As far as I know, many famous schools put forward and began to solve the problem of mathematics convergence between junior and senior high schools very early, and also developed specific school-based textbooks. Why is the connection between junior high school and senior high school so important? Obviously, senior one has really become a "difficult period" for students to learn mathematics, and mathematics is seriously polarized. A considerable number of students may lose confidence in mathematics for the first time in their lives! For the first time, I feel that I am a "poor math student". We can't take it for granted that "learning math well in high school" is only the privilege of the top students in the class. It is imperative to solve the problem of mathematics connection between junior high school and senior high school!

Second, where is the root of the problem?

(1) Objectively speaking, there is a gap in mathematics knowledge between junior high school and senior high school, and it is precisely because of this gap that many students find it difficult to adapt to the study of senior high school mathematics in a short time.

According to the concept of new curriculum reform and the requirements of curriculum standards, junior high school mathematics textbooks have been reduced in difficulty, depth and breadth, which embodies the characteristics of "shallow, few and easy". Some of the knowledge often used in high school learning has been deleted, while others have diluted the requirements and increased the burden of high school mathematics. Some students think that the teacher speaks too fast in class, and the capacity of each class is too large and the requirements are too high. Some junior high schools have no knowledge and methods at all, and they are directly applied to senior high schools, which makes students very confused.

For example: 1. The cubic sum and difference formula has been deleted in junior high school, but the operation in senior high school is still in use.

2. Factorization in junior high school is generally limited to quadratic polynomials with coefficients of "1", rarely involving coefficients other than "1", and there is almost no requirement for factorization of cubic or higher polynomials, but it is widely used in senior high school textbooks, such as factorization to solve equations and inequalities, and applying factorization to reasonable deformation. After the third year of senior high school, most students still use the root formula to solve the quadratic equation of one variable, which is not only inefficient in solving problems, but also low in thinking level, so they will not use some root analysis with parametric equations. )

3. There is no requirement that both numerator and denominator are rational in middle school, but numerator and denominator are common problem-solving skills for high school functions and inequalities.

4. The requirements for quadratic function in junior high school textbooks are low, but quadratic function is an important content throughout senior high school. Formulas, sketches, defining domains, solving quadratic inequalities (unfamiliar to students), judging monotonous intervals, finding the maximum and minimum values, and studying the maximum value of functions in closed intervals are the basic questions and common methods that senior high school mathematics must master.

5. The relationship between quadratic function, quadratic inequality and quadratic equation, and the relationship between root and coefficient (Vieta theorem) are not required in junior high school. This kind of topic is limited to simple routine operation and application problems with little difficulty, and the mutual transformation of quadratic function, quadratic inequality and quadratic equation is an important content in senior high school, but no special lectures are arranged in senior high school textbooks.

6. The middle school only briefly introduces the symmetry and translation transformation of images, and after the teaching function in high school, the images are up and down; The symmetry of origin, axis and straight line must be mastered in left-right translation. Take the left-right translation of an image as an example. When students talk about the vertex of a quadratic function, they only feel the law of left-right translation through the change of fixed-point coordinates, but they don't really understand the essence of function translation. Take the left-right translation of a function as an example. Most students can't, and junior high school teachers can't speak! This does not belong to the examination content, which directly leads to the confusion of the relationship between f(x) and f(x+a) after high school, not to mention the combination of numbers and shapes.

7. Functions, equations and inequalities with parameters are not needed in junior high school, but only quantitative learning, and this part of senior high school is regarded as the key and difficult point. The comprehensive examination of equations, inequalities and functions often becomes a comprehensive question in the college entrance examination.

8. Many concepts (such as center of gravity, vertical center, etc. ) and theorems (such as the proportion theorem of parallel lines and line segments, projective theorem, intersecting chord theorem, etc. ) geometry is mostly not learned by junior high school students, but is involved in high school.

(2) Compared with junior high school mathematics, the presentation mode and thinking mode of senior high school mathematics have changed greatly.

1. In terms of presentation, the introduction of new knowledge in junior high school mathematics textbooks is closer to the reality of students' daily life, more vivid, and follows the law of rising from perceptual knowledge to rational knowledge, which is generally easy for students to understand, accept and master. At the beginning of senior high school mathematics, the concept is abstract, the theorem is rigorous, the logic is rigorous, the textbook narrative is rigorous and standardized, the ability of abstract thinking and spatial imagination is obviously improved, the knowledge is more difficult, and there are many types of exercises. In this way, it is inevitable that students will not adapt to high school mathematics learning.

2. The thinking method of high school mathematics is very different from that of junior high school. In junior high school, many teachers have established a unified thinking mode for students to solve various problems, such as solving fractional equations in several steps; Factorization depends on what you look at first, and then what you look at. Even for plane geometry problems with very flexible thinking, their respective thinking routines are determined for equal line segments and equal angles. Therefore, junior high school students are used to this mechanical and easy-to-operate stereotype, and even have a dependence psychology. Great changes have taken place in the thinking form of senior high school mathematics, and the abstraction of mathematical language puts forward high requirements for thinking ability. This sudden change in ability requirements has made many freshmen feel uncomfortable, leading to a decline in their grades. Of course, if we look at this problem dialectically, the sudden change of senior high school mathematics thinking mode conforms to the law of students' psychological development. Senior high school students are basically mature, and they also need to transition from empirical abstract thinking to theoretical abstract thinking, and finally need to form dialectical thinking initially. The key is how teachers guide students to achieve a smooth transition.

(3) The above two reasons lead to students' learning difficulties, thus changing their mentality, and even some students have the idea of breaking the jar and breaking the fall. In addition, the teacher's psychological counseling is not timely enough, and his self-adjustment ability is too weak, which leads to a vicious circle.

