Tall, Dark, and Mysterious

8/21/2005

In which the iPod renders high school math obsolete, and other stories

File under: Sound And Fury, Those Who Can't, Queen of Sciences, Home And Native Land. Posted by Moebius Stripper at 12:25 pm.

The - count ‘em - eleven authors of MATHPOWER 12, the text currently in use in high schools all across British Columbia, are in a pickle. On the one hand, they know that the young people these days are more interested in the rock and roll music and the reality teevee than the book-learning, and that they’d rather play the video games than do the mathematics. Unfortunately, however, there’s not a whole lot about graphing conics that’s fun and exciting. So our intrepid authors, fingers firmly on the pulses of the jaded young members of their audience, settle for making half-assed connections between analytic geometry and Real LifeTM in the hopes that their teenaged readers will realize just how kewl math can be. Check out the hook they use in Chapter 3.3 - The Circle, to make sure that their readers are intrigued enough stick around until the end of the unit:

The compact disc player is everywhere these days. Developed initially by Philips and Sony, it first came on the market in 1983. By 1986, over one million CD players were being sold each year. Because of the low-cost laser components, the CD player has become one of the most successful electronic devices to date.

If you trace around the outside of a CD, the result is a circle.

A circle is the set or locus of all points in a plane which are equidistant from a fixed point. This fixed point is called the centre. The distance from this centre to any point on the circle is called the radius.

CD players! Cuz, like, music is cool, and think of how much cooler it will be once you know that the outline of your CD is all x^2+y^2=25 and shit.

Chapter 3.5, on the hyperbola, kicks it up a notch by getting all sonic on us:

The Concorde, a supersonic aircraft developed by Britain and France, first began passenger service in 1976. The Concorde travels at twice the speed of sound at an altitude of 17 000 meters. As a passenger on the Concorde on a flight from London to New York City, you would cross five time zones in 3.5 h. Your arrival in New York would be 1.5h prior to your departure London time.

As you saw in the chapter opener, at speeds greater than the speed of sound, air pressure disturbances accumulate in front of the aircraft and a conical shock wave forms. When the Concorde is travelling parallel to the ground, this conical shock wave intersects the ground in the shape of one branch of the hyperbola.

Right below is a graph of a hyperbola, and a description of its focal property.

Actually, this is quite interesting. I know I saw some of this back when I was an undergraduate, but I forgot it, and now I’m curious about the physics of sound. If I were a high school student reacting in what I assume is the authors’ intended way - “hey, cool, sonic booms give hyperbolas, I wonder how that works” - I might be inclined to flip through a few pages in order to learn more.

Unfortunately, there’s nothing more on sonic booms. Nothing. We don’t get to learn about the significance of the foci in the picture, nor are we even told the most basic fact about the sonic boom hyperbola - namely, that the sonic boom is heard simultaneously at all points on the hyperbola. We do not even, in fact, see how the focal property of the hyperbola allows us to derive the equation for the hyperbola, period - that formula is just handed to us as-is, and we’re left to trust that it gives us the same shape as the one whose focal distances have constant difference. Actually, we can forget about the focal property entirely: it never comes up in the exercises or in the tests. (We can even, to a certain extent, forget the equation for the hyperbola, and rely on our graphing calculator to remind us when the need arises.) But, check out the funky photo of the Concorde! (Aside: the textbook authors don’t even provide a picture of a sonic boom.)

I have tutored high school math on and off for more than a decade, and during that time, I have noticed a trend toward stripping high school mathematics texts of logic and proof - and, for that matter, of anything that requires sustained and focused attention - and filling in the gaps with pretty pictures, long-winded examples, and graphing calculator applications. No wonder my students regard mathematics as a disjoint collection of facts: their textbooks give no indication otherwise.

