At Choate, students are fortunate to have a wide and diverse collection of courses to choose from — they are largely given the power to explore their own interests by taking more advanced courses in whatever subjects they prefer. Choate’s math courses are no exception — and yet, I believe there is room for improvement.
At times, the math sequence can seem awfully linear, giving students the false impression that the study of mathematics is a simple series from geometry to algebra to calculus. Since most students only have space in their schedule to take one math course a year, those who are genuinely interested in mathematics have to sit through the entire required sequence before being able to delve into advanced topics such as linear algebra or multivariable calculus. The reality is that there is much more to math than these topics. Take, for instance, differential equations.
What exactly is a differential equation? For students that have taken calculus, the answer is somewhat straightforward; a differential equation is an equation that relates some function with its derivatives. This begs the question: what is a derivative? Put simply, the derivative of a function f(x) is another function, f’(x), that corresponds to the instantaneous rate of change at each point in f(x). In essence, a differential equation is an equation describing the values of a function in terms of its own rate of change.
This winter, I had the amazing opportunity to take a directed study in differential equations, led by Mr. Guelakis. Our course reviewed differential equations and a range of methods used to solve them, including undetermined coefficients (basically a glorified guess and check system). I took this course as a second math class, and the work proved extremely difficult. But if I had the opportunity to do it again, I would jump at it.
I found that this course allowed me to gain a better understanding of physical systems and how nature behaves. One of the most important exercises we did was finding the general solution to a linear second-order ordinary differential equation, which is the type that governs the motion of pendulums and the voltage in an RLC circuit, an electrical circuit containing a capacitor, resistor, and inductor connected in parallel or series. Many of the techniques used to simplify the resulting expression, such as certain substitutions for constants, are not at all obvious, but they carry an important significance in the physical behavior of the system, helping me to better understand why the laws of nature work as they do.
Learning about differential equations is not only important for its own sake, but it is also important for students who aim to pursue studies in STEM beyond high school. The central ideas behind differential equations provide a framework for students to analyze real-life systems as well as a solid base for students planning to pursue studies in STEM fields, especially engineering. As Steven Strogatz, a professor of applied mathematics at Cornell University, has written, even since Newton, “mankind has come to realize that the laws of physics are always expressed in the language of differential equations.” The laws of electromagnetism, waves motion, heat distribution, fluid flow, oscillation, and even quantum mechanics are all written in terms of differential equations.
And so Choate should offer differential equations as a course, exposing more students to the math behind the fundamental laws of nature. The theory of differential equations is more than just a theory, and it has numerous applications in the physical world — in physics, chemistry, biology, and beyond. Five students signed up to learn about differential equations in a single term; surely more student interest exists. I hope that before long, many more students will be able to appreciate the math that so beautifully explains the laws of the physical universe.
