Student and teacher working math equations on a glass wall

By Jennifer Sutton

 

There are several qualities that Mona Zhang ’19 admires in her math teacher, Abby Ross. The coolest one, perhaps, is that Ross can dream in code. “You know how people can sometimes dream in another language?” Zhang asks. “Abby told me how she could be thinking about a problem, and when she goes to sleep, she sees the code and figures out the solution. She wakes up, and she’s got it.” Zhang sighs. “It’s my goal to dream in code, too.”

The way Zhang is going, that goal seems eminently achievable. Last spring, she completed an independent study with Ross in discrete mathematics — a way of looking at numbers that typically is introduced in college. The two of them embarked on another independent study this fall, in linear algebra, another advanced topic. 

“Mona is so self-motivated and driven to learn more,” Ross says. “She created an independent study because she was excited about problem-solving and didn’t want to wait until college to get to these topics. But even if I had said no to working with her, she would have been learning all of this by herself. I have no doubt about that.”

Teacher Abby Ross points to the problem on the board

Teacher Abby Ross didn’t hesitate to work with Mona Zhang ’19 on independent studies in discrete mathematics and linear algebra.

Each year, NMH approves about a dozen independent-study proposals. American Sign Language, portraiture, differential equations, organic chemistry, Korean language, the literary elements of hip-hop — these independent studies have all proven to be “good opportunities for students to follow their interests and demonstrate their ability to complete high-caliber work,” says Academic Dean Sarah Warren. To get the go-ahead, students must submit an extensive application, be in good academic standing, and have a faculty mentor committed to volunteering their time.

What Zhang wanted to tackle in her independent study isn’t a genre of math like calculus or algebra or geometry. Discrete mathematics is a way of thinking about numbers; it’s based on integers, which are whole and finite numbers, with no fractions. It’s like a digital watch: The time jumps from one number to the next, with nothing in between. What Zhang likes about discrete math is that it’s used to solve real-world problems in which things change clearly from one way to another. Google Maps, for example, uses discrete math to figure out which driving routes are most efficient. Zhang says, “I appreciate math as something abstract, but I understand that it doesn’t appeal to everyone, or feel useful to everyone. Math is much more exciting when it is applied. I see it as a toolbox.”

“I could see Mona was diligent, excited about learning, and curious about different ideas, so I kept going where she wanted to go.”

Zhang, a day student, grew up 30 minutes from NMH. Her mother is a professor of computer engineering at UMass-Amherst and her father is a computer programmer. Quantitative thinking clearly runs in the family, and Zhang took to it at a young age. “It just snapped in my brain and I understood,” she says. Her affection for the subject deepened as she got older, even as the concepts grew more difficult. “Math can be subjective, complex, and tricky,” she says. “That’s what I love so much about it.”

Math formula

When Zhang went to what she calls a “math-y” camp one summer during middle school, she realized for the first time that there was a world of people like her — a “math community,” she says. As a ninth grader at NMH, she took Precalculus, a class that typically is for 11th graders. She doubled up her sophomore year with AP Calculus and AP Statistics. Multivariable Calculus followed — that was last fall — and just a few weeks into the school year, she was asking teachers and administrators about doing an independent study in the spring semester. As a kid, Zhang had spent hours working on math problems, so “the idea of working on my own wasn’t strange to me,” she says.

Mona Zhang, NMH math student

Zhang developed a successful algorithm that “optimized the collective happiness” that students felt about their class schedules.

Despite Zhang’s drive, she’s no tunnel-visioned math geek. She plays field hockey and tennis at NMH, and violin, both in the school orchestra and on her own — she likes to busk on the streets of Amherst, Massachusetts, her hometown. She’s part of NMH’s Rhodes Fellowship Course in Social Entrepreneurship, where she’s developing an initiative to help elementary-school teachers get kids more connected to and enthused about their math work. 

