1. 1 the characteristics of high school mathematics curriculum
Senior one should learn set, logic, function, sequence, triangle and plane vector. Compared with junior high school mathematics, the proportion of theoretical components in these contents is unprecedented. Compared with junior high school mathematics, the requirement of thinking level can be said to be "climbing over the hill", whether it is the abstraction of concept, the logic of argument, the flexibility of method or the universality of application.
Senior two and senior three should learn system theories such as inequality, analytic geometry, solid geometry, permutation and combination, probability and statistics, limit, derivative and complex number. Compared with senior one mathematics, these contents are more theoretical and more methodological.
Based on this, learning high school mathematics should first master the ins and outs of mathematical theory comprehensively, systematically and profoundly, and at the same time analyze and understand the rich connotations of various mathematical knowledge points and thoroughly understand their ideological essence. With such a solid theoretical foundation, we can "use theoretical thinking" when solving problems, that is, observe, analyze and solve problems with the mathematical theories and methods we have learned, which is the fundamental way to learn high school mathematics well. As a teacher, we should seriously study how to teach students to thoroughly understand the theory, how to teach students to "think with theory" and guide students to constantly sum up their experience in this field. Otherwise, it will inevitably fall into blindness, engage in "problem teaching" and even slide to the edge of "subject teaching". This will bring serious consequences to students' study.
Three years in senior high school is a three-year period of dramatic development and change of human organs, and so is the development and change of psychological characteristics.
1.2 psychological characteristics and learning strategies of senior one students
Psychologists' research tells us that senior one is a turning point: students' abstract thinking slowly begins to change from experience-oriented to theory-oriented, and will quickly enter a critical period of theoretical development. At this time, students began to have "personal opinions", their independent consciousness and ability to solve problems independently were obviously enhanced, and they felt that they had "really grown up".
At this time, a problem worthy of our high attention is that educational research shows that if the knowledge learned is challenging (challenge is motivation) and the way of education and training is appropriate, the level of thinking will be "miraculously developed"! On the other hand, if the educational content is boring and the measures are ineffective or improper, it will delay or even destroy the development and leave lifelong regrets for students. Long-term teaching practice and systematic study of law education have also led us to a very important discovery: a senior high school student's three-year development, whether it is the acquisition of knowledge, the cultivation of personality or the improvement of ability, follows such a law-"Three-year development depends on Grade One, and the key to Grade One lies in Grade One (me)". In other words, the psychological condition, learning style, thinking habit and knowledge structure formed in the first semester of senior one will have a great or even decisive impact on the development of senior three. A considerable proportion of outstanding students, intermediate students and underachievers will continue until they graduate from high school or even university. This discovery further strengthens the understanding that the last semester of senior one, especially senior one, should be a "critical period".
The negative lesson should arouse our vigilance:
There are many middle school students. It is precisely because senior one didn't realize this turning point that their mathematics learning methods and habits haven't been able to adapt to senior high school mathematics learning. Their grades have dropped again and again, and finally they even lost the confidence to learn math well, which brought heavy mental pressure and pain to themselves and their parents! This is what no one wants to see.
A serious and important topic is before us: in this critical period, it is really important to do a good job in education and training! But how should we catch it?
(1) We should face up to the "turning point" and guide students to realize the "transition" consciously.
It is necessary to tell students the characteristics of high school mathematics, encourage them to keep pace with the times, seriously study and understand the theory of mathematics learning and mathematics learning methods based on theoretical abstract thinking level, consciously and as soon as possible complete the transition from junior high school to high school in accordance with the "basic structure of mathematics learning" and form good mathematics learning habits and methods.
(2) Cherish the precious "critical period" and strive for a better development of thinking level.
The critical period is also the best period for development. As the saying goes, "an inch of time is an inch of gold." Only by grasping the critical period can we better develop our talents and benefit for life. Otherwise, "it is hard to return to school after learning" (Xue Ji), because the development of various organs and abilities of people has obvious stages.
