Jason Dyer passed me Richard Feynman’s essay, Judging Books by Their Covers, via email.

Which of the two parts of our working definition of pseudocontext does it exemplify? Justify your answer.

[BTW: I’d say this exemplifies both definitions nicely. I’ve highlighted the passages.]

Anyhow, I’m looking at all these books, all these books, and none of them has said anything about using arithmetic in science. If there are any examples on the use of arithmetic at all (most of the time it’s this abstract new modern nonsense), they are about things like buying stamps.

Finally I come to a book that says, “Mathematics is used in science in many ways. We will give you an example from astronomy, which is the science of stars.” I turn the page, and it says, “Red stars have a temperature of four thousand degrees, yellow stars have a temperature of five thousand degrees . . .” — so far, so good. It continues: “Green stars have a temperature of seven thousand degrees, blue stars have a temperature of ten thousand degrees, and violet stars have a temperature of . . . (some big number).” There are no green or violet stars [def’n #1 – dm], but the figures for the others are roughly correct. It’s vaguely right — but already, trouble! That’s the way everything was: Everything was written by somebody who didn’t know what the hell he was talking about, so it was a little bit wrong, always! And how we are going to teach well by using books written by people who don’t quite understand what they’re talking about, I cannot understand. I don’t know why, but the books are lousy; UNIVERSALLY LOUSY!

Anyway, I’m happy with this book, because it’s the first example of applying arithmetic to science. I’m a bit unhappy when I read about the stars’ temperatures, but I’m not very unhappy because it’s more or less right — it’s just an example of error. Then comes the list of problems. It says, “John and his father go out to look at the stars. John sees two blue stars and a red star. His father sees a green star, a violet star, and two yellow stars. What is the total temperature of the stars seen by John and his father?” — and I would explode in horror.

My wife would talk about the volcano downstairs. That’s only an example: it was perpetually like that. Perpetual absurdity! There’s no purpose whatsoever in adding the temperature of two stars. Nobody ever does that [def’n #2 – dm] except, maybe, to then take the average temperature of the stars, but not to find out the total temperature of all the stars! It was awful! All it was was a game to get you to add, and they didn’t understand what they were talking about. It was like reading sentences with a few typographical errors, and then suddenly a whole sentence is written backwards. The mathematics was like that. Just hopeless!

Here’s what I’m trying to say. It’s easy enough to write the following pseudocontextual problem:

Dan shot a basketball from the three-point line. The ball followed the path given by the equation:

.

How many units high will the ball be at its highest point?

Try to commit that pseudocontext to a photo, though. It isn’t impossible, but it’s much, much harder.

That’s all I meant. Not that I think Star Wars is real.

I owe Brian Caine a debt of gratitude for flipping my switch on the question of “what is multimedia doing for us, anyway?”

Multimedia makes it really, really hard to lie.

Witness David Cox’s toaster regression. It doesn’t work. We thought it was linear. It isn’t. It isn’t worthless for classroom inquiry. Maybe it’s exponential. But the linear model is a dead end.

If you’re writing the problem in a textbook, though, it isn’t a dead end. You grab some clip art of a toaster. You create a table with values that are linear because who’s going to stop you? Even though the real context isn’t linear, you’re the god of your textbook’s pseudocontext.

Then you fabricate a conclusion that supports the pseudocontext.

For whatever other good it does for problem posing, multimedia keeps you honest. How do you (easily) film pseudocontext? How do you take a picture of a premise that is false? Even harder, how do you take a picture of the conclusion of that false premise in a way that doesn’t belie the premise itself?

Arla, a Swedish yogurt producer.

Translation:

You catch a pike but the scales are broken. The pike weighs two kilograms plus half its weight. How much does it weigh?

Pseudocontext

At this point I’m comfortable with two definitions of pseudocontext:

1. context that is flatly untrue: “a basketball team scores two points every minute for the duration of the game.”
2. operations that have nothing to do with the given context: “the age of Mark’s dad is three more than four times Mark’s age.”

We’ve worked hard for those two categories. We’ve digested some really untasty mathematics in their development. They indicate problems that aren’t just boring or irritating but problems that are actually alienating, problems that disrupt a student’s innate and true sense of the world.

Pseudocontext Saturdays will run their course eventually. For now, though, the intellectual challenge of identifying different levels of badness (and, in many cases, redeeming it) is too invigorating to give them up. Those of you who have invested time and effort on these features in the comments and in your submissions – thanks.

