The anatomy of a mathematical proof: Implications for analyses with Toulmin’s scheme

Abstract

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Prospective mathematics teachers’ argumentation structure when constructing a mathematical proof: The importance of backing

“…In this case, backing as one of the components in the Toulmin scheme plays a vital role in determining the truth of claims. Despite backing is crucial, we argue that it is still under-researched since limited prior research focused on the case (e.g., Ubuz et al, 2012;Simpson, 2015). Some researchers (Inglis, Mejia-Ramos, & Simpson, 2007;Chen & Wang, 2016) consider backing as a definition or theorem.…”

“…Numerical backing help students to produce a different counter-example (rebuttal) but still yields a valid claim because the students simply provide an example of denial to indicate that the mathematical statement is false. Simpson (2015) explicates that backing supports the warrant and it will be relevant if the warrant is challenged. Three roles of backing are proposed, i.e., (1) backing for the warrant's validity which relates to definition, (2) backing for the warrant's field which associates with a field on data, and (3) backing for the warrant's correctness which attempts to show that the warrant is correct.…”

“…Some researchers (Inglis, Mejia-Ramos, & Simpson, 2007;Chen & Wang, 2016) consider backing as a definition or theorem. Simpson (2015) highlighted three types of backings which relate to warrant, i.e., backing for the warrant's validity, backing for the warrant's field, and backing for the warrant's correctness. Ubuz et al, (2012) analyzed collective argumentation constructed by students and teachers in learning.…”

“…The current research pursued the investigation of the backings as a small number of prior researches did (e.g., Ubuz, et al, 2012;Simpson, 2015). However, it differs from the types of backing and their relation to rebuttal and qualifier in mathematical proof.…”

When students offer explanations without the teacher explicitly asking them to

“…Studies have examined the criteria students use when accepting a proof as an explanation or not, with proofs including a narrative, deductive reasoning or the possibility to be understood in a variety of ways being more often accepted as an explanation than for example visual proofs (Healy & Hoyles, 2000). Krummheuer (2007) and his colleagues have focused their analysis on a particular type of explanation, argumentation, which draws on Toulmin's scheme of argumentation and can lead to the structuring of proof (Simpson, 2015). The focus here is on the elements of the emerging argument, which broadly can include backings and warrants.…”