Brandon C. (grade 10), Amy C. (grade 12), and Evan S. (grade 11) debate different mathematical methods after the latest math symposium.
It’s 3:27 on Wednesday afternoon and one last student group is slated to present in today’s math symposium. Math teacher Ted Theodosopoulos tells students that they can leave whenever they need to. Still, the majority of the class stays put, patiently listening, extending their class day to watch the final student presentation. When it ends, a group of students rush over to the whiteboard. The conversation is not over. There are still problems to wrestle with; today’s presentation sparked new ideas that demand more discussion.
Today’s class marks the third round of math symposiums held in Ted’s multivariable calculus and differential equations classes. These sessions — open for any member of the Upper School community to attend — showcase a series of video presentations, where students explain their thinking and demonstrate their methods for approaching different math prompts. Video presentations are followed by a live Q & A among classmates and other attendees.
“There’s a misconception out there that math classes have no room for conversation — there’s a right or wrong answer. But that doesn’t ring true to me as a mathematician,” said Ted. “I wanted to create a space for honest conversations about math, for sharing different points of view, for disabusing students of the notion that there is always one right answer that only an expert can give us.”
I wanted to create a space for honest conversations about math, for sharing different points of view, for disabusing students of the notion that there is always one right answer that only an expert can give us."
For senior Merritt V., the idea of presenting her approach to an open-ended mathematical prompt in front of her classmates and faculty was uncomfortable at first.
“I worried, ‘Did I do this wrong? I bet someone else is going to share a much better method. I don’t know if this is right,'” she said. “But over the course of these sessions, I realized that this is really about learning to think like a mathematician. Instead of teaching me content that I will forget in a few years, it’s about giving me different ways to approach problems.”
Over the course of the semester, Ted designs modules across different mathematical topics such as curves and trajectories, surfaces and optimization, and 3D geometry. For each of these modules, students pursue multiple prompts that offer different ways to engage with the mathematical concepts. They range from the more concrete (problem sets and computational challenges) to the more open-ended (construction projects, writing, and research exercises). Ted asks students to vary their activities and offers them the flexibility to create final symposium presentations individually or in groups.
The symposium — and the presentation artifacts that students produce for it — are part of a larger effort to make mathematical learning and thinking more visible in the school community.
“We watch math like we’re going to the movies,” said Ted, who believes asking students to make videos in advance leads to better presentations and Q & A. “Students have more time to polish the quality of their presentation. Then, during class, they’re more relaxed to listen and ask questions instead of waiting nervously for their turn.”
Over the last month, students in Ted's multivariable calculus class have been studying curves and trajectories. A combination of lectures, collaborative work sessions with classmates, and external reading and videos strengthened their understanding of concepts like parametric equations and toroidal knots, which they called upon to explore their most recent symposium prompts. Here is an example of the most recent writing prompts centered on the torus (the mathematical term for a 3D object shaped like a donut):
How might one wrap a string of fixed length around a donut (torus) so that it comes as close as possible to all points on the surface?
How might we use this approach to detect if our universe is analogous to a 3D version of the donut (torus) surface?
Here are three examples of student presentations:
Cory T., grade 12, decided to work backwards. To think about the first question, he imagined the string already wrapped around the torus and decided to think about an equation to solve for the length of the string.
Merritt V., grade 12, built upon her understanding of toroidal knots to think about the first question. She contemplated how she might think of this problem with an infinite length of string. What type of equation would ensure that a string would not continually retrace the same surface area?
Sebastian D., grade 11, pursued both questions, with an emphasis on the second. He saw the latter question as an invitation to model a four-dimensional torus.
“Often, the first time you read a prompt you don’t understand it,” said Sebastian. “You read it again, don’t understand it. Then you read it again, don’t understand it. And then you read it again and have an aha moment!”
Sebastian thought back to recent lessons on the concept of surface and dimension he explored in last month’s symposium. What would it mean for a three-dimensional object like a donut to have a three-dimensional surface? Wouldn’t that be more like a donut with a fourth dimension? He quickly began researching the concept online. He discovered theories posited by scientists that the universe is like a donut (torus) in four dimensions.
“The reason I cared about this prompt was because I was interested in thinking about a 4D torus. How could you model that?” said Sebastian, who believes the omission of more information in the prompt was intentional on Ted’s part; it was an invitation for students to seek out additional information and discover exciting connections.
“The goal with this work is making sense of math. What drives the sense-making is the caring,” said Ted, who says teaching math this way enhances his own joy in the classroom.
“The way most of us learned math when we were in school doesn’t generate true mathematical thinking; it doesn’t produce inquiry or passion; it taught you how to do an algorithm,” said PreK–12 Math Coordinator Danielle McReynolds-Dell. “How do we get students to a place like multivariable calculus? By laying the groundwork early on, introducing transferable math skills and ways to think about math, so that they have the skills to jump into those classes.”
Sebastian transferred to Nueva last spring from a more traditional high school, where he experienced math as “cookie-cutter AP calculus.” This semester he is taking both differential equations and multivariable calculus, and he finds that the concepts from both classes are constantly melding.
“You make all these back-and-forth connections. It’s like in humanities, where everyone can have the same prompt but attack their theses in divergent ways,” said Sebastian. “That idea is cropping up in math now — the beauty of different approaches.”
It’s like in humanities, where everyone can have the same prompt but attack their theses in divergent ways” —Sebastian D., Grade 11
At the end of Wednesday’s symposium, Upper School students Amy C. (grade 12), Brandon C. (grade 10), and Evan S. (grade 11) are still huddled at the whiteboard talking through the most recent prompts — so focused, they seem unbothered by the fact that the classroom lights turned off 15 minutes ago.
“My method failed,” said Evan S. “But seeing everyone present all these different approaches gave me an idea for a whole new method for solving the problem that no one has tried yet.”