An Example of Inquiry in Linear Algebra: The Roles of Symbolizing and Brokering

Abstract

In this paper we address practical questions such as: How do symbols appear and evolve in an inquiry-oriented classroom? How can an instructor connect students with traditional notation and vocabulary without undermining their sense of ownership of the material? We tender an example from linear algebra that highlights the roles of the instructor as a broker, and the ways in which students participate in the practice of symbolizing as they reinvent the diagonalization equation A = PDP⁻¹.

Keywords:

• Inquiry
• linear algebra
• symbolizing
• brokering

FUNDING

This material is based upon the work supported by the National Science Foundation under collaborative grants DRL 0634099 and DRL 0634074 and collaborative grants DUE 1245673, DUE 1245796, and DUE 1246083. 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

¹ Task 1 and Task 2, in Figures 1 and 8, respectively, have been condensed to conserve space. The handouts of Tasks 1 and 2 used in class with students can be requested at iola.math.vt.edu.

Additional information

Funding

This material is based upon the work supported by the National Science Foundation under collaborative grants DRL 0634099 and DRL 0634074 and collaborative grants DUE 1245673, DUE 1245796, and DUE 1246083. 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 on contributors

Michelle Zandieh

Michelle Zandieh is Associate Professor in the Faculty of Applied Sciences and Mathematics in the College of Integrative Sciences and Arts at Arizona State University. She received undergraduate degrees in Mathematics and Geology from Northwestern University, a master’s degree in Mathematics, and a Ph.D. in Mathematics at Oregon State University. Her research focuses on the learning and teaching of undergraduate mathematics, with a focus on student reasoning in courses such as calculus, linear algebra, geometry, and transition to proof.

Megan Wawro

Megan Wawro is Associate Professor in the Department of Mathematics at Virginia Tech in Blacksburg, VA. She received an undergraduate and a master’s degree in Mathematics from Cedarville University and Miami University, respectively, and her Ph.D. in Mathematics and Science Education jointly from San Diego State University and Univ ersity of California, San Diego. Her research focuses on the learning and teaching of undergraduate mathematics. Her current work explores student thinking and instructional design in linear algebra, as well as methodologies for documenting student reasoning at both individual and collective levels.

Chris Rasmussen

Chris Rasmussen is Professor of Mathematics Education in the Department of Mathematics and Statistics at San Diego State University. He received an undergraduate degree in Mechanical Engineering, a master’s degree in Mathematics, and his Ph.D. in Mathematics Education at the University of Maryland. His research focuses on the learning and teaching of undergraduate mathematics, with a focus on calculus and courses that serve as a transition to more formal and abstract ways of reasoning.