The mathematical thinking methods infiltrated in primary school mathematics textbooks mainly include: combination of numbers and shapes, set, correspondence, classification, function, limit, reduction, induction, symbolization, mathematical modeling, statistics, hypothesis, substitution, comparison and reversibility. In teaching, it is necessary to clarify the significance of infiltrating mathematical thinking methods and realize that mathematical thinking methods are the essence and essence of mathematics. Only when students master methods and form ideas can they benefit for life.
Below I will illustrate how to infiltrate these mathematical thinking methods into students with examples.
First, the thinking method of combining numbers and shapes
1. Form first, then number. Primary school students in grade one have just started to learn mathematics. They begin to know numbers from concrete objects and abstract from concrete images.
2. Count first and then form. For example, the teaching queue: first-year students line up to do exercises, from front to back, Xiao Ming ranks fifth, from back to front, Xiao Ming ranks fourth. How many people are there in this pair? Pupils can easily add up 4 and 5 and get the wrong result. If students are asked to draw pictures and use "△" to represent the children in line, this problem will be easily solved.
Second, the corresponding ideas
For example, find the quantitative relationship of one application problem with more (less) numbers than another. It's more abstract for second-year students. I designed it like this: there are 8 apples and 6 pears. How many apples are there than pears? Students use learning tools such as ○ and △ instead of apples and pears, or draw a picture to solve the problem.
For another example, the one-to-one correspondence between points on the number axis and real numbers makes the quantitative relationship of abstract content intuitive, concrete and visual, and turns abstruse into simple. At the same time, encourage students' innovation and make them willing to participate in such mathematical activities.
Third, the idea of classification.
Classification is to classify teaching objects into different categories according to their essential attributes, that is, according to their similarities and differences, classify those with the same attributes into one category and those with different attributes into another category for analysis and research. Classification is an important means of mathematical discovery. In teaching, a lot of complicated knowledge can be organized if the knowledge is properly classified. The general classification requires the principles of mutual exclusion, no omission and simplicity. 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 "classification of angles", 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. Triangles can be divided into equilateral triangles and equilateral triangles, and equilateral triangles can be divided into equilateral triangles and isosceles triangles. Through classification and knowledge network construction, different classification standards will have different classification results, thus generating new mathematical concepts and the structure of mathematical knowledge.
Fourth, convert to ideas.
Reduction is the most commonly used thinking method in mathematics. It transforms the problem to be solved into the solved problem through deformation, thus solving the original problem. Its basic idea is to simplify the problem A to be solved into the solved or relatively easy-to-solve problem B through a certain transformation process, and then get the solution of the original problem A by solving the problem B. This conversion idea is different from the general "transformation" and is irreversible and one-way. Its basic forms are: turning difficulty into ease, turning life into maturity, turning complexity into simplicity, turning the whole into parts, turning music into straightness and so on. In primary school mathematics, there are all kinds of contents that can be answered by induction, so that students can learn inductive thinking methods initially. For example, to teach the calculation method of circular area, the formula of circular area should be deduced here. In the process of derivation, the circle is divided into several equal parts and then spliced into an approximate rectangle, thus the area formula of the circle is derived. The process of splicing circular scissors into approximate rectangles here is the process of turning curves into straight lines.
Another example is the derivation of parallelogram area. When I create a situation to make students have an urgent need to find the parallelogram area, I directly throw "how to calculate the parallelogram area" to students, so that students can think independently and freely. This completely unfamiliar problem requires students to mobilize all relevant knowledge and experience reserves to find possible ways to solve the problem. When students convert the area calculation of unpaved parallelogram into the area of learned rectangle, they should be clear about two aspects:
Firstly, in the process of transformation, the parallelogram is cut and spelled, and the area of the rectangle finally obtained is equal to the area of the original parallelogram (that is, equal product transformation). On this premise, the length of a rectangle is the base and the width is the height of the parallelogram, so the area of the parallelogram is equal to the base times the height.
Second, after the transformation is completed, remind students to reflect on "Why should it be transformed into a rectangle?" Because the area of the rectangle has been calculated before, the unfamiliar knowledge has been transformed into the learned solvable knowledge, thus solving new problems. In this process, the concept of transformation will also sneak into the hearts of students. The same is true for the teaching of other graphics.
Fifth, gather thinking methods.
There are a lot of stereotype thoughts in primary school mathematics textbooks, and stereotype thoughts and concepts permeate all stages of mathematics teaching. It is necessary not only to impart knowledge to students, but also to consciously infiltrate the established ideas contained in textbooks, which is conducive to cultivating students' abstract generalization ability and improving their ability to analyze and solve problems. The textbook adopts intuitive means and uses the thinking method of infiltration and collection of graphics and objects. For example, when seeking the greatest common divisor of 8 and 12 in teaching, you can make courseware or slides, so that students can clearly and intuitively know from the pictures that the common divisor of 8 and 12 is 1, 2 and 4, and the greatest common divisor is 4, thus breeding the idea of intersection.
In addition, there are analogy thought, modeling thought, combination thought, limit thought and so on. , not listed here. In primary school mathematics teaching, we should pay attention to purposeful, selective and timely infiltration. There are many strategies to infiltrate mathematical thinking methods. I think:
1. Penetration in the process of knowledge formation.
Mathematical concepts, laws, formulas, properties and other knowledge are clearly written in the textbook, with a "shape", while mathematical thinking methods are implicit in the mathematical knowledge system, without a "shape", and are scattered in all chapters of the textbook systematically. Therefore, mathematical thinking method must be realized through specific teaching process. In teaching, we should pay attention to the formation process of concepts; The process of guiding students to explore, discover and deduce theorems and formulas; Finally, guide the students to summarize.
2. Infiltration in the process of solving problems.
Mathematical thinking method exists in the process of solving problems, and the gradual transformation of mathematical problems follows the guidance of mathematical thinking method. Mathematical thinking method plays an important role in solving mathematical problems. Infiltrating mathematical thinking methods can not only speed up and optimize the process of solving problems, but also achieve the effect of knowing one problem and knowing all the way. Through infiltration, let students internalize mathematical thinking methods as much as possible and improve their ability to acquire knowledge and solve problems independently.
3. Infiltrate in the process of repeated use.
In grasping the key points of learning, breaking through the difficulties of learning and solving specific mathematical problems, mathematical thinking method is the essence of dealing with these problems. The process of solving these problems is always a process of repeatedly applying mathematical thinking methods. Therefore, it is conditional and possible to pay attention to the application of mathematical thinking methods from time to time, which is an effective and universal way to carry out the teaching of mathematical thinking methods. Mathematical thinking method can only be consolidated and deepened through repeated application.
In short, paying attention to the infiltration of students' mathematical thinking methods is not only conducive to improving classroom teaching efficiency, but also conducive to improving students' mathematical cultural literacy and thinking ability. However, the infiltration of students' mathematical thinking methods can not see the improvement of students' mathematical ability overnight, but a process. Therefore, in the teaching process, we should organically combine the contents of mathematics knowledge, so as to achieve persistent, gradual and repeated training, so that students can truly understand mathematical thinking methods and achieve a qualitative leap.