The one nagging question I continue to have about WCYDWT is…what exactly does it accomplish?
Yes, it appeals tremendously to our intuition. Students are looking at the world and designing questions about curiosities. This certainly appears to be the kind of skill we want to teach. But research is at best divided about the kinds of gains similar projects have had in the past.
Dan has pointed out that his students out-performed his entire department, that they showed real and true gains according to every measure we currently use.
But still I wonder. How much of those gains were attributable to the actual WCYDWT lessons? How much of it was attributable to his skill and highly developed craftsmanship as a manager, questioner, evaluator? WCYDWT doesn't strike me as terribly efficient. And I know that's not the point, but where is the research showing that these kinds of problems are effective?
I think Dan also mentioned that he would do these kind of lessons bi-weekly (about 1 in 10 days). To which I say, fair enough. He was efficient and skilled enough in those other nine days to experiment with something like this.
I don't know, though. These problems appear to make students conceptually flexible, which is brilliant. To make them procedurally flexible- arguably more important and more difficult to teach- is probably what Dan was doing the other 90% of the time.
I know the focus of this blog is WCYDWT and debunking conventional textbook wisdom. But when there's no coke or sprite, escalators, three-pointers, or cheese, how do you do the other stuff?
Archive for March, 2011
How'd he do? Too big? Too small? Just right? If he's wrong, can you redesign it?
[h/t Steve Phelps, via e-mail]
Christopher Danielson finds a text in his college library called How to Solve Word Problems in Algebra: A Solved Problem Approach (Johnson, 1976).
A sample problem:
Mrs. Mahoney went shopping for some canned goods which were on sale. She bought three times as many cans of tomatoes as cans of peaches. The number of cans of tuna was twice the number of cans of peaches. If Mrs. Mahoney purchased a total of 24 cans, how many of each did she buy? (p. 14)
From Johnson's preface:
There is no area in algebra which causes students as much trouble as word problems…Emphasis [in this book] is on the mechanics of word-problem solving because it has been my experience that students having difficulty can learn basic procedures even if they are unable to reason out a problem.
And here is the crux of the matter. I have already argued that the very nature of word problems is such that people’s actual experience has no bearing on solving them. But in this preface is the rarely stated truism that we can train students to work these problems even when we cannot teach them to think mathematically. Entire sections of textbooks are devoted to the translation of word problems into algebraic symbols and Ms. Johnson has written the book on it.
2011 Mar 07: Christopher Danielson responds to some of our commentary at his blog.
This problem nearly ripped the Meyer household apart tonight:
Which glass contains more of its original soda?
Justify your answer.
Download the video.
2011 Mar 04: Updated to add the goods.
2011 Mar 13: 71 comments as of today means we've struck a nerve. Many commenters have put their mark down with an algebraic proof. More interesting to me are those who have included devices for illustrating the proof to their students. That's harder stuff. See MPG's comment:
Consider a similar problem using discrete objects (e.g., playing cards. Take 10 red cards and 10 black cards face down in separate piles. Take four at random from red pile; mix into black pile. Shuffle. Return four random cards face down to red pile. Ask: more black in the red pile or red in the black pile. Try this several times. If you’re not convinced, do it with the faces showing. Apply principle to soda problem.
Ben Blum-Smith, on what provoked him to inquiry this week:
This is admittedly abstruse, but honestly what’s had my notebook out all week is trying to figure out what the Riemann surface of y^3-xy-x looks like. I think I’ve figured out how the sheets permute at the branch points, but I don’t think I’ll really be satisfied until I’ve in some way drawn a picture of the mapping of the y-plane to the x-plane, and I don’t see that happening without computational help.