The Mathematics Curriculum Standard for Nine-year Compulsory Education Full-time Primary School (Trial Draft) puts forward: "Students can acquire important mathematical knowledge and basic mathematical thinking methods necessary to adapt to future social life and further development through learning." Therefore, consciously infiltrating some basic mathematical thinking methods into students in primary school mathematics teaching can deepen their understanding of mathematical concepts, formulas, theorems and laws, is an important means to improve students' mathematical ability and thinking quality, is an important way to realize the transformation of mathematics education from imparting knowledge to cultivating students' ability to analyze and solve problems, and is also the real connotation of quality education in primary school mathematics teaching. In primary school, mathematical thinking mainly includes symbolic thinking, analogy thinking, classification thinking, equation and function thinking, modeling thinking and so on.
First, symbolic thought.
Western countries introduced symbols into mathematical research earlier. /kloc-in the 6th century, the mathematician Veda made many improvements to mathematical symbols. He was the first to consciously and systematically express known numbers, unknowns and their powers with letters, which brought about a significant expansion of algebraic research and laid the foundation for symbolic algebra. Later, the great mathematician Descartes improved the letters used in Vedas. Using symbolic language (including letters, numbers, graphics and various specific symbols) to describe the content of mathematics is symbolic thinking. In mathematics, all kinds of quantity relations, quantity changes, and the derivation and calculation between quantities are represented by lowercase letters, and a lot of information is represented by the condensed form of symbols, such as the law of multiplication and division (A+B) × C = A× C+B× C, where A, B and C can not only represent 65, 438+0, 2, 3, but also. For another example, in the division teaching with remainder, a thinking question finally appeared: At the June 1st party, Xiaoming strung balloons to decorate the classroom in the order of 3 red balloons, 2 yellow balloons and 1 blue balloons. Do you know what color the 24th balloon is? Students have many ways to solve this problem. For example, the letters A, B and C represent red, yellow and blue balloons respectively, which can be converted into the following symbol forms according to the meaning of the question: aabbc aabbc aabbc aabbc aabbc AABBC .......................................................................................................................
These are the concrete manifestations of symbolic thought, which integrates all data examples into one, and expresses complex languages and characters with simple and clear letter formulas, which is easy to remember and use. As Hua said, "the characteristic of mathematics is abstraction, and because of this, symbolic representation has wider application and superiority." This symbolic mathematical language is a worldwide language and a comprehensive expression of a person's mathematical literacy.
There is a process from concreteness to representation to abstract symbolization when objective things and phenomena and their relations are summarized as mathematical symbols and formulas. Pupils will encounter more difficulties in mathematics learning. From acceptance to application, teachers need to introduce the history of letter use, take good advice and strengthen training.
Second, analogical thinking
Mathematical analogy refers to the idea that according to the similarity between two kinds of mathematical objects, the known properties of one kind of mathematical object can be transferred to another kind of mathematical object, which can solve some seemingly complicated and difficult problems. As far as the migration process is concerned, some analogies are very obvious, direct and simple, such as the migration from learning additive commutative law A+B = B+A to learning the multiplication distribution law A× B = B× A; But some analogies can only be realized on the basis of establishing abstract analysis, which is more complicated.
For example, there is a Mathematical Olympiad: a scientific investigation team will climb a mountain for scientific investigation. Go up the mountain at 8 o'clock in the morning, walk 3 kilometers per hour, and rest at the top of the mountain 1 hour. When going downhill, walk 5 kilometers per hour and reach the foot of the mountain at 2 pm. Whole journey 19 km. How many kilometers is the journey up and down the mountain? Analysis: On the surface, this problem seems to be a trip problem, but in essence it is just a typical change of the problem of "chickens and rabbits in the same cage". Characterized in that:
(1) The single value of two things is known: the uphill speed is 3km and the downhill speed is 5 km.
(2) The total number of these two different things is known: 5 hours does not include the remaining 1 hour; Whole journey 19 km.
