Students turn hard-to-grasp calculus concepts into handheld 3D models for educators around the world

Patrick Naylor says it’s the best money he’s ever spent.

He turned 20 cents worth of plastic into a manipulative – a 3D model of an abstract math concept. That 20 cents also led to a research project for undergraduate students Sarah Asrani and Hannah Nardone,

The calculus concept that Patrick turned into a manipulative became far easier for students to grasp once they held it in their hands. Patrick, an assistant professor in the Department of Mathematics & Statistics, kept the manipulative on his desk for an entire term and handed it to every student who dropped by during office hours.

He says the manipulative proved to be an effective and efficient teaching tool. “I could explain the concept or show it on a screen. But when students picked up the manipulative, they immediately understood the concept. I didn’t know exactly how or why it worked – I just saw that it did over and over again without fail.”

He talked about his low-cost, high-impact teaching tool with assistant professor Caroline Junkins, not knowing she’d used manipulatives while at Harvard. A former colleague made them for her – Patrick also had a former colleague at Princeton who was a technical expert when it came to manipulatives.

Assistant professor Lee van Brussel joined the conversation – he’s working on a project with McMaster’s Accessible Education Fellowship to make math courses accessible for students with visual impairments. Manipulatives are one of the ways to achieve that goal.

Patrick saw an opportunity to put students to work. He approached the Faculty of Science’s Office of Undergraduate Research and they offered to partially fund and promote the project to students. The department helped cover the rest of the cost.

Sarah and Hannah saw the job posting and applied. Both were looking for research experience – Sarah, who’s done a lot of tutoring, has an interest in understanding the different ways that students learn math.

While they had some coding experience, neither had done any 3D printing. That worked in their favour, says Caroline. “It can be hard for undergrads to put themselves out there and do something they haven’t done before. Hannah and Sarah were so enthusiastic and willing to try new things even if there was no guarantee of success.”

It also helped that they’d been standout students in courses taught by Caroline and Patrick. Sarah had also been a mentor in MacPRIME, a four-week online summer program for incoming first-year students created by Caroline.

The manipulatives project kicked off with the students talking with Patrick and Caroline’s former colleagues for lessons learned and best practices when it comes to 3D printing manipulatives. Lee also connected the students with experts in elementary to secondary school math.

And then Sarah and Hannah got to work, with Patrick asking them to focus on concepts from the second-year multivariable calculus course that had tripped up other students.

They quickly figured out how to use introductory and then advanced modelling software and started printing their models at the makerspace in the Thode Library.

For her first manipulative, Sarah stood over the 3D printer in the Thode Library watching her coding slowly turn into a plastic model. There was also one concept that took the better part of a week to troubleshoot – the model kept coming out wrong. She eventually figured out and it became one of her favourite manipulatives.

By the end of the project, Sarah and Hannah had created dozens of manipulatives for everything from a Riemann sum and line integrals to vector fields and torus knots. Their creations line the shelves in Patrick’s office – there’s now lots to choose from when students drop by for office hours.

Hannah and Sarah’s creations will eventually make their way out into the world. Patrick plans to upload the students’ files to a repository that educators can download free of charge. Sarah and Hannah’s names are embedded in the code and each manipulative prints with the McMaster University logo on its base.

“It’s pretty cool knowing that we’ve made a contribution to the world of mathematics and we’re helping students learn,” says Hannah. Working on the project also forged a friendship between the two students.

Caroline is looking at how to give all first-year math and stats students the same experience as Hannah and Sarah. Having every student code and print their own manipulative would make for an early introduction to the Thode Makerspace. And Patrick says manipulatives make for excellent holiday ornaments.

While the project’s wrapped up, Sarah and Hannah continue to help other students by serving on the McMaster Math & Stats Society – Hannah’s the VP Academic while Sarah’s the VP of Communications.