How to cultivate students' mathematical thoughts and methods in primary school classroom
Primary school mathematics knowledge can be divided into explicit knowledge and tacit knowledge. The primary school mathematics textbook is the explicit knowledge system of mathematics teaching, and the mathematical thinking method is the implicit knowledge system of mathematics teaching. The most important knowledge of primary school mathematics is the knowledge of mathematical thinking methods, which is the ability of 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. It is an important part of mathematical spirit and scientific world outlook, which needs long-term cultivation and frequent application. In detail, the commonly used mathematical thinking methods in primary school mathematics are: corresponding thinking method, hypothetical thinking method, comparative thinking method, symbolic thinking method, analogy thinking method, transformation thinking method, classified thinking method, set thinking method, number-shape combination thinking method, statistical thinking method, extreme thinking method, alternative thinking method, reversible thinking method and reduced thinking method. This paper talks about how to cultivate transformation thinking method based on my own teaching practice. The so-called "conversion" means transformation and belonging. When solving mathematical problems, people often attribute problem A to a problem B that has been solved or is relatively easy to solve through some transformation process, and then return to the original problem A by solving problem B. This is the basic idea of transformation method. The essence of changing 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 study hard at ordinary times, but when they encounter new problems, they don't know where to start. The root cause of this situation is that they can't use what they have learned flexibly. First, build a bridge to return to what you have learned for new problems. Example 1. Calculate+= =? Students have just begun to learn fractional addition with different denominators. How to make peace with them? This is an unknown problem to be solved. In order to solve this problem, the teacher bridged: we have not learned such 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 adding fractions with the same denominator that students can solve. That is, through the total score, the scores with different denominators are added to the scores with the same denominator, so that the original problem can be solved. That is,+(new question) = (transformed 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 simple problem, in fact, the mathematical thinking method of induction has been sublimated, strengthened and consolidated in this problem. 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, when learning plane graphics, teachers can guide students to summarize how the formula for calculating the area of plane graphics that we learned in primary school is derived. That is to sum up the application of inductive thinking method in knowledge construction in the same kind of knowledge structure. Question: What area formulas of plane figures have we learned? Summary: rectangle, square, triangle, trapezoid, circle. Enlightenment: Students, think about it. How are the areas of these plane figures derived? In what way? After giving students enough time to think independently and explore cooperatively, it is summarized as follows: squares are counted by grid, and the area of squares is equal to side length × 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 the rectangle is the length of the parallelogram, the width of the rectangle is the height of the parallelogram, and the area of the rectangle = length × width. Then, the area of parallelogram is equal to the length multiplied by the height, and the conclusion is drawn that the area of parallelogram is equal to the bottom × the 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 (transformed into) rectangle (or square), thus the area of trapezoid = (upper bottom+lower bottom) × the area of circle with height ÷2 is deduced. We use the methods of cutting, spelling, rotating and translating to classify circles into figures similar to rectangles. It is found that half the circumference is equivalent to the length of a rectangle, the width is equivalent to the radius of a circle, and the area of a parallelogram is equal to the length. Area of a circle = half of the circumference × radius = ×r=π× r2. So the area of the circle is equal to π× r2. The formulas for calculating the area of plane graphics we have derived are all 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 mathematical thinking method of conversion not only plays an important role in primary school learning, but also is an important thinking method in middle school and high school learning, and it is also a thinking method of our lifelong learning. At the end of primary school study, teachers should also guide students to summarize the application of mathematical thinking methods in calculation, geometry and application problems.