## Blown Out of Proportion

In preparing a workshop about proportional reasoning for the 2011 NCTM Annual Meeting, I came across the following from “Learning and Teaching Ratio and Proportion: Research Implications” by Cramer, Post, and Currier, which appears in Research Ideas for the Classroom, edited by Douglas T. Owens. The authors discuss the following problem, which they presented to a class of pre-service elementary teachers:

Sue and Julie were running equally fast around a track. Sue started first. When she had run 9 laps, Julie had run 3 laps. When Julie completed 15 laps, how many laps had Sue run?

The authors claimed, “Thirty-two out of 33 pre-service elementary education teachers in a mathematics methods class solved this problem by setting up and solving a proportion: 9/3 = x/15.”

I wanted to be surprised. Sadly, I was not.

Participants in my workshop (all current middle or high school math teachers) were asked to solve the same problem. Several incorrect answers were suggested, among them 3, 15, 27, and 45. Less than 40% of the attendees obtained the correct answer, 21.

This is frustrating, as proportional reasoning is extremely useful in analyzing real-world phenomena. In fact, it’s even applicable to language arts, as evidenced by the following graphic from my presentation:

The teachers in my workshop aren’t the only ones who have difficulty with proportional reasoning. Students have endless trouble, too…

Teacher: Today, we will discuss inverse proportion. Here’s an example: “If it takes 6 days for 2 men to finish a task, how long will it take 3 men to complete the same task?” The number of men needed is inversely proportional to the number of days required. Consequently, 3 men will be able to complete the task in 6 × 2/3 = 4 days.

Student: Oh, I see! I think I’ve got a real-life application of this. If it takes 6 hours for 2 men to hike to the top of a hill, then it will only take 4 hours for 3 men to hike to the top!

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• 1. Joshua Zucker  |  April 19, 2011 at 5:15 pm

Nice problem! A little bit of a trick question for a proportional reasoning session, but good to see if people are paying attention and distinguishing these different kinds of reasoning.

I suspect it’s the kind of problem where people who know less math would do better, kind of like 2nd graders beating 6th graders on this one:

Three ships leave Boston simultaneously, heading for New York. The first ship takes 11 hours, the second ship takes 12 hours, and the third ship takes 14 hours. How long does it take for all three ships to arrive?

• 2. venneblock  |  April 19, 2011 at 8:35 pm

I used the problem as an example of a problem that is not a proportion, and we examined the graph of the situation. It’s a straight line with slope 1 (they run at the same speed) and x‑intercept 6 (since Sue is 6 laps ahead of Julie). Since the graph does not pass through the origin, it’s not a direct proportion.

• 3. Joshua Zucker  |  April 20, 2011 at 6:51 pm

I’ve been surprised, working with 9th graders, how they tend to assume that any linear relationship is a direct proportion. They want to find the slope with y/x for one data point, if you ask them about it outside the context of their lessons on lines.

• 4. xander  |  April 21, 2011 at 4:20 pm

In general, I see students attempt to solve every problem with the techniques from whatever chapter we are currently studying. In a sense, it is the “if you only have a hammer, every problem looks like a nail” situation.

• 5. Joshua Zucker  |  April 21, 2011 at 4:28 pm

I think this is the opposite of the problem I was complaining about.

When in the chapter on lines, students “know” that they are supposed to take two points and use a slope formula to determine the slope.

Outside of that line, if you tell them it’s a linear relationship and give them one data point (a,b), they will assume the problem is solved: y = b/a * x. They won’t say it in those terms, of course, but they’ll act as though it’s true until you finally make them try to write an equation and they realize that they need another point in order to find the relationship.

In other words, what I’m saying is that they never really even learned how to use the hammer in the first place. Or, maybe I should say that they actually found a nail outside of the hammer chapter, but they’re trying to put it in by using a screwdriver.

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