Mathematics teaching is not only to impart existing knowledge, but more importantly, to cultivate students' innovative thinking ability with mathematical knowledge as the carrier. In teaching practice, teachers should not only cultivate students' innovative thinking habits, but also coordinate the order, flexibility and criticism of thinking and improve students' innovative thinking ability. Teaching innovation is an important subject that every educator must study deeply. Based on the author's teaching experience for many years, this paper expounds his own methods and viewpoints on how to reform classroom teaching and cultivate students' innovative thinking.
First, cultivate innovative thinking in mathematics.
"Innovation is the soul of a nation and an inexhaustible motive force for a country's prosperity." The generation of a person's innovative consciousness and the acquisition of innovative ability mainly depend on his creative enthusiasm and interest. Diligence, tenacity and initiative are all important qualities to develop creativity. In classroom teaching, innovative education is an educational process of bilateral activities between teachers and students. Not only teachers should teach from the perspective of innovative education, but students should also study actively and creatively, which determines whether teachers' innovative teaching can be transformed into students' innovative spirit and ability. On the one hand, students should have the initiative to study, explore and innovate, actively participate in classroom teaching, think positively, dare to express their different opinions, have a strong thirst for knowledge, be strict with themselves, creatively complete learning tasks, and pay attention to cultivating and developing their hobbies. On the other hand, the above-mentioned learning qualities of students need the guidance and cultivation of teachers. Students' interest in learning is a necessary condition for the development of students' creativity, so that students can sprout their desire to create in their rich imagination. When students imagine something, they often show "whimsical", "unconventional" and "whimsical", which is a simple expression of students' creative thinking. With the correct guidance and help, their wisdom will sprout and grow, and they will have unique opinions when solving a certain problem.
Once the concept of mathematical innovation is formed, it is difficult to change, and it will affect the creator himself stably and permanently; It is a stable and positive psychological tendency of innovation, internalizing mathematical innovation into a need of creators, forming inertia and forming nature. It can be said that the establishment of mathematical innovation concept marks the formation of mathematical innovation consciousness. However, the establishment of the concept of mathematical innovation can not be completed overnight. It is gradually formed through long-term accumulation and long-term infiltration under the influence of mathematics education. The middle school stage is the key period to cultivate students' innovative thinking in mathematics. Middle school students are in the golden age of intellectual development, and it is also the most important period for physical and mental development and the formation of world outlook, outlook on life and values. Therefore, teachers should make use of the situation, teach students in accordance with their aptitude, create good educational conditions, mobilize various positive factors, and promote the formation of students' innovative thinking in mathematics.
Second, the organic connection between mindset and innovative thinking
Thinking set is to think, analyze and solve problems according to certain habits, conventions and fixed concepts, which is what we often call conventional concepts. In the process of solving problems, there are specific ways, the most prominent performance is the procedural nature of ideas and methods, and the process and steps are carried out in a standardized way. For example, when solving solid geometry problems, the space problem is transformed into a plane problem, and the format is reasonable step by step. The biggest feature of mindset is that it is easy to start with and standardized thinking, which is the basic way for most students to solve problems. Its disadvantage is that once the problem is solved, it is difficult to deeply understand the essence of the problem, so as to obtain new knowledge and new conclusions.
Innovative thinking is a thinking process in which individuals discover the inevitable connection between things, discover and understand the new relationship between problems, and realize new answers to organize certain activities and solve certain problems. Its main performance is the epiphany stage, when all parts of the mind seem to suddenly connect, discover new connections, form new images and assumptions, draw new conclusions, and finally verify the resulting thinking results.
Thinking set is the main form of concentrated thinking, the premise of logical thinking activities and the basis of innovative thinking. The mindset has its limitations in the thinking space, and because of this, the mindset strives to expand the application scope of existing experience and conceptual understanding. When the mindset is preserved to a certain extent, it will form a qualitative leap and transform from mindset to innovative thinking. Although mindset has a certain negative effect on the formation of innovative thinking, we can't ignore or even exclude the cultivation of mindset because we emphasize the cultivation of innovative thinking, otherwise the result will go from one extreme to the other. Innovative thinking has strong flexibility, which refers to the flexibility of intelligence in thinking activities, which is manifested as divergent thinking. Divergent thinking is the core of innovative thinking. In teaching, cultivating students' divergent thinking can generally be started from the following aspects: (1) linking various conclusions with the same condition; Change the angle of thinking, solve more questions and change one question, and carry out variant training; Design an open proposition. Through training, help students overcome the mindset and enhance the flexibility of thinking.
Thirdly, the optimization of classroom environment.
