Last year I wrote a piece, “On Learning Middle School Mathematics”; the discussion below can be taken as a supplement to it. In the ending of that piece, I gave the so-called very “origin” kind of math problem, and this discussion picks up right from there.
What follows is an excerpt from my chat log with UNIC. I’m too lazy to rewrite it all over again, so I’ll just file it away here as a record. The only help I can offer you is this sort of empty talk, because since mathematics is a kind of technical art, the mathematics “coach” really has to teach hands-on, step by step; teaching math through QQ chat cannot bring out my value. It is a pity that I have never systematically served as a math “coach” even once. Perhaps if, friends of mine, you have a next generation and remember me, come ask me to teach them math—I’d be very happy to do it.
……
What I’m showing you—so-called origins and the like—are not knowledge, but a demonstration of technique
The technique of mathematics is the same when dealing with any problem
Knowledge is what a trigonometric function is, the definition of a trigonometric function, what a rectangular coordinate system is, and so on
When training technical skill, it does not matter what kind of problem you are facing. With complex, advanced problems, there are inevitably many knowledge points mixed in, so the technical component is not so distinct; but when doing those most basic, “origin” problems, there is hardly any knowledge you need to have mastered in advance, so what is being tested is pure technique. That was exactly my original intention in beginning with those problems, to let you experience this.
For example, if you want to become a table tennis player, then you may need to systematically learn some methods, such as how to play a loop, how to play a better loop, and so on. But if you just want to play casually, you don’t need to learn anything at all; keep playing over and over, keep playing long enough, and you’ll get the hang of it.
But if you don’t play, and only study, then no matter how you study you still won’t be able to play
Repeated practice can make up for a lack of knowledge learning, but the reverse is not true: learning knowledge cannot make up for a lack of practice.
Moreover, learning technique is more about imitation and adaptation than about memory
What you need to train is not a rich accumulation of knowledge, but a “conditioned reflex”
When you pick up a problem, you immediately know how to do it. What this relies on is not entirely some mathematical intuition coming from talent; in short, it is a conditioned reflex. And it can be trained.
/——But I always feel that different problems have different methods, don’t they?
Fundamentally, they are all the same
The way every chess game is played is different, right
But how did you learn to play chess?
In hindsight, one can classify and describe certain methods—for example, chop, drive, and so on—but there are no sharp boundaries between various techniques. In actual play, some balls are more likely to be returned without following any preset pattern, and this way of hitting may be one you have never learned before, but once you see the opening, you send it back; after you’ve played it, you reflect and feel through it afterward—for instance, “Oh, this is a loop.” The first time you produced it, you feel out the sensation from then on, and next time you may be able to play another one. That is how you learn it.
So things like rationalizing denominators and the like really do not need to be learned first and used later; they can all be used first and learned later.
/——But that would take such a long time~~ sigh~ then I’ll learn olympiad math in my next life too… ordinary exam-oriented education rarely gives such opportunities
It won’t take very long
Olympiad math is different. Since it is named after the Olympics, the people who study it are in fact similar to athletes, and the training methods are different too. Ordinary mathematics learning does not require boot camp.
How long did it take you to go from never riding a bicycle to being able to ride one?
It’s very easy to get the hang of, but becoming a bicycle athlete is another matter
Learning to ride a bicycle is also just imitation plus adaptation
Learning mathematics is about the same
Ultimately, it is a technique, not something like English where you have to memorize vocabulary and so on. Of course, English can also be learned entirely as a technique; it’s just that the efficiency is slower that way, though it can still succeed.
Learning knowledge is only an aid to learning technique, serving roughly the role of signposts and the like; in the final analysis, knowledge is also implemented within skills.
You can also see the status of “example problems” in science textbooks. Although humanities courses also have exams and also have questions, their textbooks do not need to demonstrate “example problems.” Yet in any science textbook, especially a mathematics textbook, the largest portion is made up of example problems.
What are example problems? What are they for? Example problems are for “showing you.” First he tells you how to play table tennis, then demonstrates it for you to see, and finally lets you get on the table and practice yourself. Example problems are exactly this kind of demonstration.
From this, we can see the technical nature of mathematics.
Some knowledge points are given before the example problem, some are given through the example problem, and some are given by reviewing afterward. In other words, practice is basic, while knowledge is in fact obtained through reviewing and reflecting on practice.
/——Technique and observation come before knowledge?
There was always artillery before ballistics
/——Then how should this be specifically carried out?
Practice
When I had you reading books back then, I also said that when reading, you should always have paper and pen in hand and keep solving problems nonstop—that is what counts as reading. Mathematics books cannot be read lying in bed, held casually in your hands.
/——A digression~~~ what do you think philosophy study is like?
Philosophy? Hard to say.
