Monday, December 04, 2006
Monday Musing: Aptitude Schmaptitude!
Like most people, I have no special gift for math. This doesn't mean, however, that I am mathematically illiterate, or innumerate, to use the term popularized by John Allen Paulos. On the contrary, I know high school level math very well, and am fairly competent at some types of more advanced math. I do have a college degree in engineering, after all. (There is no contradiction in this--pretty much anyone can be good at high school math.) While the state of mathematical incompetence in this country has been much lamented, most famously in Paulos's brilliant 1988 book Innumeracy, it is still tacitly accepted. Around the time when Paulos was writing that book, I was an undergraduate in the G.W.C. Whiting School of Engineering at Johns Hopkins University, and I soon noticed that to get help with mathematics, one generally had to consult with Indian, or Korean, or Chinese graduate students. (The best looking women happened to be in Art History though, so I very quickly developed a deep fascination for Caravaggio!) Some of the engineering departments (like mechanical engineering) did not have a single American graduate student, and since that time things have only grown worse, with much of the most important technological and scientific work in this country being performed by immigrants. (About a quarter of the tech startups in Silicon Valley are owned by Indians and Pakistanis alone.)
Being incompetent in math has become not only acceptable in this widely innumerate culture, it has almost become a matter of pride. No one goes around showing off that he is illiterate, or has no athletic ability, but declarations of innumeracy are constantly made without any embarrassment or shame. For example, on a small essay that I wrote here at 3QD about Stevinus's beautiful proof of the law of inclined planes, my extremely intelligent and accomplished friend, and frequent 3QD contributor, Josh (now teaching and studying writing at Stanford) left an appreciative comment, while adding, "I couldn't math my way out of a paper bag." (Sorry to pick on you, Josh, the example just came handily to mind!) Confessing confusion about numbers is taken to display not only an endearing honesty in self-regard on the part of the confessor, but is also frequently taken to hint at a fineness of sensibility and high development in other areas of mental life. Alas, (Josh notwithstanding) there is no evidence of any such compensatory accomplishment in those who are innumerate. Not knowing high school level math is not easily excusable. But reader, if you are innumerate, it may not be your fault and I will not scold you. In fact, I'm going to try and pin the blame on American culture.
The way I see it, there was a one-two cultural punch which has knocked out numeracy in this country: first, there was a devaluing of mathematical competence in and by pop-culture; second, justification was provided for not learning mathematics to those already disinclined to do so by the devaluation. That's it. The rest of this column is an attempt to flesh this out a little bit.
Just like learning to read (or for that matter, learning to play the piano) mathematics is something that it takes years to learn well and develop a good feel for. Reading, writing, playing the piano, and doing math are highly unnatural activities (unlike speaking, say) which we are not naturally evolved to do. Instead, we take abilities we have evolved for other purposes and subvert them because it is so useful to learn these things. And the price we must pay is that they are not always a great joy to do. Just as one must learn one's ABCs or practice one's scales, one must also memorize one's times tables, and I cannot think of a way to make that particularly interesting. It just has to be done. In fact, young students have to be disciplined into learning these things. But before anything else, it must be made clear that while learning math requires no special abilities, it is different than learning some other things in one crucial way: the study of math is (at least up to the high school level) very hierarchical and cumulative. While one may suddenly do very well in a European history course in high school while having paid no attention to any history in junior high, it is not possible to do well in Algebra in high school without having learned the math one was presented with in junior high. I sometimes tutor students for graduate admissions tests like the GRE or GMAT, and the first time I meet with them they often show me algebraic word problems they got wrong in a practice test. I ask how their junior high math is, and no one ever admits that they can't do 7th or 8th grade math. Then I ask them to subtract one number from another for me, using a pen and a piece of paper I hand them: say -2and7/8ths minus 1and3/17ths. You'd be surprised how many of them are tripped up and make a mistake in a simple subtraction that any 8th grader should be able to do. The problem is they really cannot do ANY algebra until they are consistently and confidently competent in such simple tasks as adding, subtracting, multiplying and dividing numbers, and yes, this includes fractions, decimals, and negative numbers, but even these college graduates generally are not.
When I was a young child in Karachi, I liked reading Archie comic books, the hero of which is a bumbling, freckled, red-headed student at Riverdale high. He and his slightly evil schoolmate Reggie have a rivalry over class-fellows Betty and Veronica, who in turn are rivals for their attention. A slew of hackneyed characters rounds out the cast of this teenage-hormone-drenched-yet-wholesome comic book sit-com, including the glutton Jughead, the jock Moose, and others, but one of the least attractive characters serves as the pop-cultural stereotype of the math prodigy: Dilton Doily. Ridiculously and alliteratively named for a small ornamental mat, poor Dilton is smart but must pay the price. He is a small, unattractive, unathletic and insignificant nerd, complete with coke-bottle glasses and a pocket protector. No one in his right mind would or could look up to Dilton as a role model. Rather, he almost seems to be there as a warning of what might happen to one if one doesn't watch out and avoid math. The rather stupid everyman Archie is, of course, glorified as an ideal and it is he who usually gets the girl. This is just one of a million such stereotypes in movies, TV shows, books, cartoons, and a zillion other things in which being mathematically literate is equated to basically being, at best, impotent and insignificant and, at worst, a sideshow freak. This is in part because geniuses in math, like in everything else, are sometimes eccentric, and in crudely contemptuous caricature, this eccentricity is easily exaggerated into freakishness. (In fact, I think that Stephen Hawking captures the popular imagination precisely because with his computer-generated voice and his sadly twisted pose in his wheelchair, he looks freakish to people and this so conveniently fits in with the popular prejudice about mathematical genius. It makes people feel good about being innumerate if being numerate is going to make one into a physical Stephen Hawking. The public even exaggerates his mathematical and scientific ability in a twisted sort of sympathy: in a poll of professional physicists, Hawking did not even make the twenty top living physicists, though popular polls would probably place him at number one; and probably number two, after Einstein, of all physicists, living or dead.) I could really go on forever providing examples of cultural hostility to mathematical literacy (and an argument could even be made that this is part of an overall anti-intellectual trend in America in the last few decades) but I am not interested in doing that here. My point is that it ain't cool to be good at math.
