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Topic: How do we do proofs?
Replies: 6   Last Post: Nov 11, 2009 7:11 AM

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Bob Burn

Posts: 48
Registered: 12/3/04
Re: How do we do proofs?
Posted: Oct 22, 2009 9:23 AM
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Joel
I listened to your screencast with interest, and am sure the students who came found it profitable.
I have two comments.
The major one is that I recognise an assumption behind the presentation that the university maths course in Nottingham consists of lectures in a "Definition, Theorem, Proof, Exercise" format. Nothing unusual about that. But there are alternative ways of running a course in which the problems of proof come up rather differently. I think, for example, of the recent publications of the M.A.A, on the method of R.L.Moore. There are also courses which consist of problems and there are undergraduate courses in which a thesis forms a significant part.
The minor one is that asking a group of maths undergraduates why they thought they should do proofs, a researcher got the dominant answer "because we have to do proofs in the exam". Surely the motivation can be better than that. The first question in your presentation was "If n is an odd integer, prove that n^4 - 1 has a factor 8." The same mathematics could be rephrased in the form:
3^4 - 1 = 80
5^4 - 1 = 624
7^4 - 1 = 2400
Do you think 1 less than the fourth power of an odd number must have a factor 8 ?

This gives a different slant, and a slightly different motivation. The distinction between evidence, conjecture and proof emerges.

[By the way I was a bit puzzled why the simpler question: if n is an odd integer prove that n^2 - 1 has a factor 8, was not offered. This has a geometrical solution as well as modular arithmetic and factorisation solutions.]

Thank you for the screencast. It made me think.
Bob Burn
University Fellow, Exeter University
Sunnyside
Barrack Road
Exeter EX2 6AB
01392-430028
________________________________________
From: Post-calculus mathematics education [MATHEDU@JISCMAIL.AC.UK] On Behalf Of Joel Feinstein [Joel.Feinstein@NOTTINGHAM.AC.UK]
Sent: 20 October 2009 19:36
To: MATHEDU@JISCMAIL.AC.UK
Subject: How do we do proofs?

The second of my screencasts on how and why we do proofs,
"How do we do proofs? (Part I)", is now available at http://wirksworthii.nottingham.ac.uk/webcast/maths/G12MAN-09-10/How-Proofs-I-0910/
The final episode of the trilogy will be in two weeks time.
I should, perhaps, clarify that I am running these as optional workshops for the second-year maths students at Nottingham: About 30-50% of the students came.

Joel Feinstein
http://explainingmaths.wordpress.com/



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