The knowledge system and its structure of high school mathematics have formed a relatively complete system. It can be seen from the guiding ideology and key points of the reform of mathematics textbooks in senior high schools that we should pay attention to the background, process, history, thought and culture of mathematical knowledge produced by problems in mathematics teaching, and finally implement it as an important link in the application of mathematical knowledge. Therefore, in mathematics teaching, teachers should cultivate students to form an overall understanding of mathematics from the basic ideas, methods, concepts and basic attitudes towards mathematics, so that students can form an overall cognitive structure of mathematics.

In order to make students have an overall cognitive structure of mathematics, improve students' mathematical ability, innovative consciousness and rational spirit, and focus on students' lifelong development, teachers should also conduct mathematics teaching from a systematic and holistic perspective. The author will talk about some thoughts based on teaching practice.

First of all, look for connections in the formation of mathematical knowledge from a holistic perspective.

If teachers can consider the problem from the perspective of overall mathematical knowledge and view the problem from the perspective of connection, they will find that the formation of basic mathematical knowledge often contains rich educational value, which is an important way to cultivate students' mathematical concepts, improve students' mathematical quality and form students' overall mathematical cognitive structure.

For example, the definition of "functional parity" in the new high school mathematics curriculum standard is put forward as follows: firstly, the concept of functional parity is introduced from the symmetrical phenomenon in students' daily life and two functional images about the origin and y axis symmetry respectively, and then their image features are transformed into algebraic features f (-x) = f (x) and f (-x) =-f (x), thus the definition of functional parity is obtained. This reflects the mathematical thinking method of changing "unknown" into "known", "shape" into "number", and the combination of shape and number is also in line with the cognitive law of students from familiar to unfamiliar, from special to general, from intuitive to abstract.

In view of this process, we can also think deeply from the overall point of view, and further ask such a question from how to stimulate students' cognitive needs: Why should we study the parity of functions? Why should we learn the definition of functional parity? How to embody the learning style of independent inquiry, hands-on practice and cooperative communication advocated by the new curriculum standard of senior high school mathematics? Therefore, on the basis of symmetry in daily life, teachers can let students observe their familiar proportional function f(x)=kx(k≠0), inverse proportional function f(x)=(x≠0) and quadratic function f(x)=ax+c(a≠0). The teacher then asked this question: What are the advantages of such a symmetrical function image? (Stimulate students' interest in thinking) It can be concluded that these function images not only have the beauty of morphological symmetry, but also can know the image on one side of the origin or Y axis to draw the image on the other side.

After introducing the image characteristics of function parity, teachers can let students judge the parity of the following functions first: ①f(x)=x, x∈[0, +∞); ②f(x)= x； ③f(x)= x+2x+； ④f(x)=。 For the function diagram of ①, students can easily answer; For the function image of ②, it is not difficult for students to draw an image by tracing method and then answer it. Students will find it difficult to draw functional images of ③ and ④. It can be explained that the use of image features to judge the parity of functions has its limitations. Even if some function images can be drawn, there will still be problems such as accuracy and visual reliability. Therefore, students can have cognitive conflicts, thus stimulating students to transform "form" into "number", intuition into abstraction, and sensibility into rational cognitive needs. In this way, it is more in line with students' psychological needs to further explore the definition of functional parity.

Through the above process, the old and new knowledge related to functions can be organically linked, on the one hand, it can stimulate students' cognitive needs, on the other hand, it can strengthen students' intuitive understanding of function parity, and at the same time pave the way for the formation of the definition of function parity, so that students can naturally master the image method and definition method of judging function parity. In this way, we can reveal and study the parity of functions from the overall perspective, and also enable students to form a complete cognitive structure for the parity of functions.

Second, from the overall point of view, find the connection in mathematics problem-solving teaching.

From the perspective of generalized mathematical knowledge, mathematical thinking method is universal and implicit knowledge in a certain range, the essence and soul of mathematical knowledge, the link for students to form a good mathematical cognitive structure, and the key to knowledge formation ability. Teachers should attach importance to the mathematical thinking method contained in the teaching of mathematical problem solving, guide students to seek the connection between mathematical knowledge from a holistic perspective in the process of discussing mathematical problems and their solutions, so that students can form a good mathematical cognitive structure and improve their mathematical ability through problem solving teaching.

For example, suppose that the maximum value of the function y=+ is m and the minimum value is m, then the value of is ().

A.B. C. D。

In the teaching to solve this problem, if the teacher only explains the method directly, the first one is: first, square the two sides of the function to get y=4+(-3≤x≤ 1), and then convert it into finding the maximum value of the quadratic function in a given interval; Solution 2: from -3≤x≤ 1, we can get 0≤x+3≤4, let x+3 = 4cosβ (β ∈ [0,90]), which is converted into finding the maximum value of trigonometric function; Solution 3: Let u= and v=, then u+v=4(u≥0, v≥0), u+v=y, and then find the maximum value by analytical method. It seems that the problem is easy to solve, but the students' reaction is still at a loss. What puzzled them was how the teacher thought of doing this. In order to avoid this phenomenon, teachers should pay attention to guiding students to explore the relationship between mathematical knowledge and mathematical thinking methods from a holistic perspective, reveal the essence of mathematical knowledge, and optimize and improve students' mathematical cognitive structure.