Enough about the conics. Let’s skip ahead to a section on combinatorics (7.4 - Pathways and Pascal’s Triangle), because there are enough elementary applications of that material that there’s no need to provide any disjointed, contrived ones. But reading the introduction to this section, I find that an apology is due - to the kid I’m currently tutoring, and to all of the students who perplexed me with their dogged refusal to read the questions before answering them. Because, see, when I (and my readers) ranted about students who expect to be able to solve word problems without reading them, we were assuming that the authors of those word problems were not insufferable windbags who are being paid by the word. And by “insufferable windbags who are being paid by the word” I mean insufferable windbags who are being paid by the word, because I can’t think of any other explanation for this atrocious introduction to path counting, which I swear to God I’m not embellishing:

The destiny of Ottawa changed forever when Queen Victoria chose it as the capital of the province of Canada in 1857. Construction of the Parliament buildings began in 1860 and was completed approximately six years later, just in time to be reaffirmed as the nation’s capital in 1867.

Prior to European settlement, various aboriginal groups occupied the region, including the Ottawa nation from which the city took its name. With the arrival of the French in the 1600’s, and later the British, the fur trade became the mainstay of the economy in the Ottawa River valley. Despite this fact, Europeans did not settle the Ottawa area until 1800.

Not long after the timber trade began, and with the completion of the Rideau Canal in 1832 by Lieutenant-Colonel John By of the Royal Engineers, a community called Bytown was firmly established as a centre for timber trade. This thriving town was incorporated as the city of Ottawa in 1855.

The simplified street map shows a portion of downtown Ottawa, close to the Parliament buildings. If you start at the corner of Bank and Laurier, and travel only eastward or northward, there are two routes you can take to walk to Corner B. How many routes are possible to walk from the corner of Bank and Laurier to the corner of Rideau and Elgin?

I’m no longer going to tell students to read the entire question before answering it. A clueless student who reads only the last sentence of this question has a fighting chance of answering it correctly. One who reads the whole tome, on the other hand, has a non-negligible chance of processing the quantitative data in a way familiar to everyone who’s graded word problems: Number of pathways = 1857+1860-1867-1600+1800-1832+1855 = 2073.

21 Comments

  1. Maybe that last problem was to compensate for the non-Canadian content in the first two that you mentioned?

    - Rudbeckia Hirta — 8/21/2005 @ 3:07 pm

  2. Perhaps those particular insufferable windbags are history textbook writers manqué(e)s? I can just imagine them thinking “Gawd, this math stuff is boring. If only we could write the definitive history of our nation’s capital instead!”

    But whatever the reason, people who can’t get excited about math qua math, or who can’t imagine students getting excited about it, have no business writing math textbooks.

    - Q. Pheevr — 8/21/2005 @ 3:43 pm

  3. I know that only some provinces require Canadian history to be taught in high school. Maybe the textbook’s authors are really historians bureaucrats/politicians with a topical interest in history who want to get history into classrooms through subversive tactics like using math classes as a conduit. This theory actually fits the model perfectly, as in both the cases (history and math) the authors na├»vely think a disjoint collection of facts constitutes the subject.

    - Nicholas — 8/21/2005 @ 3:48 pm

  4. The answer is 56 (8C3), is it? Interesting set of problems that I’m sure I never grasped as a kid.

    I think it’s reasonable to have some asides to read while studying. You can’t spend the whole time concentrating on maths… if this presents an alternative to climbing the walls, then that’s positive.

    - Ronald — 8/21/2005 @ 4:10 pm

  5. I’ve heard of putting in extraneous information (I remember coming across “red herring” in bold print defined in Larson/Hostetler, with some indication that it would feature in many of their word problems), but that seems a bit extreme. It can always be used for those silly problems where you weren’t supposed to actually answer the question, just cross out all the information in the problem that was irrelevant, provided your pencil doesn’t run out of lead along the way.

    - tabstop — 8/21/2005 @ 4:31 pm

  6. Ronald, I’m climbing the walls just reading that text, and I have yet to hear a student of mine comment favourably on these huge tangents. (Teenagers aren’t stupid: when they see that “Your CD Is A Circle” is in the intro to the section on circles, they’re less likely to think “now I’d like to learn more about circles” than “THIS is how they justify teaching this irrelevant crap? This is the most relevant ‘application’ they can think of?”)