(3) What is required is the number of these two different things: what is the time for going up the mountain and going down the mountain? It can be seen that the solution to this problem is exactly the same as the solution to the problem of "chickens and rabbits in the same cage". Assuming that all five hours are uphill time, the walking distance of * * * is 3× 5 = 15 (km), which is less than the actual walking distance 19 15 = 4 (km). Because the downhill time is also regarded as uphill time, the downhill time is 4 ÷. Therefore, it can be concluded that the downhill distance is 5× 2 = 10 (km) and the uphill distance is 19- 10 = 9 (km). Of course, we can also use this analogy, assuming that these five hours are downhill time to solve. The application of all formula theorems in mathematics is the direct embodiment of analogical thinking.
At present, there are many analogies in primary school mathematics textbooks, and the extension, inference and extension of some theorems and problems published in magazines are also reflections of analogies, which need teachers to explore and implement. For example, the area formula of a rectangle is length× width = a× b, and so on, and the area formula of a triangle can also be understood as length (bottom )× width (height) ÷ 2 = a×. Similarly, the formula of cylinder volume is bottom area × height, so the volume of cone can be understood as bottom area × height. The idea of analogy not only makes mathematical knowledge easy to understand, but also makes the memory of formulas natural and concise, which can stimulate students' creativity. As the mathematician Paulia said, "We should discuss these processes of generalization, specialization and analogy, which are great sources of discovery."
Third, the idea of classification.
Every concept in mathematics has its unique essential characteristics, and it changes according to certain laws, and there is a relationship between them from qualitative change to quantitative change. To understand these concepts correctly, it is necessary to analyze specific concepts according to specific standards. This is the classification idea of mathematics, which is to divide the mathematical objects in the research area into several parts for analysis and research according to certain standards.
Generally speaking, we need to conform to the principles of mutual exclusion, no omission and simplicity when classifying. For example, if it is divisible by 2, integers can be divided into odd and even numbers; If we classify natural numbers by divisors, they can be divided into prime numbers, composite numbers and 1. Classification in geometry is more common. For example, when learning "angle classification", many concepts are involved, and the relationship between these concepts is permeated with the law of quantitative change to qualitative change. Several angles are classified according to the degree, from quantitative change to qualitative change, and it is inferred that the largest angle in the triangle is greater than, equal to and less than 90, which can be divided into obtuse triangle, right triangle and acute triangle. Triangle can be divided into equilateral triangle and equilateral triangle, and equilateral triangle can be divided into equilateral triangle and isosceles triangle. Different classification standards will have different classification results, resulting in new mathematical concepts and mathematical knowledge structures. Due to the classified discussion, firstly, students are subtly inspired by dialectical materialism in the process of learning mathematics; The other has an obvious distinguishing effect on students' ability. As a mathematical idea, the universality of classification research in the real world will inevitably attract people's attention.
For example, after teaching the multi-digit reading and writing method, design an open question: the following five cards are written with the numbers 0, 0, 1, 2, 3, which can be used to form many different five digits and find the average of all five digits. Analysis: Based on the highest digit, divide all the five digits that can be composed into three categories, and then arrange them in descending order as follows.
( 1) 10023 (2)200 13 (3)300 12
10032 2003 1 3002 1
10203 20 103 30 102
10230 20 130 30 120
10302 2030 1 3020 1
10320 203 10 302 10
12003 2 1003 3 1002
12030 2 1030 3 1020
12300 2 1300 3 1200
13002 2300 1 3200 1
13020 230 10 320 10
13200 23 100 32 100
The average of these 36 numbers, the number on ten thousand digits is 2, which can be determined by (1+2+3) ÷ 3 = 2, and the number on other digits is 1, which can be determined by (1+2+3) × 6 ÷ 36 = 65438. The average value is 2111.
Fourth, the idea of equations and functions.
The process of establishing equations between known numbers and unknown numbers and "translating" life language into algebraic language is equation thinking. Descartes once imagined that all problems were classified as mathematical problems, and then mathematical problems were transformed into equation problems, that is, through the mathematical relationship between known quantities and unknown quantities in problems, they were transformed into equations (groups) by mathematical symbolic language, which is the origin of equation thought.
