Directionality of the Equals Sign: A Case Study Using Logarithmic Properties

Abstract

Mathematics educators view the equals sign as a bidirectional relation symbol, but the authors have observed that students might not have such flexibility in their understanding of the equals sign. The authors have observed that students have a hard time viewing mathematical properties (for example, $\text{log}\hspace{0.17em}(\mathit{MN})=\hspace{0.17em}\text{log}\hspace{0.17em}(M)+\hspace{0.17em}\text{log}\hspace{0.17em}(N)$) with bidirectional flexibility, and subsequently wondered if the “direction” that students are presented mathematical properties has an impact on their ability to use the properties successfully. The authors studied this phenomenon in 11 undergraduate, introductory-level mathematics courses (College Algebra and Precalculus) by presenting logarithmic properties in different “directions” and seeing what effect that had on students’ ability to apply the logarithmic properties successfully. Our results indicate that, despite careful classroom discussion of the meaning of the properties and sustained problem solving activities, the “direction” that the properties are presented to the students does have an effect on their ability to use the properties successfully.

Keywords:

• Equals sign
• logarithms
• college algebra
• precalculus

ACKNOWLEDGEMENTS

This research was partially supported by a 2015-16 Scholarship of Teaching and Learning Grant from the Center for Advancing Teaching and Learning at the University of Wisconsin–La Crosse. We thank Robert Allen and Matt Chedister for helpful comments on our manuscript while it was in preparation. Finally, we thank the referees for their valuable insights and suggestions.

Additional information

Notes on contributors

Tushar Das

Tushar Das is an associate professor of mathematics at the University of Wisconsin-La Crosse. He received his Ph.D. from the University of North Texas in Denton and works at the interface of dynamical systems, fractal geometry, and Diophantine approximation.

Whitney George

Whitney George is an associate professor of mathematics at the University of Wisconsin-La Crosse. She received her Ph.D. from the University of Georgia and her main research area is contact topology.

Nathan Warnberg

Nathan Warnberg is an assistant professor of mathematics at the University of Wisconsin-La Crosse. He received his Ph.D. from Iowa State University and currently defines himself as an anti-Ramsey theorist.