I had wanted to write this article before the summer vacation, but I never got around to it……
A teacher once said: if you can’t learn math well, you’re stupid; if you can’t learn English well, you’re lazy. I am obviously a lazy person, but apparently not stupid……
But in fact, although doing mathematics does indeed depend heavily on a certain kind of talent, and the fact that men are better at math than women is hard to deny, just as women are better at language than men, the influence of talent may show up more clearly in top-level mathematics competitions. For the mathematics required in elementary and middle school, however, the bar is very low. To muddle through the level of the college entrance exam in math does not require much genius at all; as long as you grasp the right methods, I believe ordinary people can handle it with ease.
But then why are there still some people who seem to be born unable to learn math? Why is it that some people, who never touch mathematics except in class, do better than those who struggle day and night in an ocean of problems? Is this imbalance between input and reward not a matter of talent?
I think talent is not the key factor here. The reason why some people find math so “much effort for little gain” probably lies in the fact that from the very beginning they never really grasped the “method” of doing math.
Mathematical knowledge is a chain, link after link, from addition and subtraction in elementary school all the way to analytic geometry in high school. There is an entire logically connected thread running through it, and every new topic is built on mastery of the knowledge and methods that came before. Take analytic geometry, for example: it is built on functions and the like. Functions are built on solving equations; solving equations is built on word problems in fifth grade; word problems are built on arithmetic in second grade; arithmetic is built on counting in first grade. If you never learned functions well, then even if you temporarily learn how to do some analytic geometry problems, the foundation is not solid, and once you forget it, you cannot find your way back.
The bad thing is that even if the earlier foundation has not been fully mastered, math can still be studied on, if only barely. Such learners then feel that from elementary school to high school, mathematics keeps presenting them with new territories, and that they must keep learning brand-new theorems and methods and readjust every time. But in fact, the whole of mathematics from elementary school to high school is the gradual unfolding of one road, the successive stages of one and the same staircase. One person is constantly confronted with new situations and overwhelmed; another simply sits on a boat moving with the current and browses along the whole line, and then understands everything at once.
A great deal of futile labor in math teaching is this: right now I’m teaching X; I’ve taught it once, and you still haven’t fully grasped it, so I’ll teach X again. You can’t do it, so I’ll hand-hold you and let you do it a few more times. Class time isn’t enough, so you go to tutoring and study it over and over again. You study until “practice makes perfect,” memorizing all the possible question types through repeated drills, and then at least you can muddle through the exam. But perhaps the real problem in your failure to learn X is that you never mastered Y, the foundation of X, which should have been grasped even earlier. If you keep grinding away at X without consolidating Y, then either you barely get by in an “much effort for little gain” way, or you end up making the student lose confidence, convincing himself that he has no talent for math and giving up in self-destructive resignation.
Most tutoring classes, even one-on-one tutoring, do nothing more than help you repeat over and over the thing you are currently learning. There is no one who would say that when tutoring a high school student in math, they would help with middle school, or even elementary school, problems. This is the reason general math tutoring “treats the symptoms but not the root cause.” If I were doing tutoring, whether for a middle school student or a high school student, I would start by explaining elementary school material!
How many major kinds of methods are there for solving math problems? Assumption? Proof by contradiction? Backward reasoning? Combining numbers and shapes?…… Leave all those various methods aside for the moment. In the end, all methods return to the same source: in solving math problems, one could say there is only one method—the method of “proof.” Proof is the most fundamental method and means; one might say it is your own household “martial art.” The other concrete problems are the opponents you must face one by one. According to the different circumstances of the opponent—for example, whether he is strong or agile, whether he uses a knife or a spear, and so on—of course you need different strategies. But in the final analysis, your own martial art is the foundation of everything. If your own martial art is not well trained, you will tremble at the sight of anyone; whereas if your own martial art has been refined to perfection, you can face even an unknown opponent with complete confidence.
The method of “proof” is consistent from elementary school to high school. If someone cannot write a rigorous, impeccable “proof” for a high school math problem (even if he can “do” the problem), then he certainly cannot write a rigorous, impeccable proof for any elementary school math problem either. The converse is also true.
So, if you want to understand what a rigorous, impeccable “proof” really is, it is enough to do elementary school or middle school problems. Once you truly master the method of “proof,” then go on to find different specific opponents to practice against. Otherwise, if you have no martial arts skills at all and just plunge into an ocean of problems, it would be strange if you did not end up battered black-and-blue!
There were once a few middle school and elementary school netizens (actually, probably only three people) who came to me to chat about math. Some were asking for advice, and some were a bit self-righteously asking me to set problems for them to test them. I “hit” them hard all the same—I made them do proof problems for me, and I said to them: “Is this what you call a proof?” In some people’s proofs, obvious loopholes can be found; in others, what they gave me simply did not count as a “proof” at all.