Specifically, theoretical components occupy a large proportion in the mathematics content of senior one, which provides an opportunity for the development of theoretical abstract thinking level. In the whole process of teaching every theory (definition, theorem, formula and law) (experiment → conjecture → demonstration → analysis → example → application), students should actively participate in mathematics activities under the guidance of teachers, strive to achieve "four advances" and solve problems independently, and promote the development of students' abstract thinking ability.
1.3 psychological characteristics and learning strategies of senior two students
Psychologists' research tells us that the abstract thinking level of senior two students has entered a mature stage of theoretical development. At this stage, if the education and training are proper, the thinking level can be greatly developed and the thinking ability will be further improved. However, this cycle is generally only one or two years. After this mature period, the development of theoretical abstract thinking ability will slow down and gradually stabilize (that is, the later, the smaller the development space). Instead, it will be the development of dialectical logical thinking ability.
It should be the starting point and the end result of mathematics teaching in senior two to do everything possible to seize the golden period of "maturity" and strive for the great development of mathematics ability.
(1) First of all, we should mobilize students' thoughts. It is necessary to tell students the law of "maturity is only one or two years" to arouse their sense of crisis in developing their thinking level. It is easy for students to move.
(2) High (2) The theoretical and methodological nature of mathematics is high-the further improvement of mathematics provides sufficient spiritual food for the great development of mathematical ability. As a teacher, we should not only deeply study and develop the intellectual function of each chapter, section and example exercise, but also study and pay attention to each student's thinking characteristics, carefully design and operate, help students learn mathematics well, and strive to promote the development of their thinking level.
(3) the focus of the guidance of learning the law is still:
① How to improve the understanding level of mathematical theory;
② How to improve the consciousness and level of "thinking with theory".
Grasping these two points will grasp the foundation of learning and using mathematics well.
2 the scientific view of mathematics learning
It is a universal fact that a good creative idea can save a factory, develop an enterprise and revitalize a nation!
Similarly, a good learning concept can make a student who has suffered setbacks repeatedly in his studies move towards academic success and embark on the broad road of life. What I recommend to readers here is such a scientific concept of mathematics learning. To understand this problem, we need to understand the following questions first: What is real mathematics learning? What is its essence and core?
As we all know, the knowledge points in mathematics are not isolated, but closely related. People call several interrelated digital knowledge points mathematical knowledge structure. Mathematics learning is a process in which learners constantly construct (construct, construct) and improve the mathematical knowledge structure in their own minds. Psychologists call this process "internalization" of mathematical knowledge. As a result of internalization, learning is successful if a coherent, rich, closely related and experienced knowledge structure can be gradually formed. On the contrary, learning is unsuccessful or even a failure. Reflection on this internalization process can lead to the following two conclusions:
First, the process of learning mathematics is essentially a process of understanding mathematical knowledge and its relationship. A thorough, profound and comprehensive understanding will lead to a high quality of internalization. It can be seen that understanding the core of mathematics learning. Chen Shengshen, a contemporary American mathematician, said: "Mathematics is understanding!" He said this because mathematics has three characteristics-"high abstraction", "strict logic" and "extremely wide and flexible application". Without in-depth understanding, it is impossible to learn mathematics well, and it is impossible to learn mathematics well. Therefore, understanding is extremely important for mathematics learning, and mathematics learning in the true sense is unbreakable. Put understanding first and do everything possible to improve it. The scientific way of learning mathematics must be based on the way of deepening understanding, otherwise it will deviate from the real sense of mathematics learning and it is absolutely impossible to learn mathematics well.
Second, understanding is constructed by learners themselves. This kind of understanding can't be given by others, and can only be realized by participating in mathematics activities in the learning process. The preface of the new American mathematics series says "the best way to learn mathematics is to do it", which is the reason. In order to clarify the principle and make perception reach the operational level, it is divided into four links:
(1) Participation
Participation in mathematical activities is the premise of obtaining mathematical understanding, and participation can be divided into two forms: active participation and passive participation. Some students "listen first and try to keep up with the teacher's ideas" in class. Although he also participated, the content and intensity involved in this participation are very useful. Some students are not satisfied with the understanding in class, but try to solve problems by themselves like mathematicians. This strong sense of autonomy mobilized his whole body and mind to devote himself to mathematical creation. This kind of participation is qualitatively different from before in both content and intensity. The experience he gained is naturally much richer and deeper.
