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Eric Knuth

February 2005

Many consider proof to be central to the discipline of mathematics and the practice of mathematicians; yet surprisingly, the role of proof in school mathematics has been peripheral at best. Fortunately, the nature and role of proof in school mathematics is receiving increased attention with many math educators advocating that proof should be a central part of the math education of students at all grade levels (see "Recommendations," below).

UW-Madison education professor Eric Knuth agrees. If students are to develop their competencies in proving, then proof must play a more meaningful role in their school mathematics experiences. Enhancing the role of proof in the classroom requires substantial effort by teachers to ensure that students have the means and the opportunities to engage in proving. This will be no easy task, Knuth says. Previous research has found that many students find the study of proof difficult and that many current and future teachers themselves have inadequate conceptions of proof as well as limited views of the nature and role of proof in school mathematics.

Knuth points to two ways to help teachers enhance the role of proof in the classroom. One is to help teachers learn to recognize and to capitalize on classroom opportunities to engage students in proving. Another is to design curricular materials that both support teachers' efforts to enhance the role of proof in the classroom and provide opportunities for students to engage in proving activities.

To learn more about middle school students' conceptions of proof, Knuth recently asked about 400 students to generate justifications about the truth of several mathematical propositions or statements. Some examples:

• If you add any two consecutive numbers, the answer is always odd.
• If you add any three odd numbers together, the answer is always odd.
• If you add any three consecutive numbers, the answer is always equal to three times the middle number.
• Take any number and multiply it by 5 and then add 12. Then subtract the starting number and divide the result by 4. The answer is always 3 more than the starting number.

Consistent with results from previous studies, the majority of students in Knuth's study tended to rely on lists of examples to demonstrate and verify the truth of a statement or proposition. For the students who did attempt to produce general arguments, Knuth found that the number and success of their attempts at generality increased with grade level. Students' expectations regarding the need to produce a mathematical justification increased with grade level as well. To some degree, these results might be a positive indication of the influence of reform; by eighth grade, the students in this study had entered their 3rd year of experiencing a reform-based curricular program-a program with an explicit emphasis on reasoning and proof.

Thus, as the curriculum provides middle school students with opportunities to engage in proving activities, some students develop an awareness of the need to justify and the need to treat the general case.

Knuth's study raises several questions regarding students' understandings of proof that deserve continued research:

1. To what extent do students recognize that a proof treats the general case?
2. To what extent can plausible proof learning trajectories be identified?
3. To what extent can critical transition points in students' understanding of proof be identified?
4. What is the relationship between students' proof production competencies and their proof comprehension competencies?