In mathematics classroom teaching, we teachers often encounter such a situation: when teachers ask students to describe the definition of concepts, they often give fluent and satisfactory answers, but they often cannot use them correctly to solve related problems. In teaching practice, the author has encountered a similar situation. For example, when learning quadratic function, he can accurately tell several forms of analytical formula, but he can't flexibly use which analytical formula in specific topics to solve problems, and he won't use the method of combining numbers and shapes to draw sketches for analysis. Students' correct and fluent answers only cover up the essence they don't understand, which is a common phenomenon in middle school mathematics teaching practice, and we call it superficial understanding. The reason, the author thinks, is that most students simply don't understand or deeply understand the essential connotation of mathematical concepts, theorems and laws. , and blindly rote, set questions to do. This is related to teachers paying too much attention to "analogy" and "high-density training" in the teaching process and ignoring students' deep understanding of mathematics knowledge. In view of the above problems, this paper makes an in-depth analysis and talks about some measures to promote junior high school students' mathematical cognitive understanding.

First, provide students with rich perceptual materials in various ways.

Mathematical concepts, properties, theorems and so on are highly abstract and generalized. If junior high school students can understand it directly, there will definitely be great difficulties. Therefore, in mathematics teaching, teachers should provide students with rich mathematics learning materials such as objects, models, teaching AIDS and teaching software. Let students have enough time to operate specific things, and let them gain the specific experience needed to learn new knowledge through their own thinking activities and form an understanding of concepts. Instead of mechanical repetition, remember the ready-made concept explanation given by the teacher, so that students can acquire comprehensive, clear and solid knowledge. In the teaching process, the following measures can be taken:

1, let the students operate.

For example, when teaching the axiom of judging the congruence of triangles, you can ask each student to draw a δ△ABC on white paper with a ruler and protractor, so that =60, ab=5cm, bc=3cm, and then cut the triangle with scissors, and then compare it with the triangles drawn by other students to see if they can overlap. At this time, students will find that it can overlap. Then let the students change the angle and length before cutting the triangle, and then do it again. At this point, the teacher inspires the students again and draws the conclusion that if the two sides of two triangles and their included angles are equal respectively, the two triangles are congruent, that is, the edge theorem. This teaching method not only enlivens the classroom atmosphere and stimulates students' interest in learning, but also makes abstract mathematical knowledge contained in simple experiments and makes students easily accept new knowledge.

2. Illustrated with pictures and texts

For example, solving linear inequalities in middle school mathematics is a difficult point. In the teaching process, teachers can design a composite slide of 1 Figure 4, and analyze and summarize the pictures one by one, so that students can have a clear understanding of the solution of linear inequalities.

3. Using modern multimedia technology

For example, in the section of "Similarity of Figures", we can make two maps of China with different scales by using computer-aided teaching, find Changsha, Wuhan and Shanghai from them, connect these three points to form a triangle, and then calculate the length of the corresponding sides according to the scale to find the properties of similar figures. Students find it easy to understand. In this way, abstract mathematical concepts become tangible things, internalized into students' knowledge structure, thus achieving better teaching results.

The application of modern teaching methods can make "dead" graphics come alive in teaching and "dead" book knowledge come alive, which can provide vivid and intuitive materials for students, thus broadening their horizons and expanding their knowledge structure.

Second, cleverly set the problem situation

When setting the problem situation, you can start from the following aspects:

1, let students know what they will learn.

It is the best "temptation" to let students consciously participate in learning. For example, the first lesson of factorization of formulas-square difference formula, teachers can create problem situations like this: in an intelligence contest, the host provided 1 topic "2009 -2008 =". As soon as the host's voice fell, some students stood up and answered, "It is equal to 40 17." The students answered. Students, do you know how he worked it out?

At this time, the students began to be silent and think about the problem, but they never came to any conclusion. ...

Teacher: After learning the square difference formula today, we can solve this mystery, thus creating a problem situation, so that students can have the desire of "I want to be a quick responder like him" and actively participate in learning.

2. Construct cognitive conflict

When there is a cognitive conflict between new learning and students' original knowledge level, this conflict will become the driving force to induce and promote students' thinking development, and make students have the desire to make up for the "psychological gap". For example, in the teaching of "the middle vertical line of a line segment", teachers can create problem situations like this:

As shown in Figure 5, there are villages A, B and C3. Now we will dig a well P for them, so that the distances from P to A, B and C are all equal. So where should P be located?

Then the teacher tied one end of three rubber bands together as point P, and fixed the other end at point A, point B and point C3 respectively. While moving point P, the teacher asked, "Are the lengths of pa, PB and PC equal?" After several attempts, students will think that observation alone is inaccurate and measurement is not feasible. At this time, the teacher pointed out: "As long as you master the knowledge of the midline of the line segment, this problem is a piece of cake." At this time, students have a psychological gap-how to accurately determine the position of point P? In this way, students will actively participate in the study of new knowledge.

3. The problem situation is familiar to students.

When setting the problem situation, it is best to start from the perspective of students' familiar life situation and production practice, which can not only ensure students to understand the relevant concepts of the problem, but also enable students to actively construct their own mathematical cognitive structure. For example, when a math teacher talks about merging similar items, he can introduce a new lesson like this: an individual breeder wants to sell a batch of chickens, ducks and geese, where A is the price of chicken, B is the price of meat and C is the price of fish. He wrote down 3.5 kilograms of a chicken, 4 kilograms of a piece of meat and 5.5 kilograms of a fish in the ledger, and also wrote down 3 kilograms of a chicken and a piece of meat 1. By solving this practical problem, the principle of merging similar items is naturally derived. This lecture is not only vivid and easy to understand, but also makes students feel that the mathematics knowledge learned in class is very close to real life, thus improving the sense of value of knowledge.

