The meaning of mathematics.

1. Mathematics is the core for human beings to explore the world and study anything in nature.

2. Mathematics gave birth to physics, chemistry and biology, and mathematics constantly promoted the development of mankind;

3. Mathematics is the fulcrum of axioms and conventions. With mathematics, research can continue;

4. Mathematics deduces two-dimensional, three-dimensional and high-dimensional, which are the basis of these things.

First, what is the use of middle school mathematics?

1, what do you study in junior high school mathematics?

We take the current junior high school mathematics textbook (system 63) as an example:

Grade 7 (1): rational number; Addition and subtraction of algebraic expressions; One-dimensional linear equation; Preliminary geometry;

Grade 7 (below): intersecting lines and parallel lines; Real number; Plane rectangular coordinate system; Binary linear equation; Unequal and unequal groups; Collection, arrangement and description of data;

Level 8 (1): triangle; Congruent triangles; Axisymmetric; Algebraic expression multiplication and factorization: fraction;

Grade 8 (below): secondary radical; Pythagorean theorem; Parallelogram; Linear function; Data analysis;

Grade 9 (1): a quadratic equation; Quadratic function; Rotating; Round; Preliminary probability;

Ninth grade (below): inverse proportional function; Similar; Acute angle trigonometric function; Projections and views.

The contents of these six books can actually be rearranged into three modules according to the research content.

Algebra module: rational number; Addition and subtraction of algebraic expressions; One-dimensional linear equation; Real number; Plane rectangular coordinate system; Binary linear equation; Unequal and unequal groups; Algebraic expression multiplication and factorization: fraction; Quadratic radical; Linear function; One-dimensional quadratic equation; Quadratic function; Inverse proportional function.

Geometry module: preliminary geometry, intersecting lines and parallel lines; Triangle; Congruent triangles; Axisymmetric; Pythagorean theorem; Parallelogram; Rotating; Round; Similar; Acute angle trigonometric function; Projections and views.

Statistical module: data collection, arrangement and description; Data analysis; The possibility is preliminary.

The sudden increase in the difficulty of mathematics is usually in the second semester of junior high school. In this period, whether geometric proof or algebraic simplification, problem-solving requires high pattern recognition and skills, and students need certain training, which is boring; At the same time, some observation is needed. It was at this stage that the grades started, and many students lost interest in mathematics.

2. What do you study in high school mathematics?

Original New Curriculum Standard High School Textbook:

Compulsory part:

Compulsory 1: set; Functions (concepts, properties, linear functions, quadratic functions); Basic elementary function I (exponential function, logarithmic function and power function)

Compulsory 2: Preliminary solid geometry (space geometry, positional relationship); Preliminary analytic geometry (plane Cartesian coordinate system, linear equation, circle equation, space Cartesian coordinate system)

Compulsory 3: preliminary algorithm; Statistics; Probability; possibility

Compulsory 4: Basic elementary function II (trigonometric function); Plane vector; Angle conversion formula

Compulsory five: solving triangles; Sequence; inequality

Elective course 1 series (liberal arts):

Elective course 1- 1: common logical terms; Conic curves and equations; Derivative and its application

Elective courses 1-2: statistical cases, reasoning and proof, number series expansion and complex number introduction, block diagram.

Elective 2 Series (Science):

Elective course 2- 1: common logical terms, conic curves and equations, space vectors, solid geometry.

Elective 2-2: Derivative and its application, reasoning and proof, number system and the generalization of complex numbers.

Elective 2-3: Counting Principles, Probability and Statistical Cases

Other elective courses

3- 1 History of Mathematics, 3-3 Spherical Geometry, 3-4 Symmetry and Group Theory, 4- 1 Selected Lectures on Geometric Proof, 4-2 Matrix and Transformation, 4-4 Coordinate System and Parametric Equation, 4-5 Inequality Selected Lectures, 4-6 Elementary Number Theory, 4-7 Optimization Method and Experimental Design, 4-7

The college entrance examination questions in many provinces are selected from three parts: 4- 1 geometric proof, 4-4 coordinate system and parameter equation, and 4-5 inequality. It should be said that it is more suitable for the study of higher mathematics in colleges and universities, but it is still a little regrettable that the matrix is not selected.