(2) reflect on the problem
Frieden Thiel, a master of international mathematics education in the Netherlands, believes that "reflection is the core and motive force of mathematics activities" and that "without reflection, students' understanding cannot be sublimated from one level to a higher level". It can be seen that he attaches great importance to reflection! So, what is reflection? Generally speaking, it is "looking back at the footprint", that is, "thinking deeply and repeatedly" about the whole process of mathematical activities and the relationship between old and new knowledge, so as to discover the true meaning of mathematics. Therefore, if you want to learn mathematics well, you must learn to reflect and develop the habit of reflection, which is the basis of learning mathematics well.
(3) Summarize the problem
The process of transforming perceptual knowledge gained in the process of participation and reflection into rational knowledge, from which we can discover laws, gain insight into essence and improve the level of understanding mathematics. Research shows that this process is particularly important for learning and understanding mathematics, and this is exactly where students are very difficult. Therefore, it is more necessary to learn to generalize.
(4) migration.
The so-called transfer means that learners apply the acquired experience, methods, ideas and concepts to the situation. This is a kind of creation in itself.
To sum up, in order to get a high level of understanding, we must grasp the four steps of "participation-reflection-generalization-migration", actively participate, strengthen reflection, learn to generalize and try to migrate. This can be regarded as a microscopic process of learning mathematics. Obviously, in this process, no part of learning is incomplete learning, and it is impossible for incomplete learning to obtain a high level of understanding.
3. Scientific methods of mathematics learning
3. 1 Strive to realize the "four advances" in the classroom
1. think ahead: after the teacher puts forward the topic, try to come up with ideas and answers before the old explanation.
2. Do it in advance: After the teacher writes the example, before the teacher explains it, try to find ideas and even make a result.
3. Summary in advance: After the teacher answers, before the teacher explains, try to reflect, summarize and summarize the answering process.
4. Beyond the premise problem: after the teacher makes a summary, before the teacher explains, he should try his best to find the problem, ask the question and study the problem.
"Four Ahead" is first put forward for the teaching of theoretical courses, and it is also applicable to the mathematics of example courses. Its basic idea is to keep one's thinking in a very positive state in class, and actively collect, analyze, synthesize and transform information from all directions, so as to obtain new guesses, new ideas, new insights and new creations from this process.
The proposal and implementation of the "Four Progresses" have injected vitality into the mathematics classroom, completely ended the situation of passive class attendance, and strengthened the consciousness of independent thinking and problem solving. Practice has proved that this kind of consciousness plays a very important role in realizing the great development of students' mathematical ability and cultivating their innovative spirit. Moreover, if the "four advances" are achieved, it is possible to find out the gap by comparing the teacher's explanation with the students' discussion and exchange, and the study will be more targeted.
3.2 Learn "three kinds of review" after class.
3.2. 1 Review in time-after class every day, you should read textbooks and arrange your notes to accomplish two tasks:
(1) Digging theory (concept, theorem, formula, rule).
Mathematical concepts and theorems have three characteristics of mathematics, and it is difficult to understand and master them without in-depth excavation. Digging deeply mainly finds out the following four aspects:
① Background and process of the theory. (Why do you want to put forward this concept? How was the theorem discovered? How to prove it? How is the formula derived? )
(2) Conditions for theoretical application. Under what conditions can this theory not be used? )
③ The structural features of the theory. The structural characteristics of numbers and formulas, figures, propositions, etc. ).
④ The essence and function of the theory. Look at the essence through form and pay attention to function. )
(2) Digging examples
We divide the learning of examples into three levels: how to do it (learn how to do it), how to think (learn how to think, the core is to learn to think with theory), why to think like this, and how to think like this (really understand). You know, "learning to do doesn't mean thinking, and thinking is not necessarily reasonable." Only when you can think and reach the level of understanding can you be considered as "knowing why, knowing why", and you can draw inferences from others.
