Tracking changes in teaching and learning abstract algebra: Beliefs and ability to abstract.

摘要

The first purpose of this study was to investigate recent changes exhibited in pedagogy, content and textbooks in abstract algebra over professors' teaching experiences. Professor beliefs about student misconceptions, and difficulties in learning abstract algebra were examined as well. To investigate these changes, seven mathematics professors with various experiences in teaching abstract algebra were interviewed.;The mathematics professors purported minor changes in pedagogy, content, and textbooks within their teaching experiences. The professors believed student difficulties in abstract algebra ranged from lack of knowledge of abstraction, to student beliefs as to what constitutes mathematics.;The second purpose of this study was to determine to what extent student beliefs about the nature of mathematics would be changed over the course of enrollment in an abstract algebra class. To achieve this, three belief surveys were completed by thirteen students, pre- and post-abstract algebra. The three surveys were designed to track: student changes in self-reported definition of mathematics, student changes in rankings of mathematician's definitions of mathematics, and student changes in an attitude and beliefs survey.;Self-reported definitions of mathematics did not change dramatically over a semester enrollment of abstract algebra. Student rankings of mathematician's definitions of mathematics reported inconsiderable changes. One question from the attitude and beliefs survey demonstrated statistically significant differences (n = 13, alpha = .05).;The third purpose of this study was to establish changes students exhibit in problem solving behaviors when confronted with problems designed to track changes in abstraction both pre- and post-abstract algebra. Nine students participated in problem solving sessions both pre- and post-abstract algebra.;Changes post-abstract algebra included, but were not limited to: student ability to prove; student ability to determine if algebraic structures represented groups, subgroups, or rings; student ability to solve word problems using abstract objects; and student ability to define similarities in sets under their operations by using the set's binary operation.