3.2 The theories underpinning this study
3.2.2 Toulmin’s argumentation theory
3.2.2.1 The six constituent components of Toulmin’s argument pattern
Toulmin’s argument pattern is a model that decomposes an argument into six constitutive elements and describes the relationships between them: claim, data, warrant, backing, rebuttals, and qualifiers. The elements are further categorised into two triads. The first triad, deemed necessary to make a good argument, comprises claims, data, and warrants. As an example of a mathematical argumentation, Figure 3—2 describes an argument relating for the proposition that
“The sum of the interior angles of a triangle equals 1800”.
Figure 3—2. The diagram used to make a claim of an argument
The adequacy of Toulmin’s model for Euclidean geometry is depicted in Figure 3—3; an illustration of elements of an argument and their relationships in a Euclidean geometry context. In
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trying to argue their case, a learner’s argument structure could take the following form. They make a claim (assertion or conclusion) that “angle b = angle d”. In this case, a claim is regarded as the point an arguer wants an interlocutor to accept. Asked “How do they know that?”, they respond by saying that they observed the given data (Figure 3—2 and its labels) and saw that parallel lines were cut by a transversal.
When the arguer began to link the claim with data and mention the theorem that “If parallel lines were cut by a transversal, then alternating angles were equal”, that showed that the data warranted their claim. In other words, the learner warranted their claim by coordinating evidence (data) with their claim. Thus, the warrant performed a linking function and is typically implicit and therefore often left unstated; hence the dotted ellipse in Figure 3—3. The warrant needs to be a universal proposition and therefore shared among members of the field of Euclidean geometry.
When a warrant is unstated, it is the interlocutor’s responsibility to recognise the underlying reasoning that led to the claim in light of the data on which the claim is based. In general terms, both the arguer and the interlocutor were engaged in an argumentation process that used perspectives of a mathematics community to which they both belonged. The claims, data, and warrant constitute a primary or basic argument.
Toulmin (2003) adds three more elements of an argument to supplement the primary elements constitute the second triad: backings, rebuttals, and qualifiers. In an argument, the level of confidence with which the claim is made can be indicated; using terms such as “probably”,
“possibly”, “I think” or “perhaps”. Also, when the claim is challenged, a warrant, which is the logical connection between the data and the claim, is provided to support it and thus strengthen its validity. In an attempt to provide additional information to support the warrant, a backing is provided. Toulmin (2003) defines backings as the ‘other assurances, without which the warrants themselves would possess neither authority nor currency’ (p. 96). Put another way, a backing is used to justify why the warrant is a rational assumption.
In this example (Figure 3—3), to support the warrant and answer the question “How do you know that your reason is correct?” The learner could defend the warrant by appealing to a
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proposition that is commonly shared by indicating that, “The theorem states that alternating angles are equal”. Then, the backing is a statement derived by appealing to an axiom, definition, principle, or theorem to support the warrant. However, the claim or warrant could be challenged by a statement that showed exceptional circumstances under which it may not hold, and this sort of statement is referred to as a rebuttal. For example, the warrant provided by the learner could be challenged to show exceptional circumstances under which it did not hold true: “Does your reason that alternating angles are equal work on a sphere?” Then, the response from the learner may be that “My reason applies to plane geometry only”.
Rebuttals are necessary to include because they make an argument more nuanced and complete as they demonstrate that the arguer took opposition to his or her claim (or warrant) into account. In addition, rebuttals force the arguer to think beyond their claim as they anticipate potential challenges to their claim or ground. Toulmin (2003) points out that rebuttals not only challenge claims but also warrants by showing exceptional circumstances under which the warranted conclusions were incorrect, in which case the warrants has to be set aside. It is therefore incumbent upon the arguer to anticipate any challenge to the generality of their statements, that is, to leave very little room for a statement that may collapse the structure of their argument.
The elements of Toulmin’s (2003) scheme considered in this study are set out and put to use in Figure 3—3. Altogether, the structure of the argument presented here can be summarised as follows: (1) Given that BC is parallel to DE (D), and since parallel lines cut by a transversal make alternating angles equal (W), so (Q), angle b is equal to angle d (C) on account of the theorem stating that “If two parallel lines are intersected by a transversal, then alternate interior angles are equal” (B) unless the surface is hyperbolic or spherical (R). Essentially, these are the elements of Toulmin’s model that rationally stand against scrutiny.
