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From: Herman Rubin on 13 Oct 2009 11:11
In article <5234166d-9e05-4f23-9dad-0ab503f2f3b2(a)m11g2000yqf.googlegroups.com>,
Tim Norfolk <timsn274(a)aol.com> wrote:
>On Oct 9, 3:34=EF=BF=BDpm, hru...(a)odds.stat.purdue.edu (Herman Rubin) wrote=
>:
>> In article <gerry-1BAEBD.13035509102...(a)feeder.eternal-september.org>,
>> Gerry Myerson =EF=BF=BD<ge...(a)maths.mq.edi.ai.i2u4email> wrote:

>> >In article
>> ><3fb1bc8f-73c3-41ff-b2d7-1b7d371f2...(a)a6g2000vbp.googlegroups.com>,
>> > pubkeybreaker <pubkeybrea...(a)aol.com> wrote:
>> >> OTOH, =EF=BF=BDI would expect that a high school teacher would know th=
>e Peano
>> >> axioms.
>> >I'd be very surprised if any of my high school (math) teachers knew
>> >the Peano axioms (and I went to a good high school).

>> You may be both partly right. =EF=BF=BDSOME high school math
>> teachers do learn the Peano axioms, but probably most
>> do not.

>The ones that we graduate here have seen them, and done some small
>problems. Whether they remember and understand is something else. On a
>related note, I have to disagree with your contention that the 'good',
>but not excellent, student can learn analysis before the calculus. I
>am teaching a complex analysis course right now, and the homework that
>I am grading suggests that the students cannot easily 'see' that 2(3x)
>=3D 6x, because their manipulative skills are so weak. How can they
>follow a good proof?

If they UNDERSTOOD algebra, instead of just learning how
to solve problems, they would be. Also, if they had the
"Euclid" geometry, they would be. Every manipulative course
makes their ability to see relations worse.

If you asked them to simplify 2(3x), they would give you
the same expression which they cannot recognize as equal
to it. I would teach arithmetic from the Peano postulates
in the beginning, including some proofs.

I was considering real analysis. If taught without assuming
calculus, it can be understood. But by the time students
have had all that instruction in doing manipulations, it
gets much harder.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin(a)stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
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