What’s your favorite subject in school and why is it math?
I remember enjoying math far before I remember admitting it to myself. Be it writing out strenuous long division problems in elementary school, learning to factor quadratics in 6th grade, constructing polygons with a compass in 7th grade (okay, kind of kidding on that one), or solving deriving trig identities in 10th grade, it not only came easily to me, but also seemed fun and satisfying. But if someone asked me what my favorite subject was, I don’t think I ever would have said math, though it may have been the honest answer. It somehow seemed shameful to like something that didn’t seem to be all that important in everyday life. After all, in real life, you have a calculator (keep that in mind, it’s going to come back)
The idea that the sort of math you learn in school is hardly practical isn’t a ridiculous viewpoint. Most people are not required to do integrals or solve algebraic equations in their jobs. Mental math is a useful practical tool, but ironically, that’s not usually what’s emphasized in school. It helps to understand some math when you’re dealing with personal or business finances, but that sort of math is hardly taught either. I’m sure there are other examples, and undoubtedly there are ways we can improve how math is taught in schools, but right now I want to explore a few possible arguments for why the math we are taught in school may be helpful.
Here’s a common comeback to the claim that math taught in school isn’t practical: It’s not so much the specific math tools you learn to use, but the way that math classes teach you to think analytically and systematically about problems. I understand the inclination to immediately ignore this argument, as it comes off sounding a bit vague and non-substantive. However, I do think there is something here. The essence of mathematics comes back to consistent logical reasoning based on certain rules (axioms), and assuming we hope to operate as logical people in the everyday world, math is a great way to train your brain. In math, we usually know the right answer, so you have immediate feedback on whether your reasoning was flawed or not. It’s certainly still reasonable to ask why other classes in the liberal arts or sciences might not be able to teach analytical thinking just as well. For me, math was the best way for me to learn this, but I don’t think that necessarily applies to everyone.
Here’s an argument in favor of learning many different subjects in general: When going through elementary school, junior high, and high school, you’re not likely to know exactly what you want to do for a career. It’s best to learn at least a bit of everything so you can decide what you like best and what you excel most at. I’m quite convinced by this argument in general, but I do recognize that there might be a middle ground where students might be more free to stop taking certain types of classes once they decide they really dislike that area and won’t be needing it in their careers. For my part, I know that continuing to take math and science classes was instrumental in helping me decide what direction to go in college.
Here’s a wishy washy academic answer: Everyone should have some basic knowledge across different disciplines to be a competent, knowledgeable citizen in today’s society. This logic is a bit circular, and this doesn’t really answer the problem about math not being required for everyday life. I think the people arguing this are those who already share an academic mindset (like myself), but I’m willing to accept that this is not the case for everyone.
A lot of math is actually much more useful than you may have thought.
I saved the best for last there to hopefully catch you off guard. I want to spend the rest of this post clarifying what I mean by that. I think that one problem with how school math works is that we are taught how to do the math, but are seldom taught why it might matter for anything in particular. This doesn’t answer all the concerns I raised initially, but I think it is a missing piece to the puzzle.
I’ll illustrate this with one example that continues to astound me (and it comes back to calculators): Taylor series. I originally learned the concept of Taylor series in high school for the second calculus class I took. They were presented immediately after we went through the painful ordeal of proving whether different series converged or diverged (it wasn’t too bad for me, but for most people it was difficult, and made much worse by the fact that no one had any idea why it could ever be important). Taylor series, in turn, seemed to be another useless concept, and hardly seemed to catch anyone’s interest, not even mine.
For those who may not know (or may have forgotten) what Taylor series are, I’ll offer a brief description for context. A series in general just refers to summing up a bunch of terms. Like 1+2+3+4+5+6+… or 1+1+1+1+1+1+… The summation can go on for a finite number of terms or for an infinite number of terms. A “Taylor series” is a summation of polynomials (e.g. x, x^2, x^3, etc.) which can approximate practically any function. To perfectly match a function, the summation would need an infinite number of terms, but you can also approximate a function by just using a few terms over a small section of the function’s domain. If that doesn’t quite make sense, check out the visual below (Fig 1). The sine function is a well-known “wave”, but over a small section, we can approximate it using a series of polynomials. For instance, around x=0, we could approximate sin(x) with the first order polynomial x. If we use higher order polynomials (higher “n” in the figure below), the approximation becomes better and better, as shown.
The function sin(x) can be approximated by polynomials - higher order polynomials get closer and closer to the function itself.
Okay, so who cares? It’s hard to immediately think why this might be useful, and I don’t remember our calculus teacher offering an explanation. Even getting to college calculus, I learned some tricks you could use Taylor series for, but I still wasn’t sure how they could be important in a practical sense. I remember talking to a computer science student in one group I had joined during my Freshman year at UC Davis and complaining that I probably would never use all this stuff I was learning about series in my calculus class. He recognized my concern, but told me that Taylor series were actually incredibly important for a wide range of applications. For example, Taylor series can allow your calculator or computer to calculate values for practically any function at almost any value. Why? Well, it’s relatively easy to use electronics to compute values of polynomials, as this simply involves some combination of multiplication and addition, which is easily carried out by circuit components. However, there’s no general way to immediately compute values for other functions, like trig functions, exponential functions, logarithmic functions, etc. But if you use enough terms in your Taylor series, you can calculate the value of the function to arbitrary precision*.
But this was just the start of my realization that Taylor series are actually important. In a biophysics class my third year, which just happened to be taught by the professor I work with, Dr. Volkmar Heinrich, we used Taylor series in most, if not all, of his lectures. In fact, I imagine that would be the case for many upper division courses focused on some branch of physics. I routinely encounter Taylor series in my research, and recently, I started reading some of Einstein’s seminal papers, and found that Taylor series pepper some of his most impactful papers on special and general relativity!
Admittedly, most of the Taylor series examples above apply primarily to the world of academia. But the lesson is more general - you might actually use some of that math you never thought you’d need to use. For instance, for everyday life, it really helps to understand how exponential functions work - this gives you some intuition for population growth and disease spread, not to mention compound interest (that matters!). And yes, you’ll always have calculators, but calculators wouldn’t be possible without seemingly “abstruse” mathematical concepts like Taylor series, so math may just be more important than you thought.
*Quick technical note: Usually, other methods are actually used to compute function values in computer science because they converge faster. Also, note that to write out a Taylor series, it must be centered on a certain x value (where x is the independent variable of the function). This should be a point where we know the value of the function, and if we choose an x value close to that we are evaluating at, we don’t need to use as many terms in the Taylor series to calculate the value. For instance, if we want to evaluate sin(x) at x = 0.5 radians, we could easily center our summation on x = pi/6 radians, where we know sin(x) = 0.5.