A Three-Fold Approach to the Heat Equation: Data, Modeling, Numerics
James Puckett and Kimberly Spayd have authored a wonderful article in the current issue of PRIMUS with full citation:
Spayd, K. and J. Puckett. 2016. A Three-Fold Approach to the Heat Equation: Data, Modeling, Numerics. PRIMUS. 26(10): 938-951.
The abstract to begin says it all with respect to what happened, "This article describes our modeling approach to teaching the one-dimensional heat (diffusion) equation in a one-semester undergraduate partial differential equations course. We constructed the apparatus for a demonstration of heat diffusion through a long, thin metal rod with prescribed temperatures at each end. The students observed the physical phenomenon, collected temperature data along the rod, then referenced the demonstration for purposes in and out of the classroom. Here, we discuss the experimental setup, how the demonstration informed practices in the classroom and a project based on the collected data, including analytical and computational components."
Moreover, the article is rich in detail on what and how they and their students built a set up in a lab, collected data and modeled. Simply wonderful piece. Puckett is in the Physics department at Gettsyburg College and Spayd is in the Mathematics Department. In both cases these faculty bring their students in touch with their research interests as well as offer them a chance to actually build models of the mathematics they are learning.
I confess to a bias for the journal, PRIMUS - Problems, Resources, and Issues in Mathematics Undergraduate Studies, for I founded the journal over 25 years ago as a quarterly (now with 10 issues per year and each packed with great ideas) where colleagues could share ideas about teaching undergraduate mathematics. This piece is a sort of ultimate article from my point of view as founder of the journal. We congratulate James and Kimberly.
The conclusion to end, says it all with respect to student experiences, "Overall, we believe this entire experience was greatly beneficial for the students. It was clear that the students enjoyed being in the lab for the demonstration; it was a camaraderie-building experience and an active component in an otherwise traditional math class. It appeared to us that the novelty and physicality of the demonstration increased the students’ enthusiasm for the material. This naturally led to more engagement, discussion, and insight than we would have expected otherwise. This approach to the heat equation, from the lab to the project write-up, allowed the students to experience mathematics as an applied and interdisciplinary field. The data, modeling, and numerical components all worked together to significantly enhance the learning that took place."
ADVISORIES: If your institution is NOT subscribing to PRIMUS, although most colleges and universities do subscribe through Taylor & Francis' (the publisher) bundling of journal subscriptions to libraries in higher education, then you need to tell your librarian to subscribe to PRIMUS, for you, for your colleagues, and most of all, because PRIMUS is ultimately about them, for your students.
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Finally, if you are James or Kimberly we invite you to write up a Modeling Scenario so ALL can do what you have done with their students to include information about how to set up the experiment and collect data and your collected data for those who cannot do the set up.
Again, congratulations to James and Kimberly, for we are all the better for your efforts to reach your students with meaningful mathematics. Thank you.