An Inquiry-Oriented Approach to Span and Linear Independence: The Case of the Magic Carpet Ride Sequence

Abstract

In this paper we present an innovative instructional sequence for an introductory linear algebra course that supports students' reinvention of the concepts of span, linear dependence, and linear independence. Referred to as the Magic Carpet Ride sequence, the problems begin with an imaginary scenario that allows students to build rich imagery and formal definitions. The approach begins by focusing on vectors, their algebraic and geometric representations in and , and their properties as sets. Samples of student work are provided to illustrate the variety of student solutions and how these solutions lead to the creation of formal definitions.

Keywords:

• Linear algebra
• research in undergraduate mathematics education
• student thinking
• linear independence
• span

ACKNOWLEDGEMENTS

This material is based upon work supported by the National Science Foundation under grants DRL 0634099 and DRL 0634074. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Notes

¹The most recent text by Strang [23] includes a section within the first chapter that introduces linear independence and dependence through two examples in , and Gaussian elimination appears in Chapter 2.