On the View of Mathematics Learning in Senior High School
Give it to Biya.
What is mathematics?
Diversity of Mathematical Definition of 1
Mathematics is a quantitative science Aristotle
Mathematics is the bacon of science, and the quantity it deals with is completely divorced from the axioms of matter and natural philosophy.
All the sciences that study order and measurement are related to mathematics, Descartes.
Mathematics is a scientific Engels who studies the relationship between spatial form and quantity in the real world.
Mathematics is a science dealing with infinity, Hilbert.
Mathematics is the study of abstract structures, bourbaki.
Mathematics is a scientific study of world relations.
Mathematics has always been called the science of patterns, and its purpose is to reveal the structure and symmetry observed by people from the abstract world of nature and mathematics itself.
Mathematicians have given different definitions in different periods of mathematical development. However, it can be seen that the definition of mathematics always follows the development of mathematics, which is one-sided, only reflects one or several aspects of mathematics, and there is always a feeling that the blind touch the elephant, which also illustrates the powerful vitality of mathematics from another side.
2 mathematical way of thinking
First of all, let's look at a little humor: one day, a mathematician felt that he had had enough of mathematics, so he ran to the fire brigade and wanted to be a fireman. The fire chief said, "You look good, but I have to do a test first." The captain took the mathematician to the back alley of the fire brigade. There is a warehouse, a fire hydrant and a hose in the alley. The fire chief asked, "Suppose the warehouse was on fire, what would you do?" The mathematician replied, "I connected the fire hydrant to the water pipe, turned on the tap and put out the fire." The captain said, "Exactly! One more question: What if the warehouse doesn't catch fire? " The mathematician thought for a long time and replied, "I'll just set the warehouse on fire." The fire chief shouted, "That's terrible! Why? " The mathematician said, "I always turn a problem into a problem that I have solved." Isn't it interesting? Think about it. This is the way of thinking in mathematics. The thinking methods in mathematics include set correspondence, logical division, functional equation, combination of numbers and shapes, etc. Its core idea is transformation, and the specific method is reduction. Combined with examples, example 1 cut a triangular iron plate into a square iron plate as large as possible. How to cut? Firstly, it is simplified to a geometric problem: finding the largest inscribed square of a known triangle. In order to find this square, let's analyze the relationship between the square and the known triangle:
Let BC= a, height AD= h, square side length x,
Then = that is, x=
This can make an inscribed square EFGH.
According to the nature of a triangle, if the three sides are A, B and C, and the three high lines are all 0, and a≤ b ≤c, then a = b = c =2s.
a + ≤b+ ≤c+
Therefore, the inscribed square on the smallest side has the largest area, and its thinking process is:
Mathematical algebra
Find a solution
As can be seen from the above example, we turn the problem to be solved into another problem, and then apply the solution result to the original problem by solving the latter problem, so that the original problem can be solved. This method of solving problems is collectively called reduction method. Judging from the process of thinking, it is an "aging" process, which boils down unfamiliar problems to well-known problems that have been mastered and finally solved through a series of aging.
Professor Wang Qiuhai of Siping Normal University redefines mathematics from its core ideas, leading methods and subject objectives: mathematics is based on transformation into basic ideas, with reduction as the main method, and mainly studies quantitative and modeled science. This definition reflects the development direction of mathematical definition research, clarifies the basic thinking mode of mathematics, and puts forward the basic essentials that should be grasped in learning mathematics, which is of great practical significance for students to establish a correct view of mathematics learning.
Second, the requirements of mathematics learning ability
Because of the particularity of mathematics itself: highly abstract, strict logic and widely used. Decided that mathematics learning has the following ability requirements.
1 ability to explore and discover
Senior high school students mainly rely on indirect experience, mainly through the guidance of teachers, to understand the truth discovered by predecessors. Therefore, in the process of learning, middle school students study in the spirit of exploring and discovering the truth, and regard learning activities as a kind of creative labor and the joy of continuous success. For example, the relationship between quadratic equation of one variable and coefficient, proportion and golden section, Pythagorean theorem, Yang Hui triangle and binomial theorem, ancestor principle and the volume of a ball, Euler formula and topology, binary and computer, mechanical motion and analytic geometry, there are many interesting knowledge that we need to explore and discover in the study of middle school mathematics. In this process, we need to explore and discover the ability, and at the same time, we will also cultivate this ability.
2 abstract generalization ability
Mathematics is highly abstract and must be accompanied by a high degree of generality. Especially, highly generalized symbolic languages are used, such as:,, f (x) ξ ~ n (μ, σ) and so on. In the process of learning, students should always adhere to the thinking methods from concrete to abstract, from special to general, attach importance to the thinking methods such as experiment-induction-analogy-association, attach importance to the occurrence process of knowledge, and clarify the mathematical thinking methods involved in this process.
