on May 14, 2003
The role of proof in mathematical content knowledge
When I think about proof especially in the K-10 arena I think more of
an informal proof than the traditional proofs of higher mathematics. I
believe that it is important to engage students from the youngest ages
to make conjectures which ideally would be posted on a conjecture board
and for which the students and teacher would strive to prove or
disprove. Proof by counter example or by negation is an informal
method of thinking and reasoning.
The content a teacher needs to facilitate such a process includes not
only an understanding of the mathematics being taught but also the
fundamental concepts upon which that mathematics is being taught. For
example, a group of sixth graders were trying to develop an algorithm
for division of fractions. One boy suggested getting the common
denominator for both fractions then dividing the numerators and the
denominators. The denominators will always = 1 so the result of
dividing the numerators is the answer.
To facilitate the proof of this method the teacher needs to be fluent
in his/her understanding of the algebraic processes that go on in
dividing fractions. Children would be encouraged to try to find
examples where this 'conjecture' do not work if there are any and it
would stay on the conjecture board until it could be disproven.
It so happened that in that same class a student had been shown the
traditional invert and multiply method. The challenge the teacher then
faced was two-fold. Does the common denominator method always work and
if so how does it relate to the invert and multiply method? I think
that even in grade six the use of an algebraic proof could be shown(it
probably would not be understood by all) but could show the link
between the two methods.