The Unreasonable Ineffectiveness of Mathematics Education

– October 19, 2012 refsmmat.com

Revised October 23 based on comments on Hacker News and Reddit.

In American schools, mathematics is taught as a dark art. Learn these sacred methods and you will become master of the ancient symbols. You must memorize the techniques to our satisfaction or your performance on the state standardized exams will be so poor that they will be forced to lower the passing grades. Never mind the foundational principles, proofs, or derivations – you’ll learn those in due course.

Why? Why do math? Because you’ll need it, that’s why. You’ll use it in your physics classes. And I’m sure I can think of examples of how you’ll use math in “real life”, whatever your chosen career may be. Right? Right. I hear engineers have to know how to solve differential equations, for example, and before you can do differential equations you need to learn logarithms. So get back to chapter 14 and get working.

This is the message we’re giving our children, and it’s no wonder so few students develop an interest in mathematics. Ask any math major: Math isn’t about memorizing some formulas and learning how to factor polynomials. It’s… well, it’s something much deeper. It’s fascinating. But what is it exactly?

Rules rules rules

The universe appears to work on rules. Gravity tends to pull us to the Earth in the same way every day. Light has behaved exactly the same way for millions of years. Objects in motion tend to stay in motion unless acted upon by an external force. Magnets and electricity obey the same laws now as they did in the 1860s, when Maxwell discovered the laws.

Much of the human world obeys rules as well. Interest and fees accumulate in bank accounts according to rules set down in extremely small font on pieces of paper immediately discarded by account-holders. Internet traffic piles up in buffers and gets routed to its destination according to complicated sets of standards. Engine control modules execute millions of lines of code to read sensor data and process the driver’s instructions to keep pistons flailing in low-emissions synchrony. Aircraft autopilots use clever mathematical algorithms to decide how to keep long metal tubes with wings from falling out of the sky.

I could go on, but you get the idea. Many things follow predictable sets of rules. How can we precisely express these rules in a useful way?

We could use English (or Spanish, or whatever chosen language), but languages are notoriously ambiguous and tricky to work with. It would take many pages to precisely describe Maxwell’s equations of electrodynamics in English, and more importantly, the result would be impossible to work with.

After all, once the rules are set down you’d like to put them to use: make predictions about the behavior of reality, invent new devices, or calculate how many more boxes of ramen noodles you can afford before your paycheck arrives.

How do I move from general rules about the world to specific rules that describe what will happen in one particular situation? With rules written in English, I’m limited to one method:1

1. Write down the rules and the situation.
2. Think very hard.

There’s no system by which I manipulate a rule written in English and arrive at new facts about a specific situation. If I gave you a Monopoly rulebook, could you deduce the best properties to invest in, based on the expected likelihood of a player landing on each property? (If you must resort to written mathematics, you have proved my point.)

Rules aren’t useful if we can’t use them. Fortunately, we have mathematics.

Math is just a bunch of Lego bricks

Forget the mathematics you learned in school. Let’s think of mathematics in the abstract. Mathematics, at its most basic, is a very simple set of very well-defined rules. The rules describe the behavior and interaction of certain completely imaginary objects. Upon these rules, mathematicians have built others. By combining rules, mathematicians demonstrate certain facts about these imaginary objects: when certain objects are arranged in certain ways, the rules show they must have certain properties.

On top of all these rules mathematicians have built a universe. Inhabiting this universe are various objects – tensors, matrices, groups, Hilbert spaces, ordinary numbers, complex numbers, and so on and so forth – which are defined by mathematicians by the sets of rules they follow. Many of these rules are in fact defined in terms of combinations of much simpler rules. If a mathematician wishes to know how a certain mathematical object behaves under certain circumstances, he must simply apply the simple rules in creative ways to discover what must be true.

It’s much like being handed an extraordinarily large and complex Lego brick creation and figuring out what it does and how it works. Lego bricks are exceedingly simple, and you understand exactly how they work. The ungainly creation you’ve been given is made of Lego bricks, so you must simply take what you know about individual bricks and work out what happens when they’re put together. Soon enough you sort out what the stubby-looking bit on the left side does, and you no longer have to worry about the individual bricks: you just know that it’s a frobnulator, and now you understand how frobnulators work. Eventually, with much work, you can deduce what the entire machine does, and write down a set of rules describing its behavior. You can forget about the individual bricks and worry only about the entire creation.

Objects of mathematical construction are much the same, although they are much less painful to step on.2

Glued Legos

High school mathematics doesn’t focus on the very basic rules and constructions of mathematics; they are very abstract, rigorously defined, and difficult to connect to physical reality. Our curriculum instead focuses on certain constructions that relate to reality. Geometry, for instance, is based on a very basic set of rules, but allows us to prove facts about real objects in three-dimensional space. A clever mathematician can wield the rules and basic facts to learn about all sorts of complicated shapes without ever leaving the two-dimensional world of a sheet of paper.

But that skill is not taught in high school. School mathematics is about memorizing the constructed mathematical objects, not learning how to wield the simple rules to build new objects and dissect their behavior.

High school mathematics, then, is much like being given a set of Lego airplanes which have been carefully glued together. You may learn how they work, but you do not have the tools to disassemble them and you haven’t the faintest idea how to build a new one. Should you ever meet these airplanes again, your knowledge will be useful; otherwise, what’s the point? It’s just like an 8th-grader complaining that he’ll never use the quadratic equation again in his life.

But what can you do with the basic rules of mathematics? Why do we care?

We’ve already discussed how many familiar parts of reality appear to follow rules. For many, like electromagnetism, we have no idea how the universe “knows” to follow these rules or why it must use these rules and not some other set. But we can construct complicated mathematical objects that in some ways are exactly analogous to physical objects. We can construct a mathematical object which represents an electric field, specify the rules it operates by, and use the rules to sort out what happens when a mathematical object representing an electron wanders by.

Or we could devise mathematical rules to represent the wobbling behavior of a plate spun and thrown in the air.3

Or we could devise rules describing the grip of a car tire on a road surface in different atmospheric conditions.

Or rules describing how furniture can be arranged in the available space of your living room.

Or rules describing your retirement plan.

Or rules describing… well, anything you can think of that seems to follow rules. Perhaps mathematicians haven’t yet devised mathematical objects that behave in the right ways, or perhaps they have but do not understand how to manipulate them. Perhaps you can write down a set of rules but it’s so horribly complicated that it would take a computer years to determine what will happen ten seconds after you flip a switch. But the rules exist, and mathematics, one way or another, can probably describe them.

Some justify mathematics education as a way to teach students to think critically; that’s good, but they could also analyze essays and literature or study logic for that. We should not teach our students so they can be human calculators and perform simple arithmetic; that is important, but it is hardly the most useful part of mathematics.

We should teach our students mathematics because they can use it to describe reality. They can use it to discover facts about the universe. Facts about their retirement funds, their living rooms, and the rate of fish food consumption in their fish tanks.

Mathematics is a tool to explore reality. We should teach our students to use it.

1. This method is also known as the Feynman Problem-Solving Algorithm, but it only works if you are Richard Feynman.

2. Though I once sprained an ankle stepping on a perturbed Hamiltonian.

3. Where by “we” I really mean Richard Feynman.