Has a nice example of fantasy context.

Making good textbook problems is hard, and worth celebrating when you see one.
Of the things that bug me when I see a bad math problem, the issue of pseudo context is a very big one, and worth articulating. Dan’s work on this has been really great, and I appreciate the attention it has gotten.

There is a survival skill that kids pick up very early, and they learn to be really good at it. Without it, they would never survive school. I call it “figuring out what the teacher wants”, but it doesn’t only apply to teachers. The Mommy rules are different from the Daddy rules, and the way you behave at church is different from how you behave at the supermarket.
What you do in one of these pockets would look totally ridiculous from the perspective of the other. When you run track, you pretend that it is important to get to the end of the track fast, but if you could get to the finish line faster by running in the opposite direction, you agree that that doesn’t count. In math class, you want to get the answer to 23×45, but if the best way to get there is using your calculator, we all agree that this doesn’t count. In history class, if the best way to find out who “the Red coats” were is to look it up with Google, we all agree that this doesn’t count.

I don’t particularly mind that math class is its own universe with its own rules. Yes, the rules are arbitrary, but so are the rules of running track.

What really bugs me is when I see someone spending her break between classes solving SuDoKu puzzles with gusto and on her own accord, then walking into the high school math classroom and telling her friends “I hate math!”
I bet she’s doing more “math” thinking in the five minutes of SuDoKu puzzle than she and her class mates do together in the whole hour of math class.
What have we done to “math class” so that it reliably wipes out any interest and joy the students might have in problem solving?

“A woman race car driver (because we need to placate the publisher’s mandates against men race car drivers) travels 500 miles on a 1/4 mile race track in 20 min. How fast is she going?”

Pretty simple to fix, no?

J Chak Don’t waste time trying to come up with more realistic numbers. Spend that time trying to come up with more realistic problems.

This seems like another false choice. Why can’t we have both? I know my students would prefer it.

So fine, for tennis, use the tiebreaker example. For skydivers, do a bit of estimation but get them to calculate something you might have to calculate anyway (such as how many seconds to freefall until you should pull the cord).

In that sense, I agree with curmudgeon. Don’t waste time trying to come up with more realistic numbers. Spend that time trying to come up with more realistic problems. If the student questions the numbers, just say you’re a 800lb robot parachuting out of a plane in Mars!

One other cool thing about the tennis problem is there’s a rational function whose input is p, the probability of winning a point, and whose output is the probability of winning the full tennis game:

G(p) = [p^4 (1 – 16(1-p)^4)] / (p^2 + (1-p)^2)(p^2 – (1-p)^2)]

It’s not easy to come up with this, but it’s possible by analyzing the different ways to win (better to use “q” for “1-p” initially).

And G has the “right” behavior, with domain 0 <= p <= 1: G(0) = 0, G(0.5) = 0.5, G(1) = 1… it actually looks kind of logistic.

A nice example of a messy function with behavior that matches its context.

Maybe the World Series should become "win by 2". That would be sweet.

Yikes. It’s kind of a depressing mindset that sees inclusive textbooks and real context as mutually exclusive.

Well, I’m sure that’s just because, as Stephen Colbert often points out, Reality has a well-known liberal bias.

– Elizabeth (aka @cheesemonkeysf on Twitter)

Bowen: It led to a better problem: while still about tennis, either player wins when they win 4 points, with the rule that they must win by 2. This is much more understandable and has less “insider knowledge”.

FWIW, this is called no ad scoring, which is legitimate, so you’re totally in the clear here.

I’m curious what you mean by “insider knowledge”. Recently, I tried to write a problem about the probability of winning a game of tennis given the server’s was 60% likely to win any given point. The problem included all the details about the point scoring of tennis (0, 15, 30, 40, game, except 40-40 is deuce… etc). Another author pointed out this didn’t make much sense to him, because he had no experience with tennis, and would be forced to learn the ridiculous scoring rules of tennis in order to solve a problem about probability. For him, tennis was a useless context (not really a pseudocontext, since it was being used legitimately).

It led to a better problem: while still about tennis, either player wins when they win 4 points, with the rule that they must win by 2. This is much more understandable and has less “insider knowledge”.

So that’s what I agree with, but I couldn’t disagree more that books can’t have questions that require thought, or that we can’t use realistic contexts. I’d like to think every problem in a good text requires thought — otherwise, why is it there? And I can only speak for our series, but we use many real contexts legitimately. We take students through the entire process to construct and understand the formula for the monthly payment on a car loan, starting from numeric example and working across different chapters of different books, including topics such as recursive rules, generalizing from repeated calculation, and sums of geometric series. (Our books use other contexts, but that’s the one I think is really great.) Inclusive textbooks and good, quality mathematics are not mutually exclusive.

I am happy that we published a full high school series without ever once writing 80 problems for a section. 10-15 good problems is more than enough, and the best number of problems to work through is still 1.

– Bowen