A mathematician argues that we need to represent math as it once was: an art.
The American educational system is broken. It’s been broken for a long time, and we all know it. Everything points to it. The United States ranks 36th in mathematics. This is below Vietnam, where there are 17,000 deaths per year from tuberculosis; and yet they can educate their students better than the U.S. But even more simply than that, we feel it in ourselves.
Let’s do an experiment shall we? Use yourself as a sample data point and think of the last math course you took. Can you remember what the title of the course was? Maybe? OK. Now, can you remember anything from the course content? Kinda? Something about how to solve for an unknown? Maybe something about angles of triangles? Or was it rates of change or something?
Let’s be serious: Unless you pursued a career in science, there is almost no chance you managed to keep anything from those many hours you spent sitting in math classes, and the many more you spent pouring over math books in hopes of absorbing some modicum of understanding that would safeguard you against a failing grade. This is clearly a travesty. But it really is much worse than that.
Do you have any idea what those things you studied were? Regardless of whether you can remember how to use those tools or not, were you ever told why they were interesting, or what they could be used for? Who decided math was important?
Perhaps you can recite the quadratic formula. OK. Perhaps you even know how to use it to solve a quadratic equation. But can you say why you would want to do so, or what it means when you “solve” a quadratic equation? What can you do with this knowledge? Why did anyone care enough to figure it out? Why has it been retained in the collection of human knowledge, and valued so greatly that is has been passed down over hundreds of years, and finally crammed down the throats of unsuspecting students?
These questions all have answers, and good ones, but they are never
explored in math classes.
Why are we not asking questions more relevant to our humanity? What does a quadratic look like? What does it feel like? What character does it have? How does it change, what are its moods and personality quirks? Should we make friends or guard against it?
We need to talk about the “why” much more than the “what” or else we are just programming our students to perform simple tasks that we could program a computer to do. This is a waste: computers are faster and more consistent if we want to do that, and in the balance we deny the humanity of the student. The thing humans can do that computers can’t is appreciate the beauty of an idea. This is what we need to focus on.
I remember when I was in middle school there was a heavy push by the teachers to elucidate that mathematics was crucial to every profession, and that that was why it was important to study it. The problem was that almost everything they said was a lie.
As a math teacher, when I get asked in the classroom “When are we going to use this in real life?” I make it a point to answer honestly: “You won’t.”
It is just as absurd to suggest that studying mathematics as it is currently taught will garner tools that are going to help you with basic tasks as it is to suggest the same thing about studying literature, poetry, art, or music. These things do not make you a better cup of coffee in the morning, help you change a flat tire, or convince your boss to give you a raise.
Should we really just stop teaching math?
Yes. Stop teaching math.
Now, arithmetic, OK, we should certainly have people who can count, add, subtract, multiply, and divide. But beyond that, it really becomes a matter of enrichment, instead of necessity.
Couldn’t we just restructure the way that math is taught? No, we are past that. Not only is the system broken beyond repair, but the subjects that are not being taught, are mostly useless.
Some people will read this, and wonder why I say this as a mathematician. Do I not love math anymore? On the contrary: I love it as I always have. But I do not love it because I feel it is necessary. I love it as a poet loves poetry. I love it for its beauty and its elegance; I love how it makes me think, and how it makes me feel. Much as I’m sure poets would agree that we do not get more thoughtful people by making them memorize great poems, I believe we do not get more thoughtful people from making them memorize mechanical techniques for solving abstract problems.
We need to represent mathematics again as it once was: as an art.
Math is not like the other sciences, it is closer to philosophy. Science usually is about looking at the physical world in quantifiable ways that are
reproducible, and looking for patterns. Yes, many things that have come out of mathematics are used to look for these patterns, but that’s not what mathematics is.
Mathematics is about starting with an empty universe, and building abstract structures from scratch. These can be shapes, or ways to count things, or ways of guessing what will happen when you do things randomly, but it’s not looking for patterns; it’s about creating them. Most often mathematics is a matter of building structures so abstract that they need new names, because they do not have any concrete analog in the real world.
So if mathematics is not useful for everyday things, what is it useful for? It is useful for enriching life. It enables one to appreciate things in life in new ways that any new perspective gives. The same way studying a new language, or the history of a foreign culture does.
