TOWARDS A FRAMEWORK FOR UNDERSTANDING WAYS OF KNOWING MATHEMATICS: SIX STUDENTS IN FINITE MATHEMATICS AND A LINKED SUPPORT COURSE
by Gideon L. Weinstein
This study investigates college students' intellectual
development in mathematics. The theoretical background is provided by
Baxter Magolda's student development theory (1992) and influenced by Perry
(1970) and Belenky, Clinchy, Goldberger, and Tarule (1986). These theories
are concerned with students' beliefs about knowledge, learning, and authority,
and state that college students typically begin with narrow, black-and-white,
uncomplicated views but slowly develop more complex, contextual, shades-of-gray
views. In this vein, I focused on three particular "ways of knowing
mathematics": beliefs about mathematical knowledge ("What am I studying?");
habits in learning mathematics ("How should I study it?"); and sources of
conviction ("Why should I believe in it?").
I conducted a multiple case study of six students
enrolled in an introductory course, Finite Mathematics, at large midwestern
research university. These students considered themselves at risk of
doing poorly in mathematics and consequently decided to concurrently enroll
in an elective study skills course, Learning Strategies for Mathematics, that
I taught. My role as study skills teacher was beneficial in that it
helped me build rich descriptions and gain deep insights into the students'
learning. I kept a teaching fieldnotes journal and interviewed each
student four times throughout the semester and once during the next academic
year. I used the information I collected to create descriptive portraits
of the students as mathematics learners, and I classified each student in
terms of his or her intellectual development in mathematics. Using
these results, I sketched a prototype developmental scheme for describing
how students progress from naïve to sophisticated "ways of knowing mathematics."
The framework produced has two categories of development (Learning Mathematics
and Verifying Mathematics), each with five levels of sophistication.
Low levels describe students with procedural and absolute views of mathematics
who tend to passively receive knowledge and mimic their instructors.
Students at slightly higher levels can at least choose among different procedures
and check answers a variety of ways. Students at even higher levels
begin to feel that mathematics is about concepts and underlying patterns,
and they believe their understanding must be actively constructed to fit
existing mathematical structures, which are the results of replicable and
verifiable social agreements.