New Curriculum Standard New Textbook for Senior High School

Mandatory version 1 * * * two volumes:

Volume 1: Set and Common Logical Terms; Univariate quadratic functions, equations and inequalities; The concept and properties of function; Exponential function and logarithmic function; trigonometric function

Volume two: plane vector and its application; Complex number; A preliminary study of solid geometry: statistics; Probability; possibility

Mandatory version b * * * Four volumes:

Volume 1: Set and Common Logical Terms; Equality and inequality; Function;

Volume 2: exponential function, logarithmic function and power function; Statistics and probability; Plane vector preliminary

Volume 3: trigonometric functions; Vector product and trigonometric identity transformation;

Volume 4: Solving Triangle; Complex number; A preliminary study on solid geometry

Optional compulsory * * * Three volumes:

Volume 1: space vector and solid geometry; Equations of straight lines and circles; Conic curve equation

Volume II: Sequence; Derivative of unary function and its application

Volume III: Counting principle; Random variables and their distribution; Statistical analysis of paired data

To sum up, high school content can also be roughly summarized into three modules:

Function and Algebra Module: Set and Common Logical Terms; The concept and properties of function; Elementary function (exponential function, logarithmic function, power function, trigonometric function including trigonometric identity transformation); Plane vector (preliminary of plane vector, quantitative product of vector, triangular solution); Equality and inequality; Sequence; Derivative of unary function and its application

Geometry module: 1) solid geometry-space geometry; Spatial positional relationship; Space vector and solid geometry; 2) Analytic geometry-rectangular coordinate system; Equations of straight lines and circles; Conic curve equation

Probability statistics module: statistics and probability (data collection, characteristics and representation, sample estimation population; Random events and independence, classical probability); Counting principle (permutation and combination, binomial); Random variables and their distribution (random variables and conditional probability); Statistical analysis of paired data (correlation and regression)

3. The connection between middle school curriculum and university curriculum:

According to different research objects, mathematics can be divided into four simple parts:

The research object of algebra is algebraic structure and algorithm;

The research object of geometry is the change of graphic attributes and spatial relations;

The research object of analysis is the nature of function, that is, the relationship between variables;

The research object of number theory is the properties of integers.

It is not accurate because as a category, all parts of mathematics are closely related, and various professional fields learn from each other, so it is difficult to distinguish them strictly. For example, function images in junior high school mathematics, trigonometric functions, analytic geometry and vectors in senior high school mathematics are typical manifestations in this respect.

Generally speaking, if you are not a college student majoring in mathematics, the most important mathematics courses in the undergraduate stage are advanced mathematics, linear algebra, probability theory and mathematical statistics, which are also the main contents of postgraduate mathematics. Advanced mathematics belongs to the category of analysis, linear algebra belongs to the category of algebra, probability theory and mathematical statistics belong to the category of applied mathematics, but both need analytical tools and algebraic tools. Geometry and number theory are generally only studied by mathematics department and a few majors.

Middle school mathematics knowledge is the basis of learning college mathematics knowledge and the significance of learning middle school mathematics. Let me sort out the relationship between middle school mathematics knowledge and how they form the basis of college mathematics learning.

Let's talk about algebra and analysis first:

When we were in primary school, all the calculation problems we did were the operation of numbers, and the result was a number, so we all learned the arithmetic of numbers. In the senior grade of primary school, we began to learn to use letters to represent numbers, which is called algebraic expression.

Algebra was introduced to China by Li Yijie, a mathematician in the late Qing Dynasty, with the meaning of "substituting words for mathematics". Algebraic expression is a transformation of language system. We can construct the formula in this way, generalize the operation and get the general solution. When faced with a specific problem, a specific number can be substituted into the formula to solve the problem; The purpose of algebraic research is to find the general solution. In 820 AD, the Persian mathematician Huala Mozi published a monograph in the field of algebra, expounded the general solutions of the first and second equations, clearly put forward some basic concepts in algebra, and developed algebra into an independent discipline comparable to geometry. Al jabr is used for the first time in the title of the book, which means "regrouping", that is, moving items and merging similar items. After being translated into Latin, it became algebra, and later it entered English. This is the etymological meaning of the word "algebra"

After the introduction of algebraic expression, the expansion of number system appeared. With the increasing complexity of digital processing, four operations of addition, subtraction, multiplication and division can't get the result of natural numbers. A-B (1

Then I began to learn the addition, subtraction, multiplication and division of algebraic expressions (algebraic expressions with letters without denominators, including monomials and polynomials), and learned the inverse operation of algebraic multiplication-factorization, that is, how to transform a complex polynomial into a simple polynomial multiplication; And from another main line, we also learned the whole equation, that is, one-dimensional linear equation, two-dimensional linear equation and inequality. Algebra can also be divided into fractions, and he can also be a fractional equation. However, when solving the quadratic equation with one variable, we encounter the problem of roots. This operation is different from the four operations, and the result is not necessarily rational, so we accept the existence of irrational numbers and extend the number system to real numbers. There are some special arithmetic laws in the root operation, for example, negative numbers cannot be squared, and this law should also be observed in algebra. This is the root. With these foundations, the problem of quadratic equation in one variable can be solved, and we get the general solution of quadratic equation in one variable-the formula for finding the root.