Similar to the way you can never look at big bugs the same way after reading The Metamorphosis, or how hearing Italian opera elicits a different feeling once you have studied the language, studying math gives a new perspective on life. When I saw Yayoi Kusama’s Fireflies on the Water it looked different to me than it does to someone who has never studied infinity. When I hear Bach’s Well-Tempered Klavier it feels deep to me because I know about the density of the rational numbers. When I look at La Grande Arche de la D’efense in Paris, it looks different to me, because I can see that it represents an object from a higher dimension.
These things are not important to practical matters, they just make life more fun. Why else would you ever study anything?
So, if I am proposing that we remove mathematics as it is currently taught from the curriculum of high school and college general educations requirements, what should it be
Much as art, or music, there should remain a basic “math appreciation” course required for everyone. This would give a generalized overview of what mathematics does, and can do, and why it is fascinating in and of
But the essence of mathematics—at the true deepest core of it all—is reasoning.
Reasoning should be taught instead of calculative techniques.
Reasoning is the foundation upon which mathematics is based. It is the essence from which it stems. Mathematics is essentially curious minds mucking about on the playground of reason. It’s the business of making connections, and knowing what is true depending on what you assume. This is a critical skill for life, and applies everywhere. Every time we serve on a jury, every time we vote, every time we make a consumer choice, we make these choices based on a reasoned analysis of the information we have.
The study of reasoning can be present to support both the sciences and the arts. How often have fledgling students found themselves chasing their tails when trying to compose a persuasive essay, because they could not parse the logic of the complex arguments at play? How often have young physicists been left to memorize techniques, because the underlying concept of the tools seemed too complex to deconstruct? How often have students of history been left confused because they couldn’t dissect the political situation of a certain period of time? How often have young chemists come to accept a method merely because they were told “this is how it is done” and then failed to master the underlying principle?
Ultimately if someone wanted to learn the calculative tools being taught these days, people with an understanding of reasoning could easily pick up a book of mathematics and teach themselves.
If we traded calculative math for reasoning, we could perhaps get back to the days when schools were designed to teach people things, and not merely trade the commodities of knowledge and credentials for money.
It seems peculiar that testing of calculative techniques is so ubiquitous. The PSAT, the SAT, the GRE, the MCAT, the LSAT, and even the standardized test for admission to dental school all have calculative math sections. But we all know, with the exception of those pursuing one of the harder sciences, precious little of what is learned will ever be employed.
Why aren’t more relevant skills being tested? It is specifically because math is not really taught. No concepts are given, only a set of abstract procedures. The only way to succeed in this is to build a robust medium-term memory, and store the procedures carefully, and then on the day of reckoning, recall them with perfect fidelity, and perform the instructions for the given situation.
What does this prove if it can be done? It proves that one can operate well in a void of understanding, under difficult and stressful conditions. Because of this difficulty, it has been turned into a tool for separating out those who cannot function under such conditions.
We have come to a point where we are teaching math specifically to make it harder to learn so we can use it as a barrier to entry for other things. Of all the objectionable things about schooling, this is the worst. It would be bad enough if we were merely failing to teach, but that all this is specifically done poorly in order to make things more difficult is a crime.
How did we end up in this mess? We can go back to the history of the development and see that education became a standard right for everyone. Teachers then had large classes instead of the small handful of rich children to groom to be gentlemen scholars. The systems of standardized testing came into being, and were then used as a mechanism for evaluating teacher performance. As soon as this occurred the teachers in question had to revamp their methods from teaching the concepts behind the material, to coaching for test-taking.
However another question looms: Since it is recognized by so many as the mess it indeed is, why has it never been reversed?
Here comes a much more delicate point: While the current system is bad for the student, and also bad for the teacher who has been turned into a mere “instructor” issuing instruction, it is good for a class-based society. Those who have the ability and resources, to teach themselves the conceptual basis for the material, will succeed and make the cut. Institutions seeking leadership material can merely pluck them from the top of the heap, while those who could have been taught but were not will still come out the other end trained to be good worker bees.
If we want to rekindle the American tradition of egalitarianism, and train creative free-thinkers to innovate and break paradigms, we need to remove the current so-called “math” education, and get back to reading, writing, and reasoning.