How to Become Good at Math

Math can be a challenging subject for many students at King's. Professor Phil Williams, math professor at King's, provides helpful perspective in approaching mathematics courses.

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Math can be a challenging subject for many students at King’s. Professor Phil Williams, math professor at King’s, provides helpful perspective in approaching mathematics courses.

By Professor Phil Williams

When I think about how people struggle with learning math, I often find myself contemplating some of the other skills, besides math, that I’ve learned to do in my life. One example I find instructive is the example of juggling. Juggling is something that I personally have no natural reason to be good at it. I may have a natural talent for math, but I can claim no natural talent for juggling—none of my faculties are suited for it. I was never good at sports, or physical challenges in general. My hand-eye coordination, for instance, is certainly average or below average. Nevertheless, one summer after college I decided to learn how to juggle. I learned the basic, three-ball juggling pattern in a few days, putting in a few hours each day. I had some assistance: a guidebook. I took the guide seriously and didn’t compromise about what it asked me to do. I fully believed the author of the book knew what juggling was, and how to learn it.

So, how does one juggle? To start, you hold three balls: say, two balls in your right hand, one in your left. Then:

Toss one of the balls in your right hand upward towards your left hand.
While that ball is in the air, toss the ball in your left hand towards your right hand.
Catch the ball that is now coming down towards your left hand; meanwhile, toss the ball in your right hand towards your left hand, and then catch the ball coming down towards your right hand. There is now a ball in the air going towards your left hand. Return to step II.

No one can do what is described here on their first try. It will seem impossible. There is too much to keep track of, and too much that can go wrong. Be that as it may, it was not hard for me to learn how to juggle when I received clear instructions, not simply on what juggling is, but on how to go about learning how to juggle. It took real engagement with the tasks that my guide asked me to do. Each task was a manageable, but non-trivial, accomplishment in training my muscle memory. The tasks were all challenging enough to be interesting. For example, one of the first tasks was to just practice throwing a ball from one hand to the other with enough precision that the catching hand did not need to move at all to catch it. This is not easy at first. And you really must master it before you move on. The next step is a similar challenge, and so on, with manageable steps, until one has mastered the basic juggling pattern.

So, juggling is an example of a skill that requires the right sort of engagement if one hopes to learn anything at all. If you aren’t properly engaging in the tasks required to learn it, you aren’t going to make much progress. This is true of many things. In chess, for example, getting good requires you to play against skilled opponents. Playing a relatively weak opponent in chess teaches almost you nothing. Playing someone even slightly better than you will teach you a lot. Driving is a common task that requires the right sort of practice. I spent years as a passenger in cars (including long road trips), and learned virtually nothing about driving. And, of course, I learned nothing; the driver was doing all the driving! And, it would have made no difference if the driver had narrated to me his thoughts while driving, or occasionally asked me to reach over and press the peddles with my hands or hold the wheel with my chin. That wouldn’t have been the right sort of engagement. To learn to drive one has to actually drive.

The basics of mathematics, the stuff it takes to be comfortable in a college math class, are on par with other simple skills like those I’ve mentioned. If you can learn how to drive, you can learn math. But, like these other skills, you need to engage it in the right way if you hope to learn. The biggest trouble is that, with math, it is often not as clear as it is with other tasks when you are doing something that is not math at all. When you are not driving, it will be clear that you are not driving, when you are not juggling it will be clear that you are not juggling. But, when it comes to math, sometimes you don’t realize that you are the passenger rather than the driver, or that you have broken the juggling pattern, thrown all of the balls in the air and have no hope of catching any of them.

What do I mean by “doing something that is not math at all?” Well, many students who think they are naturally “bad at math” are just misconceiving of mathematics, engaging in activities that resemble math but will not result in actual progress in their mathematical understanding. For many students who struggle with math, all of the hours that they put into math, and all the habits that they develop along the way, are, in some sense, responsible for their ongoing difficulties with math.

What do such students get wrong about math? Well, I believe that students who struggle in math are usually focused too much on how they will be tested. They are preoccupied with what they might be asked to do and remember. They generally believe that what is presented as the actual body of ideas in course—the technical sounding “definitions” and “theorems” in the textbook, along with whatever things that the professor rambles on and on about in class—constitute an inaccessible body of knowledge, something reserved for “math people” to understand. So their focus is on the “practical”: what it will take to pass that test.

The better students, on the other hand, seek first to make sense out of the actual content. Do you want to know the most important thing such students do? They ask, over and over again, a simple question, the same question philosophers ask: “why?” Why does the formula work? Why is the theorem true? Why do we do algebra this way, rather than a different way? Why do we add the exponents when we multiply the variables? Math is all about understanding the ideas. You must ask “why” about things that you may never actually be asked to explain.

I know the strong temptation to ignore this advice and simply learn for the sake of the test. I’ve done it myself. And it sometimes works, at least partially. But it doesn’t build a good foundation. You don’t attain an understanding that is deep, or flexible. You might be able to handle the problems on, say, an exam review sheet, but if I throw something comparable but different at you (which I will do!), you are going to have trouble. In addition, I believe this approach will ultimately cause you a great amount of anxiety, always unsure of what you might be asked to do. You will come to see math as both complex and unpredictable. The more you study in this way, the more the material appears to be complicated, the more it seems like there are more and more special cases to consider, more rules, more exceptions, more things to remember.

But if math was truly like this—a convoluted mishmash of rules, exceptions, and arbitrary instructions—then would anyone ever learn it, much less teach it or invent it? The math that we are supposed to learn in college is part of the curriculum precisely because it is possible for the average student to obtain a clear picture of it. You just have to have faith that there is something there, something that, with some effort, can be grasped.

So my advice to you, when you encounter math in college, is to not settle for half-measures; seek genuine understanding. When you seek to understand the ideas, it will still take work; practice is important in math, and good students practice a lot. I don’t want my advice to seem like some sort of paradoxical “Zen” thing, where you attain enlightenment by finally choosing to see things in the right way. It’s not like that. There will be time spent in confusion, even frustration. Ideas in math can be subtle and intricate, and many require sustained deep thought, and guidance (which is why I am here), to understand. The good thing is, you’ll know when you are doing it right, because you’ll feel it—the same way you feel it in your muscles and your lungs when you truly exercise. You’ll know that you’re mind being stretched, as you seek to wrap it around something real and deep. And trust me that it is worth it to stretch your mind like this.

Professor Williams is the Associate Professor of Mathematics at The King’s College.

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