       # Discussion: Keynote Part 1   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 K-10 arena I think more ofan informal proof than the traditional proofs of higher mathematics. I believe that it is important to engage students from the youngest agesto make conjectures which ideally would be posted on a conjecture board and for which the students and teacher would strive to prove ordisprove. Proof by counter example or by negation is an informalmethod of thinking and reasoning. The content a teacher needs to facilitate such a process includes notonly an understanding of the mathematics being taught but also thefundamental concepts upon which that mathematics is being taught. Forexample, a group of sixth graders were trying to develop an algorithmfor division of fractions. One boy suggested getting the commondenominator for both fractions then dividing the numerators and thedenominators. The denominators will always = 1 so the result ofdividing the numerators is the answer. To facilitate the proof of this method the teacher needs to be fluentin his/her understanding of the algebraic processes that go on individing fractions. Children would be encouraged to try to findexamples where this 'conjecture' do not work if there are any and itwould stay on the conjecture board until it could be disproven. It so happened that in that same class a student had been shown thetraditional 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 thinkthat even in grade six the use of an algebraic proof could be shown(itprobably would not be understood by all) but could show the linkbetween the two methods.       © TERC 2003, all rights reserved Home • Keynote • Poster Hall • Panels • Discussants Reflect • Resources • Lounge • Info Center