Re: A question about s paper in Jour ofResearch in Math Ed 34(2003)4 (fwd)

[Jack Uretsky <jlu@HEP.ANL.GOV>]

Hi all-
        The above paper reports on the comments made by eight students who
were asked to validate purported proofs of a simple theorem (n a positive
integer, n^{2} is divisible by 3 -> n is divisible by 3, any integer can
be written as 3n, 3n+1, 3n+2 is also given).  What grabbed my attention
was that only two of the validators were able to prove the theorem.
        I have permission of one of the authors (Selden and Selden) to
forward the below exchange to this net for comment.
        Note my presumptuous remark about Phys-L members.
        I suggest that Annie be made a copy recipient of any comments.
                                                        Regards,
                                                                Jack

--
        "What did Barrow's lectures contain?  Bourbaki writes with some
scorn that in his book in a hundred pages of the text there are about 180
drawings.  (Concerning Bourbaki's books it can be said that in a thousand
pages there is not one drawing, and it is not at all clear which is
worse.)"
                                        V. I. Arnol'd in
                                Huygens & Barrow, Newton & Hooke

---------- Forwarded message ----------

> On Sat, 26 Apr 2003, Annie Selden wrote:
>
> > Dear Jack,
> >
> > We separated the validating of purported proofs from
> > the construction of proofs for the purposes of our study.
> > It seems clear to us that future secondary mathematics
> > teachers should know both how to make proofs (of a reasonable
> > degree of difficulty) and should also be able to validate
> > purported proofs (in particular, to be able to judge whether
> > student-generated proofs are correct).  Indeed, we stated
> > this in our JRME paper.
> >
> > Proof has not vanished from mathematics education.
> > The NCTM Standards have reasoning and proof
> > as one of  its process standards across the
> > prekindergarten through grade 12 curriculum.  However,
> > the NCTM Standards are a vision of what should be.
> > What actually is the case and what the push toward
> > state standards (an entirely different thing) and testing
> > are doing needs to be investigated.  Our study was just
> > one small effort in that direction.
> >
> > In practice, at the lower-division college level, I would
> > say (from my experience) that there is little emphasis
> > on proof in such courses as calculus and differential
> > equations that are primarily service courses.  Math.
> > depts. generally regret this, but there is a push from
> > engineering and science depts. to have math. depts. teach
> > the students to learn to apply formulas and solve problems.
> > However, the situation is quite different for upper-division
> > math courses, which are taken mainly by math majors.
> > Abstract algebra, advanced calculus, and real analysis
> > (amongst others) require students to prove lots of theorems,
> > and students' competence is judged primarily by their
> > ability to prove theorems on their own.  Little emphasis is placed
> > upon students' regurgitating theorems from textbooks or lectures.
> >
> > Annie
> >
> > At 02:31 PM 4/26/2003 -0500, you wrote:
> >       Dear Annie,
> >       Thank you for your prompt reply. Am I being utterly
> >       naive in
> >       expecting that a "validator" of proofs (I take it that this
> >       means a future
> >       teacher) should be someone with expertise in constructing
> >       proofs?
> >       I think that the members of the physics-L net would
> >       generally
> >       agree that the concept of proof has all but vanished from
> >       mathematics
> >       educattion. Do you have reason to disagree?
> >                                                   Best,
> >                                                       Jack