Taking it to the Next Level with Linear Algebra
A Collaborative and Dynamic Classroom for Five Seniors and their Teacher
When students and teachers are equally excited by the subject matter and students themselves are inspired to push the boundaries of learning, there’s magic in the classroom. This year’s Linear Algebra class of five seniors was, for Upper School math teacher Julie Plunkett, just such a moment and a true career highlight. “In speaking to the students at the end,” said Julie, “they felt the same way. It was just an incredible learning and intellectual environment.”
In its fourth year as a full course offering at Friends’ Central, Linear Algebra is open to students who have completed Calculus II. In this year’s class were seniors Ethan Cohen, Kai Davidson, Sean McGoff, Ben Meyers, and Lavinia Wang.
“Linear Algebra is sometimes taught after Calc III,” explained Julie. “At Friends’ Central, we like to do it after Calc II so that our students see that there’s more to math than Calc. There are a couple of different ways to approach the course; there’s a more technology-based approach and then there’s a more abstract approach. This year, the more abstract it was, the better the students liked it. It was incredible. Before I could ask them any sort of theoretical questions, they were asking me themselves!” The students also enjoyed the applied aspect of the material which the course covered, she explained, “but to find a group of students who are actually interested in math theory and abstraction was just such a gift!”
“This year, the more abstract it was, the better the students liked it. It was incredible. Before I could ask them any sort of theoretical questions, they were asking me themselves!”
Julie Plunkett, Upper School Math Teacher
“Out of the Calc I through III sequence, Calc II is the hardest,” shared Julie. “It’s quite literally second-year rocket science. So these kids have been through trial by fire. Another piece of this that was so lovely for me this year was that they didn’t just take the course because it’s the next available course, they took the course because they love math.”
Linear algebra is typically taught in sophomore year of college for math, engineering, and physics majors. Defined in simple terms as the branch of mathematics aimed at solving systems of linear equations with a finite number of unknowns, linear algebra is all around us, crucial to a wide range of fields, including data science, structural engineering, machine learning, image processing, linear programming, fluid mechanics, control theory, network flow, and engineering design. Julie herself described it as “the study of linear systems.”
“We had two major coding people in the group: Sean McGoff and Kai Davidson. Sean’s going to Trinity College in Dublin to study computer engineering, and Kai’s starting out as a computer science major at Worcester Polytechnic Institute. Ben Myers is interested in so many things, but it sounds like he’s going to be taking at least enough math to minor in it at the University of Southern California; he’s very interested in economics, as is Ethan Cohen, who will be going to the University of Pennsylvania in the fall. And Lavinia Wang loves to pair her analytical abilities with her artistic abilities. She will be studying at UC Berkeley.”
“Our final unit was on Markov chains,” shared Julie. “I wanted students to do projects that they were interested in.” For his project, Kai created a program that solved Sudoku puzzles. He wrote all the code, with Julie providing suggestions and guidance here and there. “It was a beautiful, complex program,” Julie said. Sean and Lavinia both independently studied Google’s PageRank algorithm. “And it’s not just about the web page; one of the mathematician’s behind it is named Larry Page!” explained Julie. “Sean wrote code to simulate Google’s algorithm, which included a good chunk of network analysis – something with many real-life applications from traffic control to neuroscience. Lavinia addressed the theory behind the mathematics of PageRank. Sean and Lavinia’s projects were so nice and complementary; their presentations were on the same general topic but entirely different in approach. Ethan did a market resources survey, and he gave a presentation that was so easy to understand; he made it so clear how linear algebra is used in economics. And Ben, in his project, looked at probabilities that, if a word starts with a certain letter, it’s going to be this word, or if a sentence starts with this word, what’s the probability that you end up with a particular sentence. All of this is really complicated stuff, and they all worked completely independently.”
“The Linear Algebra class was so exciting and successful because we explored concepts that approach math from another viewpoint than you traditionally do in high school.”
Kai Davidson ’22
The students were clearly just as enthusiastic as their teacher about the class. “The Linear Algebra class was so exciting and successful because we explored concepts that approach math from another viewpoint than you traditionally do in high school,” Kai Davidson shared. “Ms. Plunkett did an amazing job teaching the class; it was about learning the material and making sure we all understood it rather than strictly following a curriculum. This gave us the flexibility to ask questions and to tailor the curriculum around what the people in our class wanted to learn about. The small group also helped to cultivate this environment, and we all became close while learning the material and worked together to learn the subjects. Linear Algebra serves as a refreshing introduction to a subject many people will explore in college, connects well with topics learned in other classes, and Ms. Plunkett made it a fun and exciting class.”
“We had to visualize different types of planes and almost inconceivable changes to these planes,” said Ethan Cohen. “Ms. Plunkett did a fabulous job of helping us achieve this, whether it was looking at models or doing 3-D demonstrations in class. She creates an atmosphere within her classroom that invokes out-of-the-box thinking and building on each other’s ideas. In our last unit, we were working with eigenvalues and had to complete presentations on this. Every time someone presented, we found everyone in the class connecting each topic back to their own and building onto it.”
“(Ms. Plunkett) listened to every student’s concern and question and answered patiently. She gave us problems to challenge us, but at the same time, she always gave us encouragement to support us and make us confident without questioning ourselves. I absolutely loved this class!”
Lavinia Wang ’22
“The Linear Algebra class was the best class I had in Upper School,” said Lavinia Wang. “Although the class only had five people, everyone was so supportive and respectful. It didn’t feel like ‘teacher and students’ were having ‘classes.’ Instead, our class was more like a discussion about math among friends. Ms. P is also super caring. She listened to every student’s concern and question and answered patiently. She gave us problems to challenge us, but at the same time, she always gave us encouragement to support us and make us confident without questioning ourselves. I absolutely loved this class!”
“These kids are whiz kids; they blow me away,” Julie said. “I’m not a coder myself, but I’m a good debugger, and I was able to help them with the logic. When they go off to college, they will have such an advantage. And just having their minds open to the way ideas can connect in the world of mathematics is so important.”
When Julie first started teaching at Friends’ Central, Calculus BC and Calculus AB were on offer. In her second year, former math department chair Brad Morris asked Julie to start teaching a Calculus II class. More recently, when the suggestion of teaching linear algebra came up, Julie was happy to take on the challenge.
For Julie, the opportunity to continue to challenge herself as well as her students is clearly central to her teaching approach. Recently, in 2016, she received her Master of Arts in Mathematics from Villanova. This was, she said, a “lovely complement” to the Master of Education in Secondary Education, Math, and the Bachelor of Arts in Math which she’d previously earned (both from Temple University).
“One of the things I’ve always been so proud of about the Friends’ Central’s math department is that, for a small school, there’s so much flexibility and so many options,” Julie said. “Because this particular group of students was so advanced, I had a lot of freedom to write different kinds of tests and different kinds of assignments, and I felt proud of myself as an educator because I had to come up with some pretty interesting assessments and assignments, and the students rose to the occasion every single time.”
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