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.