Devlin's Angle

February 2010

Is Math a Socialist Plot?

Last month, on the most recent of my occasional appearances on NPR as The Math Guy, I talked about a formula developed by a British mathematician to determine the minimum space required between parked cars in order that you can park your car between them with a simple one-two, reverse-in-and-drive-forward maneuver. You can listen to the piece, and read about it, at here. You can also find the mathematical paper that led to the story, by Professor Simon Blackburn of Royal Holloway University, London, at here(pdf).

The NPR website also has the listeners' discussion thread generated by the program. This is what I want to talk about today. Glancing at it a few hours after the program aired, I noticed something of interest regarding the popular conception of mathematics. So much so, that on a couple of occasions I threw in a response myself, to try to dig a bit deeper.

The result is, of course, hardly a scientific survey. The total number of contributions was only 69, including repeat submissions by some listeners. On the other hand, the contributors were individuals who (1) prefer to get their news, or at least some of it, from NPR, and (2) are sufficiently interested in a story about mathematics to take the time to sit down at a computer and write in a comment.

Apropos my mention of NPR listeners, I note that the NPR audience is about a mere 8% of the U.S. population, and is significantly more educated than the population as a whole. Which is why I think it is worth taking at least some account of the responses to my parking piece.

Over-educated? A brief aside

Talking of the educational level of NPR listeners - and now I may appear to be straying slightly off my main topic, but as you will see this ties in with the thrust of this month's column, namely national education standards and U.S. economic competitiveness - did anyone else notice the article on Yahoo! News on January 29, written by Sam Stein, which carried the headline "Obama Still Loved by the Over-Educated"?

"Over-educated"? I did a double-take. Is it ever the case that a person can have too much education? Surely it's a misprint. No, there it is again in the article's very first sentence,

"President Obama's popularity has slipped among a wide swath of the population. Among the nation's overeducated, however, he continues to do just fine."

Good Lord. What can Mr. Stein possibly mean? Our reporter goes on to say,

"Gallup surveyed more than 25,000 voters over the past calendar year and found that the president remains well-liked among those with multiple degrees."

Ah, there we have it. According to Mr. Stein, and the editors at Yahoo! News who published his educational wisdom, anything beyond a bachelors degree amounts to over-education. Not so Gallup, I'm relieved to say. The research organization itself published the result of its survey under the deadline "Americans With Postgraduate Education Still Back Obama". No, it's Mr Stein and his editors at Yahoo! News that think anything beyond a first degree amounts to over-education. Folks, if this (presumably under-educated, or so he comes across) Mr. Stein is at all representative of the U.S. population, and it is indeed generally believed that anything beyond a bachelors degree is superfluous, then we may as well just throw in the towel when it comes to international competitiveness and start to learn Chinese and Hindi. (I only hope for his sake that Mr. Stein never gets sick. He's going to have a devil of a time trying to find a physician who is not over-educated. And we can assume he never uses Google in his research, right? That, after all, came right out of a post-graduate research project at Stanford - you know, the superfluous stuff we don't really need.)

Enough of this anti-educational ignoramus. What about parking?

I first came across Professor Blackburn's car parking formula from an MAA News Alert. "Aha!" I thought. "This might make a nice piece for the Math Guy to talk about on Weekend Edition." I knew that a story that links math and car parking was sure to grab listeners' attention.

A quick Google search (gosh, wasn't life much better before we were over-educating all those people like the Google founders?) took me to Blackburn's university homepage, along with his paper. The mathematics he used turns out to involve nothing beyond Pythagoras' Theorem. Strike two! I would not have to tell listeners, as I often have to, that "the math is pretty complicated, and can be understood only by people with a Ph.D. in math." True, Blackburn has to use Pythagoras' Theorem in a way that is a bit more complicated than many high school students are used to seeing, but nothing that a good high school student could not follow - or even work out for themselves.

So now I have a story with an attractive audience hook where the math is something everyone saw at school. The one other thing you need to make a science news story work is to be able to answer the question "What is this good for?" Why this is an important question, has always baffled me. After all, the news media are full of reports about sports, music, movies, entertainment, and the arts, none of which are "good for anything" in the sense that science stories are supposed to live up to. "People enjoy it" or "Entertainment is a good thing in itself" or even "People are just naturally curious and want to know stuff" (though not too much according to Mr. Stein) are generally regarded as sufficient justification for most things the news media report on. Still, my colleagues in the media tell me that a science story won't work unless it gives an indication of a possible "application". And I know from many years of experience that, as in any other profession, the professionals in this case do know what they are talking about. (The story I am about to tell you provides further confirmation of the science-story-application dictum.)

Who needs a math formula for parking?

