The operative questions in this post are two. First, can we teach all students math in the way we teach elite students math? Second, if we can, why don't we?
Math can be about intuition and the spark from an idea that has potential. Math can also be drudge, slugging through a lot of notation that is not enlightening, perhaps learning algorithmic procedures with no other motivation to do so other than that it will be on the test. One might think there is a clear division between the two, though I believe otherwise. While on occasion results may appear immediately to the bright student, verification is a staged procedure and one needs to work through the steps. More often, even the bright student discovers the result only after working through several preliminary results. The intuition is at hand at the beginning, to suggest this is a good path to try. Or the intuition may appear only after some faulty alternatives have been attempted. Elite students learn this linking between spark and process and develop intellectual confidence this way. If it is well learned it becomes enjoyable to do and the student wants to continue to employ the approach even in non-math pursuits. One reason I like to write blog posts is that the exploration of ideas that appear as I compose, Donald Murray calls it writing to learn, feels like working a math problem, as does the pre-writing I do before getting to the keyboard.
In the nature/nurture debate about what drives individual performance, I don't want to rule out the nature component entirely, but here I do want to focus on the nurture bits. I was an elite math student and I will use my recollections to illustrate, both how that came about and the advantages that education conferred. Much of this happened outside of regular math classes, either in entirely social situations or in academic situations that were available only to elite students.
The earliest memories are from grade school and I have two distinct ones that come to mind. Students who bought the hot lunch ate in the cafeteria. Students who brought their lunch ate in the auditorium. I don't recall which grade this was but I sat together with a few classmates in the auditorium and one of them would quiz the rest of us as follows. x plus y equals some number. x minus y equals some other (smaller) number. What are x and y? We did these for the fun of it only, at least as far as I could tell. And I was definitely a follower here. I don't remember what got me to sit with these boys, nor what drove the leader to ask these questions. But what is obvious to me now is that this social interaction was early preparation for algebra, which I take makes for a stumbling block for many kids. For me it was a breeze.
The other memory is of my sister getting tutoring at home. She is five years older than I am and so was many grades ahead of me. The way our house was set up, the logical place for the tutoring session was in the dining room, which we otherwise didn't use too often. Sometimes I would sit in the living room and listen to the tutoring. For a while this was science tutoring. Later it was math tutoring done by a different tutor. I don't remember specific math I picked up this way, but it was again early exposure to things I would see later.
Perhaps the bigger deal is that the math tutor was Mrs. Joffe, who would become my eighth grade math teacher. When I was in her class she remembered me from earlier and was insightful enough and kind enough to suggest I join the math team. I wouldn't have done this on my own. At the time I thought of myself as a social studies guy. (My dad was a lawyer.) The school had some current events magazine that I did join myself as an extracurricular activity. Some of the other kids on the math team had done it in seventh grade too (and I recall them eventually going to Bronx High School of Science, even though it was a schlep from where we lived in Bayside).
The math team was a world unto itself, with a culture of its own and a linkage to people who were a year ahead of me in school. (My school was converting from a junior high school to an intermediate school. So I graduated from there after eighth grade. The other school we had our math meets with was still a junior high school at the time.) This exposure to bright kids a year ahead indirectly is like getting a big gold star. It really conveys a sense that you belong there and creates an expectation that you should follow a similar path to what these people were doing. Also, the math team established for me a connection between doing math and playing chess, which would become important in high school.
I next did math team in eleventh grade, with the coach my ninth grade math teacher, Mr. Conrad. Ultimately, I took two other classes from him, analytic geometry and trig, which most college bound students take, and math team workshop, a specialty class aimed at preparing us for the competitions and for working exotic problems. Near the end of one marking period, I recall playing him a game of Twixt for my course grade in the math team workshop. I'd get a 90 if I won but only a 75 if I lost. Not that long ago I found this Math League Web site. My teacher was one of the co-founders of Math League. So I contacted him via that site and asked if he recalled the Twixt game. He did and said he didn't pull any punches in playing me. He was a tournament bridge player and quite competitive about these sort of games. By the way, I did get a 90 in the class.
Prior to and during math team there was another activity that was similar, but not done in a timed way, called the Problem of the Week. A non-typical algebra or geometry problem was posted on the bulletin board outside the Math Department office. Students were invited to submit their proposed solution. This was done for the fun of it, not for the credential. Doing these, in my view, is similar to working a very hard Sudoku. The procedure to solve the problem is not automatic. One has to discover it. My favorite one of these that I recall is the modified donkey theorem. Two triangles are congruent if angle, side, side, equals angle, side, side as long as angle is the largest angle of the triangle.
How does one go about proving such a result? There is a trick, of sorts. Someone on the math team is apt to figure this out on his own but many other students would not, because it simply wouldn't occur to them to do this. The trick is to draw the triangles adjacent and sharing a common side (one of the sides that are equal). It turns out that using the side that is opposite that largest angle is what you want. That's not a full construction, but it is enough that a bright student should be able to figure out the rest. I've written a chapter of my book Guessing Games entitled Guessing in Math that argues students should be taught to work such problems. Many people should be able to do so. Alas, those who see math only as a drudge would view such a goal as outside their capabilities or as a painful thing to do, rather than a reward in itself, which it rightfully is.