Third, some concrete suggestions on the implementation of junior high school mathematics convergence.

1. On the basis of fully understanding students' learning situation, write "cohesion teaching materials" and try to be targeted. In the process of implementation, it should be regarded as the real teaching content, which should not be underestimated! Of course, you can also gradually infiltrate according to your own needs!

2. Teaching in the first year of high school, try to have a low starting point, small steps, slow and stable slope; Consolidate the foundation and reduce the difficulty,

3. Strictly control the difficulty and mobilize the enthusiasm of each student to the maximum extent. After all, senior one is different from senior three, so it is necessary to cultivate students' good study habits step by step. The difficulty of each exam can be controlled at around 0.65.

3. Provide timely guidance and psychological counseling for senior high school mathematics, so that students can quickly adapt to the learning mode of senior high school mathematics.

4. Teachers should have a correct attitude, not be impatient, and be patient and meticulous in teaching concepts and methods! We should encourage students with learning difficulties in time. As I mentioned at the beginning, some students with learning difficulties may be hit like this for the first time in their lives and feel that they are "poor students in mathematics" for the first time. If the teacher encourages in time, it is likely to save many students who used to be brilliant but are now poor!

Appendix: What needs to be supplemented or strengthened?

The operation of 1. number and formula: the cubic sum (difference) formula is supplemented, and the binomial sum (difference) cubic formula (is the best contact point of binomial theorem, that is, the most advanced development field of binomial theorem. ), the derivation and application of the square formula of the sum of three numbers (positive and negative); Strengthen the operation and simplification of roots and fractions. (Quadratic root: appropriately supplement the equivalent operation. Such as overall operation, etc. )

2. Factorization: cross multiplication, grouping decomposition and addition and subtraction; Enhanced formula method. Cross multiplication and group decomposition. Very skilled. Especially cross multiplication, it is the fastest way to solve the quadratic equation of one variable, and of course, it is also the fastest way to solve the quadratic inequality of one variable. )

3. The reinforced discriminant of roots of quadratic equation with one variable and its application: it supplements the relationship between roots and coefficients of quadratic equation with one variable.

4. Solution of supplementary inequality: including one-dimensional quadratic inequality and its solution; Solving simple fractional inequality: solving inequality with absolute value.

5. The reinforced collocation method can find the fixed point and symmetry axis of quadratic function, strengthen the image and properties of quadratic function, and supplement the maximum problem of quadratic function in a given interval. This is a very important basic problem in the whole high school. It can be said that the solution of many comprehensive problems can eventually be transformed into the maximum value problem of quadratic function in a given interval. )

6. Supplement the distribution (interval root) of the roots of a quadratic equation with one variable.

7. Complement the simple solution of binary quadratic equation. The understanding of ternary linear equation and binary quadratic equation has been deleted from the mathematics textbook under the new curriculum standard of junior middle school. Of course, the basic idea of understanding the equation: elimination and reduction is deleted. These thinking methods are essential in high school, and the requirement of high school is that students can list them and solve them. )

8. The solutions of fractional equations and irrational equations that can be transformed into quadratic equations of one variable are supplemented (the junior high school textbooks delete the fractional equations and irrational equations that can be transformed into quadratic equations of one variable, and at the same time delete the idea of solving fractional equations and irrational equations with method of substitution; Deleted the important thinking method of fractional transformation into algebraic expression and unreasonable transformation into rationality).

9. Complement the definition and geometric properties of the triangle "four centers".

10. Complement theorems and properties related to plane geometry, including equal ratio theorem and combination ratio theorem; Proportional theorem of parallel lines; Theorem of bisector of triangle interior angle; Theorem of bisector of triangle outer angle; Projection theorem in right triangle; Properties of trapezoid midline.

1 1. Supplementary theorems related to circles: including quadrilateral inscribed in a circle and its property theorem, vertical diameter theorem, tangent angle theorem, chord truncation theorem and tangent theorem.

12. Complement the relationship between the side length, radius, vertex and central angle of a regular polygon inscribed (circumscribed) by a circle; Especially the relationship between the side length, radius, vertex and central angle of regular triangle, regular quadrilateral and regular hexagon.

(B) Need to supplement or strengthen mathematical thinking methods

The main mathematical methods are: (1) matching method (it plays an important role in high school, but it is also involved in junior high school, but students need to master the basic process of matching method skillfully).

(2) method of substitution (also one of the most basic mathematical methods) plays an inestimable role in solving mathematical problems. The training of this method has been greatly weakened in junior high school, but it is often used in senior high school mathematics.

(3) undetermined coefficient method (as a basic mathematical method, the requirements for junior high school are obviously reduced, and senior high school teaching can be systematically taught and trained). (4) reduction to absurdity.

Mathematical thoughts mainly include: function equation, combination of numbers and shapes, classification and discussion, and transformation.

Among them, the key contents of cohesion teaching are: cross multiplication, grouping decomposition, addition and subtraction decomposition to decompose factors; The relationship between roots and coefficients of quadratic equation in one variable: one-dimensional quadratic inequality and its solution: the solution of simple fractional inequality: the solution of inequality with absolute value; The maximum value of quadratic function in a given interval; Distribution of roots of quadratic equation in one variable: definition and geometric properties of "four centers" of triangle. The difficulty lies in: (1) factorization by adding and decomposing terms; Solving simple fractional inequality: solving inequality with absolute value; The maximum value of quadratic function in a given interval; Distribution of roots of quadratic equation in one variable: the theorem of bisector of triangle inner (outer) angle; Theorem related to circle and its application.