Promoting Linear Algebraic Reasoning among Students: Affordances and Challenges

Abstract

As students transition from the mathematics they learn in school years, including their first-year calculus courses, to the first course in linear algebra, they experience discontinuities in their perspective of what mathematics is. Their propensity to continue applying the same habits of learning in the face of this change leads to failure and frustration. The failure manifests itself in the quality of understanding basic concepts as well as in the lack of linear algebraic reasoning. Instructional treatments applied in my teaching experiments to foster students’ ability to reason linear algebraically resulted in mixed success – some of the treatments were successful, others less so. The latter are accounted for by the structural complexity of the subject matter and students’ background knowledge. The pedagogical approaches offered in this paper are oriented within a particular theoretical framework for the learning and teaching of mathematics, called DNR. Reflections and broader implications are addressed through the lenses of this framework.

KEYWORDS:

• Linear algebra
• linear-algebraic reasoning
• DNR-based instruction

DISCLOSURE STATEMENT

No potential conflict of interest was reported by the author(s).

Notes

1 The deficit language I use in the paper to describe students’ current knowledge of and performance in linear algebra by no means suggests that the blame is on the students or their teachers. There are confounding factors – institutional, financial, social, and cultural – contributing to the status of students’ unsatisfactory performance in undergraduate mathematics courses. There is often a tendency to put the blame on schoolteachers for the performance of their students, to which I vehemently oppose. The process of becoming a teacher is complex and includes various stages. The central and most demanding stage involves us, the college instructors. In this stage, teachers fulfil the course requirements and complete the teaching credentials their higher education institution places on them. And we, their educators, lead them to a “successful” completion of these requirements. Equipped with the knowledge they acquired from us through our textbooks and our teaching methods, the teachers embark on their teaching career. Their students, upon graduating from high school, come to our institutions assuming they are prepared for what awaits them in undergraduate courses.

2 All the classroom observations and instructional approaches discussed in this paper are taken from my teaching experiments in undergraduate linear algebra.

3 The DNR-based instruction theoretical framework (DNR, for short) has been discussed extensively in various publications (see, for example, [10,11,16]). The initials D, N, and R stand for the three central pedagogical principles of the framework: Duality, Necessity, and Repeated Reasoning. The reader might wonder, why the DNR theoretical framework appears at the end of the paper rather than at its start, as conventionally is the case. The choice to do so is deliberate because this order is consistent with DNR principles. One of the implications of DNR is that introducing formal statements of definitions and theorems after students have understood and appreciated their meanings has a positive effect on student learning. This pedagogical norm applies to any form of communication aimed at conveying new ideas more effectively. I hope the reader will find the paper fulfilling this goal through its choice of DNR-based presentation.

4 A detailed account of this approach appears in Ref. [12].

Additional information

Notes on contributors

Guershon Harel

Guershon Harel is a distinguished Professor in the Mathematics Department at the University of California, San Diego. His research interest is in cognition and epistemology of mathematics and their application in curriculum and instruction.