Many people have written about how writing can enhance the learning and doing of math.  One of the first books I read on the subject was “Writing to Learn Mathematics”, by Joan Countryman. The Problems of the Week at the Math Forum were started back in 1993 with the explicit goal of giving students incentive to write about their math problem solving, using the newly-available Internet.

In the Problems of the Week project, we emphasize two purposes for writing when we are communicating with students.  The first purpose is to write so that other students could learn from their work.  We reason that giving them an audience for their work – their peers – will help them decide how much detail to include in their explanations. The second purpose is to write so that a mentor, be it one of us at the Math Forum, their teacher, or one of the volunteers who work with us, will know enough about what they were thinking to be able to help them make progress towards a mathematically sound solution.

The third purpose, which we probably should emphasize more frequently, is so that students might find their own mistakes in their work.  In reading the solutions to the Math Fundamentals PoW I Get a Kick Out of Soccer, which asks how many full laps a player would have to run to be sure they ran at least a mile, there were a number of instances where I wondered, “What if they had to explain that step or had included more information?  Would they have noticed their mistake?”

For example, here’s one submission:

My answer is they had to run 15 laps to complete a mile.

I figured this out by first figuring out how many yards are in a mile, which 1760. Then i divided it by 115. I used guess and check to get my solution. Also it makes sense that 15 laps are in a soccer field that big.

Imagine the student being required to finish this sentence: “I divided 1760 by 115 because….” What would they say? What if they said, “…because 115 is the length of one side of the soccer field.” Would that make sense to them? Would it seem like a reasonable step to take? Or what if students were required to explain everything in terms of quantities instead of the values of those quantities? So in place of, “I divided 1760 by 115″, they instead wrote, “I divided the number of yards in a mile by the number of yards in one side of the field.” Would they then scrunch up their face and say, “Hey, wait a minute. That doesn’t make sense!”

I knew that there was 1,760 feet were in a mile so I did 1,760 divided by 4 to get 44.

I did 1,760 divided by 4 to get 44. I kept trying so I did 30*44 but it was wrong, so I did 44*40 which I got 1,760 to get my answer.

How would they complete, “I divided 1,760 by 4 because…”? What would the “quantities” version look like? “I divided the number of feet in a mile by the number of sides of the soccer field to find the….oh….wait”? (I’m just guessing why they might have divided by 4!)

Here’s another one:

10 laps.

I got this answer by quickly drawing myself a rectangle. On one side i put 75 yards and on the other 115 yards. Then I went on the internet to find out how many feet in a mile,5280. I then did 75 times 3equals 215 ft, and 115 times 3 equals 335. I did 215 +335=550. Finally, I took 5280 divided by 550, which is 9.6. I rounded up and got 10. So 10 laps around the field is more than a mile.

Instead of, “I did 215 + 335 = 550″, what if they wrote, “I added the number of feet on the short side of the field to the number of feet on the long side of the field to find the perimeter.” Would they notice it’s not really perimeter?

One more:

You would have to run 40 laps in order to run a mile.

75 yards wide and 115 yards long

115

-75

=

40 laps

If they are reading the problem closely enough to know what the 115 and 75 represent, would they really write, “I subtracted the length of the short side from the length of the long side to find how many laps they need to run to cover at least a mile”? I know they’re not likely to write that much starting from where they appear to be now, but consider it a goal.

It’s all well and good to remind kids to always say “why” they did each step. Supporting the idea by always asking “why?” in classroom conversations is a good start. But some teachers like to provide a little more structure to the students to remind them of their responsibilities. One method I’ve seen used in elementary classrooms is a simple t-chart that encourages students to write down an explanation for each step they took.

Here’s another chart that’s very similar, but uses labels focused on what you “thought” and what you “did”.

How often do your students find mistakes when they’re trying to explain their work in writing? How often does it happen when they’re talking out loud? What strategies are you using to try to make these things happen more often?

Some “I Get a Kick Out of Soccer!” links in case you are interested: