UNDERGRADUATES' REPRESENTATION SCHEMES IN MODULAR ARITHMETIC

Jennifer C. Smith

University of Arizona

Jsmith@lnx.math.arizona.edu

In this exploratory study, we classify four types of representation schemes useful in solving problems involving linear congruences, division-remainder, divisibility, geometric, and equivalence. The schemes were presented both formally (direct instruction) and informally (outside of class) to students in an undergraduate number theory course. After a midterm exam covering linear congruences, students completed a questionnaire, on which they first described all of their representation schemes for ("ways of thinking about") the statement a ≡ b (mod n), and then solved two problems, noting which representation scheme they had used. These two items were specifically chosen to be awkward to solve using the students' observed favored representation schemes, division-remainder and divisibility.

In spite of this awkwardness, most students chose to use these favored schemes to solve the problems. On both items, this resulted in several students solving incorrectly or making inaccurate statements in their explanations. In addition, the representation schemes chosen caused most students to use tedious and unnecessarily complex methods to solve the problems. The students' approaches contrast strongly with the flexible use of representation schemes by experts (graduate students and professional mathematicians) on the same items.

This poster reports on the results of this exploratory study and presents examples of activities which may enable students to develop the flexibility and understanding of this topic required for further study in algebra and number theory.

References

Janvier, C., Girardon, C., and Morand, J. (1993). Mathematical symbols and representations. In P. S. Wilson (Ed.), Research ideas for the classroom: High school mathematics (pp. 79-102). New York: Macmillan Publishing Company.

Lesh, R., Post, T., and Behr, M. (1987). Representations and translations among representations in mathematics learning and problem solving. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 33-40). Hillsdale, NJ: Lawrence Erlbaum.