Two Writing Assignments in Proofs Class

Pam Crawford

Jacksonville University

All mathematics majors and minors at Jacksonville University are required to complete the writing intensive course MS 220WI Mathematical Reasoning. In this course, we look at methods of proof and mathematical writing through study of the fundamentals of mathematical logic, set theory, relations, functions and their limits, and cardinalities of sets. Students submit weekly homework assignments for grading but this past semester (Spring 2003) I developed ideas from Margaret Robinson and Annie Seldon into writing assignments. This talk will report on those two assignments and their results.

After their investigation of direct and contrapositive proofs, my students are introduced to proofs by contradiction. Students first work their way through several simple proofs involving even and odd integers. Then, students work with me in-class to develop the proof of the irrationality of the square root of two. We discuss the ideas behind the proof, why certain statements are needed, when they are needed, what our goal is in the proof and when we know we have reached our goal. Students have several opportunities in the following classes to question their understanding of the proof of the irrationality of the square root of two. After assurance from my students that they understood the proof, I gave my students the assignment developed from an idea of Margaret Robinson's and reported in a draft version of the MAA's CUPM Curriculum Guide. Students were required to write out and explain the proof of the irrationality of the square root of two to another person – and to submit a signed statement from the other person stating that she/he understood the proof! This other person could be anyone but a math faculty member or someone who previously had taken MS 220WI. Many of my students chose either family or friends as their other persons; some chose science faculty on campus. When asked to comment on the experience, several of the students reported that it took two or three (or more) attempts before their other person would agree they understood the written proof, including any notations the student had used.

On their subsequent examination on proofs by contradiction, I then asked my students to prove the irrationality of the square root of two. Many students commented to me outside of class that they were sure they knew the proof because they had had to explain the proof several times to another person. Other students who apparently were successful at explaining the proof failed to remember much about it, however.

As a final writing assignment in MS 220WI, I used Annie Seldon's idea of asking students to write about their favorite proof, including a discussion of what aspects made it their favorite proof. Some students wrote that the first proof they could do on their own was their favorite proof, often because it gave them a sense of empowerment and understanding of the structure of a proof. Other students stated that particular types of proof appealed to them, and then went into thoughtful discussions of why they preferred the structure of one type of proof to another type of proof.

I believe that both these assignments helped students write about mathematics, and further developed their understanding of mathematical proofs and notations and their ability to communicate mathematics.