Remember: what kind of math education do we need?

Remember: what kind of math education do we need?

Note: This article has a lot of personal views, with a very strong subjective color. Some of these ideas are not necessarily correct, and there are some words that I am not qualified to say. I just want to share some of my thoughts with you. People remember to keep their opinions. Please also keep this passage when you reprint.

I am not a mathematician. I'm not even a math professional. I am a pure soy sauce of mathematics enthusiasts, just more than the average enthusiast more persistent, more crazy. Middle School, high school all the way, the university is not mathematics major, which let me not to test for the destination to learn their own interest in mathematics knowledge, let me have such a strong interest in mathematics. Since the founding of this Blog in 05, every time I see an astonishing conclusion or a wonderful proof, I am busy taking a moment to record it for fear of forgetting. However, I know that these simply astounding tricks is not really the beauty of mathematics, mathematics really profound thoughts I am afraid I have not had the slightest experience.

I have told people many times, my ideal of life is, I hope one day to learn the various branches of mathematics, and then stand at a high point, overlooking the whole field of mathematics, really appreciate the beauty of mathematics. However, it is difficult to achieve this. The biggest difficulty is the lack of a way to learn maths. Read the textbook? This is what I want to say today-textbooks are extremely unreliable.

I have a deep understanding of this. In the last two years, I have been doing junior high school math training, have some of their own views. The mathematics education is roughly divided into three stages, see Mountain is the mountain to see the water is water, see the mountain is not the mountain to see the water is not the water, see the mountain is the mountain to see the water is water.

The first mathematical education is to teach you a few theorems, to tell you how they are proven, and to show you some new theorems.

Later requirements changed: optical math is not enough, but also to use mathematics. Mathematics education has risen to a level: we should use mathematics in life to explain the phenomenon of life. For a time, the textbook is good, the examination questions, all is the actual life closely related to the mathematics application problem, as if looking around really everywhere is the mathematics. Shopping malls, bookstores selling books, farmers cultivated land, workers paving bricks, once again emerged in textbooks, teaching aids and test questions. In fact, mathematics can explain life, but we do not do so. There are too many variables in life, and it is impossible to take everything into account in a powerful mathematical model. For the average person, really can use the place of mathematics, also only calculate the accounts.

One day, mathematics education will be elevated to the third level: return to basics, math real cow B or it itself. You will find that the great mathematical ideas, the new mathematical theories, the motive of the initial study is not to be eager to explain the strange phenomenon around us, but its own wonderful. The emergence of linear algebra owes much to the magical Cramer paradox, and the birth of group theory is also the product of the structure of the solution of the Galois. Euler's founding graph theory derives from the problem of Königsberg egg pain without any practical value; the emergence of non-Euclidean geometry, is entirely due to the charm of the question itself. What about calculus? It does have a very wide range of practical values, and the various definitions of physics depend on calculus, but unfortunately it is not a disruptive mathematical idea.

When the textbook is negative, repeat the actual meaning of negative numbers, such as altitude, score, temperature, revenue and so on, to turn negative numbers into a real existence. In fact, this is not the main motive for people to use negative numbers. The value of negative numbers is that it can be reduced by a number to add a negative, many Gaga reduction complex to even need to discuss the classification of things can be unified with a formula. For example, the profit and loss of primary school: if each person divided 3 apples more than 8, if each person divided 5 apples 2 more, ask how many people how many apples? The solution is that the two different kinds of apples out of 6, which is each person more than two apples caused by a total of 3 people, so can be calculated to have 17 apples. However, if the problem is changed to "3 per person will be 8 more, each person divided into 5 fewer 2" What should be done? The formula above is changed, 8 can not be reduced by 2, to add 2. Therefore, the primary school said the profit and loss problem will be divided into "profit and loss", "yingying", "loss" three kinds of situations discussed. In fact, if the "2 less" is understood as "more than 2", the problem is exactly the same, the previous formula also applies. Negative numbers This new idea immediately unifies the three situations, and their essence becomes identical.

This is the example I must tell when I give negative numbers to the students. This is the meaning of negative numbers. This is the textbook should be repeated examples of emphasis.

Some people asked in the forum, what is the meaning of group theory? Someone reply, group theory is very interesting ah, just the textbook write it boring, for example, how can group theory do not say Rubik's Cube? I don't agree with the reply. The place where mathematics attracts people lies not in its application in life, but in its beauty. Why not talk about the Lagrange theorem? Why not talk about the Sylow theorem? For me, the most attractive to me to study a mathematical subject is a series of extraordinary conclusions and the wonderful proof of it.

At the end of the science fiction "sad", there are many mathematical theories that have not been used in practice for a long time, but they do not say a more extreme example. The Crown of mathematics--Number theory--2000 has been without any practical application for years, is the purest mathematics. It was not until the advent of computer, especially modern cryptography, that number theory first came out of mathematics and into people's lives. What is the research that supports number theory? It can only be math itself.

