Concord Academy’s Mathematics Department offers a variety of courses for students of different interests and backgrounds. To ensure equity, the department provides options for both advanced or introductory and passionate or indifferent students. For those who have already surpassed CA’s standard and accelerated math paths, Advanced Topics courses are often their go-to. Topics rotate in a two-year cycle and this semester the course offered is Advanced Topics: Vectors and Variables (multivariable calculus) taught by Mark Engerman.

Advanced Topics courses began more than 20 years ago. Back then, CA offered these courses after school for the convenience of Concord-Carlisle and Middlesex students—who at that time did not have the opportunity to take college-level math courses. As those students got alternatives at their own schools, Advanced Topics was moved to the school day. Over time, an increasing number of students finishing calculus before graduation demanded implementing more frequent advanced classes. Thus, the department redesigned the course nine years ago to take place every semester, rather than every other semester. 

Advanced Topics courses aim to expose students to college-level math classes in no particular sequence. This way, students can take different Advanced Topics courses in different orders. 

This year is only the second time that Vectors and Variables has been offered. Engerman proposed this course having been inspired by his challenging experiences in college. When he studied multivariable calculus freshman year, he found it much more difficult than AP Calculus. Engerman wanted CA students interested in pursuing math-related fields to get an early grasp of these difficult concepts and to therefore experience a smoother transition into college-level multivariable calculus. 

The units incorporate three-dimensional geometry, parametrically defined vector values and functions, derivatives, partial derivatives, maximization, and integral and multivariable calculus. According to Engerman, the most difficult part of this course is that many concepts are extremely hard to visualize and are counter-intuitive. He hopes to teach it more on an intuitive rather than rigorous side by doing more applications. 

Lisa Liu ’22 notices the same things and agrees with Engerman on the utility of this course. Having taken three other Advanced Topics courses, she observes that this particular course is less proof-based and involves more problem-solving skills. She also thinks that it is much more applicable than other topics courses. “Normal vectors, velocity and acceleration equations, and just integrals in general connect back to what we have been learning in physics,” she says.

While Advanced Topics courses are generally recommended for students who have excelled in Calculus courses and are highly interested in math, there are a myriad of other options to choose from, such as economics and statistics. Ultimately, the department hopes that students will be able to use math in different ways that can still be interesting to people at different levels.