How to integrate mathematical thinking methods into the discovery, exploration and research of mathematics and how to integrate mathematical thinking methods into mathematics teaching are the lifelong goals pursued by countless scholars and teachers who love mathematics research and mathematics education. Many famous works of world mathematics, such as Goldbach conjecture and Fermat conjecture, have gone through the mathematical thinking process of observation and experiment, induction, analogy and association, intuition and conjecture, reasoning and proof.
The fact that American mathematician and math educator Paulia (1887 ~ 1985)' s three books, How to Solve a Problem, Mathematical Discovery and Mathematics and Guess, have been popular all over the world for a long time fully shows that people no longer think that the process of mathematical discovery and creation is just the mathematical game of the world's top mathematicians, and they don't want to cheer for those "abstruse" mathematical theories and discoveries. This shows the important significance and role of "reasoning and proof" in mathematical discovery and exploration.
Through the in-depth study of the problem-solving process, especially the successful practice, Paulia found that there is no "universal method" that can be mechanically used to solve all problems; In the process of solving problems, people always put forward enlightening questions or hints to themselves according to specific conditions, thus starting and promoting the thinking process; Therefore, he tried to summarize the general methods or models, which played an important role in enlightening and guiding the future problem-solving activities. Paulia has long noticed that "mathematics has two aspects: mathematics proposed by Euclid is a systematic deductive science; But mathematics in the process of creation is an experimental inductive science. " Therefore, he clearly put forward two kinds of reasoning: perceptual reasoning and deductive reasoning. Deductive reasoning can be used to determine mathematical knowledge, and perceptual reasoning can be used to provide a basis for conjecture. Moreover, in the process of solving problems, perceptual reasoning has the functions of guessing and finding conclusions, exploring and providing ideas, which is conducive to the cultivation of innovative consciousness.
Many mathematical problems and conjectures, including world-famous puzzles, are often solved by direct observation, induction, analogy and conjecture of logarithm, formula or graph, and then verified by logic; At the same time, with the solution of the problem, mathematical methods have been refined, the scope of mathematical research has been expanded, and mathematics has been continuously promoted and developed. Fermat boldly put forward Fermat's conjecture by studying Pythagorean theorem! In order to find the proof of this conjecture, many mathematicians devoted their whole lives to it. In the last century, the British mathematician wiles proved this conjecture, and finally formed Fermat's Last Theorem. This old hen, which was called "golden egg" by mathematician Hilbert, was put forward through reasonable reasoning. In centuries of exploration, mathematicians' creative process contains reasonable reasoning. Therefore, to some extent, perceptual reasoning promoted the discovery and development of mathematics, and eventually formed many wonderful works in the history of mathematics in the world, such as euler theorem's conjecture, Goldbach's conjecture, and the four-color problem.
Goldbach conjecture is a jewel in the crown of mathematics. It has been explored for two and a half centuries since 1742 was put forward. Although the correctness of this conjecture has not been confirmed, and no one can deny it, the research around this conjecture has accumulated a lot of data and achievements. It can be said that the research on Goldbach's conjecture has reached a very profound level.
/kloc-one day in 742, Goldbach wrote a series of equations on paper:
6=2+2+2, 7=2+2+3, 8=2+3+3, 9=3+3+3, 10=2+3+5, 1 1=3+3+5…
Finally, he couldn't help it, and wrote to Euler, saying that he wanted to risk publishing the following conjecture: "Any natural number greater than 5 can be written as the sum of three prime numbers." Soon, Euler wrote back that he thought: "Every even number not less than 4 can be written as the sum of two prime numbers."
This is the famous Goldbach conjecture.
After 200 years, no one can prove this conjecture.
At present, the best result in the world is proved by Chen Jingrun, a famous mathematician in China, in 1966, which is called Chen Theorem.
This famous mathematical problem has attracted the attention of thousands of mathematicians all over the world. This is the great value of a good question and the historical significance of a good guess.
1in August, 900, Hilbert, a master of mathematics under 40 years old, gave a classic speech on "mathematical problems", raised 23 famous mathematical problems, and left a famous saying about problem (conjecture) developing mathematics: "As long as a branch of science can raise a large number of questions, it is full of vitality; There is no problem, indicating the decline or suspension of independent development. Mathematical research also needs its own problems. "
Conjecture not only guides the research goal, but also shows the cognitive needs of social development. The history of mathematics is full of conjectures. It can be said that mathematics develops with the conjecture about mathematical propositions.
In a sense, a history of mathematics is a history of guessing and verifying guessing. There are both great guesses and trivial guesses; There are conjectures that are finally proved and conjectures that are finally denied; There are conjectures that have been solved quickly, and there are conjectures that are still "hanging" today. Many mathematicians are guessing. They have extraordinary intuitive ability, leaving an interesting and attractive guess for future generations. In particular, the process of solving major conjectures will often bring great impetus to the development of mathematics.
Guess makes people's understanding get rid of the passive state of passive waiting; Conjecture plays an important role in the development of human cognition. No wonder scientists always feel: "Every great success of mankind begins with bold speculation."
The process of conjecture is a process of reasonable reasoning between observation and experiment, induction, analogy and association, intuition and conjecture. The essence of rational reasoning is "discovery", that is, discovering new relationships, new laws and new methods. In mathematics learning activities, perceptual reasoning not only plays an important role in discovering new propositions, but also is an important method and means to explore problem-solving ideas, summarize and explain new mathematical facts and laws, expand cognitive fields, promote the mastery and transfer of knowledge, enlighten thinking and develop mathematical ability.
If deductive reasoning can cultivate students' computing ability, spatial imagination ability and rigorous academic attitude, then reasonable reasoning can cultivate students' innovative thinking ability, creative imagination ability and innovative practice ability. Therefore, it can be said that reasonable reasoning is the basis and necessary condition for developing and cultivating students' innovative ability, and it is the quality that new talents should have in 2 1 century.
Second, the practice and exploration of reasoning and proving the educational value
Paulia, a famous American mathematician and mathematics educator, said: "For students studying mathematics and teachers engaged in mathematics work, guessing is an important (but usually neglected) aspect, because: before proving a mathematical theorem, you have to guess the content of this theorem; Before you make a detailed proof, you should guess the idea of proof; You should synthesize the observed results and then compare them; We must try again and again ... The proofs (or solutions) we usually get are discovered through reasonable reasoning and conjecture. "
In particular, creativity is an important criterion to measure talents, and it is also the requirement of quality education for ability training, and the cultivation of creativity depends on the training of thinking methods that pay equal attention to argumentation and reasonable reasoning in teaching.
Paulia, an outstanding mathematician and mathematical educator in the 20th century, spoke highly of the rational reasoning model and the role of observation, experiment, analogy, induction, reduction and conjecture in mathematical discovery and innovation, and formed extensive knowledge all over the world. In the "discovery learning" in Brussels and the "research learning" carried out by Shanghai Academy of Educational Sciences, the teaching of rational reasoning has been highly praised. Reasonable reasoning teaching meets the requirements of quality education in China.
Let's look at an algebraic example of inductive and analogical reasoning:
We use the sum of the first few natural numbers to represent the sum of the squares of the first n natural numbers, and so on:
Find the sum of squares first, and try the following methods:
Add the left and right sides separately, and you get
We didn't get it, but we got it:
This attempt gives us an inspiration, which is summed up from the process and method. So can we find it by analogy in the investigation process? Now, let's follow this conjecture and give it a try. ...
Add the left and right sides separately to get:
Therefore:
At this point, we are more convinced that the cubic sum of the first natural number can be obtained through analogy and association.
Available overrides:
Therefore, there are:
From the process and result of summation, we can see that:
It's about quadratic forms,
It's about the cubic formula,
It's about the quartic formula,
So can we guess the formula five times before asking?
In fact, many properties in plane geometry can be extended to solid geometry by analogy, such as triangle analogy in plane geometry to tetrahedron (or triangular pyramid) in solid geometry and so on. Let's look at another geometric example of inductive analogical reasoning.
As we know, a straight line divides the plane into two parts;
Two straight lines (as long as they are not parallel) divide the plane into four parts;
Can three straight lines generally divide a plane into eight parts?
Is this conjecture correct?
If three straight lines are parallel to each other, the plane can only be divided into four parts;
If only two of the three lines are parallel, they divide the plane into six parts;
If three straight lines intersect at a point, they also divide the plane into six parts;
Generally speaking, three straight lines are not parallel and are not the intersection of three lines. Therefore, it is necessary to discuss under the condition that three straight lines will intersect at three points and form a triangle with these three points as vertices.
As can be seen from the figure below, three straight lines divide the plane into seven parts: A, B, C, D, E, F and G.