For someone who does not even understand what a “proof” is, to still think that his math level is pretty good—that is truly sad. In fact, his level may indeed look pretty good; he may still be able to do some difficult problems. Many multiple-choice and fill-in-the-blank questions do not require strict proof and can be “seen” through. However, if one is not skilled in the method of proof, then even the problems one answers correctly cannot be said to be truly understood.
Some people complain: “The teacher never taught us the format for writing proofs!” In other words, they are shifting the blame onto the teacher. But the format of proofs is not a matter for some special topic class, and I have never taken any course on “how to write proofs” either. There is no need to offer such a course. Because the format of writing proofs has been taught continuously since elementary school. The solution to every example problem in the textbook is generally rigorous enough. How do you read textbooks? Do you think some example problems are simply too easy and skip over them? In fact, example problems are teaching you, step by step, how to write proofs! “Because,” “therefore,” “let,” “suppose,” “it suffices to prove,” “if and only if”…… The use of these words has not gone untaught; on the contrary, it has been taught countless times already.
I don’t want to just talk in empty abstractions, so let me give a few examples:
Set three proof problems:
1. Prove the Pythagorean theorem.
2. Prove the formula for the area of a triangle (not some complicated Heron formula or anything of that sort, just “base times height divided by two.”)
3. This one is of middle school difficulty: prove that the three medians of a triangle intersect at one point.
Additional requirements:
a. At the beginning of the proof, first restate in rigorous language the problem to be proved, for example, what exactly is meant by the Pythagorean theorem.
b. Clearly explain the meaning of every concept and every step in the proof, especially for the first two problems (too simple, are they?), giving an explicit definition for every mathematical concept used, and clearly stating whether each step is based on some axiom or theorem. — It is not required, for example, that you follow Euclid’s system of axioms and definitions; the axioms and definitions may be invented by yourself, but everything must be stated clearly.
Completing these problems perfectly is much harder than one might imagine. However, writing a mathematical proof is different from writing an essay. As long as you truly grasp the relevant knowledge and methods, you will certainly be able to produce an answer so impeccable that even the most demanding teacher who has a grudge against you would have no choice but to give you full marks. For such simple problems, my requirement is that they be written so well that even the most demanding person could find no objection at all—other than, say, your handwriting being too ugly~.
First, let’s look at an obviously noncompliant proof:
Prove the Pythagorean theorem:
It’s just a square~ with a right triangle cut out on each side, (with the hypotenuse as the outer side of the big square), leaving a square in the middle; let the hypotenuse of the triangle be B, the short leg be A, and the long leg be C. The area of the big square is: b×b = (a—c)×(a—c)＋4×0.5×a×c = a×a＋c×c
……Leaving aside the issues of proof format and standards, this cannot be counted as a “proof” at all; it can only be counted as an “explanation.” Indeed, this explanation is enough to “introduce” a correct proof method, but it is far from rigorous.
……Leaving aside the fact that it does not explain what a square is, what a right angle is, what a triangle is, and so on—requiring all of these concepts to be defined one by one seems too harsh; I am not yet so cruel. It is normal for ordinary middle school students not to think of defining them, but precisely here lies the difficulty of this problem.
— What does “cut out one on each side” mean? How do you cut? Can you just cut however you like?
— Is it really always possible for any right triangle to be “cut out” like this on the side of some square so that four of them appear?
— What is the “outer side” of a square? Is there also an “inner side”?
— How do you “leave a square in the middle”?
— How can you guarantee that after cutting away the four triangles, a square can definitely remain in the middle?
— Which side is the “short leg”? What if it is an isosceles right triangle?
— The letters used for the variables are nonstandard. According to unwritten convention, lowercase English letters (a, b, c…) are used to denote side lengths, while uppercase letters (A, B, C…) are used to denote vertices. And in a right triangle, it is customary to label the hypotenuse with c; generally, the convention is to let the side opposite angle A (that is, BC) be a……
Never underestimate those customary conventions. Developing good habits will greatly help mathematical study. Behind these habits lies the orderliness and concision of mathematics, and they are closely related to the mathematical way of thinking through “symmetry.” For example, the notation for the three sides of a triangle embodies rotational symmetry. In other contexts, one uses p, q, r here instead of x, y, z; here one uses a1, a2, a3 instead of a, b, c; here one uses P, P’ instead of P, Q…… There is often some reason behind these habits, and if you pay close attention in ordinary life, you will surely come to understand them inwardly.
An “explanation” is not a “proof.” Ordinary middle school and high school students can certainly write such an explanation, but if they can only write to this extent, then they have not mastered the method of proof. If one has truly mastered the ability to deal at any time with a tricky, exacting teacher who has a grudge against you, then there is no point in fussing over these things anymore. Sometimes the “proofs” I write are even more concise than the one above, and words like “easily proved” and “obvious” are used very often. But the premise is that I can at any time expand an incomplete argument into a complete one; only then do I have the qualification to use “easily proved” and “obvious” to save ink.
October 5, 2006
Translated from the Chinese original with AI assistance. The original text is authoritative.
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