Values in caring for proof. (English) Zbl 07907745
Larvor, Brendan (ed.), Mathematical cultures. The London meetings 2012–2014. Basel: Birkhäuser/Springer. Trends Hist. Sci., 235-257 (2016).
Summary: In response to [W. T. Gowers, in: Mathematical knowledge. Proceedings of the conference, Cambridge, UK, June 30–July 2, 2004. Oxford: Oxford University Press. 33–58, 175–183 (2007; Zbl 1328.00088)] who sought qualities of proofs that make them memorable, we suggest converting this into qualities that make proofs reconstructible. But Re-constructibility is a property of the person, situation and proof. This leads us to consider what values are being displayed when students are offered proofs, and the notion that, as a caring profession, there are tensions between showing care for students and care for mathematics. We offer interpretations of the commonly used words appreciation, comprehension and understanding in relation to proofs, and use these to analyse and develop three proofs of the irrationality of \(\sqrt{2}\), in order to suggest that in addition to immersion in the mechanics of proof such as justifications, warrants, sentence structure and the like, what students need in order to be able to re-construct proofs is to internalise the act of personal narrative or self-explanation concerning not simply the steps and stages of a proof, but their appreciation, comprehension and understanding of the key ideas, including conceptual insights and technical handles.
For the entire collection see [Zbl 1347.00069].
For the entire collection see [Zbl 1347.00069].
Keywords:
mathematical thinking; mathematical reasoning; monic polynomial; personal narrative; opposite cornerCitations:
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