Third, pay attention to the application of variants.

1, highlighting the essential attributes of the concept through nonstandard variants.

In the object set of concepts, although logically, all objects are equivalent, in fact, their positions in the students' concept system are different. This is because some of these objects have a "standard" form, or are influenced by students' perceptual experience, and so on. Although the standard form helps students to grasp the concept accurately, it is also easy to limit students' thinking, thus artificially narrowing the extension of the concept and making it impossible for students to understand the concept thoroughly. One way to solve this problem is to make full use of non-standard forms: by transforming the non-essential attributes of concepts, the essential attributes are highlighted.

In geometry teaching, many teachers often use the most common and familiar graphics for teaching. Some students understand that it can change with a constant, but some students are limited by this "standard graph" and have difficulty in understanding it. Therefore, in geometry teaching, paying attention to the diversification of graphics, that is, there are many changes in the shape and placement of graphics, can enable students to form a correct representation quickly and broaden their horizons. For example, when explaining vertical, triangular height and parallelogram, we can compare standard shapes with non-standard shapes to help students understand.

2. By comparing conceptual variants with non-conceptual variants, the extension of concepts is clarified.

Mathematical concepts are usually not isolated, but exist in a conceptual system composed of many concepts. Therefore, to clarify the extension of concepts, we must distinguish the relationship between concepts and related concepts, which is the premise of understanding concepts. We can use so-called "non-conceptual variants", such as non-conceptual graphics in plane geometry, to help students understand the essential attributes of concepts by comparing non-conceptual variants with conceptual variants.

There are many forms of non-conceptual variants, among which "counterexample variant" is commonly used, which is what we usually call conceptual counterexample. Because counterexamples have distinct intuitive characteristics, they are easy to attract students' attention and are accepted by students. Therefore, counterexample teaching is one of the effective methods to promote students' deep understanding. For example, when learning a diamond, it is important that the diagonal lines are perpendicular to each other, but many students mistakenly think that a quadrilateral with perpendicular diagonal lines is a diamond. At this time, teachers can use the counterexample diagram in Figure 6 to help students clarify their misconceptions and thoroughly understand the essence of the diamond.

Fourth, guide students to sum up what they have learned.

Learning mathematics can not isolate and separate knowledge, but pay attention to the "horizontal" and "vertical" connection between mathematical knowledge. In mathematics teaching, teachers should guide students to sum up what they have learned.

1, vertical summary

After learning each unit and chapter of knowledge, guide students to summarize and sort out the internal relations, logical order, master-slave position, problem-solving skills and skill structure of what they have learned; When reviewing, we should pay attention to the induction and generalization of the mathematical thinking methods we have learned, so that students can try to write their own learning experience in this respect, or write a chapter. Of course, to summarize and sort out knowledge is not to list the definitions, theorems and laws learned, but to establish the internal relations and differences between knowledge. By drawing the block diagram of knowledge structure, the relationship between knowledge is clear at a glance, which can help students form a good cognitive structure.

2. Horizontal summary

Horizontal summary is to systematically connect and contact all kinds of knowledge and methods scattered in each unit but solving similar problems, thus providing various methods for solving similar problems. For example, the following methods can be used to prove that two straight lines are vertical: vertical definition, the unity theorem of three lines of isosceles triangle, the judgment and property theorem of right triangle, the related properties of square, rectangle and diamond (the four corners of square and rectangle are right angles, and the diagonal lines of square and diamond are perpendicular to each other), and the vertical center property of triangle. In the teaching process, teachers should be good at using opportunities to consciously exercise students, so that their cognitive structure can be gradually improved.

Fifth, pay attention to mathematical communication and improve students' mathematical language expression ability.

1, strengthen the training of mutual translation of graphics, symbols and characters.

Mathematical concepts, theorems, formulas, rules, etc. It is often expressed in only one mathematical language, and students must use three mathematical languages (written language, graphic language and symbolic language) flexibly to truly understand, master and use them. For example, the theorems in geometry are all expressed in written language, but the arguments in proving problems need to be expressed in symbolic language, and graphic language, as a supplement to written language and symbolic language, provides an intuitive model for mathematical thinking. Therefore, it is necessary to communicate and translate the three languages well in geometry teaching.

2. Carry out group cooperative learning.

In the classroom, teachers should appropriately change the form of teaching organization, carry out group cooperative learning, and provide students with a relaxed and free learning environment, so that they have enough independent space in the learning process. Group communication should provide every student with equal opportunities to participate, so that students can cooperate with others, communicate with each other, listen, explain, think about other people's views and reflect on themselves on the basis of independent thinking. Through this process, the original vague understanding can be clarified. In group learning, teachers should give full play to their guiding role, which requires teachers to do the following: First, design questions that students are interested in. When solving problems, students should use their hands and brains, wholeheartedly serve others and cooperate with other students, otherwise they can't finish; Secondly, teachers should actively patrol and grasp the trend of students' discussion, make further comparison and evaluation of students' different opinions, and guide students to discover the possible logical relationship of various answers; Third, teachers should also inspire and encourage those students who are not good at words and have poor grades to speak boldly and express their views.

Whether students understand and digest mathematical concepts depends on the gradual infiltration of teachers in classroom teaching, which cannot be achieved overnight. Only by using a variety of methods, forms and means to fully mobilize the enthusiasm of students can we achieve the best teaching effect.