Obviously, the process of in-depth excavation is the process of "reading deeply" advocated by Professor Hua, the process of revealing the rich connotation of theories and examples in depth, and the process of fully absorbing intellectual nutrition. This process is an indispensable basic project to learn mathematics, an extremely important step to improve the level of understanding, and a necessary condition and magic weapon to abolish the tactics of asking questions about the sea.
3.2.2 Unit Review-After reading each unit, conduct unit review and complete the following tasks:
(1) organizes knowledge points in series to form a theoretical system of units.
After the knowledge points are connected in series, the ins and outs of theoretical development are clear at a glance, and its main branches and latitudes are clear, so it is easy to see the guiding role of basic mathematical ideas, so that you can take the overall situation "from the height of the system" and even grasp the direction of theoretical development.
(2) Summarize the basic ideas, central topics and mathematical methods of unit theory, so as to reach a higher level of understanding.
(3) Select typical examples and exercises in the unit for in-depth study and review.
Obviously, this method of systematically organizing knowledge is the "thin reading" method advocated by Professor Hua. This method can string scattered knowledge into a string, form a chain and form a system, which plays a great role in further thinking and understanding the connotation transmission of unit knowledge and improving ability. Moreover, once the theory forms a system, it not only gives birth to the overall function of the system, but also can remain in memory for a long time because of its logic and visualization.
At this point, some students may ask, is it necessary to make such great efforts in after-class review and unit review?
Our answer is very affirmative. There is a simple reason. In senior high school, rational thinking (thinking under the guidance of mathematical theory) plays a decisive role in the process of mathematics learning and problem solving. Therefore, first of all, we must work hard on theoretical study, thoroughly understand the spirit, and exert our strength on the blade. This will improve the level of understanding of the theory, and the thinking will have the right direction when solving problems. Otherwise, thinking will inevitably fall into a passive water situation, which is the fundamental reason why many students have not learned mathematics well in high school.
3.2.3 Pre-test review and post-test summary
Many students will not review math before the exam, but will only find a problem to do. This will often make the knowledge system incomplete, forgetful, and even unable to remember the questions in the usual exams. So it is of practical significance to learn to review before the exam. The task of reviewing before the exam is within the scope of the exam:
(1) Close the book and say it from beginning to end. When you can't go on, open the book and have a look, and continue talking until you can make it all clear. This is the learning method of Professor Ding Zhaozhong, a China scientist who won the Nobel Prize in Physics. Only by reviewing in this way can you be complete, focused and truly understand your mastery of theory. This "method of teaching mathematics" is very effective and worth popularizing. You might as well try!
(2) Repeat the central topic, mathematical ideas and methods of unit review according to the above methods, so as not only to fully grasp the topics and thinking methods of mathematical problem solving, but also to be focused and targeted, saving time and effort.
(3) Analyze or do typical examples and exercises.
Make a summary after the exam, not only to summarize the successful experience, but also to summarize the places where you lost points. The scores are divided into four categories: ① theoretical errors; ② skill operation errors; (3) Understanding the mistakes in ideas and methods; ④ Errors caused by psychological factors.
We should find out the reasons, find out the improvement methods, and strive for "no mistakes" in the future.
Professor Hua advocates "repeated revision" in mathematics learning, and the above is the operation method of implementing repeated revision.
3.3 The "three requirements" should be met in the operation.
(1) Review before doing your homework (only by mastering the teaching materials comprehensively can you understand the purpose of each exercise, and do your homework in a time-saving and labor-saving way with high quality and efficiency).
(2) When doing homework, pay attention, write beautifully, operate normally, calculate correctly, and try not to alter it (paying attention and doing things in an orderly way is an excellent psychological quality, which is of great benefit to success. Some students usually do not pay attention to training, and it is difficult to correct problems. )
(3) If there is a wrong question, do it again and find out the reason.