So how can we learn high school mathematics effectively?
More than 2,000 years ago, Confucius said, "Knowing is not as good as being kind, and being kind is not as good as being happy." Here, "good" and "happy" are learning interests. How can we cultivate interest in learning?
1. 1 Understand the importance of mathematics
Mathematics is known as the queen of natural science. In modern society, digital trends such as digital campus, digital hospital, digital TV and digital survival make the slogan of "popular mathematics" sweep the world. Some people think that future jobs are for those who are ready to study mathematics. "Preparing for mathematics" here is not only to understand mathematical theory, but more importantly, to use mathematical knowledge flexibly to solve practical problems in life. For example, strengthen the connection between mathematics and traditional physical and chemical disciplines, and strengthen the application of mathematics in economy, management and industry.
1.2 Appreciating the Beauty of Mathematics
When it comes to beauty, people are most likely to think of beautiful natural scenery, or pleasing pictures, beautiful music, exquisite poems and so on. These are all artistic beauty. However, mathematics, the queen of natural science, contains a more beautiful realm than poetry and painting.
The famous philosopher Russell said: "Mathematics, if viewed correctly, not only has truth, but also has supreme beauty." Indeed, where there is mathematics, there is beauty. When we step into a mathematical world full of vitality and beauty, the extensive abstraction and application of theory, the rigor of logic and the coordination of structure, the symmetry and harmony of form, the richness and profundity of content, the beauty and strangeness of methods, etc. , all give people the enjoyment, encouragement and pursuit of beauty. There are many forms of mathematical beauty.
1.2. 1 concise and beautiful
Einstein said, "Beauty is simple in essence." . He believes that only with the help of mathematics can we achieve simple aesthetic standards. Simplicity and simplicity are its external forms. Only simple, delicate and profound can be called the most beautiful. For example, formulas such as circle circumference, sector area and Pythagorean theorem all contain the beauty of simplicity in mathematics.
1.2.2 formal beauty
(1) The Beauty of Words in Mathematics
Characters in mathematics are the universal language in the world, ranging from 0 to 9, from "+,-,×, ⊙" in four operations to ",=" with relatively large size, as well as square brackets (), brackets [], braces {} and so on, which change the operation order. These words are of moderate size, symmetrical up and down, and written in the same way.
(2) the beauty of symmetry in mathematics
The beauty of symmetry in mathematics reflects the harmony of nature, which is widely used and manifested in middle school mathematics. Among geometric figures, the most typical one is the axisymmetric figure. Round, square, equilateral triangle, isosceles trapezoid, etc. They are all axisymmetric figures, and the circles with numerous symmetrical axes are known as "the most beautiful figures on the plane".
1.2.3 harmonious beauty
In geometry, squares, isosceles triangles and circles are beautiful pictures; The triangle is the epitome of the pyramid, and the circle is the symbol of the sun. The vivid fan-shaped, plum-petal-shaped combination figure and the copper coin-shaped round China edge all show the harmonious beauty of geometric figures.
1.2.4 rigorous beauty
Rigorous beauty is the unique inner beauty of mathematics, and we usually use "watertight" to describe mathematics. It is manifested in the rigor of mathematical reasoning, the accuracy of mathematical definition and the coordination of mathematical structure.
2. Learn the spirit of delving into mathematics
Most of the contents of junior high school mathematics are perceptual and concrete, which also conforms to the age and psychological characteristics of junior high school students, so imitation is the main way to learn mathematics. In senior high school, after the implementation of the new curriculum reform, teachers have increased a lot of inquiry teaching, infiltrated a lot of mathematical thinking methods, and paid attention to the analogy, so learning mathematics is mainly based on imitation and independent inquiry. In the process of independent inquiry, it is necessary to cultivate the spirit and patience of research. You can train patience by increasing the time you spend doing math problems. For example, it usually takes 5 minutes to give up the fill-in-the-blank question temporarily, and it usually takes 10 minutes to give up the answer temporarily. After all the exercises are finished, do the questions you didn't do just now again. If you have no ideas, ask your classmates or teachers again. Don't give up easily.
3 thinking about the process of effective learning
3. 1 Do a good job in pre-class preparation to improve the pertinence of lectures.
"Know yourself and know yourself, and you will win every battle". Preview is to prepare for "Know yourself and know yourself". High school math class has a large capacity, so it is very necessary to preview before class. The difficulties found in the preview should be marked out, and those are the key points of the lecture; It is necessary to make up the old knowledge that has not been mastered in the preview in time to reduce the difficulties in the course of attending classes. This will not only help to improve the efficiency of class, but also improve your thinking level and self-study ability.
3.2 Listen attentively and focus on taking notes.
Pay attention to every concept and example in class, and pay attention to the beginning and end of the teacher's lecture. The beginning is generally to summarize the main points of the last lesson and point out the content to be talked about in this lesson, which is a link to link old knowledge with new knowledge. The conclusion is often a summary of a lesson's knowledge, which has a strong generality and is an outline for mastering the knowledge and methods in this section on the basis of understanding. If there is a conflict between taking notes and attending class, attend class first and make up your notes through after-class memories.
3.3 Review the knowledge of the day and repeat it twice in three days.
Ebbinghaus, a famous German psychologist, made a famous experiment and got the following conclusion: The knowledge just learned.
time span
Overmemory
In 20 minutes.
