“When are we ever going to use this?”
When I was in high school, this sort of question was a frequent occurrence in math classes. As I grew older and the math we were learning grew increasingly esoteric, this line of questioning cropped up more and more frequently. Students wondered why we weren’t being taught something actually useful, like how to do our taxes. The responses we got were usually some variation of “well, suppose you’re walking home one day and you see a car traveling north at 37 miles per hour…”. I don’t think anyone found this very convincing. Even worse were the textbook’s promises that we might one day use our skills to do fun activities like 3D model platonic solids.
Despite the fact that math’s frontline defenders were not particularly sturdy, I was never worried. It seemed obvious to me that, however discontented my fellow students might be, the responsible adults in charge of things would never seriously consider a plan to replace math with tax class. Recently, I’ve become aware of a disturbing trend in the opposite direction. Pundits are questioning whether the mathematical learning lost during the pandemic was really so bad. Parenting blogs are suggesting kids learn math “naturally” by doing things like counting their steps. College professors are writing editorials with titles like “Is Algebra Necessary?”.
I can’t help but to feel like math could use some defending. With that in mind, I’ve summarized what I think are the best arguments for advanced high school math education below. I’ll conclude with a few suggestions about how our current educational paradigm might be improved.
Math is beautiful.
A proof is a beautiful thing. You start with some premises, meditate for an hour or so, and on the other end you have a piece of new knowledge about the world grown entirely from your own mind. This chain of events can’t help but make you feel closer to the universe. Anecdotally, it seems like this has resulted in math departments being the happiest and most self-actualized groups on campus. There are a number of math professors at Yale whose passion and will to live probably outshines the entire chemistry department.
The actual process of working through a proof is meditative in the extreme. It requires a level of inward-focused concentration that we rarely achieve when doing anything else. In my opinion, denying this sort of experience to students in favor of teaching them to fill out a tax form would be akin to replacing literature with a vehicle operator’s manual.
I’m aware that many people won’t find an aesthetic argument for math education particularly convincing. That being said, there are a lot of less-practical subjects we justify based on their aesthetic value to students, so this shouldn’t be entirely overlooked.
Math is hard.
Of all the core subjects we learn in high school, math is almost certainly the hardest to self-study. There are increasingly online resources available to help teach mathematical concepts, but learning a paradigm-shifting concept like algebra from YouTube videos is not easy to do. Tax documents, on the other hand, have the instructions written on them.
Why anybody thinks we should cut back on the hardest subject to teach oneself rather than one of the many classes which consist of some variation on being told to read books1 is beyond me.
Math is useful.
Over the last couple of years, I have come to the conclusion that I and other students grossly underestimated the amount of math that gets used in people’s real world jobs. I actually chose not to enter college as a math major because I thought my only professional option would be teaching math to high schoolers2. This could not be further from the case. As I wrote in my post on the elite job market, a huge proportion of students from top universities go into consulting, finance, and tech. Almost all of these positions rely heavily on math. Some machine learning and quantitative finance jobs are more or less all math, all the time.
I’ll concede that not everyone will use calculus or discrete math every day in their job, but the highest paying jobs almost universally do. I’ve got a hunch that Harvard-Westlake and Exeter aren’t cutting their math programs to teach students how to do taxes. If you want to ensure that public school students never compete for top positions again, slashing math department budgets would be the way to do it.
Mathematicians make good citizens.
We live in an increasingly complex world, where policy-makers rely more and more heavily on statistics and mathematical models. We are rapidly approaching a world where decisions are actually made by the mathematical models themselves, in the form of neural networks. It is in this context that the high school students of today will be expected to cast informed votes.
A frequent talking point on this subject (and in fact one brought up by the anti-algebra editorial which originally drew my ire) is that this implies schools should focus on teaching students statistics instead of algebra and calculus. I don’t entirely disagree with this; AP Statistics is the high school class whose knowledge I apply the most today, and I think it is crazy that statistics isn’t offered at more schools. That being said, the idea that we can just replace most advanced math classes with a single statistics course is ludicrous.
First, a lot of statistical concepts are built on concepts from calculus; if you don’t teach the underlying math and just expect students to take the argument on faith, you are defeating part of what makes math so powerful in the first place. Second, there are a lot of important policy decisions that hinge on mathematical concepts like financial models, which are outside the scope of statistics.
If we’re ok with living in a world where citizens vote for the candidate with the best hair, then cutting math classes makes sense. If we want a functioning democracy, advanced math is a must.
What now?
Hopefully you’re convinced that advanced math education is something to be cherished and preserved. Unfortunately, our current math curriculum3 seems to leave something to be desired. The aforementioned editorial contains some anecdotes to the effect that math is the most hated and dropout-inducing high school subject, and, unfortunately, this seems to match my own experience. Is there any way we can do math classes better?
I’m not a pedagogical expert, but I think the arguments above suggest at least two major improvements that could be made to the state of American math education. The first is simple: let students write proofs! The fact that our math curriculum so carefully shields students from what mathematicians actually do boggles my mind. Proofs are more engaging4 and teach problem solving skills that, unlike memorized formulas, are not going to be forgotten after the next exam. As a bonus, proof-based classes can build from the ground up in a way that makes students less reliant on undisrupted learning in years past.
The second major improvement would be to spend more time reading cases built on applied math, like scientific research using statistical tests or econ papers using mathematical models.5 This is the math skill that is most key to skepticism and functioning democracy. It would also help students appreciate the many ways in which more complex math gets used in the real world.
I believe that with these changes, math education could live up to its full potential as a beautiful, meditative, social-mobility-increasing, democracy-preserving force for good. But hey, why do all that when you could be teaching kids what line of the tax form they’re supposed to be writing their name on?
Books are great! This is not intended to be anti-book.
Math teachers are great! This is not intended to be anti-math teacher.
When I make this bold, universal statement, I am actually referring to the math curriculum in my state three years ago.
Yes, this value judgement is true and I refuse to perform a randomized controlled trial to prove it.
As a bonus, throw in some papers that got the math wrong and see if anyone can pick out how.