Let's Get Real About Math Word Problems

OK, here is a math word problem for you: How many different ways can you arrange five keys on a circular ring?
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OK, here is a math word problem for you: How many different ways can you arrange five keys on a circular ring?

Before you read any further, you might want to try to solve it.

This very problem came up on my Twitter feed some while ago. (I follow a number of sources that generate this kind of thing.)

If you are well trained in Pre-Common-Core school math, you probably jumped right in and came up with a number, most likely 12. That's the answer I would surely have come up with when I was at school, having been well trained to perform like a mathematical circus animal.

But today, as a professional mathematician who uses mathematics to try to solve real life problems, my reaction was very different, and so was my answer: 3,840. The kind of thinking I brought to the problem was exactly what the Common Core is supposed to promote: sensible, reflective thinking that can be of real use.

(But note my use of that word "supposed" in that last sentence. Though rejection of the Common Core mathematics standards would leave US children unable to compete with the rest of the world in science and technology, there is much to worry about when it comes to some early attempts to implement the standards.)

My first reaction to the key ring problem was this:

What a nice little problem to form the basis of a class discussion. Sure, it is a bit contrived. How often has anyone needed to know how many different arrangements there are?" But that aside, it has a number of features that give it great pedagogic potential:
  • it is simple to state
  • it is easy to visualize
  • a teacher could get the class to each bring in a key ring and some keys, and begin the investigation experimentally, first with one key, then two, etc.
  • the question evidently has multiple solutions, depending on how you interpret what is being asked, making it a super exercise in mathematical modeling of a real-life situation
  • those different solutions are in some cases orders of magnitude different
  • the simplicity means that students should be able to formulate their own models and put forward clear explanations of why they chose the particular model they did.

Just what the (eight) Common Core Mathematical Practice standards ask for!

The Tweeter who posed the question posted a subsequent tweet giving the answer 12. That individual presumably interpreted the question as (in math-problem-speak): "How many ways are there to order five points on a non-oriented circle having no distinguished point?" If that is the question, the answer is indeed 12.

But that was not the question as posed, which was about keys on a key ring. If you stop yourself from switching off your common sense and going into math-class-mode, looking for the fastest way to put numbers into a formula and get a number out, and instead approach the problem for what it is, a straightforward question about a real-world situation, you will find yourself on a very different trajectory. A trajectory of thinking as opposed to (mindless) computing.

Here is how I tackled the problem, approaching it as I would any mathematical problem I have been hired to solve.

First, I reflected on what the problem is about. Well, it's about keys and keyrings. I know about those. In a drawer at home, I have several key rings I have acquired over the years. So many that I suspect that, from that one source alone, a future historian might be able to retrace many places I have visited and institutions I have interacted with.

Those keyrings all have one feature in common, namely a topologically circular ring, and the majority share a second feature, namely to the ring is attached some kind of object, often a badge or shield, sometimes a ball-shaped object, an occasional pen or flashlight, etc.

I'll add that some of those badge-like attachments have two distinct faces and are rigidly attached to the ring, some have two distinct faces but the entire badge can be rotated on an axle, and some are symmetrical so that the entire keyring looks the same from both sides. ("What difference can these features make?" a student might ask. If they don't, a teacher should ask them.)

I also have an even larger collection of keys. (Why do we keep old keys, by the way? I have no idea what the majority of them lock, and for sure they no longer fit anything I still have access to.) A very small number of those keys are symmetrical: when I turn them over, they look identical. (Again, is this significant to how we model the problem?) But the vast majority do not have such symmetry.

In my case, when I modeled the keyring probem, I based my model on my nostalgia-based collection of real keyrings and my irrational collection of old keys. In other words, I did what I think any sane, sensible person would do outside the math class: I took the problem at face value, as asking about real keys on real keyrings. I based my solution on what I most commonly saw in my drawer.

I then had to decide what constitutes an "arrangement". Again, I interpreted this word (I almost said "key word", but decided that would be too tacky, though puns are their own reword) in the way I thought most natural. The problem asked about different arrangements of keys, not about permuting them around the ring. In particular, if I change the orientation of one of the (asymmetrical keys), I change the arrangement. Again, there is a story to be told, and a decision to make, when symmetrical keys are involved. And I wondered for a while what to do about having two or more identical keys on the same ring, when the mathematical model you choose could depend on why the question was asked in the first place.

In the end, I came out with 120 x 32, namely 3,840. A far cry from the "expected", pre-Common-Core classroom answer of 12. Which of us is right?

Neither. There is no right answer for the problem as posed. It's a modeling task -- using mathematics to analyze something we encounter in the world. As such it's a super example of how mathematics is used in the world on a daily basis. (What people like me get paid to do.)

The fact is, when used wisely, "simple" word problems provide great exercises in the kind of mathematical thinking that the world needs. The kind of thinking we should want our kids to master: thinking that provides reliable answers to real questions.

As a nation, we should stop the current suicidal cry to turn back the clock to a form of math teaching that did no one any good and which those of us who became professional mathematicians first had to unlearn, and focus on making good, practical, sensible use of centuries old teaching tools (such as word problems) to produce a generation well equipped for life in the Twenty-First Century.

NOTE: Much of this article is abridged from a much longer post, which first appeared in my Mathematical Association of America "Devlin's Angle" column in May 2011.

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