3 Logical reasoning ability
The concept theorems and laws of mathematics are developed under the logical system, and each branch forms a scientific system through deduction and axiomatization, forming a well-structured logical system. This characteristic determines that mathematics learning must have strong logical reasoning ability. Therefore, in the process of learning, we must first have a clear understanding of the concept, so that our thinking can be based. Secondly, it is necessary to clarify the process of knowledge generation and master logical reasoning methods through the proof of theorems or the derivation of formulas.
Three principles of mathematics learning
1 initiative principle
Hua is a famous mathematician, born in a poor family, and had a hard life as a teenager. He was unable to pay his tuition, dropped out of school and suffered many misfortunes and blows. He was even seriously ill in bed for half a year and hurt his leg. But he still overcame one difficulty after another with amazing perseverance, insisted on self-study and became a famous mathematician. This fact is enough to illustrate the importance of being proactive in mathematics learning. Learning should be an active and purposeful activity. Through continuous exploration and discovery, we strive to cultivate a positive attitude towards life, face difficulties bravely, and then face life bravely.
2. The principle of combining learning with thinking
Learning without thinking is useless, thinking without learning is dangerous. Learning is to receive and store; Thinking is comparison, judgment and processing, which are transformed into each other, just like an endless spiral. The combination of learning and thinking requires students to combine accepting knowledge with consolidating knowledge and combining learning with thinking. Students are required to consciously be diligent in thinking, be good at thinking, be brave in innovation, strive to independently know and understand mathematical knowledge, and independently discover and solve mathematical problems.
3 the principle of gradual progress
The famous physiologist Pavlov's hope for young scientists is: step by step, step by step, and then step by step. Mathematics is a highly logical and systematic discipline, and its contents are interrelated and internally regulated, forming a strict logical system. Generally speaking, the former knowledge is the basis of the latter knowledge, which is the inevitable development of the former knowledge. Therefore, students should gradually deepen their understanding from known to unknown, from simple to complex, from concrete to abstract, from perceptual knowledge to rational knowledge.
4 the principle of timely feedback
The principle of timely feedback requires students to check the learning situation in time, find out what knowledge and methods are misunderstood, and take corresponding measures to solve the problem in time. Students' self-feedback is conducive to improving their confidence and interest in learning, enhancing their ability to distinguish and improving their learning efficiency.
Four methods of mathematics learning
The learning activities of middle school mathematics mainly include three aspects: the learning of mathematical concepts, the learning of mathematical propositions (theorems, formulas, rules and properties) and the learning of mathematical problem solving. We mainly study concepts, propositions and basic methods of solving problems.
1 senior high school mathematics concept learning
Concept is the starting point of mathematics learning, the basis of thinking, the basis of solving problems and the basis of reasoning. It is very important to master and apply mathematical knowledge and form correct concepts. How can we learn mathematical concepts well?
(1) Understand the concept deeply
Understand concepts through their formation. The concept of mathematics is mainly introduced through the calculation of examples and models. Strengthening the understanding of concept formation can enhance the intuitive effect and help to understand the concept correctly.
Understand concepts through hierarchy. When learning concepts, we should analyze them from three levels: literal language, symbolic language and graphic language. For example, the theorem in solid geometry can be understood in different levels.
Understand this concept through its variants. For example, the geometric concept should draw its variant figures (standard and non-standard), and the deformation application of formulas in trigonometric functions.
Understand concepts by comparison. Such as the contrast between old and new concepts (exponent and exponential function); Comparison of opposing concepts (exponent and logarithm, function and inverse function); Comparison of similar concepts (line angle, line angle, dihedral angle), etc.
Understand the concept through the concretization of special cases. For example, the image of trigonometric function is used to understand the periodicity of function, and the image transformation of trigonometric function is used to understand the image transformation law of abstract function. Logarithm and general logarithm are often used.
(2) firmly grasp the concept
Grasping concepts firmly is the premise of flexible application of mathematical concepts. Incorrect understanding and confusion of concepts can easily lead to problems or mistakes in solving problems. If the product of two non-zero vectors is less than zero, it is concluded that the included angle of two non-zero vectors is acute, which obviously leads to the misunderstanding of the concept of vector included angle. The intercept of a straight line on two axes is equal and the slope of a straight line passing through a fixed point (2, 1) is equal to 1 or-1. The above solution confuses the two different concepts of intercept and distance. The slope of the straight line is k and the inclination angle is α. If k ∈ (- 1, 1) is α ∈ (-,), the above solution has a wrong understanding of the concept of inclination of a straight line. There are many examples like this, which mainly reflect that the concept is not firmly grasped and mistakes are not allowed.
