Coffee to Go! Modeling Thermoclines in Multivariable Calculus

Abstract

Although modeling is an integral process in applied mathematics, students rarely encounter modeling opportunities in their calculus courses. We introduce a laboratory experience as a starting point for calculus students to investigate multivariable functions. A layered system of coffee and milk serves as a physical model for temperature gradients in lakes or the atmosphere, where temperature depends on temporal and spatial variables. Students collect temperature data, graph the data, develop mathematical models to fit the data, and write a report about their findings. This article includes details about the class activity conducted in two different college settings and provides our assessment of student learning and interaction with the lab.

Keywords:

• Multivariable calculus
• mathematical modeling
• thermo-clines
• data collection
• inquiry-based instruction
• project-based learning

ACKNOWLEDGMENTS

We thank David Brown (Colorado College), Miro Kummel (Colorado College), Jim Powell (Utah State University), and Marti Garlick (South Dakota School of Mines) for participating in the in-project workshops and testing labs in their classes.

Additional information

Funding

Our work was supported by the National Science Foundation through an NSF TUES grant, #1245421 and #1245407.

Notes on contributors

Andrea Bruder

Andrea Bruder is an associate professor in the Department of Mathematics and Computer Science at Colorado College. Her mathematical interests lie in the areas of differential equations and applied mathematics. Andrea Bruder received her Ph.D. from Baylor University. She was hired at Colorado College in 2009 and takes a project-based approach to teaching. When not teaching or writing she enjoys mountain biking and rock climbing.

Brynja R. Kohler

Brynja R. Kohler is an associate professor in the Department of Mathematics and Statistics at Utah State University. Her specialty areas include both secondary mathematics teacher preparation, and interdis ciplinary research through mathematical modeling of biological systems. She received her B.A. from the University of Chicago, her M.S. from the Courant Institute of New York University, and her Ph.D. from the University of Utah. Brynja Kohler spent a sabbatical at the University of Auckland in New Zealand in 2014, cultivating an appreciation of all things Kiwi, with her husband and two kids.