways of using proof for explanatory purposes (Hanna, 2000). However, this need not to be construed as seeing the explanatory function of proof as more important than the others. Hence, I used de Villiers’ (1990) multidimensional framework to organise the discussion of the literature on the functions of proof in mathematics. It is in this framework that investigations and conjecturing, enshrined in the CAPS, aligned well with mathematical inquiry which in turn aligned with argumentation. I now turn to defining what I meant by the term “argumentation” in this study.
The review of literature The notion of argumentation
propositions intended to justify (or refute) the standpoint before a rational judge. (van Eemeren et al., p. 5)
According to them, a rational judge is an authority – which could be an existent person or an abstract ideal – to which the assessment of an argument is entrusted. This definition is found not only to be compatible with the practice of mathematicians but also with the CAPS guidelines on handling prescribed tasks. In the latter, emphasis is placed on ensuring that the mathematicians’
practices were reflected in high school mathematics as well. That is, learners themselves needed to engage in a line-by-line explanation of the proof and in that process invite argumentation from their peers. Thus, argumentation is indeed an integral part of proving in mathematics.
That notwithstanding, I saw argumentation as the process of linking evidence (information, ground, or datum observed from diagram) to claim (answer to a question in Euclidean proof) where the statements which connected evidence to claims were referred to as the warrants (reasons). My understanding of argumentation is informed by that of Toulmin (2003) who describes it as a process in which substantiated (warranted) claim are made on the basis of data. Typical argumentation in everyday sense involves interactions wherein participants rely on oral or written information to make (1) claims and support them with (2) evidence, both of which can be rebutted (Berland & Reiser, 2008; Toulmin, 2003). So I saw an argument in this thesis, unlike in logic which is a deductive process involving two or more premises resulting in a conclusion, as constituted by data, claim, warrant, and a rebuttal to evaluate the strength of a claim. Thus, in this sense, a logical conclusion is a result of two or more claims. Figures 2—1 and 2—2 illustrate the similarities between deductive proof and argumentation where D = data, C = claim, W = warrant, and R = rebuttal.
I agree with Osborne, Erduran, and Simon’s (2004) distinction between “argument” and
“argumentation”. An argument is regarded as a referent to the claim, data, warrants and backings that form the content of an argument and argumentation is viewed as a referent to the process of arguing. I argue that since mathematics is viewed as a human activity whose proofs (results) require communication (interactions) among members of the mathematics community, arguments
The review of literature The notion of argumentation
are an integral part of the subject. Figure 2—2 shows how the diagram is used as an argumentation prompt (instrument), that is, to make learners engage in written argumentation.
Figure 2—2. An example of an argument in Toulmin’s (2003) sense
Argumentation as a social activity is evident in a discourse between two or more interlocutors as they defended their claims and made counterclaims when doing proof. Thus, argumentation in mathematics lessons has become a means to better understand proving processes in class (Reid &
Knipping, 2010). Drawing on Lakatos’ (1991) perspectives, proof is defined as a product of a process that entails the use of arguments to formulate conjectures that are consistent with evidence whose validity is agreed upon by the mathematics community at a given time. Also, Menezes, Viseu, and Martins (2015) define mathematical proof as a process of argumentation. These perspectives of proof as a particular kind of argument presupposes a relationship between argumentation and proof (Conner, 2007). For instance, Knipping (2003) define argumentation as
‘a sequence of utterances in which a claim is put forward and reasons are brought forth with the aim to rationally support this claim’ (p. 34).
Further, Aberdein (2012) characterises mathematical proof as an argument. In support of this standpoint, Boero, Garuti, and Mariotti (1996) argue that embedded in the proving process is some continuity – labelled as cognitive unity – which takes place between the construction of a conjecture and the construction of the proof. Before turning to the next section in which I explore
Argument Example
C: My statement is that
…
ê = ĉ
W: My reason is that … Alternating interior angles
R: Arguments against my idea might be that …
But, the lines DE and BC are not marked as parallel
The review of literature Studies on functional understanding of proof
functional understanding of proof in mathematics, it is important to end this section by providing a definition of cognitive unity as seen by Boero, Garuti, Mariotti (1996):
During the production of the conjecture, the student progressively works out his/her statement through an intensive argumentative activity functionally intermingled with the justification of the plausibility of his/her choices. During the subsequent statement proving stage, the student links up with this process in a coherent way, organising some of the justifications (‘arguments’) produced during the construction of the statement according to a logical chain. (p. 113)
In the classroom, the pursuit of cognitive unity helps learners to connect the two fundamental aspects of reasoning, argumentation and mathematical proof, at the same time. It is precisely for this reason that I claim that argumentation cannot be more than a benefit for the task of constructing a proof.