dy/dan

James McKee:

I have to disagree that this is pseudocontext under the argument that (as I posted on the original CarTalk thread) that a trucker actually came to me with this problem a couple of years ago. He wanted a dipstick method that would give him more accurate fuel readings than his fuel gauge (I guess they’re universally inaccurate?). If a member of a profession brings you a real-life problem from their “world,” I think that has to disqualify the problem from being labeled pseudocontext.

Does anybody have a problem with this? I don't have a problem with this.

Mike Manganello offers a useful critique of Car Talk, pseudocontext, and WCYDWT:

I can certainly accept working definitions that require clarification, but the Car Talk problem confuses the issue [of pseudocontext] (at least for me). I’ve only done a little tweaking to the Car Talk problem:

"The fuel gauge of an 18-wheeler is broken, so the driver decides to check the gas level of his cylindrical gas tank with a dipstick. When the level of the gas measures 20 inches high, the tank is completely full. What will the dipstick measurement be when the gas tank is one-quarter full?"

Based on the working definition of pseudocontext, this problem fails on both counts. It completely ignores reality: Why wouldn’t you just fix the gas gauge? Then the problem asks for an irrelevant measurement: Why would we need to know that the tank is one-quarter full?

I assumed the trucker wanted to know one quarter of a tank (rather than four fifths) so he'd know when he had to refuel. An arbitrary number maybe (why not one fifth of a tank?) but not irrelevant. As for ignoring reality, I know more about mid-century Russian architecture than I do about long-haul trucking, but it seemed plausible to me that the trucker couldn't waste time fixing the gauge in the middle of a run. In both of these cases, I deferred to the authority of the radio hosts. If either of Mike's complaints were valid, why wouldn't the hosts have echoed them?

Mike has also misquoted the definition of pseudocontext in small but crucial ways.

Mike's: "It completely ignores reality."
Mine: "Context that is flatly untrue."

Mike's: "Problems that ask for an irrelevant measurement."
Mine: "Operations that have nothing to do with the given context."

Mike:

Personally, I find the Car Talk problem kind of boring and not very mathematically rich.

Once again we find that a problem's basis in either pseudocontext or context has nothing to do with how much anyone enjoys or profits from it. (Seems only to fair to mention, though, that Alex's class had the opposite reaction.)

Mike:

Another word of caution: Mathematics is part utility and part artistry. By limiting mathematical study to problems related to genuine physical phenomena can only serve to retard the growth of mathematics.

It's worth clarifying my total agreement here. My blog covers math applications pretty much exclusively not because I think those are the only problems worth studying but because those problems are the easiest to create and teach poorly.

This is winding down.

I'm mostly content with the two-part definition I set up two weeks ago. I'm also satisfied that pseudocontext is much less of a problem than real context that's either a) poorly represented or b) poorly examined. Any final comments, questions, or challenges, get them on the record ASAP.

This winter I'm taking "The Design of Technologies for Casual Learning" with Shelley Goldman so I ran a quick survey of math apps for the iPhone and iPad. Scratch off all the drill and test-prep software and you have a lot of apps that are trying to make use of some kind of context. Do they represent pseudocontext or not?

iLiveMath

Could you swap out "peacocks" for, well, anything without changing the implications of the problem? It fails definition #2, "operations that have nothing to do with the given context." See another example.

The iLiveMath family includes iLiveMath Oceans, iLiveMath Animals of Africa, iLiveMath Animals of Asia, iLiveMath Speed, iLiveMath Trains, iLiveMath Entomology, iLiveMath Ford Cars, iLiveMath Farm Fresh. Each one is stuffed to overflowing with pseudocontext. Also, at $4.99 each, you'd wish they'd spellcheck this stuff even a little. (These are the promotional stills, incidentally.) Produced by a company called iHomeEducator, this stuff is obviously pitched at homeschoolers, which, as a homeschool graduate, kills me.

Princess Math

Fails definition #2 again. What does 2 x 2 have to do with a tiara?

Ninja Math

Ditto the others. You don't really "use math to defeat" the boss. You're doing math and the game rewards you by defeating the boss. I will break out the parade if someone can point me to the iPad game where you really use math to defeat the boss. That sounds awesome.

Dress Up Math

The operation, at last, matches the context. It fails definition #1, though, because there's no way that pink blouse costs $4.00. Even I know that.

Mellyme, a five-star review of iLiveMath Animals of Asia:

This app is a very good concept and it works well in practice. It's multimedia approach to word problem drills is genius. Kids can learn about various animals while doing math. The videos provided are great as well. Give at least one of this company's apps a try. I have bought two from them-my daughter loved them both.

Okay, so here we go. Students can find pseudocontext interesting. Students can find real context boring. Reasons for the latter include poor representation and poor examination. Reasons for the former include social and academic pressures, enthusiastic teachers, and idiosyncratic student tendencies from which it's really unwise to generalize to "all students" or even "my students."

The problem with pseudocontext isn't the risk of boring the majority of your classroom. It's the message that this is how the world works and this is how we apply math to it. Pseudocontext teaches students — even the students who tolerate or even like pseudocontext — a lesson that is very hard to unlearn.

Advanced Mathematical Concepts: Precalculus with Applications. Glencoe. 2006.

Pseudocontext

Amanda Dean:

It asks students to ignore reality in order to solve it. The wind would probably be moving the balloon as the balloonist tried to take these measurements, and it's unlikely that the two angles could be measured from exactly the same vantage point. The problem also assumes that Groveburg is a flat city, or at least that the elevation of the soccer fields is the same as that of the football field.

And, while not pseudocontext, we have another situation where the author asks the student to solve for inconsequential measurements and ignore the consequential one:

The questions aren't what you want to know. You want to know how high up the balloon is! It's not like you can't figure it out from the info given either. It can be done with the Law of Sines, which is the focus of the lesson.

And then we're back to the pseudocontext:

If you do calculate the balloon's height, you find that the balloon is about 1.24 miles above Groveburg, which is also unlikely since an average hot air balloon ride only goes up to 2000 feet.

Transcription:

A hot air balloon is flying above Groveburg. To the left side of the balloon, the balloonist measures the angle of depression to the Groveburg soccer fields to be 20° 15′. To the right side of the balloon, the balloonist measures the angle of depression to the high school football field to be 62° 30′. The distance between the two athletic complexes is 4 miles.

1. Find the distance from the balloon to the soccer fields.
2. What is the distance from the balloon to the football field?

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.

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!