[PS] Check For Understanding

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!

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


  1. This one is a good example of BOTH kinds of pseudocontext. It assumes that you can tell what the temperature or “color” of a star is just from going outside and looking up. Second, it assumes anyone would care about adding the temperatures of stars that are billions of miles away from each other.

  2. It’s the 2nd part. There is nothing inherent in observing stars, or the temperature of stars, that leads to addition of those temperatures.

    The green and violet stars being made-up are flatly untrue bits, but as the rest of the data is roughly true (according to the writer) it’s not as significant of a crime as the use of addition.

  3. Wow, you hadn’t read that essay before? It’s awesome. It’s just filled with awesome. You have to read the whole thing through.

    There are so many wonderful Feynman essays. There’s his essay on cargo cult science, his essay on the education system in Brazil (where students are memorizing rather than learning), his essay “What is Science? (he discusses how he learned science from his father), and so many more. I won’t include the URLs because I don’t want this comment to get marked as spam, but you should be able to find them with Google.

    And don’t miss his books, Surely You’re Joking, Mr. Feynman! and What Do You Care What Other People Think? When I was a young kid who loved science and was frustrated with school, those books were absolutely inspiring, and undoubtedly helped influence me to enter into a career of math and science. And even if only 30% of the anecdotes in his books were actually true, I don’t care, it’s still both entertaining and inspiring.

    Sorry if I sound like a Feynman fanboy. I guess I am.

  4. Michigan Tech sends their applicants a copy of “surely you’re joking.” I thought that automatically made it a good college.

  5. Well, that’s another point in Tech’s favor.

    As for pseudo-context, I just looked through the text used in first year algebra in Detroit Public Schools and I have a sick headache from all the pseudo-context it contains. I’m struggling to find a germ of a problem that can be crafted into something worthwhile. Maybe after a good night’s sleep. . . and a book burning.

  6. i agree with Josh that it is the 2nd part, but…

    i don’t see myself (or anyone else for that matter) ever having to add the temperatures of 2 stars, and i think it’s mildly sad that that was the best example they could come up with.

    how about using ‘traveling through space’ as a context? imagine being at the controls of a spaceship, coming in for a landing. “How much thrust, for what duration & at what power should you give it?”

    thinking about Apollo 13, i’m sure some arithmetic was needed to know how much O2 they had left, & how much more they would need to make it home safely…arithmetic is everywhere…

  7. @Brian, just to note the essay is from the mid-60s, before Apollo 13.

    Check out the forum for the simulation game Orbiter sometime. Hardcore: it’s got people trading physics equations to do particular maneuvers.

  8. This reminded me to discuss a type of pseudocontext for your taxonomy. You can call it wrong grain size, or wrong level. It is best expressed in the following old joke about oral examinations.

    A veterinary student has only studied one topic for the final: fleas. The examiner asks question number one, about dogs. The student answers: “Dogs are mammals covered with fur. In the fur, there are fleas. Fleas are…” The examiner asks question number two, about cats. The student answers: “Cats are mammals who also have fur. In cats’ fur, there are fleas. Fleas are…” The examiner, somewhat exasperated, asks about fish. The student says: “Well, fish don’t have fur. But if they did, there would be fleas in it, and fleas are…”

    Way too many math questions completely miss the relative importance, coolness, relevance of different features of the context. So-called “storytelling” questions are often especially atrocious.

  9. I’ve been thinking about definition 1, and I’m not sure that problems of this type should be called pseudocontext.

    There are factual errors that come from misinformation, like the the color of the stars. Those can be corrected and the problem can be salvaged. So, I would not call that pseudocontext.

    There are simplifying assumptions that are made in a problem. Simplifying assumptions are just part of math and science. Those should be stated at the beginning of the problem. For example, if you study simple harmonic motion in a trigonometry or calculus class, you use a sine function. However, that ignores the internal friction of the spring. By ignoring some details, we can focus on other details. I wouldn’t call simplifying assumptions pseudocontext.

    I do completely agree with definition 2. Clearly, a problem about adding the temperature of stars is virtually unfixable. It is not a naturally arising question from the study of stellar bodies.

    There is a natural tendency to attempt to pigeonhole every bad word problem as pseudocontext. To make the term “pseudocontext” useful, it should describe a proper subset of bad word problems.