CAREER:Understanding and Teaching the Processes Used to Comprehend Mathematical Arguments and Proofs

In both reform-oriented K-12 mathematics classrooms and in upper-level university classrooms, a significant emphasis is placed on the activities of justifying and proving. In particular, in these classrooms, we expect students to learn new mathematics by attending to and critiquing the mathematical arguments of others. For these goals to be achieved, students need to be able to understand others' mathematical arguments and to determine whether or not a mathematical argument is valid. Research has shown that students and mathematics teachers frequently lack this ability. We know little about what cognitive processes a person should use to determine whether a mathematical argument is valid or to construct understanding from reading a mathematical argument. The purpose of this proposal is to address both of these issues with the following series of studies:

(a) An expert-novice study in which undergraduate mathematics majors and mathematicians are observed as they read and evaluate a collection of mathematical arguments, the results of which will inform

(b) The development of pedagogical techniques and curricula materials using a design research paradigm to teach students the knowledge and strategies they need to comprehend mathematical arguments, and

(c) A controlled experiment in which the pedagogical techniques developed in (b) will be implemented with a group of students and these students' performance on a post-test will be compared to a control group's performance.

Intellectual merit.

The proposed work will provide insight into the mathematical practices of reading and evaluating mathematical arguments and will delineate why students find these activities problematic. The proposed work will yield documented effective teaching methods to teach undergraduates the knowledge and skills to read and evaluate mathematical arguments effectively.

Broader impact.

These results will be of interest to mathematics professors who regularly teach proof-oriented classes and teachers of other scientific disciplines, such as physics, where argumentation plays a pivotal role. These results will also be useful for educators who prepare future teachers of mathematics. The results of these studies will be presented at academic conferences for both mathematicians and mathematics educations, and these results will be published in peer-reviewed journals.

Educational goals.

I will give workshops for mathematics professors who are looking for new ways to teach proof-oriented mathematics courses and for