Thoughts on story problems

Story problems! They are the great bugaboo of math class, the problems everyone remembers hating, On the other hand, when educators think of “real-world” math, useful math, or motivated math, story problems are where they want to go. And this instinct makes sense. Story problems should be a fantastic resource in the classroom—a chance for reading comprehension, making sense of problems, modeling, and more—but instead they’re dreaded and ineffective.

The seedy underbelly of story problems was stunningly revealed recently in this video by Robert Kaplinsky.

Let’s take a minute and consider what happened here. Three quarters of the students who saw this question attacked it with a kitchen-sink strategy: just do some arithmetic with the numbers and maybe you’ll get it right. Why don’t they do what the other quarter did and say, “this doesn’t make sense”?

It’s possible that they don’t want to disappoint the questioner, and that they figure doing something is better than doing nothing. It’s possible they didn’t feel comfortable asking questions or expressing confusion. But look at that last girl describing why she decided to divide; this strategy doesn’t come from nowhere. These kids are doing story problems as they’ve been taught to do them.

In school, many teachers teach kids how to solve story problems as a sort of code. There’s a protocol:

  • Step 1. Underline the numbers
  • Step 2. Circle the important words, such as plus, minus, sum, product, difference, quotient, together, and, more, less, etc.
  • Step 3. Create an equation using the numbers and the operations corresponding to those words. (If the operation is subtraction or division, we’ll subtract the smaller number from the larger, or divide the smaller number into the larger)
  • Step 4. Solve the equation. That’s probably your answer.

This kind of rubric for solving story problems is self-defeating. We’re basically turning the intuitively sense-making project of reading a story into another kind of encoded math project, devoid of meaning. There’s a subtle line here, because teachers want to help, and underlining or noticing key words isn’t inherently a bad thing. But to begin by sweeping aside all pretense of meaning in favor of a mechanical process is bad mathematics.

Not surprisingly, it ends up being self-defeating as well. Story problems get trickier as students get older, and when they do, these kinds of mechanistic strategies backfire big time. The fact that most of the eighth graders in the video still seem to approach problems in this way spells trouble for their future in math.

So what’s to be done? Story problems have been around for millennia, and though they’ve often felt a little contrived, I don’t think they’re going anywhere. And truly, they are a relatively untapped resource. How do we use them to better effect?

Here are some ideas. I’d love to hear yours too.

Idea 1. Use story problems, but don’t teach a rubric to solve them.
Drop the story-problem “strategy” and focus on helping kids draw pictures or models of the situation instead.

Idea 2. Create story problems that resist basic strategies.
Interestingly, it’s not that hard to write story problems that can’t be cracked with the rubric-based story-problem-solving strategy. I’ve seen these pop up on high-stakes tests at the end of the year, leaving teachers to feel cheated that the test was gamed with questions designed to fool their students. But what if all the questions always resisted the basic hack of getting the answer without understanding the problem?

There are a number of ways to create hack-resistant story problems. The simplest is to create problems that don’t follow the same basic structure, but can be better solved by understanding the situation or drawing a picture. Here’s an example of this kind of problem from our Summer Staircase curriculum. This is a sheet for 2nd graders. Simply adding more steps to the problem makes these virtually impossible to solve without understanding what’s happening. On the other hand, if you draw a picture or build a model, they’re quite straightforward.

a-trip-to-the-bakeryGet the full worksheet here.
Idea 3. Create story problems with real interest as well as complexity.

The name “story problem” suggests a story. Why not tell a real story? Folks like Marilyn Burns and Greg Tang have been writing math story books for some time now (see a long list at living math.net). What’s nice is that these aren’t too hard to write, they get huge buy-in from the students, and they combine the fun of story time with a much deeper thoughtfulness regarding the math. Here’s one I wrote for our Summer Staircase curriculum, suitable for 3rd/4th graders.

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Notice that the engagement created by a good story allows for a much greater complexity in the mathematical modeling. What’s the relevant info for each question? The meaning can’t be lost, and kids are tuned into the meaning because it’s a real story. Some teachers gave kids a chance to draw their own version of the monster, creating an opportunity for an interdisciplinary lesson, involving reading, math, and art.

the-monster

Even if we skip the pictures, we get reading comprehension combined with mathematical meaning. The downside is that creating these kinds of stories is more work. But I can imagine a collectively-produced library of them.

Here are a few more examples of these long-form story problems.
Story Problem – The Ant and the Grasshopper
Story Problem – The Kite

Idea 4. Switch to Video
The 3-act math lesson is another way to grab attention and focus on meaning-making. There’s a lot to be said for this format (and a lot has already been said by others). Check out a lovely example here: The Cookie Monster.

Using video or rich images as a launch can be great. Really, though, this is a different animal than story problems, so I won’t focus on it here.

Idea 5. Use story problems as launches to complex tasks
This is another idea that stretches the very idea of a story problem. Consider a problem like the indecisive director problem.

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This is a great project, but it resists a straightforward solution, and requires less of the modeling and simple arithmetic of the story problems above, and is more about digging in to a much deeper problem. I’m a huge fan of complex tasks, and personally think they should be much more represented in math class, but this feels like a different animal to me too.

Idea 6. Have students write their own story problems
There’s nothing like standing on the other end of a process to understand its inner workings. A student who finished all the problems in The Kite asked what she should do next, and I suggested she write her own question. Here’s what she came up with—I liked it so much I converted it into a challenge problem for the other kids. (Check out the lesson to see.)

Having kids write story problems is usually a great idea. The dangers are that it becomes another unmotivated exercise, and that their story problems may not be appropriate for others to solve—it can be tough to write a story problem of the appropriate difficulty! That said, there are great opportunities when students are involved in the back end of producing problems as well as solving them.

What’s your take on story problems? It seems that story problems are an untapped resource, and with the right approach, they could be leveraged in all sorts of powerful ways. I’m still hopeful about producing a free library of good story problems in the style of The Monster. I like the prospect of combining the meaning-making involved in reading comprehension and mathematics.

Thoughts?

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