(3) Flexible use of concepts
The understanding of the concept can not be achieved overnight, but should be deepened and improved in application. The process of solving problems is the process of applying concepts, which needs to be accumulated in learning. For example, to find the distance between two points on a sphere, you need to use basic knowledge such as arc length formula of out-of-plane straight line cosine theorem of dihedral trigonometric function. For another example, the length of line AB is 4, and the sum of the distances from point P to both ends is 6. Find the maximum distance from point P to point M in AB. If the triangle is solved by cosine theorem, it can be solved, but the operation is complicated. If we use the definition of ellipse, the geometric meaning of ellipse can be easily solved.
2 Mathematics proposition learning
The study of mathematical propositions mainly focuses on the properties of formulas, theorems and laws, and the following aspects should be paid attention to in the study:
(1) Introduction to Attention Proposition
Paying attention to the introduction of propositions means paying attention to the process of knowledge generation, the process of discovery and exploration, and a good opportunity to cultivate the ability to solve problems. Mathematics comes from the spatial form or quantitative relationship in the real world. For example, the size of clothes and shoes, the load of trucks, all involve the order problem, and the installment payment problem also involves the order problem. Paying attention to these can also arouse our interest in learning and help us understand and remember the conclusion.
(2) Deduction of attention proposition
After introducing a proposition, we must prove it. The proof process often contains important mathematical ideas, and mastering its derivation is helpful to form skills. For example, the "dislocation subtraction" used in the derivation of geometric series summation formula can be used for the summation of other series. For example, there are many formulas in trigonometric function and many variants in formula derivation, which can help us improve our ability to apply formulas and use formulas to deduce formulas.
(3) Pay attention to the conditions for the proposition to be established
Any mathematical proposition can always be used in a certain range, and the proposition and its establishment conditions are inseparable. The biggest weakness of learning is to mechanically apply formulas as "universal formulas".
3 Middle school mathematics problem-solving learning
The so-called problem solving is to reveal the internal relationship between conditions and conclusions. Different problems have different solving skills, but there is always a universal law-transformation, that is, the solution of mathematical problems lies in the change of form. When solving problems, our thinking activities are generally carried out in accordance with the steps of "observation-association-transformation", and many problems will appear in the process of thinking. The following are a series of questions:
Step by step thinking program
Observation 1 What problem to solve? What kind of question is it? 2 What are the known conditions (data graph and its connection with the conclusion)? What conclusion is needed (unknown substance)? What are the characteristics of the given graphs and formulas? Can you express this problem with graphs (geometric, functional) or mathematical formulas? 4 What are implicit conditions?
Have you seen Lenovo 1 before? Have you ever seen this problem done before? What did you think at that time/have you seen some of the three questions (conditions, conclusions, formulas, figures) before? What question did you see? Which formula subgraphs in question 4 are similar to those in memory? What is the possible connection between them? There are usually several ways to solve this kind of problem. Maybe this method is simple. How about a try? 6 What known conditions can be deduced from known conditions? What do you need to know to ask an unknown conclusion? 7 What is the knowledge (concept theorem formula) related to this problem?
Can the transformation of 1 simplify the complicated formula in the problem? Can we divide the conditions and turn a big problem into several small problems? 3. Can we carry out variable substitution identity transformation or geometric transformation to make the form of the problem more obvious? Can 4-Form Number Reciprocity Solve Algebraic Problems by Geometric Method? Solving geometry problems by algebraic method? 5 Use the equivalent proposition (negative proposition) or other methods to turn the problem into a familiar one. The ultimate goal is to turn the unknown into the known.
The above question list gives the general direction, and students still have a lot to do. There is still a lot of work to be done to solve mathematical problems, as Paulia, a master of problem-solving research, said: an adequately supplied and well-organized knowledge base is an important capital for a problem solver. I hope that everyone will strive to do the following:
1 master the basic knowledge system of mathematics.
Understand mathematical concepts, theorems, formulas and rules.
Familiar with basic logic rules and common problem-solving methods, and accumulate problem-solving skills.
Master common thinking methods, such as observation, experiment, induction and deduction analogy analysis, comprehensive abstract generalization.
After solving the problem, you should reflect.
Bibliography: Introduction to the History of Mathematics, Li Wenlin Higher Education Press.
Mathematical thinking method Deng Henian Wang Yuqi Zhang Weisheng Wang Qiu Haiwang Xianchang Jilin University Press
Introduction to Mathematical Problem Solving, Luo Zengru Shaanxi Normal University Press.