Block diagram of knowledge structure:
The simplest understanding of mathematics is that mathematics is "calculation", that is, "operation". "Operation" includes two aspects, one is the object of operation, and the other is the law of operation. Numbers, letters (algebraic expressions), exponents, logarithms, trigonometric functions, vectors and so on are all operational objects. "Associative Law", "a+(-a)=0" (that is, adding one item and subtracting one item), "commutative law" and various "distributive laws" are all algorithms. "Operation" permeates almost every corner of mathematics, which is the basic clue of mathematics and the main line of mathematics curriculum, and plays an irreplaceable role in senior high school mathematics curriculum.
1. Understanding of operation
Operation is a basic content of mathematics learning. The continuous expansion of operational objects is an important clue to the development of mathematics. Since primary school, students' contact operation has been expanding, from integer to fraction, from positive number to negative number, from rational number to real number and complex number, from digital mother to polynomial. Digital operation, letter operation, vector operation, function, mapping, transformation operation, matrix operation, etc. , are all mathematical operations.
The operation from numbers to letters is a leap in operation. The operation of numbers can be used to describe the quantitative relationship in specific problems and solve specific problems about quantity. Alphabetic operation can describe a kind of implication law problem and solve a kind of problem. For example, it describes a law of number operation-associative law. At the same time, letter operation is also a tool to express functional relations and depict universal laws. From number operation to letter operation, students' mathematics learning will undergo qualitative changes, and their understanding of operation will also make a leap.
From digital operation to vector operation, it is another leap in cognitive operation. An operation is a mapping. In algebra, the most common operation is such a mapping. It is a binary mapping, and the addition and multiplication of real numbers are both binary mappings. However, not all binary mappings are operations. In fact, most binary mappings are not operations, and only binary mappings that meet the rules can become operations, that is, algebraic operations. Numeric operations and polynomial operations are all types of algebraic operations. For example, the addition operation satisfies the law of association, has zero elements, and also satisfies the distribution rate. In junior high school, all mathematical contents are inseparable from operations, such as basic algebraic formulas, factorization, equations, inequalities, functions and so on. Vector can be "calculated". The characteristic of vector addition and subtraction operation is that two vectors get the third vector through addition and subtraction operation, which also satisfies the law of association and has zero elements, so the addition and subtraction operation of vectors belongs to type algebraic operation. The characteristic of vector multiplication is that a number and a vector are multiplied to get a vector, which satisfies a series of operation rules, such as:, distribution rate:, and so on. Therefore, the multiplication of numbers and vectors is also an operation, which is an algebraic operation of types; The characteristic of vector product is that two vectors get a number by sum of quantities. Similarly, it also satisfies a series of operation rules, such as distribution rate:? And so on, so the product of vectors is also an operation, which belongs to the algebraic operation of type. The operation of vector is different from the operation of number, which covers three types of algebraic operations. Compared with the operation of numbers, the operation of vectors expands the object of operation. Vector operation shows the characteristics and functions of three kinds of algebraic operations more clearly. At the same time, vector operation has some operation rules different from algebraic operation, which is fundamental for students to further understand other mathematical operations and enhance their operation ability. Therefore, from the operation of numbers to the operation of vectors, it is another qualitative change in students' mathematics learning, and students' understanding of operations will also be improved.
Exponential operation, logarithmic operation, trigonometric operation, derivative operation, etc. Formally, they are all types of mappings, but they meet some operational rules, such as exponential satisfaction. Usually, regular mapping is called "operator", also known as unary operation. For example, derivative operation is also an operation, and the derivative function satisfying the sum of two functions is equal to the first derivative and then summed. This is an algorithm, and of course it satisfies other laws. This is a leap in the understanding of operation.
In the future research, the object of operation will be further expanded. The study of the above operations lays a foundation for students to further learn other mathematical operations, understand the significance of mathematical operations and the role of operations in constructing mathematical systems.
Operation runs through a main line of the whole mathematics course. Understanding senior high school mathematics in this way is very useful for improving mathematics literacy and problem-solving ability.
2. The role of operation
(1) operation and reasoning
Operation itself is an important content of algebraic research. Professor Xiang Wuyi thinks that algebraic problems are solved by operation and operation rules, which makes sense. In a sense, in middle school, solving equation problems, solving inequality problems and studying the properties of some functions are all algebraic problems. The basic feature of algebraic problems is not only to prove under what conditions the "solution" exists, but also to construct the "solution" concretely. This is a structural proof, and operation and operation rules are the basic elements of algebraic reasoning. For example, when discussing binary linear equations, we should not only prove under what conditions there is no solution and there is a solution, but also construct a "solution" in detail; For another example, when a problem is proved by a vector, the result of the problem to be proved can be "calculated".
