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 example, the concept of parallel lines can be illustrated by many examples of disjoint lines in students' daily life, and their * * * characteristics can be found out, so that students can form a preliminary image, then abstract it into two straight lines, and gradually move from intersection to disjoint, thus summarizing the concept of parallel lines.
Another example is problem-solving teaching, because the key is the process of solving (proving) problems, and the process of seeking often shows: "Have you put forward a proposition in the past from the aspects of knowledge, conclusion or graphics?" Wait a minute. 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.
For example, when proving the edge theorem of triangle congruence (the new textbook has been changed to axiom), the proof method in the textbook is: as shown in the figure, put △ABC on △AˊBˊCˊ, so that the longest side BC coincides with BˊCˊ, so that points A and A on both sides of BˊCˊ are connected. How did the teacher think of it? "These are all students' puzzles. If this problem is not solved well, students will only learn the concrete solution to this problem at best, and teachers will lose a good opportunity to train their thinking and cultivate their ability. In teaching, if we can fully mobilize and stimulate students' knowledge reserves through two prototypes, the effect will be completely different. The process is as follows:
(1) First Prototype Extraction
Teacher: I have learned this method of proving the congruence of triangles before. How are they different from the known conditions of this problem?
Student: I have studied edges, corners, corners and so on. One or two angles in their condition are equal, but the condition of this problem is that three sides are equal. ..... oh! We should first prove that the angles are equal. (Inspired by the prototype. Positioning thinking as "the angles of proof are equal", students' thinking has made the first leap. )
(2) Second prototype extraction
Teacher: How to prove that the angles are equal? What methods have you learned in the past?
Student: Use parallel lines; Use congruent triangles; Use an isosceles triangle.
Teacher: What method should we use for this question?
After students think, it is easy to rule out the parallel line method. After the teacher's guidance, congruent triangles's method can also be ruled out, and finally he focused on "using isosceles triangle". )
Teacher: There is no isosceles triangle in the picture. What shall we do ... If an isosceles triangle is required, two equal line segments (waist) should start from the same vertex, but the equal line segments in this problem are scattered in two triangles.
(At this time, some students have been able to think of putting two triangles together, and teachers can only improve their thinking through guidance, thus achieving such an effect. It should be said that the inspiration is successful; If the students can't come up with the idea of "jigsaw puzzle" by themselves at this time, but the teacher gives the spelling by himself, it should be said that the purpose of inspiration has been achieved. Because of this arrangement, students' thinking has gone through the process of understanding. They not only learned how to "spell", but also knew why to "spell", so as to know the method and understand the reason, which is the purpose of heuristic. )
Second, the opportunity for inspiration.
As for the timing of inspiration, Confucius has long said, "If you don't get angry, you won't get rid of it." . 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, is to do the following two things:
First of all, we must seize the opportunity. When proving the edge theorem in the above example, 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 the "equal proof angle"; When students don't know how to prove that the angles are equal in the analysis, they will extract the heuristic prototype again when they are confused for the second time. Positioning the idea as "using isosceles triangle"; When students don't know how to construct isosceles triangle, and there are three times of thinking obstacles, teachers guide students' thinking to "piece together" in time through the characteristics of isosceles triangle, which has received good results.
The second is to create opportunities. According to the characteristics of teaching materials and students' level, teachers create angry situations in time 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 and understanding situations.
For example, when teaching "factorization of addition and subtraction items", a question x6- 1 is given first, and students will get two results according to what they have learned:
X6- 1 =(x3)2- 1 =(x- 1)(x+ 1)=(x2-x+ 1)(x2+x+ 1)
X6- 1 =(x3)2- 1 =(x- 1)(x+ 1)(x4+x2+ 1)
The teacher intends to arrange two students with different understanding to perform and guide the students to analyze: the formula used by the two students is accurate, how can there be different results?
Because the students answered the questions themselves, as soon as the questions were put forward, the students' thinking focus immediately focused on "Why?" "What's the problem?" On this issue, students have a kind of anger that they can't stop, which creates a good opportunity for later teaching.
Third, the intensity of inspiration.
As for the intensity of inspiration, the ancients also discussed it long ago: "Tao leads, strong and restrained, open and ambitious", "obviously has a beginning and an end", which means: pointing out the direction of thinking for students without leading them by the nose; Strict requirements but no pressure; Remind the students but can't tell the answer directly. At the beginning of teaching, the teacher induces and prompts. When students try to get some results, teachers will correct them.
For example, when students have guessed that x4+x2+ 1 can continue to decompose, if the teacher directly gives the questions to the students to explore the decomposition method, even if the items to be decomposed are pointed out, most students may still have no way to start. This is another example of poor inspiration. In mathematics teaching, it is necessary to set some suitable steps for students to climb the stairs, jump down and reach them, so as to ensure that students' thinking experiences the process of discovery without feeling unattainable. The process is to guide students to test their guesses by polynomial multiplication (x2-x+ 1)(x2+x+ 1). Because the merged items in the calculation clearly point out the items that should be dismantled or added when decomposing x4+x2+ 1, it is not very difficult for students to try to decompose themselves after the test.
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.