Teaching Calculus through Fingerpainting
by CFT Assistant Director Derek Bruff
As part of the Center’s continuing exploration of the role of visual thinking in teaching and learning, we’re glad to share this interview with Vanderbilt mathematics professor Sal Froipol on his use of fingerpainting to teach calculus. We recognize that everyone’s teaching context is different, but we hope that hearing others’ perspectives on teaching and learning will help our readers reflect on their own teaching.
Derek Bruff: What led you to experiment with fingerpainting in your mathematics courses?
Professor Sal Froipol: As you know, Derek, there’s ample evidence that all students are visual learners. Our brains are wired to quickly and accurately make sense of visual stimuli. Given the history of visualizations in mathematics, I thought it would be a natural extension of this tradition to incorporate more visual thinking in my calculus courses.
DB: But why fingerpainting?
SF: Well, the cognitive load in a calculus course is very high. Students must learn a variety of concepts and techniques that are quite unfamiliar to them. I wanted to leverage their existing visual thinking skill sets, and I thought, Why not tap into their pre-school experience in fingerpainting? I think it’s important to recognize where your students are in their learning and try to meet them there. About the only thing I could assume was true for all my students was that they had, at some point in their lives, fingerpainted.
DB: How can students learn calculus through fingerpainting?
SF: It’s surprisingly natural. Students create visual representations of whatever the topic of the day happens to be: integration by parts, the Intermediate Value Theorem, the alternating series test, first-order linear differential equations–whatever! By using color, shape, and their fingers–of course!–to represent these concepts and techniques, they come to understand them more deeply.
DB: Is it true that you don’t fingerpaint yourself during class?
SF: As is commonly said, math is not a spectator sport. It’s important for students to be active involved in their own learning. In fact, the more active they are, the more they learn, according to research. Because of that, I decided to let my students do all the fingerpainting themselves. At the start of class, I distribute the paints and papers and write the topic of the day on the chalkboard. Since I assume they’ve all read their textbooks before class, I know they already have a good understanding of the topic. Then I turn them loose on their fingerpainting! It’s great fun.
DB: What role, then, do you play during class?
SF: The traditional role for a professor is as a “sage on the stage.” Some education reformers have argued that we should instead play the role of a “guide on the side.” I prefer the idea of a “muse in the pews.” I circulate among the students, throwing out suggestions to them as they paint. “Try using some orange!” “Paint with only straight lines!” “See what you can do with your eyes closed!” I find that these suggestions spur the students to greater creativity.
DB: Here’s the tough question: How do you grade their work?
SF: Oh, I have a three-page analytic rubric with 17 categories and four levels of quality within each category. You can grade anything with a cleverly designed rubric.
DB: Last question. What’s next for your use of visual thinking in the classroom?
SF: Well, actually, inspired by the CFT’s upcoming workshop on teaching outside the classroom, I’m planning to take the graduate students in my differential topology course to the streets with spray paint. I call it graffiti-based learning.
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