In primary school, students still stay in the method of elementary school arithmetic when solving application problems, and can't accept the idea of equations for a while, because only a specific known number is allowed to participate in the operation when solving problems, and the result of arithmetic is to find the solution of the unknown. The biggest weakness in the process of arithmetic problem solving is that the unknown is not allowed to be the object of operation, which is also the fatal injury of arithmetic. In algebra, unknowns have the same right to participate in operations as known numbers. The unknown represented by letters does not passively rest on one side of the equation, but, like known numbers, can accept and perform various operations from one side of the equation to the other, so that the mathematical relationship between the known and the unknown is very clear. If this equation idea is not infiltrated in the mathematics teaching of middle and senior grades in primary schools, it will be difficult for students to improve their mathematics level. For example, the slightly complicated application problems of fractions and percentages, trip problems, reduction problems, etc. It is relatively simple to solve it by algebraic method, that is, to assume the unknown number, because after the number is represented by the letter X, the required unknown number and the known number are in the equivalent position, and the quantitative relationship is more obvious, so it is easier to think and find a solution idea. In modern mathematics, the idea of equation is closely related to the idea of function. Based on the set, the function uses the viewpoint of movement and change to attribute the relationship between variables to the corresponding relationship between elements in two sets. Mathematical thought is the inevitable product of in-depth study of quantitative relations in the real world. Engels has clearly expounded the importance of variables in his book Dialectics of Nature: "The turning point in mathematics is Descartes' variables. With variables, movement enters mathematics; With variables, dialectical methods enter mathematics; With variables, differentiation and integration are immediately needed. "Mathematical thought dialectically reflects the changing law of quantitative relations in essence, which is an important foundation for the occurrence and development of modern mathematics. In the practice of primary school mathematics textbooks, there are the following forms:
6×3= 20×5= 700×800=
60×3= 20×50= 70×800=
600×3= 20×500= 7×800=
Some teachers ask students to finish the calculation, and the answer is correct. Experienced teachers design teaching in this way: first calculate, then check the answers, and then let students observe the characteristics of the answers (find the rules) and how the changes in the answers are caused. Then the following two groups of questions appear:
45×9= 1800÷200=
15×9= 1800÷20=
5×9= 1800÷2=
Through comparison, let the students realize that "when one number changes and the other number remains the same, this number changes regularly". The conclusion can be told by students in their own words, only for experience, not by rote. The research and analysis of the relationship between variables in specific problems are generally expressed in the form of analytical expressions. At this time, the analytical expression can be understood as an equation, and the function problem can be analyzed by learning the equation. There are many contents in middle school, such as positive and negative proportional function, linear function, quadratic function, exponential function, trigonometric function and so on. Although there are not many primary schools, there are some. For example, it is very common in fractional application problems. A specific quantity corresponds to an abstract fraction, so finding out the corresponding relationship between quantity and fraction is the key to solving problems. It is also common in practical problems, such as travel problem, where the speed and travel time of the bus correspond to the distance traveled by the bus, while the speed and travel time of the truck correspond to the distance traveled by the truck; Another example is the unary equation x+a = b and so on. Learning these functions well is a necessary condition for further study; Builders need a leap in thinking; Using the thought of function can not only meet the requirements of solving problems, but also be clear in thinking, clever and fascinating in solving problems.
Fifth, modeling ideas
At present, the idea of "realistic mathematics education" put forward by Friedenthal, a world-famous mathematician and mathematics educator, has been widely recognized by the international mathematics education community and accepted by the majority of mathematics teachers. This idea shows that a school's mathematics has a realistic nature, and mathematics comes from real life and then is applied to real life. Second, students should seek truth from facts when learning mathematics, that is, students gradually discover and draw mathematical conclusions through familiar real life. This means that the application and practicality of mathematics curriculum has become a basic trend of international mathematics curriculum reform.
For example, one of the basic characteristics of 1989 mathematics curriculum standard and American mathematics teachers association 2000 standard is to emphasize the application of mathematics; The Netherlands began the reform process of realistic mathematics education in the 1960s. By the early 1990s, almost all primary and middle school students in the Netherlands were already using mathematics textbooks based on realistic mathematics education ideas, focusing on cultivating students' mathematics application consciousness and practical ability. The mathematics curriculum in Japan has set up comprehensive subject learning, which also reflects the concern about the comprehensive application of mathematics knowledge. This series actually emphasizes a mathematical modeling idea.
