Odds and evens: mathematical reasoning and informal proof among high school students

Ten high school algebra students were asked to judge simple statements about combining odd and even numbers, stating whether they were true or false. They were also asked to give justifications or explanations for their decisions. All of the students initially reasoned inductively or empirically, appealing to specific cases and justifying their answers with additional examples. On being prompted for any further explanations, seven of the students attempted to formulate some type of non-empirical rationale. However, only three students were able to create fairly coherent arguments, none of which used standard algebraic notation. Instead, two of these original, idiosyncratic arguments were based on visual representations of odd and even numbers, and the third consisted of an informal and partial argument by cases.

Intuition comes to us much earlier and with much less outside influence than formal arguments … Therefore, I think that in teaching high school age youngsters we should emphasize intuitive insight more than, and long before, deductive reasoning. —Polya, George (1981, pp. 2–128)