I suppose many people hit a wall with math. They find themselves in a class that seems over their head and don't feel comfortable about working through their difficulties, with no confidence that they can put in sufficient effort to overcome their lack of understanding. I first experienced this sort of thing at the HCCSIM, the Hampshire College summer program in math. I was a member of their very first cohort, forty years ago. I took a class in number theory/abstract algebra done in an intuitive way sans textbook. The program was for six weeks. For the first two weeks or so I was doing fine in this class and keeping up with the daily homework. But by week three I started to find it difficult and didn't know what to do about it, so floundered thereafter. I don't believe I was alone in this. There were three or 4 geniuses in the class and a few others who were keeping up even at the end, but the rest of us were not treading water. During the last two weeks of the program I took a class on probability, which was more do-able.
Looking back at the time I think there were two things going on that explain the trouble. First, abstract algebra for me was not as much fun as geometry, because with the algebra I didn't yet have the mental equivalent of drawing pictures, so we were taught some structure without much intuition. There was some intuition - doing arithmetic modulo a prime number gives an example of a non-ordered field - but in other cases there wasn't. I don't know if I first heard this at Hampshire or only later in college taking abstract algebra, but there was encouragement to not rely on the examples because they might include features that don't generalize. That proved hard for me. I wanted the examples. Previously I had found most math fairly immediate. This was the first instance where I didn't.
The other thing was how the day was scheduled and not putting in enough time to make up for my shortcomings. Mornings were filled with class. In the afternoon you could do your homework. But it was good to get some physical activity and I often played tennis in the afternoon and there was a very popular volleyball game after dinner. Later in college I learned that figuring things out takes as long as it takes. Sometimes that can be quite a while. Hitting a wall may make you less inclined to put in the time, though I think in my case then I simply didn't understand it was necessary.
Nonetheless there were some very large positives from the Hampshire experience. It gave a much better sense of what would be next academically, much more so than high school ever could. One of my college roommates at MIT, Neil, also was in the Hampshire program. (Our other roommate was from Jamaica High School, where my mother taught; she introduced us.) It's much easier going away to college already knowing some people there before you start.
Each of us had some math inclination. We took a few classes together. MIT at the time offered some special topics courses for freshmen that were for half the credit of a regular course. We took one on Calculus Theory, taught by A.P. Mattuck, which developed our intuitions about infinite series. At the start of this class Neil was ahead of me in figuring out things. I believe that by the end of the course I had caught up. It was the first time taking a class where I was aware of my own growth in understanding things at a deeper level, the first class where my intuition noticeably improved. We took another course from Mattuck the next semester - linear differential equations. This is Mattuck 33 years later giving the first lecture in the regular differential equations class. He was a wonderful and dedicated teacher.
I started to have some emotional problems sophomore year and in the math classes I took then - abstract algebra and analysis - I found myself in a situation similar to what I had experienced in Hampshire. I had a strong feeling of needing to get into a different environment, one where there was more diversity of interest. At MIT I was too much like everyone else. I ended up transferring to Cornell for the second semester. I still was a math major there, but math occupied a smaller part of my life.
I did eventually did learn how to break through the wall in a topology class taught by George Cooke. One factor in this breakthrough clearly was his teaching and the high motivation he provided. I found his problem sets very intriguing. He didn't use a textbook at all and in class he encouraged us to talk our way through the abstract concepts, our imprecision with language a demonstration that we did not yet fully understand what we were being taught. Another factor was that I did this course entirely on my own, not knowing the other students in the class ahead of time and doing the problem sets in isolation. A third factor was that I was not overwhelmed by other school work so over the weekend I could put in a full day of thinking through the homework and doing only that. And a fourth factor was the prior experience. I believe the earlier failure was fully necessary for this later success.
When I went to graduate school at Northwestern in economics, after having very little economics at Cornell, I felt an enormous deficiency in background relative to my classmates. I was determined to make that up and worked harder than I had ever done previously during that first quarter at NU. Spending most evenings in the reserve room of the Library till at least 9 pm, I did learn I was capable of making such an intensive commitment. Yet I soon became aware that I was better prepared than most of my classmates. I could think about the economics deeply, because that's how I had been trained to think about the math.
Indeed it is because math thinking is such good preparation for other thinking that I believe we should make effective math thinking a primary goal in our education system. I'm not talking about how well students do on the math SAT or other standardized tests they take earlier in their school careers. I'm talking about whether students can creatively find the right path to solve a complex math problem. Because the SAT is essentially a speed test it can't possibly measure the ability to find a path that will only be uncovered after hours of thought. At best it can test whether a path can be found in under a minute. Certainly that is better than not finding the path at all, but this notion that path finding is always a quick hitter activity is pernicious. It is much better to know whether the student will persist till the path is found. The SAT doesn't measure persistence at all.
In teaching the student we should be encouraging that sort of effort. Elite students receive that encouragement, in a variety of different ways. Can't we find at least some ways to offer it to the rest?