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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