Wednesday, October 2, 2019

Effective Teaching of Abstract Algebra :: Mathematics Education Papers

Effective Teaching of Abstract Algebra Abstract Algebra is one of the important bodies of knowledge that the mathematically educated person should know at least at the introductory level. Indeed, a degree in mathematics always contains a course covering these concepts. Unfortunately, abstract algebra is also seen as an extremely difficult body of knowledge to learn since it is so abstract. Leron and Dubinsky, in their paper  ¡Ã‚ §An Abstract Algebra Story ¡Ã‚ ¨, penned the following two statements, summarizing comments that are often heard from both teacher and student alike. 1."The teaching of abstract algebra is a disaster, and this remains true almost independently of the quality of the lectures." (Leron and Dubinsky agree with this statement.) 2."There's little the conscientious math professor can do about it. The stuff is simply too hard for most students. Students are not well-prepared and they are unwilling to make the effort to learn this very difficult material." (Leron and Dubinsky disagree with this statement.)(Leron and Dubinsky, p. 227) Thus the question is raised: if there is something the "conscientious math professor can do about" the seemingly disastrous results in the learning of algebra, what is it that we can do? As a teacher of undergraduate mathematics, I want and need to know what these effective methods of teaching abstract algebra are. Leron and Dubinsky's paper referred to above and papers resulting from their research contain the bulk of literature that I reviewed. In this paper, they summarize their experimental, constructivist approach to teaching abstract algebra. Among the classroom activities are computer activities, work in teams, individual work, class discussion, and sometimes a mini-lecture summarizing the results of student work (which by this time is familiar to them), providing definitions, theorems, and proofs in their abstract forms. The computer activities use the ISETL programming language. As an example of its use, students write a program implementing the group axioms. They then can enter what they consider to be a group, and the computer will give as output a true or false response. They can use the same process to determine whether their proposed group is closed, has an identity, etc. They choose their answer and then let the computer respond. In this way, students  ¡Ã‚ §construct ¡Ã‚ ¨ the group process, with the view that they will also have a  ¡Ã‚ §parallel construction ¡Ã‚ ¨ occurring in their minds. Students have an experience on which to base their learning of group theory. The method proposed here by Leron and Dubinsky certainly seems patterned after Dubinsky's theoretical foundation for student learning laid out in his work  ¡Ã‚ §Reflective Abstraction In Advanced Mathematical Thinking.

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