How to create a better situation for cultivating innovative thinking is an important topic that every educator must study. Because most of the teaching process takes place in the classroom, the setting of classroom environment has become an important topic. Only by establishing the individual student view of "master, subject and protagonist", strengthening the behavior research of learning science centered on teaching students to learn, and constructing a new teaching process centered on students and based on students' autonomous learning activities can teaching activities be truly realized.
1. Create a relaxed and warm atmosphere
Teachers communicate with students through language, gestures and eyes. Teachers should have the ability to express their ideas correctly. Teachers' words and deeds are largely role models for students. Teachers' agile and innovative thinking, calm and flexible analysis, enthusiastic and meticulous discussion and vivid and humorous explanation are all good strategies to stimulate students' learning motivation. Heuristic teaching is the basic teaching method most used by teachers. In the teaching process in recent years, the author often uses the method of "initiation" instead of "distribution" to give students more room for thinking. "Distribution" is accomplished through students' thinking and even discussion. My job is to sum up as clearly and completely as possible, and then ask questions. For example, there is an example of the application of mathematical interpretation function in senior one. For example, there is an isosceles right-angled triangular iron plate with waist length A. How to cut a rectangular iron plate with this iron plate to maximize the rectangular area? There are two methods to analyze (omit) teacher-guided inquiry: from the mathematical point of view, both methods have optimal solutions, but which method will we choose in reality? Let's discuss it with each other. Student activities: Student N: (Raise your hand immediately) Method 2 is good, which is conducive to the reuse of materials and saves materials. Student O: The second method is good. It can omit the working procedure and can be completed in two steps. Student P: Method 2 is good. I just calculated that the length of the cutting path is relatively short. Because in reality, if a thick steel plate is used, not only a few scissors but also the length of the shearing track should be considered. Applause broke out in the class to show support, affirmation and encouragement to this classmate. ) the teacher guided the inquiry: everyone said it was very good. Is the first method really useless? The classroom began to quiet down. Suddenly, a student couldn't wait to stand up and say. Student Activities: Student Q: In some practical situations, you need to choose Method 1. For example, if you want to build a triangular house, the hypotenuse is something, the vertex of the right angle is due north, and the window of the house faces south, you must choose the first method. Student activities: Student R: Mathematically, the method 1 is more general. The teacher guided the inquiry: Great! Who is the optimal solution in reality can only be analyzed in detail. From a mathematical point of view, method one is indeed more universal. If you change the background of a right triangle to an arbitrary triangle, you can only use. . . . . . Student activities: Students say in unison: Method 1. The way of thinking reflected in students' answers is always affirmed, and students are constantly encouraged to analyze the problems in depth with their own ideas. These practices have achieved satisfactory results in at least three aspects. First, they can clearly expose the shortcomings of students in mastering basic knowledge and the defects in solving problems; The second is to give students more thinking space, so that they have more opportunities to play their thinking ability and express the results of their creative thinking; Thirdly, the classroom efficiency is improved, so that students can coordinate their senses in listening, doing and thinking, and it is easier to deepen their study of mathematical problems.
2. Teacher-student translocation, with discussion.
Students are the main body of the teaching process, and teachers play a leading role. However, the traditional teaching method that teachers talk, students listen and then practice can no longer meet the current teaching requirements. To cultivate students' innovative thinking, students must have time to think and the opportunity to express their thinking results. In the review of senior three, the author suggested that every student should prepare the content of the speech. In the classroom, students should first publish their own research results, or explain the problems that they think are unique in question types and problem-solving ideas. After the implementation, students have deepened their thinking, expanded their research scope and enhanced their initiative in learning. Although some students' expressions are not satisfactory, they have made me see a lot of sparks of thinking and strengthened the correctness of my own practice. Practice has proved that organizing topic explanations by students themselves has the following three advantages: first, students have more autonomy, which can give full play to students' imagination and creativity, and also encourage students to study problems more consciously and deeply; Second, students are at the same level, and their explanations and ideas are more easily accepted by other students, which has changed the disadvantages brought by "strong teaching" in the past and made the questions better and faster; The third is to cultivate students' organizational ability and expression ability. In addition, the author really realized the profound meaning of "learning from each other's strengths".
3. Three-dimensional teaching, creating scenarios
Applying mathematics is the starting point and destination of learning mathematics. The purpose of current teaching is to benefit all students, emphasizing the sense of application, which includes the application of mathematical knowledge and the ability to have creative opinions on problems. How to strengthen students' application consciousness and cultivate students' innovative thinking is the key to creating classroom situations. Generally speaking, a good question should be meaningful or practical and play a connecting role in learning. Interesting and challenging, can stimulate students' interest in learning; The condition of the question is easy for students to understand, and the scene of the question is familiar to students; The timing should be appropriate and the difficulty should be moderate. For example, the application of inequality in senior two creates situations: mathematics comes from life and production practice, and is also applied to life and production practice. Practical problems contain rich mathematical knowledge, mathematical ideas and methods. We need to observe the world from a mathematical perspective, express the mathematical relationship in practical problems in mathematical language, and seek mathematical models to solve practical problems, because practical problems rarely appear in front of us in mathematical language.
