Kachima is making triangular bandanas for the dogs and cats in her pet club. The base of the bandana is the length of the collar with 4 inches added to each end to tie it on. The height is 1/2 of the collar length. If Kachima's dog has a collar length of 12 inches, how much fabric does she need in square inches? If Kachima makes a bandana for her friend's cat with a 6-inch collar, how much fabric does Kachima need in square inches?
I'm not a pet owner so somebody please set me straight: is pet apparel a productive context for mathematical inquiry? Does PETA know about this?
Previously: Unnatural Currents
Yes, they make triangular bandanas for dogs, single ply. Usually cut with zig-zag scissors. My dog comes home from every stay at the kennel looking like a boy scout. every. time. This is still a terrible problem though.
Trying too hard:
Prentice Hall's Algebra I: California Edition:
A hot air balloon flies at a speed of (n + 8) miles per hour. At this rate, how long will it take to fly (n2 + 5n – 24) miles?
[via Matt Vaudrey]
I encounter problems like this too frequently, and my ‘put on the spot’ knee-jerk reaction when they pop up is usually something like “Oh, that’s a stupid problem. Just skip it.” Of course, the message that students get is that “Math is stupid.”
I love it when Nature has terms that factor so well.
It begs the question, who actually writes this?
The authors are all distinguished teachers and professors, many with PhDs. But would any of them stand by this? Was it handed over to an intern? Was it caving in to the “applications” lobby? Or do they consider it a good problem?
The problem with multiple authors is that none of them “own” the work; none of them consider it theirs. Its a project they are working on, but they’re just collaborators.
Given a choice, I will always choose a single author book. I know it will have been written with greater care.
It's bad enough when you're trying to gin up interest in math by way of pseudocontext. It's worse when you're trying to assess math by way of pseudocontext. If the student isn't interested in math by now, what do you think an assessment is going to do?
If your students miss these problems, how certain are you they really misunderstood the mathematics? How certain are you they weren't distracted by the problem design?
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.
Pseudocontext sends two signals to our students, both false:
- Math is only interesting in its applications to the world, and
- By the way, we don't have any of those.