Saturday, October 17, 2009

Deep inside the math box

Don't let the reality in!

This is depressing. I gave my intermediate algebra students the following problem. It's a standard distance-rate-time exercise (I added some emphasis to some important words):
Jane rides her scooter 6 miles to the mall to buy some shoes. Eager to get them home, she drives 2 miles per hour faster on the way back, traveling the same 6-mile route. The total travel time for Jane’s round trip to and from the mall is 2.5 hours. How fast did she travel on her way to the mall?
It's not a catchy, exciting, and engaging application problem, but it's comfortingly mundane. Certainly people live in a world where distance, rate, and time are not entirely foreign. Most of my students drive and know that traveling for 2 hours at 60 miles per hour equates to a 120-mile trip.

It's not scary stuff. Not rocket science.

One of my students—and not an indolent homework-shirking student either—quite innocently asked me (after she screwed up the problem), “What words in the problem were supposed to tip us off that we had to add the two times together to make an equation? How we were supposed to know that 2.5 was their sum?”

No, I didn't slam my head on the board multiple times, even though I felt like it.

How about “total time”? How about “round trip”? How about “to and from”?

Would it have helped to include “Hint: Add the freaking times!”?

This particular student (among quite a few others) has put math in a box. The real world isn't allowed to leak in. Don't think about how things operate in reality. It's not permitted! Math is a pure mind game that doesn't mean anything. It's just a formal system that you have to beat if you're going to graduate.

I answered her question with a question: “If it takes you ten minutes to get to school and seven minutes to get back home, how long did you have to travel?”

“Seventeen minutes,” she answered instantly, her expression suggesting that I had asked a dumb question.

I waited for the light to dawn.

Still waiting.


Barry Leiba said...

I thought you were going to say that she didn't know how to solve the quadratic equation.


[He-he. Fittingly, the CAPTCHA on this comment is "faliest", very close to "failest". Hm.]

Anonymous said...

I guess I'm being fairly dull this Saturday morning, but it was a minute or two before I had a solution. On the way there, I had a brief dalliance with considering average speed, before realizing too much trouble lies that way since it's weighted by the (initially unknown) times.

On first glance, the question as asked seems pretty ridiculous. However, considering that adding the times makes it much easier to deal with the problem, I'm tempted to think one might reasonably ask how to think of that approach. On the other hand, merely expressing the description as equations should work, so one who asks about "what words" ought to have a chance.

I can't even defend the second question about 2.5 by trying to reinterpret it as something else. It's a bit disappointing, though even then I guess it's a consequence of a failure to generalize. One who can deal with concrete values like ten minutes and seven minutes may be completely unable to do the exact same thing with expressions even slightly more complicated. The ten minute and seven minute situation will be handled instantly, which may be the problem; a typical student will add quickly enough, it seems, to forget that an operation took place. I've had some rather long conversations with students trying to get them to see that they were adding, and that in exactly the same situation they can add expressions other than numbers, too, though I expect that could improve with my technique. I'm reminded of the surprise expressed back when I was explaining to fellow undergraduates how they can do arithmetic (including "long division") in any base they wish, given a multiplication table and an understanding of what they're actually doing in decimal.

I would have thought you'd have been numbed to this phenomenon through constant exposure, though I suppose it's heartening to see you still care enough to comment. There are so many students who learn mathematics as the brittlest of "brittle knowledge" (which a Google search shows credited to Whitehead, though I was just thinking of Feynman's writing now), treating it as merely a collection of unrelated algorithms, to be invoked like magic rituals. I get the feeling that some are trying to spend as little time as possible on a distasteful activity, trying to become "good enough" not to have to think, but I'm not sure it's even occurred to others to look at mathematics in another way. I have to wonder what it's like to try to make sense of all the quantitative ideas we regularly encounter with such an approach to mathematics.

George said...

Did anyone answer -1.2mph?

Peter said...

And who is to blame? Why is this so difficult?
Mea culpa? Mea maxima culpa?

Zeno said...

Most of my students never even got to the correct quadratic equation on this one (which factored, so the quadratic formula was not necessary).

But no one offered the answer that the time was -1.2 mph.

Anonymous said...

...(which factored, so the quadratic formula was not necessary).

Good, because I got my answer by drawing little pictures.

Anonymous said...
This comment has been removed by a blog administrator.
Coridyn said...

Wow, I hadn't realised that my maths had become so rusty.

It kept me busy all day trying to factorise the quadratic equation (after the initial struggle of working it out).

Thanks for the motivation Zeno, I think I'm going to take up some recreational mathematics to keep on top of things.

Corey said...

I was always bad at quadratics. I tried this and failed badly. I probably messed up the fractions.

George said...

Corey -
Start with (6/x) + (6/(x+2)) = 2.5
where x = speed to the mall

Do some basic math and get -
0 = (5x + 6)(x - 4)

So two solutions x=-1.2 and x=4

I know how to sit around for 0 mph but negative walking does not work so the answer is x=4.