Holding Calculus in Your Hands

by Audrey Malagon, Lead Editor, DUE Point

Students often struggle to visualize concepts in multivariable calculus. While technology can help, virtual surfaces can still seem out of reach. Aaron Wangberg of Winona State University describes how this team created dry erase surfaces and supporting materials to enhance students’ interpretation of the mathematical objects they were studying.

NSF grants are competitive. What do you think it is that set your proposal apart and got the project funded?

We created contextualized activities with concrete manipulatives to deepen students’ understanding of key concepts in multivariable calculus. There was a clear end goal with a concrete plan to create materials that could be widely disseminated for a broad impact.  We had prototypes of the products and promised to disseminate the final materials to 30 institutions. Our proposal included letters of support and names of participants from 20 institutions.

Where did you get the idea for this project?

Teaching multivariable calculus, I realized students could use the formula for the gradient, but couldn’t tell me what was important about the resulting vector.

I spent hours taping together bowls and using paper-mache or plaster-of-paris to make surfaces my students could manipulate while exploring partial derivatives and the gradient. When I used these surfaces in class and asked students where the gradient pointed, several groups listed their vector components until one group said “It points uphill!”  This spurred a lot of activity as other groups double checked their results on different surfaces. When I asked students to re-measure partial derivatives and re-form the vector after they’d rotated the surface, there was another ‘aha!’ moment when they realized the gradient vector stayed the same relative to the surface even though the numbers had all changed.

I realized then the importance of hands-on manipulatives in helping students identify the key features of a gradient vector. I wanted other students to have the same experience and other instructors to have available the resources I had painstakingly created by hand.

How did the project evolve from taped bowls and paper-mache to the sophisticated models you have now?

Ben Johnson, a practicing mold-maker and my former student, advised me how to produce smoother surfaces.  Over the course of a year, we developed surfaces that were carved with a CNC machine out of blocks of wood and then finished with white surface that acted like a dry-erase finish.  Ben and I also developed several activities exploring multivariable calculus ideas: level curves, partial derivatives, and constrained optimization (Lagrange multipliers). He presented the project at the Joint Mathematics Meetings.  

I tried the materials in my multivariable calculus course in order to introduce new concepts and found students were exploring connections between ideas that we wouldn’t typically cover until weeks later.

What inspired you to apply to this grant program?

I knew the materials were in high demand. Every time I spoke at a conference, instructors would ask how to get the materials.  I had worked with three composite engineering students at Winona State University in order to find plastic materials and a process for turning the wood models into clear plastic dry-erasable surfaces, and then the TUES program provided funding equipment to turn that theoretical process into a concrete, cost-effective application.

Since my graduate advisors had used NSF funding to develop instructional materials for vector calculus and physics, I decided to apply here for funding to expand this project’s impact to where it is today.

Tell us about the impact this project is having.

The materials and activities have been created to help students develop productive understandings of multivariable functions, derivatives, and integrals.  In the first year of implementation, the materials have been used in over 40 courses with over 1000 students.

Because students don’t have the formulas for the surfaces, they instead have to reason using context and geometric relationships. Outside of the activities, instructors are noting how students are participating in their small groups in class when they use the materials and also noting how students are contributing ideas to the classroom.  

One instructor, who used the materials in one of two multivariable calculus courses, noted how students in the course that used the materials were much more inquisitive and asked questions during non-surface days. The materials are impacting teaching styles as well, encouraging discussion and inquiry.

Are you interested in using these surfaces? Check out their project website https://raisingcalculus.winona.edu/ for more information and access to resources.

Editor’s note: Q&A responses have been edited for length and clarity.

Learn more about NSF DUE #1246094

Full Project Name: Raising Calculus to the Surface

NSF Abstract Link: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1246094

Project Website: https://raisingcalculus.winona.edu/

Project Contact: Aaron Wangberg, awangberg@winona.edu, Principal Investigator

For more information on any of these programs, follow the links, and follow these blog posts. This blog is a project of the Mathematical Association of America, produced with financial support of NSF DUE Grant #1626337.

Audrey Malagon is lead editor of DUE Point and a Batten Associate Professor of Mathematics at Virginia Wesleyan University with research interests in inquiry based and active learning, election security, and Lie algebras. Find her on Twitter @malagonmath.