Professor Blackburn states on his webpage that his research into the mathematics of parking had been commissioned by, and paid for by, Vauxhall Motors, a division of General Motors. Now, Vauxhall did not say why they asked the professor to develop a mathematical formula for parking cars. On the other hand, they did make it clear that they now own the formula he came up with, so we can assume they think it may have commercial value to them. My own immediate reaction when I first came across the story on the MAA website was that they were planning on developing an automated parking control system. Toyota introduced a commercial automatic parallel parking system for the Prius way back in 2004, and similar systems have since been developed by Lexus, BMW, Volkswagen, and Ford. With the basics of reliable automotive engineering long since established, future competition between the remaining automobile manufacturers is likely to focus on the various extras they can offer. An automatic parking system was surely Vauxhall's motivation.

Like almost everything else in the hugely competitive, automobile industry, the design details of automated driving aids such as parking systems are closely guarded trade secrets. But it doesn't take a genius to realize that the first thing the system would need to know is whether there is enough space to even start the maneuver! People do that by experience. We become familiar with our car, and it's usually enough to eyeball the available space in order to determine whether to go ahead and park there or look for somewhere else. An automatic parking system could use sensors to determine the dimensions of the space (length and width are both important), but how would it decide whether that space is enough?

"An on-board computer could do that," I hear you cry. Indeed it could. Everyone now has grown used to the fact that computers can make simple decisions. What many people evidently do not know, however, is that they do so using mathematics, and thus they can do so only after the question has been converted into mathematical form. Give an automobile control system Professor Blackburn's formula (or else a table of values computed in advance from the formula) and it will be in business. (There are other approaches. But the simplest, cheapest, and almost always the most reliable, is the mathematical formula approach. In fact, the other approaches all involve mathematics in one way or another, though perhaps not in the form of a single formula embodied in a computer program.)

So, although I could not say to my NPR listeners with certainty that Vauxhall wanted the parking formula in order to build an automatic parking system, I could address the application issue by saying I was pretty sure that was the case. I had my story.

What is math for?

The recording went well. (Everything you hear on Weekend Edition is recorded in advance except for the interview with Daniel Schorr.) It was, we all felt, a fun as well as informative piece. It also met an additional requirement I always strive for but often cannot meet, which is to provide an example of how mathematics is discovered and used that teachers could take into their classes on Monday morning. In this case, it hit that goal in spectacular fashion, since the math was school level and the application was so obvious and self-evidently relevant both to our everyday lives and to the prosperity of the nation (or at least to General Motors, whose fortunes are perhaps no longer quite the same as those of the United States, but are still important to us all).

As often happens, when Saturday morning rolled around and the day's news came in, my parking segment had to be cut down, so the bit about the possible application of the new formula was not included in the piece that was broadcast. Doubtless the producer (decidedly not a math person) felt, as I did when I heard the piece over the air, that for this story at least, the application was blindingly obvious. Well, it turns out that to some listeners, it was not at all clear what the application was. (And remember we are talking about "well-", if not "over-educated", NPR listeners, who are interested in a math story.)

Actually, that is not quite true. They did think they knew what the application was. And that, I fear, tells those of us in mathematics education the impression many of our students have when they leave us. Several respondents said things along the lines of "I don't need a math formula to tell me how to park," or "No one has the time to acquire and plug the numbers in and do that calculation before they park." In other words, when they see a formula, they think the purpose is to put numbers into it and work out the answer.

Who is to blame?

Where on earth do they get that idea? I'll tell you where. From all those exercises in school math textbooks that present mathematics in exactly that way. Even worse, those textbooks do so under the pretext that this shows the students real life applications of math. You know the type of problem, finding out how long it takes a swimming pool to fill from a hose that delivers X gallons of water a minute, or how much concrete you need to order to lay a driveway Y feet by Z feet by H feet deep. The textbook or the teacher may say that this shows why the math is useful, but the student knows full well it does nothing of the kind, and so, I suspect, do the textbook author and the teacher in most cases. Faced with filling a swimming pool with water, what you, and I, and everyone on the planet would do, is turn on the water, watch for a minute or two to get a sense how fast the water level seems to rise, then do something else nearby, checking periodically on the progress until it's getting close to being full, and then watching it until it's done. And although I once did meet a math teacher who swore she actually did calculate the amount of concrete required to lay a driveway at her home, I suspect most people would call in a contractor who would come to the site, pace out the dimensions, and then take advantage of years of experience to say, "This will take one and a half truckloads. Tell me if there's anything else you want me to do with any that's left over."

In the real world in which real people live, even the mathematically over-educated, no one uses mathematical formulas in their day-to-day life. In their work, perhaps. But not in the daily stuff of living. What they often do find themselves doing is using a device or a smartphone app or an automobile dashboard display that depends on math. This is so common that it's not at all difficult to show students where and how math is used. Our students, and we, are surrounded by such examples. Why, oh why, resort to fake "applied problems" when there are plenty of real ones? The only difference with past eras is that, these days, it's not people who "do the math," it's the various devices we buy, use, and carry around with us. And as the parking formula shows, those genuine applications can sometimes involve very basic math.

