Mathematical proof - Rodd, 2000

[torquetum]

 Rodd, M.M. (2000). On mathematical warrants: proof does not always warrant and a warrant may be other than a proof. Mathematical Thinking and Learning, 2(3), 221-244.

 

The author defines various terms, such as justification: “rationale for a belief” and warrant: “that which secures knowledge” (p. 222). He cites Plantinga in defining “warrant” as that quality which distinguishes knowledge from belief. He distinguishes between justification and warrant in that “warrants are for truth, and justifications are reasons” (p. 223). “I may justify a proposition by reproducing a proof of it, but this may not warrant my knowledge of the proposition because the proof is not my own reasoning. Moreover, it may be possible to know without being able to justify in terms of reasons (visualization, for example, may warrant).” The author cites several very interesting studies, notably one by Coe and Ruthven (1994) where they found that students understand the role of proof, but do not tend to use it themselves, preferring empirical observations. “The students were better able to explain the nature of proof than to use proofs in their mathematical investigations: They were aware that proof is required for knowledge, but proof was not the means by which they secured their beliefs in the truth values of mathematical propositions” (p. 225).

 

The author emphasises how so many studies show that proving and knowing are not the same thing for students. (Note to self: When teaching engineering students, are we really looking for “warrants” to help them understand, where warrants could be formal proofs but could equally be something else?) More definitions: “warranting (truth exhibiting) and justifying (giving reasons for the belief)” (p. 227). The author asks “What might warranting mean in classroom practice? For example, if there is a way in which students can come to know through visualizing, how can this epistemic visualization be taught?” (p. 230). There is much in this long paper which I cannot summarise here, for example philosophy, the nature of knowledge and embodiment.

 

“It is this autonomous needer of proof that I suggest is central to a developing mathematician. This need is more than an attitude adopted for math lessons, it is a part of the student’s body–mind. “I just do not believe the result is true unless I have proved it”: In more formal analytic-philosophy language, such a student’s knowledge is dependent on the student’s own proof production. Such students exist, but, in my school and college teaching experience, they are quite rare.” (p. 234). I find this quote interestingly provocative. There’s a nice quote from Polya about proof – mainly geometric. There’s a lovely section on “visual proofs” showing how visualisation can constitute a warrant. However, effective visualisation can be hard to teach and is hard to assess. While the author never mentions engineering maths in this article, I keep finding links. This is a very interesting paper and quite different in focus to the ones I’ve read up to this point.

 

Do not treat this blog entry as a replacement for reading the paper. This blog post represents the understanding and opinions of Torquetum only and could contain errors, misunderstandings and subjective views.