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"? 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.)
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.)
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.
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.
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.
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.