We create a pseudocontext when at least one of two conditions are met.

First, given a context, the assigned question isn’t a question most human beings would ask about it.

Second, given that question, the assigned method isn’t a method most human beings would use to find it.

The dog bandana is the classic example. Given a dog, would most human beings wonder about the correct size of the bandana? Maybe. But none of them would apply a special right triangle to answer it.

Rules

Here’s the game. Every Saturday, I’ll post an image from a math textbook. It’ll be an image from one of the “Where You Will Use This Math!” sidebars.

I’ll post the image without its mathematical connection and offer five possibilities for that connection. One of them will be real. Four of them will be decoys. You’ll all guess which connection is real.

After 24 hours, I’ll update the post with the answer. If a plurality of the commenters picked the textbook’s connection, one point goes to Team Commenters. If a plurality picked one of my decoys, one point goes to Team Me. If you submit a word problem in the comments to complement your connection and it makes someone lol, collect a personal point.

Why?

1. Fun. Teaching is a pretty serious occupation. It never fails to brighten my day when you all ping me with pseudocontext.
2. Caution. My position is that we frequently overrate the real world as a vehicle for student motivation. I hope this series will serve to remind us weekly of the madness that lies at the extreme end of a position that says “students will only be interested in mathematics if it’s real world.” The end of that position leads to dog bandanas and other bizarre connections which serve to make math seem less real to students and more alien, a discipline practiced by weirdos and oddballs. Caution.

This Week’s Installment

 Loading ...

(If you’re reading via email or RSS, you may need to click through to vote.)

I’ll update this post with the answer in 24 hours.

BTW. Don’t hesitate to send me an example you’d like me to feature. My email address is dan@mrmeyer.com. Throw “Pseudocontext Saturdays” in the subject.

Answer

Polls are closed. The commenters got rolled on this one, with only 3% having guessed the actual application. So one point goes to Team Dan.

Most commenters guessed “calculating probabilities,” which likely wouldn’t have been a pseudocontext. Humans wonder lots of questions about probabilities when it comes to darts, many of which are most easily answered with mathematical tools.

But this is high-grade psuedocontext. Given a dartboard, few humans would wonder about the dimensions of a square that circumscribes it exactly. And even if they did wonder about it, none of them would name the radius r + 12. They wouldn’t even name it r. They wouldn’t use variables. They’d measure it.

The publisher included the dartboard as a means to interest students in special products. If you believe, as I do, that the publisher has done more harm than good here, positioning math as alien rather than real, what can be done? How do you handle special products?

Featured Comments

Amy Hogan:

Q #11: Pretend [certainly not a woman’s name] has no concept of darts, zero aim, and is liquored up at the bar anyway. What is the probability that he’ll hit a 20? Twice, with his eyes blindfolded?

Dennis Rankin:

Question: What percent of the dart board scoring area is red? white? blue?

Extension: Are the red, white, and blue percentages of area the same on an American flag?

Bowen Kerins:

Man, this “context” is an absolute embarrassment and wastes the time of students and teachers. This sort of thing is driven by textbook requirements for “full coverage” — some lessons have a useful “why” picture and description, therefore all of them must.

Awful.

Scott Farrand reacts to the commenters’ loss:

Now I see how to make the dartboard fit into our task. First we each randomly assign each the five options that Dan gave us to 4 of the 20 sectors of the dartboard, so that 1/5 of the sectors correspond to each option. Now all we need is a blindfold, and … let’s see if we can improve our results from 3% correct to about 20% correct.

Also, please enjoy this back-and-forth about the nature of pseudocontext between Michael Pershan, David Griswold, Sarah, and me. I know I did.

McGraw-Hill:

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

Featured Comment:

Kari:

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]

Featured Comments

Emily:

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.”

Mike:

I love it when Nature has terms that factor so well.

Scott:

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.

Danielson:

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.