Tom Peachey has followed up his recent infinity musings with a second post at his Teaching Mathematics blog. Tom’s new post is a very nice and very interesting reminiscence on induction and on fractional derivatives. Go have a read.
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This is a nice discussion of induction. The infinity part is the “there is always one more” aspect of infinity rather than the cardinality idea that “counts” a completed set.
I remember vividly when I understood induction. Previously I was hung up on the “assuming what you have to prove” perception. The problem was to prove Leibniz Rule for the nth derivative of a product of functions. I was able to do it up to n = 5, and then realized that I was doing the same sort of thing to go up the steps from 1, and induction was a matter of codifying the step. When I taught it, I used the idea of a subroutine in a computer program.
Hi Ed
Looks like we had a similar history. Perhaps teaching induction is best done by following your experience, doing a few iterations, and then codifying the step?
I will give this approach a go when I teach it to my class.
I met some PhD in maths who honestly admitted that as y10 students they could not grasp the idea.
It is similar with specialist students these days, some do, some don’t.
I remember being quite comfortable with it. It was not covered in school syllabus then, if I remember correctly. It was a good ( though late) move to use it for testing specialist VCE students – but those divisibility proofs, just terrible.
Variations to the standard Induction were a surprise for me.