Even in the case where you can't find a genuine application of some mathematics, it's not hard to imagine a plausible one. Instead of asking students to carry out the swimming pool example with an unrealistic scenario, say that their boss wants them to develop a small automatic valve that can be set to turn off the water when the pool is full, or that a local builder has asked them to develop a smartphone app or a website calculator that customers can use to determine how much concrete they need when placing their order. These are formulations that will seem relevant to the students. The students will, of course, end up doing the same math! You're just presenting it in a plausible fashion.

BTW, I'm not arguing against people learning how to solve math problems. There are several good reason why solving math problems is a valuable part of education. One is that time spent solving math problems develops analytic thinking skills that prove beneficial in other walks of life. But that reason does not work well with young people who have not yet had time to experience the benefits. (It doesn't work so well with lots of others either, though in my experience people who say "I never could do math but I did okay for myself" often go on to say something that demonstrates they, or at least those who have to put up with their ill-thought-out pronouncements or decisions, would likely have benefited tremendously from their having put a bit more effort into the math class.)

What I am saying is, please don't use unrealistic, fake scenarios and tell the students they are seeing "How math is really used." Give them realistic examples. Plainly, many of the contributors to the NPR discussion following my piece simply had no idea that mathematics was required in order to develop computer control systems. They saw a math formula and thought that we - that would be me and NPR - were implicitly claiming that it's purpose was for a human driver to calculate whether there was enough space to park. Sheesh. Give me a break.

In fact some contributors went on to say they thought the research that led to the formula was a waste of time. Just think about that for a moment. Regardless of whether this particular formula is ever used in an automatic parking system, research into the topic is clearly critical to the competitiveness of the U.S, automobile industry. Haven't we had out butts kicked too many times in that industry to think that research into ways of remaining competitive is a waste of time? What about other uses of the same kind of research, such as building robots that U.S. troops in Afghanistan and Iraq can use to search buildings for explosive devices? Is saving the lives of troops a waste of time? I could go on, but surely I've made my point.

Now I am pretty sure that not one of the NPR discussion contributors would say making our industries competitive or saving the lives of U.S. troops was a waste of time. They effectively made those claims because they have absolutely no idea how mathematics is used in today's world. Moreover, they have formed this belief after having had at least ten years of almost daily mathematics instruction. In fact, so firmly rooted is their belief that math is something you do at school to solve irrelevant problems but is of no use in the real world, that even when I jumped into the discussion and explicitly gave some examples (the ones in the above paragraphs), they remained unable to see math in a new light.

Forget whether they came out of the school system good at math or hopeless at it. I'm not talking about how good or bad they are at doing it. They don't even know what it is or what it is used for. For comparison, I don't know in much depth how an airplane flies, and I could neither fly one nor build one, but I have somehow managed to learn what it is and what it is used for. That's all I'm talking about for mathematics, for heavens sake! When ten or more years instruction fails to leave people having even the faintest idea what something is, why it is done, or what it is used for, then something is seriously wrong.

Your homework for tonight is to find out what.

So what about the link between math and socialism?

Ah yes, my title promised something about mathematics being a socialist plot. What was that about? Well, two contributors to the NPR discussion claimed that this whole math thing was just a socialist plot. Since they provided no support for this assertion, I am inclined to give mathematics the benefit of the doubt here. More likely, I think, is that the two individuals habitually see everything as a socialist plot. Which is odd when you consider that by the definition of socialism implied by such individuals, Canada and every country in Europe would classify as socialist societies, though by and large those countries all seem pretty successful capitalist economies, and very definitely free societies with democratic governments. But I digress.

Okay, I admit, the socialist connection was a pretty small part of my essay, albeit an intriguing one. (It intrigued you, right?) My main reason for choosing the title I did was to try to ensure that you read through the entire column. Hey, if Channel 5 can promote the television evening news that way, why can't I? If you did, and are still reading, my ploy was successful. I promise not to pull the same trick again - for a while.

Finally, the answer to last month's quiz

Well, it wasn't a quiz so much as a teaser. I asked why mathematicians think that this is an ideal year? The answer is that January 29 of this year marked the 200th anniversary of the birth of Ernst Eduard Kummer, who introduced the notion of an ideal. According to some accounts, he did so in 1843, when his attempt to prove Fermat's Last Theorem broke down because the unique factorization of integers did not extend to other rings of complex numbers. According to that explanation, he attempted to restore the uniqueness of factorization by introducing 'ideal' numbers. (He did of course prove Fermat's Last Theorem for a large class of exponents.) Others, however, among them Harold Edwards, have stated that Kummer was motivated by his work on higher reciprocity laws. In any event, our modern notion of an ideal in a ring was introduced later by Richard Dedekind. Happy 200th birthday, Ernst Kummer.

Devlin's Angle is updated at the beginning of each month. Find more columns here.

Mathematician Keith Devlin (email: [email protected]) is the Executive Director of the Human-Sciences and Technologies Advanced Research Institute (H-STAR) at Stanford University and The Math Guy on NPR's Weekend Edition. His most recent book for a general reader is The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter that Made the World Modern, published by Basic Books.