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First, let students experience the derivation process of mathematical formulas and theorems, which is the premise for students to understand these formulas and theorems.
Hua, a famous mathematician, said: "It's best to look for information in the mathematician's wastebasket when learning mathematics, and don't just read the conclusions in the book." In other words, the research of mathematical thinking method in the process of exploring the conclusion is as important as the conclusion itself. In fact, many teachers have neglected one of the most important problems: mathematical formulas and theorems are tools for solving problems, and correctly understanding and using formulas and theorems is the basis for learning mathematics well. In order to save teaching time, some teachers often omit the derivation of formulas and theorems. Sometimes, although the sources of formulas and theorems are shown, they are mainly taught by teachers, and students do not really participate in the whole process of discovering formulas and theorems. So on the surface, it seems to save time, but this teaching form often leaves only the shell of formulas and theorems in students' minds, ignoring the causal relationship between them and not knowing their conditions and scope of use. When it is necessary to use the formula, they can't remember it, even if they can, they don't know how to use it.
The theory of multiple intelligences requires students not to blindly accept and passively memorize the knowledge taught by textbooks or teachers, but to actively explore themselves and turn the learning process into a knowledge construction process in which they actively participate. Students' flexible use of mathematical formulas and theorems is the premise of understanding these formulas and theorems; To understand these formulas and theorems, students need to experience the derivation process of formulas and theorems themselves. Only in this process can students understand their own context, applicable conditions and scope.
Second, pay attention to the deduction process of mathematical formulas and theorems, and let students use these problem-solving tools in the deduction process.
Through observation and analysis, induction and analogy, mathematical formulas, theorems, laws and other conclusions are drawn, and then logical proof is sought; Or draw a conclusion through theoretical deduction. Therefore, in the teaching of formulas, theorems and laws, we should guide students to actively participate in the process of exploring and discovering these conclusions, constantly find out the causal relationship of each conclusion under the guidance of mathematical thinking methods, let students experience creative thinking activities, and guide students to summarize and draw conclusions.
In the past, when teaching the complete square formula (AB) 2 = A2AB+B2, in order to save time, just tell the students the conclusion and think it can be used. Ask the students to memorize the formula. As long as they practice more, they will be able to master the formula. But in fact, many students only remember the shape of the formula because they don't understand the formation process of the formula. When they used it, they either missed 2ab or misspelled the symbol of b2. So I did an experiment in two classes I taught. A class continues to give formulas directly and let them recite the questions and do them directly. One class asked them to deduce the formula by themselves.
Starting from the geometric meaning, students are required to prepare a big square, a small square and two rectangles by using the group independent inquiry learning method. The side length of the big square is the side length of the big square and the side length of the small square is the width, so that they can spell out a big square with the figures at hand. Through the method of jigsaw puzzle, let students find the law in the process of doing it.
Make a square with the four existing figures in your hand, observe the figures and answer the following questions:
(1) population: find the total area.
(2) Part: Find the sum of four areas.
(3) Conclusion (a+b)2=a2+2ab+b2
The total area consists of four parts: two squares and two rectangles with different sizes. The area of a square is a2 and b2 respectively, and the area of two rectangles is 2ab, which is an important part of the whole area. Students have deepened their understanding of 2ab in the formula through jigsaw puzzles, effectively preventing the omission of 2ab in the future.
On the basis of students' inquiry into (a+b)2=a2+2ab+b2, the question is: Can the reason be explained by polynomial multiplication rule? Let the students use the principle of polynomial multiplication to derive a complete square formula: (a+b) (a+b) = A2+AB+AB+B2 = A2+2AB+B2, and then demonstrate the complete square formula. Using the calculation process of polynomial multiplication polynomial rule, let students feel the existence of 2ab again. Prove the formula from algebra and geometry, so that students can fully understand the formation process of the formula, deepen their impression of the formula and strengthen the credibility of the formula. And let the students know that the conclusion of the conjecture must be verified. Let students understand the mathematical thought of the combination and transformation of numbers and shapes.
Then let the students observe the characteristics and memorize the formula. Ask the students to describe the complete square formula in words. Encourage students to explore the structural characteristics of the formula independently: (1) The formula is expanded into three items; (2) The two square terms are the same; (3) The middle symbols should be consistent before and after. Let students know the ins and outs of the formula. I designed four judgment questions to give students a deeper understanding of the formula structure.
( 1)(a+b)2=a2+b2()
(2)(a-b)2=a2-2ab-b2()
(3)(a+b)2=a2+ab+b2()
(4)(2a- 1)2=2a2-2a+ 1()
Let students deeply understand the structural characteristics of the formula through the first question (the first question lets them know that there must be three terms in the formula, so don't omit the second question of 2ab to let them know that the square term is positive; The third question lets them know that 2 can't be omitted in 2ab. And the fourth question let them know that A in the formula is not only a letter, but also a formula. When a is a formula, parentheses must be added.
Finally, by filling in the form of the table below, organize students to discuss, and consolidate the structural characteristics of the formula again through the table: the first and second squares must be positive, the middle of which conforms to the product of the first and second items, the same sign is positive, the different sign is negative, and the middle is remembered twice, and then the summary steps are as follows:
(a) determine that sign of the first and second sum of squares; (2) Determine the coefficient and symbol of the middle term and draw a conclusion.
After the new lesson, I asked two classes to take quizzes for five consecutive days to count the error rate of using formulas.
It is found that the error rate of the two classes on the first day is almost the same, but with the passage of time, the more formulas are learned, the greater the error rate of the formula in the class that recites the formula, especially for the middle and lower class students. Simply remembering the shape of the formula, the more vague the memory, the higher the error rate. On the contrary, after the derivation of the formula, I realized the connotation of the formula, and the understanding of the formula became clearer with the passage of time, so the error rate became lower and lower.
Through a period of trial, we found that students' mastery of mathematical formulas and theorems not only stays at the level of memory, but also can understand their connotations. Through such experiential learning, students' academic performance has been significantly improved, students' interest in mathematics has become stronger, and students' learning enthusiasm has become higher.
Practice shows that if the traditional "spoon-feeding" method is adopted in the teaching of mathematical formulas and theorems, not only can students' grades not be improved, but they will become more and more tired of learning. Therefore, we must attach importance to the derivation process of formulas and theorems, so that students can understand not only what formulas and theorems are, but also how they are formed. This kind of learning is meaningful.