It can be seen that the conjecture that "three straight lines can divide a plane into eight parts" is not valid.
Comparing plane with space, we can put forward:
A plane divides the space into two parts;
Two planes divide the space into four parts (as long as the two planes are not parallel);
Three planes can divide the space into eight parts (just like three coordinate planes in spatial analytic geometry divide the space into eight quadrants);
Can four planes divide the space into 16 parts?
When we think of how a straight line is divided into several planes, we may not make this guess again.
However, can we guess analogously that a plane can roughly divide space into 15 parts? Is this conjecture drawn from analogy correct?
Let's analyze the situation that three lines are divided into planes: what is the situation that three lines that are not parallel to each other and have no points divide the plane into seven parts? One part of it is limited, the other six parts are infinitely extended, and the limited part is a triangle a surrounded by three straight lines; Infinite parts can be divided into two types: one is the part that has common edges with triangles (that is, B, C, D); The other one has a common vertex with triangle (e, f, g).
Next, let's analyze the situation that four planes divide the space: we don't consider the special situation that there are parallel planes in four planes, three planes in four planes or four planes * * * lines, but only consider the general situation, that is, four planes can enclose a tetrahedron: the interior of the tetrahedron is a limited part; The rest of the divided parts are infinite, and the infinite parts can be divided into three categories: the first category has a common surface with tetrahedron, and the ***4 parts; The second kind is the * * * part which has a common straight line with the tetrahedron; The third type and tetrahedron have a common * * * point, * * 4 parts, so the total number is 1+4+6+4= 15.
As mentioned above, the plane is divided into seven parts by three straight lines that are not parallel and have no points. What about four straight lines? Of course, these four straight lines should also be non-parallel, and every three straight lines have no * * * points. In this way, the new fourth straight line will intersect with the original three straight lines, and it will pass through the original four parts and divide them into two parts, so * * * adds four parts, and it is known that four straight lines divide the plane into 1 1 parts.
According to similar requirements ("pairwise is not parallel, three, three, three points are not * *"), a fifth straight line is added, so the division part of the plane is also increased by five parts. We summarize the data list as follows:
Straight line numbering
1
2
three
four
Number of plane division parts
1+ 1
1+2+ 1
1+2+3+ 1
1+2+3+4+ 1
From this we can guess that the N straight lines of "pairwise nonparallel points, three or three points" can divide the plane into:
part
The research on international mathematics curriculum reform shows that there are two basic ideas in dealing with the problems of mathematics thinking methods in primary and secondary schools:
First, mainly through the study of pure mathematics knowledge, let students gradually master the ideas and methods of mathematics;
Second, by solving practical problems, students can form basic thinking methods that can promote people's quality, such as experiment, guessing and reasonable reasoning.
Compared with the two, the latter is more a general way of thinking and has a wider range of applications. Major developed countries also tend to adopt the second basic idea.
The research shows that there is a high correlation between perceptual reasoning and deductive reasoning; The development of students' perceptual reasoning is also closely related to the development of deductive reasoning. Therefore, mathematics teaching should promote the synchronous development of students' perceptual reasoning and deductive reasoning.
If deductive reasoning can cultivate students' computing ability, spatial imagination ability and rigorous academic attitude, then reasonable reasoning can cultivate students' innovative thinking ability, creative imagination ability and innovative practice ability. Therefore, it can be said that reasonable reasoning is the basis and necessary condition for developing and cultivating students' innovative ability, and it is the quality that new talents should have in 2 1 century.
The essence of rational reasoning is "discovery", that is, discovering new relationships, new laws and new methods. In mathematics learning activities, perceptual reasoning not only plays an important role in discovering new propositions, but also is an important method and means to explore problem-solving ideas, summarize and explain new mathematical facts and laws, expand cognitive fields, promote the mastery and transfer of knowledge, enlighten thinking and develop mathematical ability.
As math educators, let's imagine:
When students think Goldbach's conjecture is so "easy to understand"; When students can connect the area formula of triangle with the volume formula of triangular pyramid, they can get it through thinking experiment, data detection and adjustment; Especially when students can put forward their own prime conjecture by Goldbach conjecture analogy, get the square sum formula of natural numbers by analogy, and then get the cubic sum formula of natural numbers by analogy, students' conjecture, proof method, students' inner touch, students' harvest and sharing really make us feel the greatness of mathematics and the value and significance of mathematics education!