Sunday, August 6, 2017

Writing Word Problems

On Reddit there was an AMA request asking to hear from people who write word problems. I'm not on Reddit and couldn't participate in the discussion, but a friend forwarded me the link, so I thought I'd take up the questions here.

Although the AMA was about word problems, there is at least as much to say about writing "bare calculation" practice problems—see for example the carefully designed worksheet in this article—and about writing good conceptual questions, such as Phil Daro's eye-opening grade 3 problem of using a number line to show why 4/5 is closer to 1 than 5/4. Maybe procedural problems and conceptual problems could be topics for later posts. For now, let's talk word problems! The person making the AMA request asked some good questions.

1. How did you get into the profession?

I guess I've been doing this for a long time. As part of tutoring you have to make up problems, and I started tutoring my friends when I was in grade school. My first job writing problems per se was when I was a physics Ph.D. student, and I was tasked with designing a curriculum for introductory physics; this was part of an effort by UC Berkeley to expand on the success of Uri Treisman's groundbreaking work in Calculus. I believed then, and believe still, that problem quality is essential to the success of a curriculum. In 1992 Treisman wrote of his original project,
Most visitors to the program thought that the heart of our project was group learning. They were impressed by the enthusiasm of the students and the intensity of their interactions as they collectively attacked challenging problems. But the real core was the problem sets which drove the group interaction. One of the greatest challenges that we faced and still face today was figuring out suitable mathematical tasks for the students that not only would help them to crystallize their emerging understanding of the calculus, but that would also show them the beauty of the subject.

After graduate school I became a professor, and when you're a professor you have to make up tons of problems for your students. I collected my best problems about force and motion and included them in my physics textbook. Some of the puzzles I post on this blog could potentially be word problems in school (this one, say); others, not so much. Some of my Saturday School posts have word problems in them too.

Sometimes my job involves writing math word problems at the K–12 level. Often the point of these problems isn't necessarily to be copied down and given to students, but rather to be discussed by educators; I want to stimulate conversation about the nature of the curriculum itself. One valuable resource for that conversation has been Another has been the material on, and yet another has been Table 2 in this document, which describes the fifteen different kinds of basic addition and subtraction word problems. (Fifteen!)

At higher grade levels, here is a word problem that I recently included in a presentation:

In algebra, you often have to rewrite expressions in different forms. The point of this problem is to suggest that algebra teachers might try to include some problems in which rewriting a given expression has a rationale, however slight, within the context of the problem. My CPU problem isn't the best possible example…what I'm trying to do is encourage the field at large to work on creating more and better problems that serve the purpose. The inspiration for the CPU problem was a time last year when I was trying to speed up a loop in one of my own programs, and factoring an expression did the trick. (Math: it works!)

Writing word problems is rewarding, but I also like to find great word problems from the past and keep them alive. See my previous posts about Newton's problem of the oxen and Wells's Academic Algebra textbook from the 19th Century.

Here is a fascinating problem that was a favorite of Leo Tolstoy's:
A team of haymakers was assigned the task of scything two meadows, one twice the size of the other. The team worked half a day on the larger meadow. Then it split into two equal groups. The first group remained in the larger meadow and finished it by evening. The second group scythed the smaller meadow, but by evening there still remained a portion to do. This portion was scythed the next day by one haymaker in a single day’s work. How many haymakers were there in the team?
The Tolstoy problem isn't easy. Yet elegant shouldn't have to mean difficult. Sometimes, a simple word problem can be almost poetic in its spareness.

Here is an elegant short problem in mathematician Roger Howe's essay "From Arithmetic to Algebra":

A last brief example is this penetrating and original problem about algebraic structure, from Illustrative Mathematics:

It might be the case that in recent decades, the best minds in math education have spent too much time injecting lengthy problems into the grades K–8 curriculum, and not enough time transforming and elevating the teaching and learning of simple problems.

I wouldn't hazard quite the same generalization about high school, because the high school curriculum probably needs more high-quality application problems, which tend to be lengthy. Here is a high school application problem I wrote once:

The next high school application problem was adapted from one published by the Shell Centre.

The next was adapted from a problem developed by COMAP.

(This problem should really be formulated using a google doc or Excel spreadsheet, not a printed page. I think spreadsheets are used too seldom in high school.)

2. It's assumed that you write questions below your skill level. Do you write questions at your skill level?

Not so much anymore, but "writing questions at your skill level" is a pretty good thumbnail description of doing research—so when I was a researcher, I did this all the time. See any of my peer-reviewed publications for examples.

The saga of the points in a circle problem on this blog was another example of a problem I made up that was at my own skill level; solving it took me a considerable amount of effort, and I had to create arguments of a kind I'd never made or seen before. It might not be a research-level problem, but I think it did attract some interest from a few mathematicians out there.

Sometimes I'll write a problem that isn't research-level, yet still is "at my level" in a sense, because it solves a problem that I legitimately wanted to solve. Here was one such problem; I wrote it after stepping in a pothole full of slush.