In my junior high school children out of geometry, I try to give a general question, verify that the average length on both sides of the triangle is greater than the line length on the third side, the confirmation triangle three high reciprocal and equal to the inscribed circle radius of the reciprocal, and so on. Even for pure algebra and analytic geometry, I can always make up questions that are simple and challenging to describe. How many integer solutions are there between two numbers and equal to the product? What is the linear equation that is obtained when the straight y=x is folded along the y=2x? While feeling the beauty of the conclusion, they will also be excited about solving a real math problem independently.

However, this does not count as a major problem in education. One time with a math major classmate talk to Riemann hypothesis, the other said she never heard Riemann hypothesis. I was surprised how the math majors could not know Riemann hypothesis? It was immediately understood that this was also given by the mathematics education. Open the mathematical textbook, always complete sets of theoretical system, the first definition of re-proof, said to be well-reasoned. But how did these things come about? What detours did mathematicians take in the process of drawing these things? No mention in the textbook. Textbooks have always been about what is right, but never say what is wrong. The math test will only let you prove a conclusion, never let you overturn a conclusion.

2010 Jiangsu University Entrance Exam maths problem because "too difficult" is controversial. One of the last big questions is as follows: The three sides of the known ABC are rational numbers, (1) The Proof cos (A) is the rational number, (2) the proof to any positive integer n, cos (nA) is the rational number. In fact, this problem is a very beautiful good question, the description is simple, the problem is common, the conclusion is interesting, proved ingenious, the question should be so out. But I think if we make up such a small question, the problem is really perfect: to prove or overturn, sin (A) must be a rational number. Of course, the problem itself is not difficult, equilateral triangle is one of the simplest counter-examples. The key is to overturn a conclusion, look for a counter-example, but also a basic ability of mathematical research, and this is the middle school mathematics education very little attention.

Therefore, in the teaching of Junior high school mathematics, I decorate every homework problem to "prove or overturn" the first. Occasionally, some subjects really need students to overthrow it. For example, to prove or overthrow, the circumference and area are equal to the two triangles congruent. Different people find the counter example is not the same, some simple complex, some profound blindness. And then a whole class of time to explain and review the construction of the counter-example, to the children to bring the harvest far more than the direct title.

But I haven't talked about the main problem in math education. Some time ago to Turing's translator exchange, during and Liu Jiang teacher briefly chat a few words. Liu Jiang said a website called Better explained. He said, in fact, we do not understand the good mathematics, because the teaching is not taught well, mathematics can be more intuitive, more popular.

I very much agree with Liu Jiang Teacher's statement. Let me give you an example. If a student asks, what is prime number? The teacher will say that the prime number is no more than 1 and itself, there are no other approximate numbers. No, that's not the answer the students want. What the students really want to know is, what is a prime number? In fact, prime numbers are the number that cannot be divided, and are the basic elements that make up all the natural numbers. 12 is made up of two 2 and a 3, just as H2O is composed of two H atoms and an O atom. Just unlike the chemical world, there are infinitely many elements of the arithmetic world. All objects, theorems, and methods in the arithmetic world are made up of these basic elements, which is why prime numbers are so important.

When you learn plural in high school, I believe many people will wonder: what is the imaginary number? Why do you admit imaginary numbers? How can imaginary numbers be rotated? In fact, people are building the plural theory, not because people sometimes need to deal with the negative root of the situation, but because of the following irresistible reason: if the recognition of imaginary numbers, then the n-th polynomial will have exactly n root, the number of a sudden like a crystal ball perfect. But the plural can not be reflected in the number of the axis, not only because the real number on the axis is complete, there is another reason: there is no geometric operation two times can be achieved by the opposite number. For example, "multiplied by 3" means that the distance from the origin of the point on the axis is extended to the original three times times, "3 squared", that is, "multiplied by 3 and multiplied by 3", that is, the above operation even two times, that is, to expand to 9 times times. Similarly, "multiply by 1" means that the point is flipped to the other side of the axis, "1 squared" will turn this point back. But how to say "multiply I" on the axis? In other words, what operation can be done two times to turn 1 to 1? A revolutionary creative answer is to rotate this point around the origin 90 degrees. Turn 90 degrees turn two times, naturally ran to the other side of the axis. Yes, this extends the axis to the entire plane, which solves the problem that the plural has no place. As a result, the multiplication of complex numbers can be interpreted as scaling plus rotation, and the plural itself naturally has the representation of z = R (cosθ+ sinθi). With that in the way, everything is going to be a logical one. Complex numbers are not only geometrically interpreted, but also sometimes more convenient to handle geometrical problems.

has always been interested in linear algebra, so the university chose the course of linear algebra, the results of almost zero harvest. The reason is very simple, originally looked forward to a taichetaiwu, the results of a semester, I still do not know what the matrix is, why the matrix multiplication is so defined, the matrix is reversible and what is the determinant of what it represents.

