What is heuristic teaching? Heuristic teaching is a teaching method that makes students think positively and make their own judgments under the guidance and teaching of teachers. It can also be said that the discovery method of a cognitive program is compiled under the leading role of teachers, which is the concrete implementation of heuristic principle in teaching. In heuristic teaching, the teacher's role is external cause and catalyst, and its foothold is to induce students to think positively and make guesses or judgments by independently trying to establish the connection between old and new knowledge. Judging whether a teaching is heuristic or not depends not on whether its external form is lively or not, nor on the length of students' hands-on time. The key is to see whether students' psychological activities have reached the level of understanding and whether they have made guesses or judgments through their own attempts. So, how to do a good job in heuristic teaching? In junior high school mathematics teaching, we should grasp three problems, namely, the prototype of inspiration, the opportunity of inspiration and the intensity of inspiration.

First, the prototype of inspiration

The so-called heuristic prototype is the knowledge growth point that students should learn in their existing cognitive structure. Mathematics learning process is based on students' original cognitive structure, and new knowledge is brought into the existing cognitive structure through internalization and understanding. In this process, the teacher's role is to mobilize the students' knowledge reserves, so that the new teaching knowledge can establish substantive contact with the corresponding materials in the original cognitive structure. Therefore, in teaching, it is necessary to distinguish the relevant materials (i.e. heuristic prototypes) that can assimilate new knowledge in students' cognitive structure, and design a good teaching based on this.

For example, in concept teaching, because mathematical concepts are often abstractly summarized by some actual cases and specific mathematical textbooks, in order to let students experience the occurrence and development of concepts in teaching, we must start with the actual cases and specific mathematical materials that these students know, remove their appearances, preserve their essence and gradually form concepts.

For another example, in example teaching, because the key is the process of seeking ideas for solving (proving) problems, the process of seeking ideas is often manifested as: "Have you put forward a proposition in the past in terms of knowledge, conclusion or graphics?" The "similar topics" and "easier and more intuitive propositions" here are the enlightening prototypes at this time, and teachers should be good at using these enlightening prototypes to communicate the problems to be solved (proved). In this way, the idea of solving problems will go through a process from vague to clear, from dispersion to aggregation in students' minds, and the acquisition of ideas will be natural.

Second, the opportunity for inspiration.

Regarding the timing of inspiration, Confucius has long said, "If you don't get angry, you won't get angry." . This means that only when students are bored because they can't think, and when students want to say but can't say it, can teachers inspire them. Specific to mathematics teaching, we need to do the following two things.

First of all, we must seize the opportunity. For example, when proving the nature of "the distance between two points on the bisector of an angle is equal on both sides", let the students think for themselves first. When students understand the meaning of the problem but don't know how to start, they will extract the first heuristic prototype, thus directing their thinking to "proving the congruence of two triangles"; When students can't find the congruence of triangles in the analysis, when they are confused for the second time, they extract the heuristic prototype again and position their thinking as "how to construct two congruence triangles". When students don't know how to construct congruent triangles, and the third kind of thinking disorder appears, the teacher promptly guides the students' thinking to "piece together" through congruent triangles's judgment method, which has achieved good results.

The second is to create opportunities. According to the characteristics of teaching materials and students' level, teachers can seize the opportunity to create situations and create good inspiring situations on the basis of inspiring prototypes, so that students can actively and enthusiastically participate in trying activities in seemingly ignorant situations that they want to understand.

Case 1:

The teacher intends to arrange two students with different levels to perform and guide the students to analyze. How can two students get different results when they calculate correctly and apply correctly?

As the students have answered the questions in person, as soon as the questions are put forward, the students' thinking focus immediately focuses on "why" and "where is the problem", which makes the students feel restless and creates a good opportunity for the later teaching.

Third, the intensity of inspiration.

As for the intensity of inspiration, the ancients also discussed it a long time ago: "The Tao must guide, the strong must restrain, and the opening must reach." "Show the beginning and end." It means to point out the direction of thinking for students but don't lead them by the nose; Ask questions strictly without pressure, remind students but can't tell the answer directly. At the beginning of teaching, the teacher induces and prompts, and when the students try and get some results, the teacher corrects them.

Case 2:

When talking about the concepts of monomials, polynomials and algebraic expressions, we can take the following steps:

Show two groups of algebraic expressions and ask students to point out how these algebraic expressions are formed. The teacher wrote on the blackboard:

(1) Teacher's question: Observe the laws of the first set of algebraic expressions, the laws between numbers and letters, and the laws between letters.

(2) Through observation, thinking and discussion, students come to the conclusion that there is only multiplication between numbers and letters, and only multiplication between letters.

(3) Students sum up the concept of monomial, and teachers complement it completely.

(4) What are the rules for observing the second set of algebraic expressions?

(5) Summarize the meaning of "degree" and "term" and the concept of polynomial.

In the above process, the operation between letters and numbers, letters and related concepts is the inspiration prototype at this time. They are divided into two groups: only multiplication and both multiplication and addition. Let the students observe, summarize and conclude. On the basis of enlightening the prototype, the teaching methods are discussed from the students' understanding level.

If the algebraic expression is given again, instead of dealing with it seriously, let the students explore it, but point out: "Let's see what is the operation between letters and numbers or letters and letters?" It's a sign of excessive inspiration. Because of this problem, students' main activities have become thinking according to the teacher's requirements, being led by the nose, and there are few elements left in thinking and creating discoveries, let alone understanding and judging.

In short, to do a good job in heuristic teaching, we must take understanding and judgment as the main characteristics of heuristic teaching, take heuristic prototype as the basis of inspiration, create and seize the opportunity of inspiration in time, and accurately grasp the intensity of inspiration, so as to get "Fa" and "Fa" from inspiration.

In teaching, whether teachers explain, ask questions, demonstrate, experiment, summarize, review and solve problems, and assign exercises, they should inspire students' positive thinking in various ways, stimulate students' potential learning motivation and interest, and make students actively and enthusiastically participate in learning activities.