1. mathematical induction analysis mathematical induction is a method to prove mathematical propositions, but the teaching of mathematical induction has always been a difficult point in senior high school mathematics teaching. The reason may be that this method is not logically explained in teaching, which makes it difficult for students to understand and apply this mathematical method. Many students only know the reliability of this method by analogy with domino examples, but they don't realize the rigor of this method in logical reasoning. Many students mechanically use two steps of mathematical induction to prove mathematical conclusions without a good understanding, which leads to various expression errors in the process of proof. Semantically, there are three words "induction" in the name of "mathematical induction", so is this method "induction"? From the point of view of reasoning, induction is often used to "guess" mathematical conclusions, rather than to "prove" mathematical conclusions. So, on the one hand, what is the connection between mathematical induction and induction? On the other hand, in mathematical argumentation, only deduction can be used to prove mathematical conclusions, and mathematical induction can be used to prove mathematical conclusions, so mathematical induction should be included in the scope of deduction, but how to understand this? In addition, from the students' mathematics learning psychology, the word "hypothesis" is used in the second step of mathematical induction, and students will think, since it is a hypothesis, how can it be used as the basis for further proving the conclusion? All these doubts are the reasons why it is difficult for senior high school students to understand mathematical induction. To solve the above problems, it is necessary to properly clarify that mathematical induction is essentially a deductive method, or to be precise, it is a deductive method restored in the process of reasoning and narrative form. In fact, mathematical induction implies a series of syllogisms. The first syllogism is: major premise: If proposition P(n) holds for n=k, then proposition P(n) holds for n=k+ 1; (This major premise is proved in the second step of mathematical induction "inductive recursion") Minor premise: the proposition P(n) holds for n= 1; (This minor premise is also proved in the first step of mathematical induction, Fundamentals of Induction.) Conclusion: The proposition P(n) holds for n=2. So there is a second syllogism: the major premise: if the proposition P(n) is true for n=k, then the proposition P(n) is true for n=k+ 1; Minor premise: proposition P(n) holds for n=2; (This minor premise is the proof conclusion in the previous syllogism) Conclusion: The proposition P(n) holds for n=3. Then, there is a third syllogism, from which the proposition P(n) holds for n=4. ..... In this way, every syllogism is proved to infinity by a proposition in the proposition chain, thus obtaining the proof of the mathematical proposition to be proved (which can be regarded as a proposition composed of infinite propositions). The above explains the essential relationship between mathematical induction and deduction. The above analysis also shows that, The structural relationship of propositional series can be proved by mathematical induction: Proposition P( 1)→ Proposition P(2)→ Proposition P (3) → Proposition P (n) → The structural relationship between propositional series and positive integers is consistent: 1 → Proposition P (k) P (k) proved in the second step of induction and recursion. In fact, people often analyze and study the relationship between the first few special propositions in the proposition sequence "Proposition P( 1)→ Proposition P(2)→ Proposition P (3) → Proposition P (n) →→ Proposition P (k)", from which the general process of deducing two adjacent propositions is obtained. Whether this understanding is correct or not remains to be verified. Mathematical induction proves that the mathematical conclusion has the following logical structure diagram: with a slight change in the above logical structure diagram, several variant methods of mathematical induction can be obtained. The famous French mathematician Poincare called mathematical induction "circular reasoning", which coincides with the spiral circular pattern picture mentioned above. Poincare discussed this method in his book Science and Hypothesis: "The main feature of circular reasoning is that it can contain countless syllogisms and focus on the only formula that can be considered." "In circular reasoning, people are limited to stating the minor premise of the first syllogism and the general formula with all major premises as special cases. Therefore, this endless syllogism can be reduced to a few lines. " This is his more accurate description of the logical essence of mathematical induction. Second, a suggestion on the teaching of mathematical induction At present, the teaching of mathematical induction often makes students have an intuitive understanding of mathematical induction with the help of domino games, which is a good teaching design. In order to judge whether students really understand mathematical induction in actual teaching, from the perspective of teaching evaluation, it is suggested that the following questions can be put forward in teaching: "How do you understand mathematical induction?" To be precise, it is to ask students to "be similar to a domino game, please come up with an actual situation that can reflect the principle of mathematical induction." If students can accurately