# 3D printing creates math formulas that students can hold in their hands

# 3D printing creates math formulas that students can hold in their hands

A Penn State Hazleton professor is using 3D printing in education in a unique way: creating solid representations of mathematical formulas that help students better understand mathematics.

Alfredo Jimenez, associate professor of mathematics at Penn State Hazleton, teaches calculus to engineering and science students, and said he is always looking for new ways to teach. “I have been teaching for about 35 years, and I am always interested in learning more about teaching, as I think there's an enormous amount of knowledge that can improve my teaching,” Jimenez said. “So I think one of the theories I became hooked on is the theory of constructivism, where students construct their own knowledge, so as a result of that I have been constantly searching for ways I can help students construct their own knowledge in the courses that I teach.”

Due to his interest in constructivism, Jimenez was intrigued when he saw Joseph Ranalli, assistant professor of engineering at the Hazleton campus, give a presentation on 3D printing. This got him thinking. “I thought at that time it would be really neat if I could produce some solids with the programs I have been producing using (math computation program) Mathematica, to give students a visual aid for math,” he said.

Jimenez knew at the outset that he wanted to use 3D printing of models in his classroom, he just needed help figuring out how to get Mathematica to export some clean, printable files. With the help of Penn State’s Teaching and Learning with Technology unit, which recently partnered with the University Libraries to open the Maker Commons 3-D printing lab, Jimenez was able to accomplish this to create test models to work out any kinks. Ronald Harman Jr., information resources and services supervisor/manager from Hazleton’s campus library, did post-processing on the models to enable them to be printed properly, then printed the actual final models that Jimenez used in his class.

“It was really nice, because for the first time I was holding in my hand a solid, 3D version of something that I had been drawing on a board or showing images to my students for so many years,” Jimenez said. “That was very useful for me, because it made it easier to explain the mathematical concept, it gave me a real visual. I thought ‘whoa, this could definitely have great possibilities for use in my calculus class.’”

An example of how Jimenez has used 3D printing in his class is to help explain an important concept in calculus, the definite integral to compute volumes of solids. “So what I did was to produce a series of about seven to eight different solids,” he said. “I came to the classroom and divided the class into groups. I gave each group a solid with a simple instruction, to compute the volume of the solid.

“I think this is very interesting because normally what I would do is give the students some direction, some equations, and then the students follow those directions. But this time around, they have the solid in their hands, a visual to guide them in what they are doing. Plus now they have to find the equations to describe the solid on their own.”

Jimenez said that sometimes students have a great deal of difficulty trying to visualize a solid represented by a given set of equations. But now, he notes, they can hold the solid in their hands.

“I think that having the solid really helps them understand,” Jimenez said. “I can actually give them a representation of the approximation process because I give them the solids. This is something that is very easily done and it greatly enhances the learning process.”

Jimenez said this could be taken one step further for advanced students. He said an interesting project for higher-level calculus students would be to produce a program to model a particular solid and then print it to see if the model produces the desired solid. He said students could even propose their own program and equation, 3D print it, and see if it is what they expected.

“It is changing the parameters tremendously,” Jimenez said. “As math teachers we are teaching concepts and mathematical ways of thinking, which can be very difficult for some students. Having a concrete, tangible object that represents some of these concepts and ideas is a very positive new development that is worth exploring, because it makes mathematics a real thing that they can see before them.”