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Figure 3—3. An argument in Toulmin’s structure (Adapted from Toulmin, 2003, p. 97)
Notwithstanding that qualifiers, backings, and rebuttals are used less often in analysis of argumentation in mathematics education (Inglis, Mejia-Ramos, & Simpson, 2007), only rebuttals in this triad were considered and thus formed part of the analysis process in this study. Returning to the decision to exclude qualifiers, data, and warrants, and include rebuttals, I provide three reasons. First, from a cognitive perspective, unlike counterclaims that introduce new ideas rather than challenge a warranted claim, rebuttals provide learners with opportunities to refine their ideas.
Second, argumentations with rebuttals are of better quality than those without given that a rebuttal makes a substantive challenge to the warrant as it refutes its applicability(Osborne et al., 2004).
Therefore, challenging a warranted claim engenders learners to consider alternate frameworks that can be construed as undermining their thoughts; this improves the quality of their argument. That is, I included rebuttals as part of argumentation because they provide ground for deciding whether an argumentation is of low or high quality.
Data/Evidence Claim/Conclusion
Rebuttal Warrant (Reason)
Given that BC is parallel to DE
Unless lines lie on a spherical surface So, angle b is equal
to angle d
Since the parallel lines are cut by a transversal, then alternating angles
are equal
Definition/Axiom/Theorem On account of the theorem that states that alternating angles are
equal.
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The third reason is informed by practical circumstances as they obtained in classroom practice. I did not distinguish among data, warrants, and backings since I would have been naïve to expect learners to begin their claims with adverbs as qualifiers, assumed to be a learner’s commitment based on the strength of evidence at their disposal such as “probably”, “possibly” or
“perhaps”, without having received explicit scaffolding on mathematical argumentation.
Scaffolding takes place when the teacher guides the learner in extending their knowledge through a series of small steps which they would not be independently capable of undertaking on their own (Cakir, 2008).
As Young-Loveridge, Taylor, and Hawera (2005) argue, learners struggle to appreciate the value of reasoning and attending to the ideas of others. Hence, Mason (1996) emphasises the need to provide suitable instruction as a means to support learners in the acquisition of mathematical knowledge and practices characteristic of the mathematics community. That notwithstanding, I acknowledge the value of these adverbs; they reflect the tentative nature of all knowledge, including mathematical knowledge. As already mentioned, mathematics is a human activity and humans are fallible and that arguments are about uncertainty.
Perhaps more importantly, like Osborne et al. (2004), I found that grouping all data, warrants, and backings as grounds circumvents the difficulty for learners to distinguish among these three elements since they were unlikely to have received instruction on argumentation as a learning strategy. As a consequence, I adapted Toulmin’s (2003) argument structure (Figure 3—
4) to understand and analyse the quality of learners’ argumentation.
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Figure 3—4. Toulmin’s adapted model for written argumentation
As Forman, Larreamendy-Joerns, Stein, and Brown (1998) point out, learners see no value in argumentation since for them there is only one correct solution strategy to every problem which the teacher can or should provide in mathematics (this phenomenon is depicted in Figure 3—5).
In addition, they also point out that learners hold norms about school mathematics learning which contradict those practiced by mathematicians (for example, expecting that speed and accuracy are more important than relational understanding). Further, Osborne et al. (2004) argue that argumentation is a complex task and therefore learners need guidance and support to construct an effective argument. The situation often gets more complicated because, as Driver et al. (2000) point out, teachers lack the pedagogical skills in organising argumentative discourse within the classroom.
In summary, Toulmin and others treat argumentation in such detail that the only significant modification I could make was to see data, warrants, and backings as “grounds” to circumvent the difficulty in differentiating among them in learners’ written argumentation. I believe that Toulmin offers a helpful framework that directs attention to the application of key aspects of argumentation
Ground
Rebuttal Claim
Given that BC is parallel to DE
Since the parallel lines are cut by a
transversal, then alternating angles are equal
On account of the theorem that states that alternating angles are equal.
Unless lines lie on a spherical surface Angle b is equal to
angle d
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(that is, informal logic). I drew on Toulmin’s theory as a means to focus attention on exploring learners’ argumentation in response to a specific mathematical task. Learners’ written argumentation were mapped onto the adapted Toulmin’s argument structure, that is to say the TAP model. In this study, I was only interested in characterising learners’ argument rather than requiring them to engage in constructing proofs.