(Displayed by the projector)
As shown in figure 1, use a rectangular iron sheet with a length of 80cm and a width of 50cm to make a rectangular iron sheet box without a cover (thickness and welding loss are not included). What is the possible maximum volume of this tin box?
50 cm
80cm
Student activities: Students think or discuss independently. Teachers know the students' thinking state and preliminary results through patrol.
Most students may use one of the following methods. The teacher chooses one of them to talk about his ideas:
As shown in Figure 2, the cuboid is closed by removing a small square from all four corners of the cuboid.
The teacher led the students to think about why the square must be removed. How about a rectangle? -Let students realize that it is through on-site production to ensure that the upper mouth of the closed cuboid is flush! )
Continue to ask the students to describe his ideas.
Let the side length of the removed small square be x (cm), then
V=sh=(80-2x)(50-2x)x
= 4x(25-x)(40-x)(0 & lt； x & lt25).
Teachers can always know the basis for students to solve problems-how did you come up with this idea? )
According to the basic inequality:
v = 4x(25-x)(40-x)= 2(2x)(25-x)(40-x)= 1
Teacher's guidance: In the above process, there is a mistake that is easy to make when using basic inequality to find the maximum value-the condition of finding the maximum value without considering the basic inequality. "two fixings"-harmony should be a fixed value; "Three phases"-Can you get the maximum value? )
Give this question to students for discussion, and guide them to analyze the goal of "equality" with "basic quantity thought", and then construct it with undetermined coefficient method.
Let V=4x(25-x)(40-x) make ax=b(25-x)=40-x, that is, ax+b(25-x)+(40-x)= constant, that is, ax = b (25-x) = 40-x.
The solution is a = 3 and b = 2.
Therefore, V=4x(25-x)(40-x).
The equal sign is true if and only if 3x=2(25-x)=(40-x), that is, x= 10.
When students use basic inequalities to find the maximum value of a function, they often do not fully consider the three conditions for using basic inequalities. Through thinking, exploring and demonstrating this problem, students can deepen their understanding of it and achieve the effect of review.
The scene of creating and exploring problems undoubtedly provides us with opportunities for innovative thinking when solving problems, increases the flexibility and diversity of problem-solving methods, and further stimulates students' thirst for knowledge.
Fourth, encourage problem exploration, conclusion conjecture and thinking innovation.
Understanding the essence of a problem requires an iterative process from perceptual knowledge to rational knowledge and from special to general. In the study of the particularity of a large number of cases, it is a common method for scientific research and problem exploration to find general laws and draw correct conclusions. To cultivate students' creative ability, we must first let students have an active attitude of exploration and a desire to guess and discover. The biggest drawback of "strong teaching" is that it has created a group of problem-solving machines with strong imitation ability but lack of exploration and creativity. When faced with a new problem, they seem to be unable to cope with it, and they have insufficient understanding of scientific thinking methods such as observation, analysis, induction, analogy, abstraction, generalization and conjecture, so it is difficult to find and solve the problem. Therefore, in the teaching process, we should pay attention to encouraging students to explore, guess and discover, create some exploration topics according to the teaching priorities and difficulties, and organize students to explore and discuss. At the same time, we should also strive to create conditions for students, cultivate their awareness of problems and applications, and constantly inspire students to think and ask questions. In this process, we should pay attention to cultivating students' observation and imagination, cultivating students' divergent thinking and inducing students' inspiration.
We should not only create opportunities for students to study deeply, but also pay attention to correct guidance and grasp the general direction of problem exploration and research. In the process of solving problems, students can be inspired, induced and encouraged to discuss with each other, avoid detours, waste time or even walk into a dead end, and make exploration more effective, research deeper, guess more scientifically and think about more innovative.
Implementing innovative education requires innovative teachers. In order to cultivate innovative talents, teachers must establish the educational view and quality view of innovative education; We should have innovative consciousness and spirit; There must be teaching methods that adapt to innovative education; Using modern teaching methods and models to improve the quality of innovative education; We should have a reasonable knowledge structure and ability structure, stimulate students' interest in learning and innovative beliefs with the knowledge and problems of contemporary frontier disciplines, and develop students' innovative thinking level. At the same time, teachers should also have high political consciousness, noble ideology and morality and simple life style, so as to cultivate outstanding talents with innovative ability and both ability and political integrity.