The most important knowledge of primary school mathematics is the knowledge of mathematical thinking methods, which is a kind of ability for students to adapt to society and continue learning in the future. Descartes said: "Mathematics is a subject that makes people smart". Mathematical thinking method is the essence of mathematics and an important part of mathematical spirit and scientific world outlook, which needs long-term cultivation, frequent use and subtle influence.
Mathematical thinking methods commonly used in primary school mathematics include: corresponding thinking method, hypothetical thinking method, comparative thinking method, symbolic thinking method, analogy thinking method, transformation thinking method, classified thinking method, set thinking method, combination of numbers and shapes thinking method, statistical thinking method, extreme thinking method, alternative thinking method, reversible thinking method, transformation thinking method and invariant thinking method.
Based on my own teaching practice, this paper talks about how to cultivate the thinking method of transformation.
The so-called "conversion" means transformation and belonging. When solving mathematical problems, people often simplify the problem A to be solved into a solved or relatively easy-to-solve problem B through some transformation process, and then return to the original problem A through the solution of problem B. This is the basic idea of transformation method.
The essence of transforming ideas is to transform new problems into old knowledge that has been mastered, and then further understand and solve new problems. Its basic forms are: changing the unknown into the known, changing the new into the old, changing the difficult into the easy, changing the complicated into the simple, and changing the song into the straight.
Some students usually study hard, but they don't know where to start solving new problems. The fundamental reason for this situation is that they can't flexibly use the mathematical thinking methods they have learned to think about problems and realize the transformation of problems.
So how to train students to master the mathematical thinking method of transformation in the process of primary school mathematics teaching?
First, build a bridge to turn new problems into learned knowledge.
Example 1. Calculate+= =?
Students have just begun to learn fractional addition with different denominators. How to make peace with them? Is an unknown problem to be solved, in order to solve this problem.
Teacher Bridging: We haven't learned this kind of fractional addition, but we have learned the addition of+=. Q: What is the meaning of the formula? Can you show the meaning of the formula with a plan? Can you find a way to turn a new problem into a problem you have learned, so as to find a solution to the problem?
The teacher must put+=? It comes down to the problem of fractional addition with denominator that students can solve. That is, through general division, the addition of different denominator fractions becomes the addition of the same denominator fraction, thus solving the original problem. Namely:
+(new question) = (converted into)+(old question) = = (conclusion)
When drawing a conclusion, the teacher must ask: What do you think? What mathematical thinking method is used to solve problems?
This seemingly ordinary and simple problem, in fact, the mathematical thinking method of reduction has been sublimated, strengthened and consolidated in this problem.
Secondly, summarize the role of inductive ideas and methods in knowledge construction.
After learning a knowledge, such as decimal addition and subtraction; Or after learning a kind of knowledge, for example, the calculation of plane graphic area; Or after learning stage knowledge, for example, at the end of primary school mathematics learning, teachers should guide students to sum up what mathematical thinking methods we have used to solve this knowledge. So as to further clarify the important role of these mathematical thinking methods in knowledge construction.
For example, after learning plane graphics, the teacher can guide students to sum up how the area calculation formula of plane graphics that we learned in primary school is derived. That is, the application of inductive thinking method in knowledge construction is summarized in similar knowledge structure.
Question: What area formulas of plane figures have we learned?
Summary: rectangle, square, triangle, trapezoid, circle.
Keith: Students, think about it. How are the areas of these plane figures derived? In what way?
After giving students enough time for independent thinking and cooperative exploration, summarize:
A square is obtained by calculating the number of grids, and the square area is equal to the side length × the side length;
The area of the rectangle is the area of the rectangle obtained by the square sum grid method = length × width;
The area of a parallelogram is a figure that transforms a parallelogram into a rectangle. The length of a rectangle is the length of a parallelogram, the width of a rectangle is the height of a parallelogram, and the area of a rectangle = length × width, so the area of a parallelogram is equal to length times height. Thus, it is deduced that the area of parallelogram = bottom × height;
The area of a triangle is converted into a rectangle or a parallelogram (or a square), from which it is deduced that the area of a triangle is = base × height ÷ 2;
Trapezoid (converted into rectangle (or square), thus deducing the area of trapezoid = (upper bottom+lower bottom) × height ÷2.
Area of the circle: We classify the circle into a rectangle-like figure by cutting, spelling, rotating and translating. It is found that the semi-circumference of a circle is equivalent to the length and width of a rectangle, the area of a parallelogram is equal to the length times the width, and the area of a circle is equal to the semi-circumference times the radius, so the area of a circle = semi-circumference × radius = ×r=π× r2. So the area of the circle is equal to π× r2.
All the formulas for calculating the area of plane graphics we have derived are to classify a new graphic as a learned graphic, so that we can derive an area formula of a new graphic from the learned area formula, and turn the unlearned knowledge into our learned knowledge to solve new problems. This method of solving mathematical problems is the mathematical thinking method of reduction.
The transformed mathematical thinking method not only occupies an important position in primary school learning, but also is an important thinking method in middle school and high school learning, and it is also our lifelong learning thinking method.
At the end of primary school, teachers should also guide students to summarize the application of reduced mathematical thinking method in calculation, geometry and application problems, and tell students that the most important thing in learning mathematics knowledge is the learning of thinking method, which is the most important weapon for further learning knowledge.