3. Have you ever written a question that you thought would be simpler than it was?

In physics, you often try to give your students a nicely bounded task, but then you realize that you made some inappropriate assumptions, and the problem can't be made as simple as you were trying to make it. I think it happened more than once that I jotted down a problem on my way to class and then was unable to solve it in front of my students! Whoops. Bad lesson planning.

Sometimes a word problem is just what you meant it to be, but then the students have a hard time with it, and it surprises you. Moments like that are valuable opportunities for me to adjust my own understanding of my students' brain-state. These moments also expose weaknesses of my curriculum up to that point, which I can try to correct the next time around. I wrote about an experience like that, describing a time when I gave one of my kids a word problem she hadn't been well prepared for. You can't smooth every transition in learning, but most transitions could be smoother than they are today.

4. How do you decide whether to write about trains, apples and oranges, candy bars, etc.?

In the best word problems, the math is just right and the context fits beautifully. Writing those problems is harder than it looks. A while ago I wrote this task to illustrate a geometry standard, and it's OK—it stars a ladybug—but a colleague later told me that her word problem about that standard had starred pirates and treasure. That's way better! Not because treasure is more exciting, but because the mathematical issue at hand was about pinning down a location, which is an important goal for treasure-hunters not ladybugs. The situation matches the math.

If I'm writing a word problem about, say, speed, then sometimes I'll have the numbers in hand and go looking for a context to put them in. So for example if it's a speed of around 8 mph, then I'll spend some time thinking about what sorts of things actually move at around 8 miles per hour. I think about what sorts of things from everyday life plausibly move at a pretty constant speed for an extended period of time. Working that way takes a lot of effort, but it tends to yield better problems.

Editorially, there tend to be a lot of constraints on the contexts you can use. Often there are rules like "No candy bars"—I guess parents don't like junk food in the curriculum. Also, no military applications. Sometimes these rules seem a little silly to me, and I wonder if they contribute to students' feeling that math is useless. But the rules are usually there for a reason.

5. What's the process for writing a really good problem?

Sometimes I think we should all spend less time writing problems, and more time curating excellent problems from every century and nation. In any case, I doubt there's a reliable process, or else there would be more good problems out there. Different people also have different views about what a good problem is—sometimes incompatible views. I'm eclectic and appreciate a well made problem in any genre, just as I like many genres of art.

In one mode of writing problems, you know what math you're trying to get at, and you choose a context to match. In another mode, you start with an interesting scenario, and then the math in the problem is whatever it needs to be to shed light on the scenario. Any mode has its pitfalls. When you're hunting for a context to match some specified piece of mathematics, it's possible to end up with junk like "At the store, Mary bought 78,123 cantaloupes and 831 fewer raisins…." On the other hand, when you're starting with the context, it's possible to overshoot the expertise of the audience ("How can this airline schedule its flights so that…"), or else fail to do justice to the context ("Treating the horse as a perfect sphere…").

Matching math to context sometimes sends me to the Internet to do research on different topics. What is a realistic weight for a songbird? Could there be a county in the U.S. that's roughly triangular? In order to create the problem about oil spills shown above, I studied the body of law that establishes penalties for chemical spills in the Columbia River Estuary (see WAC 173-183-840, Calculation of damages for spills into the Columbia River estuary). Once, I called a company historian at a vehicle manufacturer to make sure that a vehicle I was writing about could go as fast as I was saying it could go. This marvelous ratio/proportional relationship problem by Richard Stanley probably has some research behind it:
Tree-measuring tape. There is a special tape measure used in measuring trees; when wrapped around a tree, it gives as its reading the diameter of the tree. Describe how the special tape measure differs from a standard tape measure. Make a diagram to scale showing the marks on the first part of the special tape measure.

The first draft of a word problem is usually too long. The language needs to be condensed. It's possible to pare down the language too much, sacrificing authenticity, but usually the reverse happens, with unnecessary language demands getting between the student and the math. This is especially important because many students in the U.S. are learning English while they're learning math. The first version of the chemical spill problem had many more words than the version shown here; experts in bilingual math education helped me reduce the language a great deal.

When I'm writing a word problem, my brain is totally engaged in the math, so in that moment I can't afford to worry about whether the person shopping for cantaloupes is going to be named Jason or something else. So, when I write a problem, I just write A, B, and C for the names. As soon as the problem is finished, I open up a spreadsheet that I downloaded a couple years ago from the New York City Department of Health, which lists the most popular baby names given out in New York. In my copy of the spreadsheet, I've added a random generator that selects one of the names every time I open the file. Whatever name comes up, I use that name. Over time, the Law of Large Numbers pretty much guarantees that my word problems will avoid bias in the names of the characters. Just now, I opened the spreadsheet and it gave me "Mya." I never would have thought of that name myself.

Part of the process of developing a problem should be testing it with students, then revising it. Some of the problems I see in textbooks don't look to me as though they had been honed to perfection over time. More generally, I fear that something in the economics of the market fails to reward excellence in math problems. What too few can see, too few will pay for. There is also a tradition in math education of using problems without attribution. The rules of scholarship don't seem apply to math problems, and the status of copyright law is unclear. I should say that the problem Tolstoy loved, the problem about the haymakers, was written by a student named Petrov. He died young, of tuberculosis.

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