But here's the devastating second part of the one-two punch combination: if you haven't learned your math, it's because you don't have an aptitude for it. (And for the reasons given in the previous paragraph, you might as well thank your lucky stars for that!) Through a complex series of events, I came to the United States as an 11 year-old boy to live with my brother in Buffalo for two years before returning to Pakistan for high school. I attended 7th and 8th grades at a suburban public school, and I loved it. To this day, I remember many of my teachers with immense gratitude and fondness: Mr. Shiloh, Social Science ("Washington, Adams, Jefferson, Madison, Monroe, John Quincy Adams... " Yes, I can still recite all the presidents, Mr. Shiloh); Mr. Schwartz, Science; Mr. Coin, Mathematics; Ms. Muller, on whom I had the biggest schoolboy crush, German; etc. But one bad thing did happen to me: I was given something called a differential aptitude test (DAT) soon after my arrival, and the results were explained to me by my homeroom teacher: apparently, while I was supposedly gifted in verbal skills and artistic abilities, I was not much good at math or music. I took this to heart, and stopped paying much attention to mathematics. What was the point, if I just didn't have the requisite ability to get it? It took my father's devoted and prolonged drilling in mathematics a few years later, back in Pakistan, to undo the damage of that test, and I eventually got 99 out of 100 marks on my boards exam there.
I am by no means alone in this experience, and I believe that these tests and the whole idea that some children are better at some things and others at other things, and that they should be told this very early, is a stupendously dangerous one. What purpose can it serve, other than to encourage kids to give up on subjects that they may not have previously done well in for a thousand different completely contingent reasons? They will naturally already be trying harder at things they are good at, so they don't need more encouragement there. This idea, that some people are good at some things, and others at other things, is fine if it is a matter of catering to children's self-esteem when they are selecting a sport to play, for example. One person can be happy playing football, while another smaller person might become good at Badminton, or whatever, letting everyone believe that they have some special ability. After all, most of them will not grow up to be professional sportsmen or women. But when it is about something as fundamental and basic to future understanding of the world that they live in as mathematics is, it is hugely destructive. I firmly believe that anyone normal can be taught to master the mathematics of high school, and that it is all that is needed to produce a profoundly more numerate society, but it is near-impossible to overcome the "I'm just no good at math" barrier. Why are people even allowed, much less encouraged, to believe this about themselves? For those students who are geniuses, as well as those that are truly handicapped in some particular mental skill, these tests are not needed. That can be tested for in other ways. It is the huge majority of kids falling under the middle of the bell curve that tests like the DATs are so damaging to, and this, I think, is the real root of innumeracy in this country.
And then there are those who feel that it is no great loss to be innumerate. In that case, I'm sorry, but you don't know what you are missing. Some of the most profoundly beautiful ideas produced in the last few thousand years are beyond you, as is the serious study of about 80% of what is taught in modern universities. Even the social sciences cannot exist without math anymore, and you cannot have any deep sense of political and economic issues if you are completely innumerate.
Let me summarize: math emphatically does not require any special ability, but it does require a lot of discipline, and if you fall behind, because of its cumulative nature, you will find yourself in a cycle of failure to master whatever you are presented with next. If you try to make the argument that math is something that only a portion of the population have the congenital ability to master, even at the high school level, then you must also make the argument that this mysterious ability, unlike any other mental ability that we know of, is also sharply unevenly spread across various countries. Japanese children have much more of it than American ones, for example, because Japanese high school students regularly trounce American students at the same level in math tests. You will also have to explain how Japanese children who have been living in America for a couple of generations lose that congenital ability. No, I'm afraid that will not do.
This essay is dedicated to my friend and greatest anti-innumeracy warrior, John Allen Paulos, whose book Innumeracy I mentioned above and recommend highly. Click it to buy it, or click here to buy his other books.
My previous Monday Musings can be seen here.
POST SCRIPT: John Allen Paulos has sent the following comment by email:
I agree that to an extent mathematics is a hierarchical subject and that a certain amount of drill is absolutely necessary to do well in the elementary portions of it. Nevertheless, it's important to realize that considerable understanding and appreciation of many important ideas can be obtained via puzzles, everyday vignettes, expository articles, and sketches of applications.
A loose analogy comes to mind: If all one ever did in English class during elementary, middle, and high school was diagram sentences, or all one ever did in music class during those same years was practice scales, it wouldn't be very surprising if one lacked interest in or appreciation for literature or song. Given suitable allowance for hyperbole, however, this is what often passes for early math preparation. The analogue of literature and song is not provided to mathematics students in their early studies, so there seems little rationale for developing the necessary mechanical skills needed.
A marginally relevant anecdote: I gave a lecture once to a very large group of students at West Point. Whether because of their military interests or their personal psychology, some were quite interested in the sequence or hierarchy of mathematical subjects. During the question and answer session after my lecture, I was told that the proper order of these subjects was arithmetic, algebra, geometry, trigonometry, calculus, differential equations, and advanced calculus and then was asked what comes after advanced calculus. The students were nonplussed at my answer of "serious gum disease."
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