tell examples of this kind of actual background and know how the phenomena in the actual situation correspond to the corresponding proof steps in mathematical induction, it means that students have a better understanding of mathematical induction. In fact, students may be able to come up with more familiar practical backgrounds, which can help them understand mathematical induction. The following questions can be used for reference. Example 1: student queuing problem: there are many students with numbers (numbers are 1, 2, 3, 4, 5, 6, etc. ) (limited number, or unlimited number), and it is required to line up in the order of number. You can follow the following two steps to achieve the goal: 1. The first person can queue up as required. Everyone abides by the following rules: If not. N students line up as required, and then the first one. N+ 1 Students will definitely line up as required. After the above two items are done, all students can queue up as required. Example 2: Conditions for the smooth start of the train (the whole train including the locomotive and the subsequent cars start smoothly): 1. The locomotive starts; 2. The locomotive is well connected with the subsequent carriages and carriages. As the saying goes, "the train runs fast all by the front", which is only half right. In connection with mathematical induction, a relatively complete statement: "The train runs fast, and the front car and the back car can keep up." Third, an idea of teaching design Classroom teaching is art, so classroom teaching can have different styles of design. In actual teaching, as far as the current understanding of mathematical induction is concerned, there are the following teaching process ideas of introducing mathematical induction into teaching. First, introduce a specific calculation problem: of course, if students are very familiar with the relevant mathematical deformation methods, this problem can not be used, and other similar problems can be considered, such as finding the cubic sum of the first n positive integers. The principle of choosing questions is to make the research of questions exploratory, from which the process of inductive and recursive analysis can be embodied, and appropriate leading-in questions should be prepared according to the actual situation of students in teaching. Here, we only use the above relatively simple question to illustrate the basic teaching process of design. Because there are many addends in the above summation formula, it is difficult to add them item by item to get the required sum. Students hope to find a regular summation method, that is, to get a general formula to solve the problem. From the analysis and induction, we can find that actually solving the original problem can be transformed into solving the following general mathematical problems with universal value: the analysis will be gradually carried out later, so that students can naturally gradually form the proof method of mathematical induction, and understand mathematical induction by domino game analogy. I personally think that this teaching process may make students more aware of the necessity of introducing mathematical induction, and may explain the relationship between mathematical induction and induction. In mathematical induction, a series of special recursive steps (generally infinite) are not made, but a general recursive step is replaced, that is, the second step of inductive recursion. The method proved by this general recursive step is often inspired by the previous special recursions, which is the application of the method from special to general and also the application of inductive thought. However, this is only a part of the process of proving mathematical propositions by mathematical induction, not the whole of mathematical induction. It also shows that mathematical induction does contain inductive elements, but it is different from general induction, so it is named "mathematical induction". Zhang Yaoguang, a teacher in the teaching and research section of Jinhua City, Zhejiang Province, pointed out that "mathematical induction is a deductive method with inductive components", which is reasonable. Fourthly, prove a finite sequence of mathematical propositions by mathematical induction. Generally speaking, mathematical induction can be used to prove the mathematical conclusion of infinite proposition involving positive integers (of course, this mathematical conclusion of infinite proposition involving positive integers can also be regarded as a single mathematical proposition, but there is no strict standard for the unit of mathematical proposition), but from the above analysis, this is not absolute. Mathematical induction can also be used to prove the mathematical conclusion of finite mathematical propositions involving positive integers, as long as two steps (especially recursive steps) are implemented in limited steps. Just as there are finite series and infinite series, the domain of a function can be limited in a closed interval, we can also prove that the positive integer n involved in the mathematical conclusion is limited in a finite set, which is easy to do. We just need to make an identity transformation on n to limit its range of change. The following are two mathematical propositions involving only a limited number of positive integers. Of course, it can also be proved by mathematical induction, but variables need to meet certain restrictions in induction and recursive steps. Example 1: Proof: Example 2: Proof: In fact, the above identity transformation method for n is universal.