After learning the basic operations (addition, subtraction, multiplication, division and square root) and equations, the language system of mathematics has reached a new level by introducing functions. The main task of analysis is to study and apply functions. As the most important concept in modern mathematics, the importance of function cannot be overemphasized. Everything in the world is generally related, but the traditional natural philosophy has made a qualitative analysis of this connection: for example, if you heat it with fire, the temperature of water will rise; The greater the force, the longer the spring stretches; However, modern science needs to quantitatively analyze this relationship and find the universal law of the relationship, which requires the use of function tools. The formula Q=Cm(T2-T 1) in junior high school physics, N=k(x-x0) in spring force and F=GMm/r2 in senior high school physics are essentially the products of this quantitative research with the help of functional tools. Function is the core knowledge of middle school mathematics. The application of middle school functions is basically to solve equations and inequalities, while high school mathematics is almost based on function theory except some geometric and statistical knowledge.

High school mathematics first introduced set language, which led to later function definition. Set theory is the cornerstone of all branches of modern mathematics, but it is hardly used in high school mathematics, just need to be able to perform simple set operations. Then there are the general properties of functions, such as monotonicity and parity, the special properties of elementary functions (exponential function, logarithmic function, power function and trigonometric function) and a special function with positive integer as independent variable and real number as dependent variable-sequence, that is, real number. Trigonometric function leads to plane vector, and the vector algebra reflected by its algorithm is also a big leap in mathematical language: we find that we can not only operate numbers and algebras, but also operate orderly numbers and algebras. Then there is inequality. You may wonder why you need to learn such complicated inequalities, but when you study real mathematical analysis in college, you will know that inequality proof skills are essential skills for learning mathematical analysis. After laying a good foundation, I began to learn limits and derivatives, and high school mathematics came to an abrupt end. Function, sequence, inequality and derivative are the most difficult parts of high school mathematics, and they are also the basis of higher mathematics. The last question in the college entrance examination is basically the comprehensive application of function, sequence, inequality and derivative.

In the university, this part is the famous advanced mathematics, and most of it is calculus. Learning mathematical analysis for mathematics majors means learning calculus with a more rigorous argumentation system. However, whether it is a high number or a fraction, the functions studied are relatively intuitive, and they are basically continuous functions, or Riemann integrable functions. However, the real function that does not meet the above conditions needs to be studied based on the real variable function theory of set theory, measure theory and Lebesgue integral. On the other hand, the variables of a function are not all real numbers. If the variable is a complex number, it should be studied by the discipline of complex variable function or complex analysis. Besides numbers, independent variables can also be functions. The function of function is called functional, and the theory of studying functional and infinite dimensional space transformation is called functional analysis, which is more abstract mathematics than real analysis and complex analysis. In addition, calculus can also be used in equations, and the field of studying how to solve equations containing calculus is called differential equations. Among them, the calculus of studying one function is called ordinary differential equation, and the calculus of studying multiple functions is called partial differential equation. All disciplines in the field of analysis are closely related to the study and research of theoretical physics.

The plane vector and space vector in senior high school are mainly used to lay the foundation for solving triangles and solid geometry proofs, and are more suitable as geometry modules from the perspective of application. After learning plane vector and space vector, the content of middle school algebra came to an abrupt end. When I arrived at the university, the linear equation returned to my field of vision. Because the image of a linear function is a straight line, linear equations are also called linear equations, and linear algebra begins with the study of the general solution of linear equations. Using N-ary vector, matrix and determinant, we finally get the general solution of linear equations-Kramer's law (but later we will know that the calculation of determinant is very complicated, and Kramer's law is far less useful than gauss elimination's. Linear algebra and advanced algebra only take linear equations as an introduction, which leads to the core of linear space. The task of solving linear equations is handed over to the numerical algebra course of computational mathematics. At the same time, the objects of our operation are also extended to vectors and matrices; We find that these operations are very similar and have similar structures. Mathematicians further abstract them into linear spaces, and take studying the properties and transformations of linear spaces as the main task of linear algebra. The three-dimensional space that we can intuitively feel is a special form of linear space. In order to study this special form, bilinear function and quadratic form are introduced to obtain inner product operation, and then linear space is specialized into metric space, so that linear space theory can be used for geometric research or solving practical problems. Linear space is the simplest research object of algebra. In addition, the research objects of algebra also include groups, rings, fields and so on. Subsequent courses to study these objects and their properties are called abstract algebra or modern algebra. We need to use the knowledge of abstract algebra to prove three unmodeled problems in junior high school geometry: angle trisection, cubic product and turning a circle into a square. High school electives are 3-4 symmetry and group, 4-2 matrix and transformation, which correspond to group theory (abstract algebra) and matrix algebra (simple linear algebra) respectively. You can read them in your spare time.

Then let's talk about geometry:

The English word for Geometry is geometry, Geo- is the root of "earth" and -measurement is the root of "measurement". Geometry directly means "land survey". Geometry originated in ancient Egypt, because the Nile River in Egypt floods periodically every year, bringing a lot of fertile soil, but the boundaries of the land will also be washed away. Therefore, the ancient Egyptians re-measured the land every year, and the measurement technology summarized in long-term practice gradually developed into the original geometry.