Making Implicit Multivariable Calculus Representations Explicit: A Clinical Study

Abstract

Reviewing numerous textbooks, we found that in both differential and integral calculus textbooks the authors commonly assume that: (i) students can generalize associations between representations in two dimensions to associations between representations of the same mathematical concept in three dimensions on their own; and (ii) explicit discussions of these representations are not necessary. For example, in the presentation of partial derivatives, textbook presentations assume that students will understand the slope in a specified direction in without it ever being explicitly presented. Our preliminary interviews indicated that such assumptions may be erroneous, so we created and tested materials that explicitly use representations commonly omitted in textbook presentations of the differential and integral calculus of functions of two and three variables. This paper discusses the classroom activities that were created to include these missing representations, as well as the results when they were implemented in classroom instruction.

Keywords:

• Multivariable calculus
• representations
• derivatives
• integrals
• slope
• three dimensions

Additional information

Notes on contributors

Daniel McGee

Daniel McGee worked for 18 years at the UPR-Mayaguez where he directed numerous projects involving developmental mathematics, high school mathematics, pre-service teacher training and materials for Pre-Calculus and Calculus classes. His materials are used in universities and high schools throughout Puerto Rico. He has recently become Executive Director of the Kentucky Center for Mathematics where he continues his tradition of innovation and leadership.

Deborah Moore-Russo

Deborah Moore-Russo, Ph.D. is an Associate Professor of Mathematics Education in the Department of Learning and Instruction and the Director of the Gifted Mathematics Program at the University at Buffalo, SUNY. Her primary research interests include spatial literacy and the use of digital and physical tools in STEM education.

Rafael Martinez-Planell

Rafael Martinez-Planell received his Ph.D. in Mathematics in 1983 from Michigan State University. Since then he has taught at the University of Puerto Rico at Mayaguez, from where he recently retired. Although originally interested in three-manifold topology, he has been working in mathematics education for the past few years.