    That said: I’d be happy if there’s as much math in the history texts as there is history in this math text.

    - Moebius Stripper — 8/21/2005 @ 4:37 pm

  7. My high school math textbook had a fondness for tropical fish, pizza, and basketball. They tried so hard to work in things teenagers would find cool - unfortunately, with current turnover, math textbooks get replaced when they wear out rather than when they become outdated. Formulas seem to stay the same, at least according to administrators, so logically, the math books are all about ten years out of date: therefore, they were trying to appeal to teenagers of the late eighties and early nineties, who apparently thought disco, day-glow scrunchies, and roller derbies were cool. Even worse were the “technology” sections: they all contained computers about the size of a room and teenagers in truly terrifying polka dots and high tops.

    All my math teachers got so frustrated that they eventually stopped teaching directly from the book: they gave notes, then we used the homework problems from the text.

    What I think someone ought to take note of is the aforementioned turnover, and write a math book with applicable real world problems. Population biology is neither cool nor appealing to most teenagers, but it doesn’t go out of date and it does pull in a real-world example. It’s not like people are ever going to stop domestic travel, so problems about groups of passengers on planes might make sense. The terminal velocity of a tennis ball isn’t going to be different in ten years. And instead of showing a photo of The Cool Teenager hitting one, perhaps they might consider a picture of a tennis ball itself: or someone associated with them. Make students figure out how to balance a checkbook, how much paint it’ll take to cover a room, and at what angle to hammer a nail in so as not to crack a beam. They may be bored, but ten years down the line, they’ll remember, and more importantly, they’ll use it.

    - Madison — 8/21/2005 @ 4:55 pm

  8. *sigh*

    I saw the CD introduction and thought that relevant problems were about to appear. Like, how much more data storage you’d get from doubling the radius of a CD, and how much faster it’d have to spin to play music without skipping. Sadly, it seems like the authors of the book didn’t have that quite in mind.

    - Dan — 8/21/2005 @ 6:43 pm

  9. Madison - yes, yes, yes, on the relevant-but-not-hip problems. I’ve never seen a textbook succeed in its attempt to be cool. The authors always come across as hopelessly out-of-touch textbook authors who are trying too hard to be cool.

    Dan - wow, I didn’t even think of that. Yeah, that would actually be relevant to Real Life, and appropriate and non-contrived. You should write high school math textbooks. (Hmm - maybe not; I’m not sure I would wish such a job on someone I cared about. What I mean is, “if you wrote high school math textbooks, I would try to get them into every high school classroom.”)

    - Moebius Stripper — 8/21/2005 @ 7:35 pm

  10. Dan — I thought that type of problem was coming too because it actually did–in my physics text book. Linear velocity as a function of distance from the centre, angular momentum, moment of inertia: we got all kinds of cool problems mechanics from that CD. Come to think of it, I don’t think they ever told us it was a circle; I wonder how we were able to solve the problems without that important bit of information….

    - Nicholas — 8/21/2005 @ 7:49 pm

  11. in my attempts to get my students to at least *recognize* the possible importance of trigonometry, i often started the unit with the question, “what moves in circles?” i got answers like: planets, wheels, motors, CD’s, etc. my favorite answer, though, was one smart-alecky kid who said, “conversations with my grandmother”.

    - Polymath — 8/21/2005 @ 10:16 pm

  12. Where introducing coolness to math would help would be in making it seem cool to be the kind of person who would look at a CD and say, “I wonder what would happen to the data capacity if I were to double the radius” or look at a room and say, “I wonder how much paint it would take to cover the walls,” or look at a crochet project and say “If yarn is going into it at the rate of X yards per hour, how fast is the side of the blanket increasing?” When I betray that I think like that to others, I am immediately branded as uncool. So maybe a math textbook that has The Cool Kids pondering stuff like that would be helpful. Otherwise, I’d say The Cool Kids should be out of a job.

    - Wacky Hermit — 8/22/2005 @ 8:25 am

  13. namely, that the sonic boom is heard simultaneously at all points on the hyperbola.