Assignment:

1. Scan an example of pseudocontext.
2. Email it to dan@mrmeyer.com
3. List the textbook title, edition, and publisher.
4. Give me your interpretation of the term “pseudocontext.”
5. Let me know if you’d like credit (name, blog or twitter) or if you’d prefer anonymity.

[see midterm]

1. Which of Will’s commenters has suggested a pseudocontextual problem?

Very few of them, it turns out. Which isn’t to say that all of the suggested problems are good problems, just that our worst tendency in these discussions is to conflate the term “pseudocontext” with “problems I don’t much care for.” Pseudocontext uses the full authority of the teacher or the textbook or of grades to force a connection between The Math and The Context that doesn’t naturally exist. This is a separate matter from “Do professionals really use math in that way within that context?” or “Will my students care at all about the math, even though it exists naturally in this context?”

Spot check me here but I find it pretty easy to divide the suggestions at Will’s place into four categories:

Valid Context, Of Inherent Interest To Basketball Professionals

These questions are ideal, and really hard to find. Will is probably better off asking a basketball player, a coach, or a sportscaster because they won’t waste his time (or, especially, theirs) with pseudocontext.

Glover:

You can probably set up different metrics – basically giving different weights to things like free throw percentage, scoring, rebounds, steals, etc. – and compare players and be able to claim “I can prove that Kobe is better than Lebron” or whoever else you want to compare. This would also extend to comparing teams and stuff.

Have you seen John Hollinger’s formula for player efficiency? It’s gargantuan. Have your students create their own, balancing factors however they choose, checking the results against their intuitive sense of the best basketball players. (ie. “I know Kobe’s the best but my formula is coming up with Lebron.” So you rebalance.)

George Woodbury:

There are some probability applications. If a free throw shooter has a 75% chance of making a free throw, what are the chances that he or she makes 2 free throws? 1 of 2? 0 of 2?

Pair with video clips of different shooters at the line. Given their overall average, place bets on which players will hit both, which will hit one, and which will miss both. Then show the answers.

Valid Context, Uninteresting To Basketball Professionals But Interesting To Students If We Develop It Well

Do basketball players need to know anything about 3D vectors? When they go for jumpshots are they converting force to acceleration and then solving the quadratic formula to see if the ball will land? No, but these problems are of interest to students. Especially if we visualize them well, if we present them in a format that appeals first to intuition (“do you think it goes in?”) that only later uses math to formalize that intuition.

Angie B:

Dan Meyer’s dy/dan blog had this on it.

wmchamberlain:

How about studying the lines and shapes on the court? A little bit of geometry there.

I like the challenge here. There is a lot of geometry in the lines and shapes on the court. But the hard work remains. What question do you ask? What problem do you pose to get students diving into the geometry of a basketball court, wondering “what kind of shape is the three-point line anyway?”

One idea: I draw two points on a piece of paper. The line between them is the baseline on one side of the court. Can you draw a scale replica of the rest of the basketball court. Bonus: do it with a compass and straightedge alone. Extra bonus: write a program that accepts two mouse clicks and does the rest automatically.

Valid Context, Uninteresting To Basketball Professionals, Also Uninteresting To Students

It’s within your professional jurisdiction to ask these questions, but these questions only appeal to math teachers. Not basketball professionals, and certainly not students. But they aren’t pseudocontextual. These problems have an inherent connection to basketball. If anybody is selling a term to describe these problems (ideally something more descriptive than “lame”) I’m buying.

Lyn Hilt:

Incorporate geography…calculating traveling distances/methods to visit arenas and attend games.

Jason Kornoely:

If it takes one player an average of 15 seconds to dribble across the field, how much time would it take for a team of x to finish?

Rhett:

[What is the] volume of skin making up ball assuming skin is 1/16 th thick?

Pseudocontext

Jason Kornoely:

Determine the hypotenuse using the distance from the free throw line to the center of the hoop and the height of the hoop.

This was the only response I could call pseudocontext with absolute certainty. The Pythagorean theorem has no use or meaning here. The teacher is imposing the theorem on a context that doesn’t want or need it.

2. Create a math problem in response to Will that would be pseudocontextual.

Chris Sears:

Kentucky and UCLA have appeared in the NCAA Division I men’s basketball tournament 80 times, with Kentucky appearing 8 more times than UCLA. How many times has each team appeared in this tournament?