The so-called mathematical model is the mathematical structure of a specific research object in the real world, which is expressed in mathematical language after some necessary simplification and assumptions for a certain purpose. The idea of mathematical modeling is a mathematical idea and method to find, put forward and understand the unsolved or unsolved problems in the real world from the mathematical point of view, and simplify them into a kind of solved or easy-to-solve problems through the transformation process, and comprehensively apply the learned mathematical knowledge and skills to solve them.
Various basic concepts in mathematics are based on their own realistic models. For example, natural number set is a model used to describe discrete quantities; Various geometric figures are also mathematical models abstracted from reality. Those basic mathematical models enable us to make inferences about practical problems related to them.
For example, in the review of the chapter on plane graphic area, I designed such a comprehensive learning topic: using the graphics I have learned independently to make a simple mosaic design for my room.
The key for students to solve problems smoothly is to clarify the knowledge relationship between various plane graphics. In teaching, the plane quadrature model S = AB can be established, and the quadrature formulas of square, parallelogram, triangle, trapezoid and circle can be derived from the right-angle quadrature formula, which communicates the internal relations of all plane graphics. At the same time, with the change of the relevant side length, it shows that these plane figures can be transformed into each other. Students have learned to model and have an epiphany.
On this basis, by exploring the mosaic of plane graphics, students can know that triangles, quadrangles or regular hexagons can mosaic planes, and then design their own room mosaic scheme. In this whole process, the basic process of "problem situation-modeling-classified solution-explanation and application" is emphasized, which guides students to actively participate, practice, think independently and explore cooperatively, realizes the change of learning style, changes the passive learning style of single memory, acceptance and imitation, and develops students' ability of collecting and processing information and communication and cooperation.
Of course, in mathematics education, strengthening the infiltration of mathematical thinking methods is not just a single thinking activity, it itself contains the influence of emotional literacy. This point is often ignored in traditional mathematics education. While emphasizing the process and methods of learning knowledge and skills, we should pay more attention to the positive emotional experience and correct values that accompany this process. The standard regards "emotion and attitude" as one of the four target areas, and compares it with knowledge and skills, mathematical thinking and problem solving, which fully reflects that the new round of mathematics curriculum standard reform attaches great importance to cultivating students' good emotion and attitude. It should include being able to actively participate in mathematics learning activities and being curious and curious about mathematics. Get successful experience in mathematics learning activities, exercise the will to overcome difficulties and build self-confidence. Understand the close relationship between mathematics and human life and its role in the development of human history, experience mathematical activities full of exploration and creation, feel the rigor of mathematics and the certainty of mathematical conclusions, and form a realistic attitude and the habit of independent thinking. On the other hand, guide students to learn cooperative learning, cultivate the spirit of exploration and creativity in the process of learning knowledge, and form a correct personality consciousness.
The connotation of modern mathematical thinking method is extremely rich, such as set thought, limit thought, optimization thought, statistical thought, conjecture and proof. These are all related to mathematics teaching in primary schools. Our primary school math teachers should be conscientious in teaching, consciously infiltrate and guide, pay attention to the infiltration of the history of mathematics, and pay attention to the summary of classroom teaching, so as to present the teaching content in a popular and life-oriented way that adapts to the age characteristics of primary school students, so that students can learn to ask, analyze and solve problems with mathematical thinking methods through practical activities, so that students' mathematical thinking ability can be effectively developed and the mathematical cultural literacy of the whole nation can be improved.
Main references:
1. Nine-year compulsory education full-time primary school mathematics curriculum standard (experimental draft)
2. Harmony and innovation, interpretation of a new round of basic education curriculum reform, teaching and management, 2002-2- 1
3. Xu, "Analysis of Teaching Mode Based on Situational Problems in Realistic Mathematics Education", Foreign Educational Materials, 2000-4.
4. Kong Qiping, "Some Trends of International Mathematics Curriculum Reform in Recent Years", Foreign Educational Materials, 2000-6