posted by:

Anne Collins
on May 14, 2003
at 3:14PM

subject:

The role of proof in mathematical content knowledge

When I think about proof especially in the K10 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 twofold. 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.

