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June 27, 2018

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Why Word Problems?

Why do we assign word problems in math classes?

They aren’t easy. My students usually think that word problems are one of the most difficult and traumatic things I ask them to do, and I doubt I’m alone in this. They take a large chunk of class time, and put a lot of stress on our students. We should only teach them if the benefits justify this effort.

They aren’t really fun. Textbook writers try to make their word problems fun, but this generally winds up looking silly and transparent. Even when the whimsy is more appealing than cringeworthy, students don’t actually enjoy them. So “fun” is a bad reason.

And word problems aren’t really applications of math. I have never in my life needed to figure out how quickly the level of water is dropping in an inverted conical water tank. I don’t think I’ve ever heard of an inverted conical water tank, outside of calculus textbooks1. This just isn’t a real, useful application of calculus.

And yet, I do think word problems are important for teaching people how to apply math to the real world. Not because the problems are a good example of applying math to the real world; they’re too obviously contrived to set up a particular type of math problem. But word problems are very useful for developing a specific skill: the skill of translation. And the fact that we as instructors don’t always understand what word problems are for means it’s much harder for us to successfully teach this skill to our students.

Mathematical theorems, on their own, don’t tell us anything about the world. Theorems are always true when the hypotheses are satisfied, but the hypotheses themselves are pure mathematical abstractions and not real-world objects2. So in the most direct sense, you can’t actually apply math to the world.

And yet, we use math to study the world all the time. We do this more or less by a three step process.

  1. Take the problem and translate it mathematical language by writing down equations that describe it. That is, we take a question about the world and turn it into a math question about mathematical objects.
  2. Apply our mathematical tools to this mathematical description.
  3. Take the mathematical answer we got and translate it back into a statement about the world.

For our purposes, step 2 is boring. It’s the same as doing regular non-word math problems. So the interesting parts of a word problem, the parts that can justify teaching them, show up in step 1 and step 3, where we translate the problem into math and back again. And these steps are interesting both pedagogically and philosophically.

Pedagogically, the translation steps are what makes the problem a word problem; but they are also the steps we need to master if we want to use math to understand anything else. Word problems teach us to represent ideas in a mathematical language.

At an elementary level, this is a straightforward exercise in exchanging English words for math equations. We ask something like

Johnny has three apples, and Susie has five apples. How many apples do they have together?

And our students have to figure out that this means they should take the numbers three and five and add them together. This is important, but it’s really an exercise in key words, and recognizing which formula from class to use.

Word problems can be much more sophisticated. In related rates problems, for instance, we have to figure out what the question is asking (“how fast is the water level dropping”) and translate that into math (“what is the derivative of \(h\)?”). We likewise have to translate the facts the problem gives us, like “when the water depth is three feet”, into equations like “\(h=3\)”. Then we do math, and at the end we get a number, and then we have to figure out what that says about the original problem.

The hard part here is figuring out what the problem is asking and how to talk about it. In my experience, when students fail at word problems, it’s not because they do the actual math incorrectly. Rather, they write down what’s essentially a random equation, and then do math to it. They’re missing the crucial skill of using the math to describe the world.

Also undervalued is the final step, of looking at the number and interpreting what it says about the world. Most students won’t blanch at telling me that a person was running a fifty miles per hour, or that a box has a volume of negative ten cubic meters. This is exactly what happens when you treat math as a numerical black box, rather than saying something about the world: there’s no way to look at a number and see whether it makes any sense.

So when we’re teaching word problems, it’s really important to focus on these steps. I tell my students I give them generous partial credit for noticing that their answers make no sense; and of course most of the points on a word problem aren’t for actually solving the system of equations, or whatever. This is also why middle schoolers often get assigned word problems that have extra or missing information that they need to identify: the goal isn’t to make them do the math, but to engage with why they’re doing math.

But all too often, we lose sight of this as instructors, and treat word problems as a sort of transparent skin on top of a math problem that will somehow make it “fun”, rather than as an important skill in their own right.

Philosophically, this layer of translation between a word problem and a mathematical expression answers the question of why math is so unreasonably effective at describing and understanding the world. It also answers the question of why it’s sometimes so ineffective.

Math is completely accurate in describing exactly one thing: math. Euclid’s theorems apply perfectly to lines and circles, but the world doesn’t contain any lines and circles; it contains imperfect approximations of lines and circles. (Or perhaps more accurately, it contains objects that are imperfectly approximated by lines and circles).

If we want to use math to study the world, these imperfect approximations are important. We build a mathematical model of the thing we care about, and then that model is a mathematical object and so we can study it with math. But it’s important to remember that our math applies to the model, not to the real thing in the world.

If our model describes the world really well—as it does in many cases in physics, for instance—then this isn’t really a problem; your math tells you about the model, and your model tells you about the world. But when the model diverges from the real world, your math stops being helpful.

Many of the worst failures of science and prediction come from people doing good math on a bad model. You can see this in election predictions, where most modelers, like Sam Wang, treated results in separate states as more independent than they were. You can see this in the financial crisis of 2008, where most banks had relied on models that assumed house prices couldn’t all drop at the same time. And you can see this in the replication crisis, largely caused by researchers using statistical tools and tests while violating the assumptions that make them work.

Math is a useful tool for studying a lot of things. But you have to remember that no matter how good your math is, your conclusions about the world are only as good as your model. And your model is always a translation, a description of the world, not the world itself. Your model is always going to lose something. The trick is to find a model that’s good enough, and loses little enough, to do what you want it to do.

  1. Disposable paper cups are the closest I can think of, but they aren’t really water tanks. And I’m still never trying to relate the speed water flows out of the cup to the rate at which the water level changes inside it. 

  2. I’d like to dodge arguments about whether mathematical objects “really exist” in this essay. There are a few different things you can mean by the claim that an ideal mathematical circle exists, with varying degrees of plausibility; but no one seriously claims that they have actually picked up an ideal circle with their hands. 

Tags: math teaching word problems homework philosophy models