    I hope I’m not being obviously dense, but I’m not understanding why simultaneous hearing is a property of hyperbolae. It seems to me that, independent of the shape of the intersection between a sonic boom and the ground, everybody standing on the intersection at the same time would hear the sonic boom at the same time. Put differently, what happens when an SST flies through the center of a circle of people floating in the air?

    - r. clayton — 8/22/2005 @ 11:49 am

  14. …assuming the cow is spherical, of course.

    - meep — 8/22/2005 @ 1:38 pm

  15. R. Clayton,

    In your scenario, all of the people would hear the blast at the same time. However, if you cut the shockwave along the plane’s axis of symmetry, you get a hyperbola.
    \
    \
    ->-|
    /
    /
    Everyone along the slashes hears the boom at the same time.

    On a more general note, if you want “real world” examples, get real world examples. They wouldn’t teach the math if it weren’t good for something. Textbooks in science and engineering classes are full of real world examples.

    - Robert — 8/22/2005 @ 2:56 pm

  16. It took the spaces out of my drawing:
    .\
    ..\
    ->-|
    ../
    ./

    - Robert — 8/22/2005 @ 2:58 pm

  17. R. clayton - you’re right; the sonic boom doesn’t necessarily have to be a hyperbola. (Though, to be fair, the example does mention that the Concorde is travelling parallel to the ground - in which case it does.) Different flight paths would give the other conics.

    - Moebius Stripper — 8/23/2005 @ 5:25 pm

  18. Math with contexts!

    It must be an attempted response to that difficult question “why do we have to learn this? when am I ever gonna need to know this?”. Context, preferably real-life context, as an introduction to the real content (like mathematics, physics, chemistry)….

    - Qulog — 8/24/2005 @ 2:56 am

  19. what then are we to do?
    — textbooks are no damn good;
    everybody knows it. but i *know*
    why the industry keeps churning out
    worse and worse editions of stuff
    that was ill-conceived in the first place:
    they’re in it for the money.
    what i *don’t* understand is how come faculty
    go along with it and keep ordering the @#$%%!! things.

    obligatory citations: my own
    http://members.aol.com/vlorbik/odc/trends.html
    and the inspiring
    http://www.maketextbooksaffordable.com/
    .

    long live MS & T,D, & M.
    yrs in th’ struggle.

    - vlorbik — 8/24/2005 @ 7:57 pm

  20. I don’t think that money is the only reason for the nascent fertility of math textbooks. I am inclined to agree with Underwood Dudley, writing about calculus texts in particular:

    Why are there so many calculus books, and why do they keep appearing? One could be cynical and say that the authors are all motivated by greed. But I do not think so. I think that authors write new calculus books because they have observed that students do not learn much from the old calculus books. Therefore, prospective authors think, “if I write a text and do things properly, students will be able to learn.” They are wrong, all of them. The reason for that is…CALCULUS IS HARD.

    (That said? I can’t think of any reason other than money for the abundance of graphing calculator applications. The only people who seem to like graphing calculators are the people who make them, and math teachers who aren’t very good at math.)

    And there’s an easy way for high schools and universities to make math textbooks affordable: stop replacing the damned things ever year. There’s no excuse for requiring students to own the updated version of a textbook for a subject that hasn’t changed in two centuries.

    - Moebius Stripper — 8/26/2005 @ 12:18 pm

  21. When we historians want to get quantitative, we can do so without long digressions which preceed the math. As it happens most historians are not wordy, we prefer a chart or two of data which we ask a half dozen questions about. Most of this stuff involves economics, census data, or voting analysis. That’s what we have historical records for. We also tend to go over the same data several different ways, trying to interogate the data.

    Its quite easy to do a lot of good analysis like this with middle school level math (mostly percentages), but after you guide a class through a lesson, they have a much better idea about women’s participation in the labor market, rates of home ownership, the uneven prosperity of the 1920’s, or correlations between occupation and political affiliation. I try to have one quantitative lesson in every unit in American history. World history lacks the quantitative data, but is ready made for science applications.

    - Kenneth — 8/27/2005 @ 12:21 am

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