## Sunday, March 07, 2010

### What's Wrong with this Math Problem?

There was a problem, which was discussed at a professional development workshop yesterday. (Yes, on Saturday. Sigh) It caused a bit of a ruckus, debate and a lot of raised voices, but not too many listeners, so I don't think it would qualify as "accountable talk".

And, keep in mind, the participants were all mature adults, not to mention math teachers.

Here it is:

### Kinesthetic Learning: Uniform Motion

A car starts from a town traveling at 40 mph. Two hours later, a second car leaves the same town and heads in the same direction at 60 mph. How many hours will it take for the second car to pass the first?

So far, so good. As a whole, we skimmed over the algebraic solution with many not even looking at that page because we were moving to an exercise that involved two number lines and paper clips.

One line was titled "First Car" and was labeled every 40 miles.
The other was titled "Second Car" and was labeled the same way, but with dashed lines marking 60 and 180. (Those numbers didn't appear.)

Teams of two did the exercise and most groups came up with the same answer: Six hours.

Except that the answer was Four hours. Or was it?
What was the frame of reference? When did the clock start running? And if the educators (excuse me, "facilitators") can't agree to the interpretation, how will the students, particularly the ELLs?

Contrary to what others believed in the class, I didn't have a problem with the answer being four. (This is what I mean by not listening.) I did have two problems:

First, the question could have been a little clearer. And I say this as someone who always tells my students to go back and reread the question to make sure you answered what was being asked. This was the minor quibble.

Second, and more importantly, the activity associated with this problem led too many people to the wrong answer. It was like a game when were moving the pieces. We were taking turns. Where did the cars meet? At the 240 mile marker. How many turns was that? Six turns, but player two only moved four times during those six turns.

My point? Other than my frustration? Make sure the activity will mostly likely yield the result that it should.

One other point: Afterward, I spoke with one educator who at first thought that the answer was an inequality, having read the question as if it had said "is past the first car", which gave t > 6 (or t > 4). When he saw his mistake, we both questioned the use of the word "pass". At the solution time (whichever one), the cars will be even with each other. The first car has not been passed yet.

This isn't an insignificant matter. A question on the 2003 New York State Math A Regents -- the exam which was so bad, the results were thrown out -- there was a question that asked to find the "break even" point. Not all students understood the meaning of that financial term. Some thought they needed to find which ticket made the first dollar of profit. Their answers were off by 1.

In this example, the opposite is true. A student looking at the words "to pass" might assume that he had to add 1. Or one half. Or 0.00001.

Anonymous said...

The problem is fine. Even better, it demonstrates that you can't just answer it with a value.

Maths is a tool for calculating results, interpreting the problems/results is different to calculating the results, but also important.

The solution, with an interpretation, requires a sentence (or a few).
"Assuming no interferences, and that the cars maintain a constant velocity (no accelerating at the start, no turning, etc), the second car would reach the first 4 hours after it leaves."

With some assumptions and precise wording, the answer defines itself as correct.

(x, why?) said...

The assumptions and (lack of) precise wording is part of the problem. With those corrected, it is a fine problem. However, as is, it would lead to confusion and cost students points on the Regents. (*I* would own up to my mistake and credit both answers depending upon the work shown.)

"the second car would reach the first 4 hours after it leaves."

By the rules of antecedents, "it" would apply to "the first", which would be incorrect. Again, I knew what you meant, but would I know if that's what the student meant? Again, depends on the work shown.

lnthomp said...

Assuming (there's that word again) the Regent's is like the state testing I'm familiar with, i.e. standardized tests with multiple choice answers that are machine-scored, first of all no-one will see the work on the scratch paper, and second, the choices available can either resolve or confuse the problem worse. Either way, there is no recourse for a student to defend their interpretation and answer.

(x, why?) said...

The Regents exam, in its current form, contains 30 multiple choice questions in Part 1, followed by nine more questions of increasing difficulty in Parts 2, 3 and 4. All work has to be shown to receive the full 2, 3 or 4 points for the question. (Partial credit is allowed, according to a rubric.)

A question such as the one asked might be a 2-pt question although it is actually a little too simple.