Anyway, it’s completely different from mathematics
October 28, 2007
Latest Comments
luxin2007-10-29 15:36:55 Anonymous 124.17.17.56 [Reply]
I’ve turned into an advanced technician…
I can’t really agree
Gu2007-10-29 17:36:30 Anonymous 125.34.49.255 [Reply]
Then what do you think mathematics is? Saying that mathematics is mathematics—doesn’t that amount to saying nothing at all? Do you prefer to say mathematics is a system of knowledge? Isn’t that even more of a denigration of mathematics?
I’ve said before that mathematics can be called the language of God, a lofty art. If you can accept the artistic nature of mathematics, then you can fully accept the claim that mathematics is technique, because technique in the first place is art.
What is the status of mathematics in the natural sciences? In fact, the true pinnacle of reductionism is physics; mathematics is not the endpoint of reduction at all, but something that permeates every link in the chain. Mathematics is the most fundamental and basic methodological thing among all disciplines, and so-called method is to say technique. Whether we call mathematics a “language,” an “art,” a “method,” or a “tool,” in the final analysis it is still technique.
You are not an advanced technician; you are an artist.
Art is the original and highest form of technique.
Gu2007-10-29 18:22:15 Anonymous 125.34.49.255 [Reply]
Luxin might as well think back a little more: just how did we learn olympiad math? Compared with ordinary classmates, what exactly did those of us who studied olympiad math learn more of? As far as basic knowledge is concerned, we didn’t learn any more at all. The concepts, definitions, and commonly used theorems we used were often also known by ordinary classmates. For some of the hardest contest problems, if you look at each step of the solution, perhaps there is nothing there that ordinary classmates cannot understand either. But we can solve them, while they have no clue—why is that? Because we were all along engaged in olympiad math “training.” How was it trained? Nothing more than watching the teacher pose problems and solve them, then going home and solving problems ourselves; from beginning to end, it was strung together by all kinds of problem-solving—history classes and the like are absolutely not taught this way.
Watching example problems and doing exercises—that kind of training is precisely the “imitation—practice” mode; this is the training process of technique. I believe you can also sense from your own experience that improvement in mathematics does not rely on increasing one’s understanding or accumulating knowledge, but on practice, on doing problems. Even for those knowledge points, formulas, and theorems, we must use them repeatedly before we can “master” them, and thereby “apply them with ease.” Ordinary people may even “know” these theorems, but without practice they cannot pull them out effortlessly and use them with ease when solving problems—that is exactly the transition, in technical training, from a “thing already at hand” to a “thing ready-to-hand.” When technique is practiced to a mastery so refined that it seems to have become something internal to the person, it can operate fluently without conscious thought. When mathematics has been trained to that level of familiarity, the moment one sees a problem one can automatically have a clever method spring to mind. This may be seen by ordinary people as an unattainable faculty of inspiration, but in fact it is also something that is cultivated; beyond talent, what remains are skills that have been brought to fluency through training. Skill is the higher realm of using technique; “practice makes perfect” is exactly this. What we often call problem-solving “techniques” in mathematics are also expressions of technique.
Gu2007-10-29 18:28:39 Anonymous 125.34.49.255 [Reply]
Do you remember the way the olympiad math teacher taught class? An ordinary math class might still have to keep emphasizing key knowledge points and tricks of method. But the olympiad math teacher came straight to the point: he’d set a few problems for us to work on for a while, and then he’d solve them for us to watch (or have a classmate go up and solve them, and so on). In short, it was all about solving problems and watching problems being solved.
Does that look like learning knowledge? Rather, isn’t it more like learning to play table tennis or learning to play the piano? “Demonstration—imitation—practice.”
Gu2007-10-29 20:19:00 Anonymous 125.34.49.255 [Reply]
When learning an instrument, say, playing the flute. In the beginning, training is done piece by piece; that is to say, after you have become proficient in one piece, if you then come across a completely new piece, you still can’t play it well and have to start over from scratch. But when you reach a higher level, your command of the flute becomes more and more free and at ease. At that point, a master flutist, even when encountering a piece they have never played before, can still pick it up and play it fluently and expertly.
In mathematics we often speak of “drawing inferences about other cases from one instance,” and some people feel that this ability is magical and hard to understand. For example, UNIC said, “I always feel that different problems have different methods, don’t they?” This is exactly like different pieces having different ways of being played: beginners can only practice one problem at a time; then gradually they can practice one class of problems at a time; and in the end, once they have become highly proficient, they will feel that math problems are nothing more than that. Even when they encounter a brand-new concept or method, they can quickly master it and use it with ease. There is not much mystery in this—it is simply that practice makes perfect.
By the way, what I’m saying here clearly bears the imprint of philosophy of technology and late Wittgenstein. As it happens, Wittgenstein worked as an elementary school teacher during the period of his intellectual turn. Although as a teacher he does not seem to have been very competent, his experience in teaching should have had some influence, however slight, on his later thought. Here, even if we do not discuss the essence of mathematics, and do not draw the extreme conclusion that “mathematics is completely and utterly technique,” just in terms of learning middle school mathematics, this way of learning is much closer to learning a technical craft, and that should be quite obvious.
Translated from the Chinese original with AI assistance. The original text is authoritative.
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