Therefore, teachers should further reveal the mathematical return thought and the combination of numbers and shapes embodied in the above-mentioned problem-solving process, so that students can understand the essence of problem-solving. At the same time, teachers can also look at the problem from a holistic perspective and guide students to further explore the above problems. For example, it can inspire students to think of using the derivative of the function to get the monotonicity of the function to find the maximum value; If we only seek the maximum value of this function, we can also inspire students to use Cauchy inequality and so on. Teachers can further put forward the following variant questions to make students think: (1), if the function is changed to y=+ or y=+, how to solve it? (it can be solved directly by its monotonicity); (2) If the function is changed to y= 1-x+ or y=x+ 1+, how to solve it? (the former can be set to t=≥0, which is converted into a quadratic function about t; The latter can be directly solved by its monotonicity. Therefore, the method of finding the maximum value of irrational function is organically linked into a whole.

Third, look for the connection in the process of mathematical inquiry from the overall point of view.

The new curriculum reform of senior high school mathematics advocates cultivating students' inquiry consciousness and rational spirit. Therefore, in mathematics teaching, teachers can guide students to explore confusing problems in mathematics learning. In the process of inquiry, teachers can guide students to find the connection between knowledge from a holistic perspective, which can enrich students' cognitive structure and create conditions for the formation of new knowledge networks.

For example, the concept of variance of random variables in high school mathematics is an extension of the concept of variance of a group of data in junior high school mathematics, and it is a quantity that describes the dispersion degree of random variables (a group of data) and mathematical expectations (the average value of a group of data). In this teaching, students can explore why the variance of a set of data x, x, …, x is defined as no. The idea of inquiry can be as follows: let f(x)=, when x==, f (x) =; Let g(x)= prove again that when ①n is odd, X is the median of data X, X, ..., X, ② When N is even, X has the minimum value of g(x). Therefore, it is best to describe the dispersion of data x, x, …, x with the average number, and it is best to describe the dispersion of data x, x, …, x with the median number. In the process of inquiry, teachers can appropriately introduce the statistical background of the famous least square method and least square method to students, so as to enrich their mathematical knowledge and enhance their interest in inquiry.

After inquiry, students can complete the following exercises: (1), and the function f(x)= the minimum value is () a.190 b.171c.90d.45.

(2) In the process of measuring a certain physical quantity, due to the errors of instruments and observation, N times of measurement respectively get X, X, …, X * * N data. The "best approximation" of the measured physical quantity X is such a quantity that, compared with other approximations, the sum of squares of differences with each data is the smallest, so it is stipulated that x=_______ is derived from X, X, ….

On the basis of the above inquiry, students can organically relate seemingly irrelevant knowledge, and it is easy to get: when x =10 (the median of12, 19) in the question, f (x) = 90; (2) x=(x+x+…+x) (that is, the average value of data x, x, …, x).

Through the above-mentioned inquiry process, the maximum knowledge of the function is linked with the error theory from the overall point of view, which deepens students' understanding of the concept of variance, broadens students' horizons, cultivates students' rational spirit, and enables students to learn to see problems from the perspective of connection and understand mathematical concepts from the overall point of view. In this way, students' understanding of mathematical knowledge is profound, and the essential things in mathematical knowledge can be obtained through the positive transfer of knowledge.

Fourth, from the overall point of view, find the connection between mathematics and mathematics.

Some mathematics educators once thought that the connection between mathematics and the outside world was more natural and important for students. The relationship between mathematics and its exterior is extremely extensive, mainly including the relationship between mathematics and other disciplines and the relationship between mathematics and real life. The new high school mathematics curriculum also advocates strengthening the communication and contact between mathematics and other disciplines and real life, so that students can appreciate the value and role of mathematics.

In the process of mathematics teaching, teachers can guide students to learn to think and solve problems in real life by mathematical thinking from a holistic perspective, and at the same time, they can interpret mathematical problems by using phenomena in real life, so that students can realize the similarity between mathematical knowledge and real life, thus cultivating students' associative consciousness and habits and cultivating students' innovative consciousness and creativity.

For example, in high school mathematics probability teaching, we can look for the connection between probability knowledge and real life from the overall perspective, create problem situations, and introduce new courses:

Teacher: In the streets of some big cities with relatively developed economy and high civilization, people often set up stalls to calculate predictions. The people who came to ask for the visa were ordinary people and intellectuals. Please think about the reason.

Student: There are different opinions.

Teacher: I think it satisfies people's psychological desire to predict the future, although many people know that asking divination is unscientific.

Teacher: We often hear people say that something is more likely to happen, so we will think about how likely it is. How to embody and depict this possibility?

Student: It would be nice if we could use concrete figures to reflect and describe the possibility of an event, because the data can explain the problem well.

Teacher: People are usually used to using numbers to explain problems, that is, quantitative analysis of problems, but it may be more difficult. But there is a kind of mathematical knowledge that can scientifically reflect this possibility with digital characteristics, and that is probability.

By guiding students to think in this way, we can cultivate their awareness of thinking in the connection between mathematics and other disciplines and between mathematics and real life, and form a natural habit.

In short, in the teaching process, we should think about teaching design from the overall perspective as much as possible, so that students can understand and learn mathematics from the overall perspective, instead of treating mathematics knowledge in isolation and artificially separating it. Looking at and understanding mathematics from the overall height will enable students to organically link their own mathematical knowledge, improve their mathematical cognitive structure, and thus improve their mathematical ability and quality.

References:

Ning Lianhua. Research on Teaching Design of Mathematics Inquiry [J]. Journal of Mathematics Education, 2006, 15 (4).

(2) Spencer Pan. Alienation in Mathematics Inquiry Teaching [J]. Journal of Mathematics Education, 2008, 17 (2).

(3) People's Republic of China (PRC) and the Ministry of Education. Mathematics Curriculum Standard of Ordinary Senior High School (Experiment) [M]. Beijing: People's Education Press, 2003.

(4) Tang Ruiguang. A proposition triggered by a new solution to a college entrance examination question [J]. Journal of Middle School Mathematics (High School Edition) 2008, 1 1