Winona State University Professor of Mathematics Aaron Wangberg thinks through a multivariable calculus problem using a model surface.

Hands-on calculus in three-dimensions



With three-dimensional physical models, students at Winona State University (WSU) are learning multivariable calculus in a new fashion that involves drawing on the surfaces to help identify and work with different mathematical concepts before formally learning about them. The homegrown models have been shared with institutions throughout the U.S. as well.

Aaron Wangberg is a professor of mathematics at WSU who collaborated with students and engineering department faculty members on the making of the models and has used them to help students learn multivariable, or three-dimensional, calculus. The surfaces are dry erasable and large enough for multiple students to use as a common workspace on which to draw as they work through math problems and interpret what factors such as the height and the slope mean, Wangberg explained.

Wangberg said the models allow students to identify and work with the ideas underlying mathematics before they know precise mathematical definitions or symbols and bring the ideas up in class themselves. He stated that students recognize the questions that are important to ask about these ideas more quickly after working with the model surfaces.

“I think a big aspect of the project is the way that student ideas and student voices are used in the classroom,” Wangberg shared. “Instead of it becoming information passed from the authority to the students, from the teacher or from the textbook … it’s the students who are finding the connections between ideas, finding the relationships, making guesses or conjectures. They’re the ones creating the mathematics, and the instructor can then use the student ideas to drive the course.”

Students have referenced the models while working through mathematical problems on days when the materials are not being used in class, Wangberg said. They may note that they thought about what a vector or a curve would look like on the surfaces, for instance, he stated.

“With Dr. Wangberg’s fabrication and development of models for calculus, he has made a significant impact on the curriculum by making calculus more hands-on and increasing the visualization of 3-D rates of change,” said Nicole Williams, professor and chair of the mathematics and statistics department at WSU. “Students have commented that these materials make it easier for them to understand calculus and they make the material more visual. With these materials, Dr. Wangberg has developed a multi-faceted approach to teaching concepts of calculus, which he has shared with instructors from other universities and at regional high schools.”

Wangberg shared that by looking at how groups are interacting with the surfaces, instructors can quickly tell with which groups they need to work. He explained that it might be more difficult for instructors to do so if students were working with paper and pencil alone.

Wangberg noted that prior to the creation of the models, he had been teaching a multivariable calculus course in which students had some difficulty grasping the meaning behind a math symbol that represents the steepest direction from a particular point and how steep that direction is. With models Wangberg made using plaster of Paris and papier-mache, students could measure how steep the surface was in various directions. He said they soon understood the mathematics underlying the symbol, and they did so by making discoveries about steepness themselves, as opposed to him explaining the mathematical ideas to them.

He recalled that the papier-mache models were rickety, and he worked with a student to create better models. Wangberg also previously used wooden models with his students and said the wooden models were heavy and required upkeep.

Wangberg explained that he teaches many first-and second-year math courses that are taken by a number of students majoring in science. He said he enjoys having these students in the courses as they are frequently able to connect the mathematics being discussed to a real-life situation or an experiment and interpret the meaning behind the mathematics.

The materials have been shipped to 63 different institutions, and more than 200 instructors from over 72 schools throughout the U.S. have participated in workshops about the materials, Wangberg said. As part of a National Science Foundation (NSF)-funded project called Raising Calculus to the Surface, the materials were created and the workshops were held.

Models have also been developed for another project, called Raising Physics to the Surface, Wangberg said. These models could be used in introductory-level or upper-level undergraduate physics courses, he stated.

The NSF grant for the Raising Calculus to the Surface project is finishing up, Wangberg noted, and he and other collaborators are trying to figure out how to help more instructors access the materials and whether they can apply for more NSF funding to run additional workshops.

Wangberg explained that he and other collaborators are also working on research regarding how the materials are helping students.

Wangberg said math may sometimes be perceived as solving for x, but this solving for x may actually be the last step of the math process. He shared that math allows one to take a complicated situation, such as a manager wishing to improve revenue or a manufacturing process, and refine it so it is a precise situation. This refining occurs in part through studying when similar cases are different and when similar cases are actually the same, he stated. From there, patterns can be recognized and hypotheses can be made. Solving for x or completing a proof might then come in to verify a hypothesis, he explained.

Finding those patterns and making hypotheses about connections between different factors, Wangberg said, are mathematical skills that are useful across the disciplines.


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