In the process of operation, every step of operation should be based on the algorithm, which is similar to the axiom in geometric proof and is the premise and basic basis of algebraic reasoning. The operation process itself is the process of algebraic reasoning. Therefore, there is a close relationship between operation and reasoning. It can be said that operation is also a kind of reasoning, and operation can "prove problems", which is the most important idea that high school mathematics learning needs to be "left to students". Therefore, operational research also plays an important role in students' logical reasoning ability.
(2) Operation and algorithm
In a sense, the algorithm is to solve the problem by computer, and the algorithm is realized by computer. The basic element of the algorithm is operation. The operations that computers can perfORm mainly include arithmetic operations AND logical operations (and, or, NOT, etc. ), relational operations (etc. ), function operation, etc. Therefore, the basic elements of the algorithm and the design of the algorithm should be based on operation and algorithm. The use of various operations and operation rules plays an important role in understanding algorithms, selecting algorithms and optimizing algorithms.
(3) Operation and identity deformation
In the process of solving mathematical problems, we need to carry out various identity transformations to turn complex problems into simple ones. For example, when solving a quadratic equation with one variable, we have achieved the purpose of reducing the power and changed the quadratic equation with one variable into a quadratic equation with one variable. The matching method is completed by identity transformation, which is realized by reusing operation rules. For another example, in the study of trigonometric functions, whether it is proof or solution, we are using various basic algorithms of trigonometric functions to carry out identity deformation, and through identity deformation, we can turn problems that we can't solve into problems that we can solve. Therefore, the research of operation and algorithm is very important for understanding the principle of identity deformation and improving the ability of identity deformation.
3. Job content design
In the high school mathematics curriculum, several parts focus on operation: exponential operation; Logarithmic operation; Trigonometric function operation; Vector operation, including plane vector and space vector; Complex number operation; Derivative operation; Wait a minute.
The contents of plane vector and space vector are arranged in compulsory 4 and elective 2- 1 in senior high school mathematics curriculum. In elective courses 1-2 and 2-2, the familiar contents of expansion and plural introduction are arranged; Related operations are also arranged in compulsory exponential functions, logarithmic functions and trigonometric functions, and derivative operations are arranged in elective courses 1 and elective courses 2.
Keeping the operation closed and establishing the basic algorithm is one of the driving forces of familiar expansion. For example, in order to keep the division closed, we are urged to extend the integer to the fraction; In order to keep the closure of subtraction, we are urged to extend positive numbers to negative numbers. Keeping the operations such as square root closed is one of the reasons to promote the expansion of real number system to complex number system. Every time we expand the number of times, we need to discuss: in the new number, does the new number keep the operation law of the original number? For example, from a positive number to a negative number, in order to maintain the distribution rate from multiplication to addition, we need to define:,,. Complex numbers keep the operation law of real numbers. However, real numbers are ordered and complex numbers are out of order.
There are some new algorithms in exponential, logarithmic and trigonometric functions. Mastering these special operation rules is the basis of understanding related mathematical concepts.
The most basic operation rule satisfied by exponential operation is that if the exponential function is expressed by, the above properties can be expressed as. This operation law shows that exponential operation turns addition operation into multiplication operation, which is the reason for the rapid growth of exponential function. The properties of exponential function, especially the growth of exponential function, are determined by this operation law. The arithmetic of exponential operation also includes:
(among them,
Logarithmic operation satisfies the most basic algorithm. If the logarithmic function is expressed as, that is, the above properties can be expressed as. This operation law shows that logarithmic operation turns multiplication operation into addition operation, which is the reason for the slow growth of exponential function. The properties of logarithmic function, especially the growth of logarithmic function, are determined by this operation law. The arithmetic of exponential operation also includes:
The algorithm explains the relationship between logarithmic operation and exponential operation, and extreme logarithmic operation and exponential operation are reciprocal operations. So exponential function and logarithmic function are reciprocal functions.
Trigonometric operation, taking sine operation as an example, satisfies the following basic operation rules:
The latter two operations are unique to derivative operations.
Learning the above operations and algorithms is helpful for students to understand the significance of operations and the importance of algorithms to operational research.