58.2%
1 hour later
44.2%
After 8-9 hours
35.8%
1 days later
33.7%
Two days later.
27.8%
Six days later
25.4%
1 month later
2 1. 1%
These data tell us that the speed of forgetting is fast, first fast and then slow. After one day, if you don't review what you have learned, you will only have the original 33.7%. Therefore, you should review on the day you finish learning. We should take retrospective review, recall what the teacher said in class without reading books or taking notes, then open the notes and books, compare what we didn't remember clearly, and make up again, so as to check the effect of listening in class that day and prepare for improving listening methods and improving listening efficiency. Review the same content twice in three days, which is referred to as "three-day twice review method"
3.4 Finish the homework independently and sort out the wrong math questions.
Tolstoy said: "Knowledge is true only when it is acquired through positive thinking". To turn them into their own knowledge and make them use it freely, they must be transformed through homework practice. When you do your homework, you should think independently. If you really can't solve the problem, ask your classmates and teachers again. Don't ask, "What is the result of this question?" Instead, I asked, "What should I do about this problem?" "Why are you doing this?" "How did you think of doing this?" .
3.5 Do a good job in unit summary and reflect on mathematical thinking methods.
Professor Hua, a mathematician, once pointed out: "The number of missing shapes is not intuitive, and the number of missing shapes is difficult to be nuanced;" A good combination of numbers and shapes is indispensable for learning mathematics well; "It can be seen that the combination of numbers and shapes is an important way of thinking in middle school mathematics. At the same time, the ideas of function, equation, equivalent transformation, analogy and reduction are permeated in middle school mathematics, and methods such as collocation method, elimination method, method of substitution, undetermined coefficient method, reduction to absurdity and mathematical induction are introduced. While learning mathematical knowledge well, we should try to reflect on the principles and basis of these mathematical thinking methods, and master the steps and skills of using these mathematical thinking methods to solve mathematical problems through a lot of practice. Practice has proved that rethinking mathematical thinking methods can have the effect of drawing inferences from others.
3.6 make a study plan, combining short-term and long-term
The ancients said: "Everything is established in advance, and it will be abolished if it is not planned." Whether two students with the same intelligence have a study plan directly affects the learning effect. Make a plan for one day every morning, make a plan for this week every Monday, and make a plan for one month at the beginning of each month, and adjust it at any time according to the implementation until the plan can be successfully completed.
4. Practice the method of memorizing mathematics
A lot of mathematical knowledge needs not only understanding, but also memorizing. So, how can we improve the memory effect of mathematical knowledge? This paper summarizes several memory methods:
4. 1 Comparative Memory Method
Some mathematical knowledge is easily confused, such as exponential function and logarithmic function, arithmetic progression and geometric sequence, solid geometry and plane geometry. We can apply the corresponding relationship of these concepts, grasp the key points in the concepts and compare them, so that we can distinguish and remember them. For example, the images and attributes of exponential function and logarithmic function can be compared as follows:
4.2 Song memory method
Compile the mathematical knowledge to be memorized into songs, formulas or jingles for easy memorization. For example, when learning the number of solutions to the transcendental equation, a rhyming sentence "You have to draw an image if the function types are different" is compiled. When memorizing the inductive formula of trigonometric function, compile it into a Song formula, such as "Singularity is constant, and the sign depends on quadrant".
4.3 Understanding Memory Method
Rich mathematical knowledge is easy to memorize, and can only be remembered by deep understanding. For example, judging the plane area represented by binary linear inequality can be understood as "straight line delimitation, origin location"
4.4 step-by-step memory method
Many mathematical problems have fixed solving steps. For example, to judge the parity of a function, the first step 1 is to judge whether the domain is symmetrical about the origin, and the second step is to judge the relationship between sum. Many students tend to ignore 1 and go directly to the second step. Therefore, when solving all problems related to the nature of functions, we should follow the principle of "domain priority"; When solving the unary quadratic inequality with negative quadratic coefficient, the step of 1 is to change the quadratic coefficient into positive. Therefore, when solving a quadratic inequality, we should follow the principle that the coefficient of quadratic term should be "positive first"; In the evaluation, proof and simplification of trigonometric functions, the effects of simplifying offspring first and simplifying offspring first are completely different. Generally speaking, simplifying first can simplify the calculation. To this end, when solving such problems.
We should follow the principle of "simplification first".
4.5 key memory method
With the growth of my age, I have learned more and more mathematics knowledge. Comprehensive memory is a waste of time and unrealistic. So learn to remember the key points. On the basis of remembering the key contents, remember other contents through deduction and association. This way of memory reduces the memory burden and improves the memory.
Efficiency. For example, when remembering the same angle of the "three relations and eight equations" of trigonometric function, we can focus on remembering the definition of trigonometric function:, and then we can deduce the above eight equations.
4.6 associative memory method
Association is more important than knowledge. You can remember one familiar thing by remembering another related thing. For example, from real additive commutative law, the law of association to complex additive commutative law, the law of association. Association can open the floodgate of memory and is an effective memory method.
Bacon, a famous British thinker, said: Mathematics is the gymnastics of thinking, and problems are the heart of mathematics. Learning mathematics well can solve many practical problems in life, and the solution of the problems promotes the development of mathematics. To this end, let's act together and make unremitting efforts to learn mathematics well.