The Value of Advanced Math when Teaching Basic Math
Earlier today while tutoring a high school student in calculus, we hit a rough patch. Implicit differentiation, with a chain rule, leading to a chain rule. The student mutters “Uhhh why did they make math like this.”
Oh man. Oh boy did he get more than he bargained for with that question. Me: “In fact this just is what it had to be, as we tried to build a mathematical theory of physics. This is the end product which started with Aristotle’s physics, and later attempts during the Scientific Revolution to overturn Aristotle’s physics with a new mathematically measurable and precise physics. …” And so on as I tried to give a standing-on-one-foot history of Calculus.
He laughed, I did go overboard. I sometimes can’t help but get chatty about things I’m passionate about, and lately I’ve probably had an unhealthy obsession with the history of math and science. But I think that actually might have been an important lesson.
Rote calculation is beloved by high school curricula, and with good reason. It doesn’t depend on the teacher’s talent—which is a good thing because if your system depends on inhuman talent then most of the time it will fail. But obviously rote calculation misses the soul of the subject. This is where it can be very valuable to have a teacher who knows 10 times what is necessary to teach the fundamentals. You can supply context, meaning, motivation.
History is in some sense unimportant—I mean, it’s done right? Whether something has a sensible historical origin or not, it should be measured by its current utility, or so a member of homo economicus would argue. But that’s just not how humans work and think. Knowing that Calculus came from somewhere and someone, to solve a problem that mattered to people, helps to breathe life into the subject.
Later in that lesson, when discussing a line perpendicular to a horizontal line, the idea of infinity became relevant. I forget how but I guess I must have mentioned it a little too much, and the student said “Huh, but infinity isn’t really useful right?” Oh man. Oh boy, here we go again.
No! Actually you could view all of calculus as just the study of the application of infinity to real world problems. The derivative is in some sense “the difference quotient, but taken ‘infinitely close’ to a zero x-differential”. The integral is even more obvious, it’s so-to-speak an infinite sum of infinitely small rectangles! But calculus isn’t the only place where infinite appears in the sciences. In computer science we still look at the asymptotic behavior of functions, which is effectively the limit of a certain thing taken at infinity. Also in theoretical computer science, we consider the space of all possible computers, which is an infinite space. Importantly, there are functions which live inside this space (so-to-speak) and functions which live outside the space—this is one application of the idea of different magnitudes of infinity.
I think the ability to gesture, as I did in these cases, to the deep principles of how the universe works, is a better motivation for a student to do their homework than grades. Grades suffer from Goodhart’s Law. They are supposed to be a measure, but they become the goal, and therefore cease to be a good measure. They also make math seem like a cruel test or game invented to bore kids while adults are at work. I don’t think anyone focuses their mind and feels like their task is a reasonable one, more than when they understand why it is what it is.