A photo of a man standing next to teenagers at a table
Marco Aldi, Ph.D., an associate professor in the Department of Mathematics and Applied Mathematics, leads students in a braiding activity. (Contributed photo)

Twisted in knots? That’s a good thing at VCU Geometry Camp for middle-schoolers

The weeklong summer day program uses hands-on learning to make math meaningful at an impressionable age.

Share this story

It’s the first day of Geometry Camp at Virginia Commonwealth University, and Allison Moore, Ph.D., is handing out brightly colored pieces of rope. Her campers – 40 middle school students from the Richmond area – have gathered in the Math Exchange, a large classroom in the new STEM Building on campus. 

Moore, an assistant professor in the Department of Mathematics and Applied Mathematics, instructs the students to create a typical three-strand braid. Then she asks them to try a more complex braid. Throughout the activity, the group counts how many unique braids they can make and what types of patterns emerge. 

Moore specializes in low-dimensional topology and geometry, especially knot theory. And though her campers might not know it, they are doing high-level math.

“In school, we learn a lot about lines and graphs, but not really this third dimension. Braids and knots give us a way to look at symmetries and patterns in ways that we understand not just with our hands but with our minds,” Moore said. “When we discover math in unexpected places, it's a delight. The math we find in knots and braids is also quite subtle, so students really hone their critical thinking and problem-solving skills while studying these unusual objects.” 

At the end of the week, most campers said the braid activity was their favorite. “I really loved the hands-on activities, where you actually build things. My favorite was probably the braids and knots. I did not expect braids and knots to have anything to do with math,” said Kendall McFadden, a student from Providence Middle School in Chesterfield County.

A teenage girl holding an object made of pipe cleaners. Next to her, a teenage boy is writing on a white board.
Students work with pipe cleaners to construct models of mathematical tangles. (Contributed photo)

The VCU Geometry Camp, a weeklong summer day camp for middle school students, was organized by faculty in the School of Education and the math department in the College of Humanities and Sciences. The camp was made possible through a VCU Breakthroughs Fund grant, which supports transdisciplinary research and innovative approaches to societal challenges. Parents who enrolled children in the camp agreed to take part in the research study, which focused on the role of experiential activities in geometry education. 

“We know from research that middle school is when struggles with mathematics tend to really escalate. That's when students tend to lose interest; math no longer feels fun,” said Rani Satyam, Ph.D., an assistant professor in the math department and one of the researchers on the grant. “Many students also have polarizing experiences in middle/high school geometry. They recall looking at a list of definitions and rules to memorize, without meaning. It's difficult all around. And so that's why we really wanted to target this age group and this subject matter.”

The camp, which ran from June 12-16, focused on three mathematical concepts – symmetry, surfaces and knots – and presented each topic with hands-on learning activities. Besides the rope activity, students used pipe cleaners to investigate braids, created Platonic polyhedra out of paper, practiced drafting regular polygons using math compasses and straightedges, and explored the role of symmetry in art by cutting and punching holes in twisted, rolled paper ribbons. 

“The most important thing is basically to get the student engaged. We picked geometry simply because it's something that we believe is intuitive, fascinating and can be learned without necessarily formulas and technicalities, which is usually the approach that is used in school,” said Nicola Tarasca, Ph.D., an assistant professor in the math department and one of the researchers on the grant. “I really think about this camp as a gateway not only to math but possibly to science in general – and also as an opportunity to support middle school students by letting them realize the power of learning something new and mastering new skills.” 

A boy srawing a circle on a piece of paper
A student uses a math compass to create a Platonic solid out of paper. (Contributed photo)

In addition to Moore, Satyam and Tarasca, the project includes Marco Aldi, Ph.D., an associate professor in the math department, and Christine Bae, Ph.D., an associate professor in VCU’s School of Education.

Faculty also partnered with Hollee Freeman, Ph.D., executive director of the MathScience Innovation Center in Richmond, to serve as an outreach and educational consultant on the grant. Freeman’s expertise on STEM middle school education in the local community helped the project group recruit students from Richmond Public Schools and refine its camp curriculum. 

“I want students to see that learning can happen and should happen in community. It's not something that you have to sit at a desk by yourself and try to slog through a problem, but you really can learn a lot by asking questions and listening and trying out ideas,” Freeman said. 

The VCU Geometry Camp was offered at no cost to the students, with lunch and snacks provided. At first the project group was worried about making their enrollment target, but the camp proved popular. In the end, 40 students were enrolled. 

Owen Marr, a student from Collegiate School, was excited to attend the camp and spend more time learning about his favorite subject. “Instead of just listing out facts, like history and stuff, you actually have to put what you learned to use in math,” Owen said. “The things that you learn, you apply it to a problem to get the answer.”

Two tenage girls holding pipes while standing in front of a table with jugs of water.
Rani Satyam, Ph.D., an assistant professor in the Department of Mathematics and Applied Mathematics, and a student tie physical knots and get ready to drop their knots into sugar water to observe how they move through the fluid. (Contributed photo)

During the week, camp organizers gathered data from the students through feedback surveys, interviews and observations. Ultimately the group hopes to produce a set of engaging activities to share across Virginia and beyond. 

“We have a nationally recognized math education program that prepares mathematics specialists –  teacher-leaders who support and coach their fellow math teachers in quality math instruction. It's something very special about VCU, and we’ve had a huge impact in training math teachers all over Virginia,” Satyam said. “Through our study of the camp, we’ll assess impacts on students’ engagement, motivation and patterns of emotions while working on hard problems. Already, students are seeing that it’s normal to get stuck in math sometimes and that geometry relates to their everyday lives.”