It was not until today that I saw this web page that someone gave away the true meaning of linear algebra (which is the direct reason I finally decided to write this article). I finally found what I was trying to find for the semester. Just like turning X into 2 x, we often need to turn (x, y) into something like (2 x + y, x–3 y), which is called a linear transformation. Then I think of the definition matrix multiplication, which is used to represent all linear transformations. Geometrically, each point on the plane (x, y) is changed to the position of (2 x + y, x–3 y), and the effect is equivalent to a "linear pull" of the plane.

Matrix multiplication, in fact, is the effect of multiple linear transformations superimposed, it obviously satisfies the binding law, but does not satisfy the commutative law. The linear transformation of the matrix with the main diagonal of 1 is actually the invariant meaning, so it is called the unit matrix. The matrix a multiplied by the matrix B is the unit matrix, that is, after the linear transformation A to do a linear transformation B and then change back to the meaning, no wonder we say that Matrix B is the inverse matrix of matrix A. Textbooks on the definition of the determinant of strange, what recursion, and what reverse order, but also to write a formula to help everyone remember. In fact, the real definition of the determinant is a sentence: the area of each unit square after the linear transformation. Therefore, the determinant of the unit matrix is of course 1, the determinant of a row is 0 is obviously 0 (because a dimension will be ignored, the linear transformation will flatten the entire plane), | A. b| Obviously equals | a| | b|. The determinant is 0, the corresponding matrix is of course irreversible, because such a linear transformation has already pressed the plane into a line, nothing can change it back. Of course, higher-order matrices correspond to higher-dimensional spaces. For a moment, everything was explained.

Incredibly, something so exciting that we didn't even say anything about the textbooks we used! Those who start on the definition of the determinant of the textbook, why not the linear transformation of the area as the determinant of the definition, and then deduce the determinant of the calculation method, and then to supplement the explanation "actually logically, we should first use this formula to define the determinant, and then say that the determinant can be used to represent the area"? For the sake of rigor and sacrificing readability, it's not worth it. Writing here, I really want to immediately pick up the linear algebra textbook, re-see all the definitions and theorems in a new light, and then re-write a real linear algebra textbook.

The high number of textbooks is equally absurd. Mainstream high-number textbooks are the first to talk about the derivative, then the indefinite integral, then the definite integral, completely reversed the order. Many people have learned calculus, although it has been used handy, but still do not understand how this is going on. The reason is still the problem of mathematics teaching.

My ideal calculus textbook should be to first talk about definite integrals, then the derivative, and then the indefinite integral. First, the definite integral, but must not use the current set of integral symbols, to avoid students mistakenly think that the definite integral is the development of indefinite integral. From ancient times, some integral ideas, the method of dividing and summing to limit, create a set of definite integral symbols. Then start from the beginning to tell the difference, talk about infinity, talk about the amount of change. Finally, with the increase in X 1.1, the area below the curve is the height of the vertical line--not the function value of the curve itself? So, in turn, in order to find the area below the curve of a function, only a new function needs to be found, so that its differential is just the original function. The calculus was born.

The derivation of the formalization of light is not useful. This is the way to really understand calculus. Strict definition and strict proof should be put into the visual comprehension. Unfortunately, I have not yet seen which textbooks are written in this way.

Said so much, in fact, summed up only a sentence: we study the process of mathematics, and human understanding of the process of mathematics. We should study maths in the order of the History of mathematics development. We should start with the ancient count, learn arithmetic and geometry, learn the coordinate system and calculus, understand the motives of each branch of mathematics, and the twists and turns of this branch of history. We should understand every bottleneck of mathematical development, experience the greatness of each new theory, and experience every mathematical crisis so that mathematicians feel the rush, experience first intuitive thinking and then give a formal description of the difficult.

Unfortunately, I did not find any way to learn maths in this way.

But good. Since there is no shortcut, let me go through the definition and proof of the formal, and then I can understand the truth of it. In this way, our education is also true: first use the exam to force everyone to learn the things learned, although they do not know what they learn, and so on one day in the future to reach a certain height, looking back at the past learning things, suddenly dawned, understand what the original learning is what. This is undoubtedly a more enjoyable thing. I hope one day, like today, can realize what the higher algebra is talking about, can realize the scope of what is the use of the theory, can realize Riemann hypothesis why so cow B, can realize what Hilbert space is what, and then write them down.

I'm afraid it's going to take me a lifetime.

Remember: what kind of math education do we need?