Last fall, Ross was Zhang’s multivariable calculus teacher. Ross had just joined the NMH faculty after completing a master’s degree in applied math and data science at the University of Vermont, where she also taught calculus to undergrads. Like Zhang, she had been a math-lover from an early age. A diary she got for Christmas when she was 7 opens with the entry: “I dream of being a math whiz.” 

When Zhang asked about doing an independent study together, Ross didn’t hesitate, even though she was only a month into her new job. “In class, Mona understood everything immediately, but she was also really good at explaining concepts to her peers. She was helpful in a kind way, not condescending at all.” Zhang had already started to hang around after class, asking about math concepts that went beyond the homework assignments. “I could see she was diligent and excited about learning,” Ross says. “She wanted to keep talking about math. I was like, ‘Great, let’s keep talking.’” 

“I appreciate math as something abstract, but I understand that it doesn’t feel useful to everyone. Math is much more exciting when it is applied.”

Each independent study at NMH works differently, depending on the teacher and student. In this case, Zhang found a textbook she wanted to use and Ross mapped a path through it, selecting chapters and problem sets she thought would be most useful. She threw in additional material that she had done herself in college and grad school, and scholarly papers she had read in math journals. They met twice a week in the dining hall. “Mona would think about the scenarios that I would pose for her, and we’d talk through the work, but it was really self-directed by her,” Ross says. “She was curious about different ideas, so I kept going where she wanted to go.” 

Zhang’s main project used discrete math to examine how students’ classes could be scheduled at NMH. She developed an algorithm and learned the programming language Python to implement it. The premise was that there is a finite number of NMH students. There are finite — though many — ways to arrange their classes. And there are varying levels of happiness that students feel about the different schedules. Zhang quantified the amount of conflict in each person’s schedule, with the most-requested courses representing the highest conflict. She also quantified the happiness of each student to determine which arrangement of classes would maximize that happiness. She used the conflict and happiness values she generated to create a theoretical model “that optimizes the collective happiness of students by getting them the classes they want.” 

Over the course of the semester, Zhang also learned about Catalan numbers, which are a series of numbers that occur in certain counting problems, and combinatorics — proof strategies, number theory, graph theory, binomial coefficients — as well as the programming and coding she needed for her project. Ross was a deft mentor. “I liked how Abby let me do most of the talking,” Zhang says. “It wasn’t, ‘Have you done this? Check. Have you done that? Check.’ It was more like, ‘What have you done over the last two days? And what do you want to do next?’”

Catalan number diagram

This fall, Zhang and Ross embarked on a second independent study, in linear algebra. NMH offers the course every other year, and when Zhang couldn’t fit it into her schedule (the irony!), she pushed to do it on her own. Guided by Ross, she’s blending pieces from the regular course with resources from Ross’s grad-school work as well as online materials published by MIT. It’s a prime example of how independent studies can “enrich and broaden NMH’s curriculum,” says NMH Registrar Jay Ward ’68.

One of Zhang’s overarching goals in studying math is to entice other people to appreciate the subject. For her Social Entrepreneurship course, which continues from junior to senior year, Zhang is developing a series of kid-friendly word problems that can be used in elementary-school math classrooms anywhere. She’s hoping to test and implement the project this year. “A lot of word problems out there don’t appeal to kids,” she says — like the classic how-many-apples-can-you-buy-in-the-grocery-store problem. How many kids, she wonders, are actually buying groceries? Why not offer problems that focus on a kid’s bus ride to school, or the trip to go pick up a younger sibling at day care, or the time it takes to
get dressed and eat breakfast in the morning?

“Math problems should be about something kids see in their own lives,” Zhang says. She’s not weighing in on the topics being taught in a classroom, or on the style of an individual teacher. “I want to offer a simple way to get kids who never thought they liked math to see that it can be applicable and relevant in so many ways.” In other words, a toolbox.

 

 

This article was featured in NMH Magazine.

 

 

 

 

 

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