Below I will talk about some personal views on how to cultivate students' mathematical thinking ability.
First, to mobilize students' inherent mathematical thinking ability
1. Set correct and appropriate learning goals to stimulate students' strong thirst for knowledge.
The setting of learning objectives should conform to the new curriculum standards and adapt to students' life reality and thinking level. Teaching should be based on students' existing experience, provide students with familiar life scenes, help students understand various quantitative relations and master the mathematical relations between various things in real life, thus stimulating students' interest in exploring the unknown world. For example, when teaching "the calculation of the area of a circle", I take the knowledge of "the calculation of the area of a rectangle" that students have mastered as the connection point between old and new knowledge, and guide students to think about whether a circle can become a square. Solve new problems through the knowledge, then divide the circle into an approximate rectangle through courseware demonstration, let students analyze that the length of this rectangle is half of the circumference of the circle, and then summarize the calculation formula of the area of the circle through reasoning and calculation.
2. Create vivid and harmonious learning scenes, so that students can learn to think scientifically. Vivid and interesting learning scenes are helpful for students to learn independently and cooperate with each other.
Equal teacher-student relationship and harmonious learning atmosphere can make students participate in every link of thinking activities easily, confidently, positively and actively. When creating problem situations in teaching, teachers should pay attention to guiding students' thinking direction. The questions raised should be enlightening, hierarchical and directional, which should be conducive to activating students' thinking, but not beyond students' cognitive level, and should actively point to the central goal of learning. For example, when I was teaching divisor and multiple, I designed the following program:
(1) Which of the following formulas can divide the dividend exactly?
22÷6=3…48÷5= 1…3 15÷5=3
38÷2= 19
(2) Give examples to illustrate under what circumstances one number can be divisible by another?
(3) Why are divisors and multiples interdependent?
(4) What is the divisor of 24? What is the multiple of 2?
(5) How many divisors does a number have? How many multiples does a number have?
Through the directional solution of this series of questions, most students can master the characteristics of divisor and multiple, and have received good thinking training effect.
Of course, in addition to directional thinking training, I pay more attention to strengthening students' reverse, horizontal, vertical and multi-directional thinking training. Application problem teaching is an effective way to train students' thinking. For example, in the middle and lower grades, "asking questions according to conditions" is taught to train students to "ask questions directly related to conditions"; In senior high school, students are trained to think from multiple angles and put forward problems that can be solved according to conditions. Students' step-by-step answer, column-by-column comprehensive formula answer and one-to-many method all reflect the gradual nature of thinking training. Under the guidance of teachers, students gradually learn to think scientifically and cultivate good mathematical thinking habits.
3. Develop rich and open classroom activities to cultivate students' mathematical thinking ability.
Carrying out rich and open classroom activities can enable students to show their personality, flash the spark of smart thinking, fly their ideal wings and stimulate their thinking potential. In teaching, as a teacher, we should gradually teach students to observe, compare, analyze, synthesize, abstract, summarize and other thinking methods. For example, when teaching "Volume Calculation of Cones", I designed an activity to provide cylinders and cones with equal bottoms, equal heights, unequal bottoms and unequal bottoms, so that students can explore the volume calculation method of cones in groups. Such teaching activities not only enable students to discover the calculation method of cone volume, but also deeply understand the volume relationship between cone and cylinder. Of course, there is no fixed model to cultivate students' mathematical thinking ability in classroom teaching activities. It is necessary to comprehensively choose the most suitable method according to students' age characteristics, knowledge level and learning content, and it is even more impossible to operate mechanically according to the designed teaching plan. Teachers should always pay attention to students' thinking, grasp the learning process according to the feedback information in the interaction between teachers and students, and adjust the learning methods wisely, so that students can acquire knowledge and develop their mathematical thinking ability.
4. Design flexible homework exercises to consolidate and deepen students' mathematical thinking.
The purpose of homework exercise is to further consolidate students' thinking, but students' minds have been exhausted through organized, hierarchical and intensive classroom learning, so we must pay attention to alleviating students' thinking tension when designing homework. We should try our best to design fun activities such as games, adventures and treasure hunting, increase oral training, reduce written training and strengthen practical operation. Cooperative exercises instead of students' individual meditation can realize the diversification, flexibility, applicability and interest of problems. This will not only help students to consolidate their knowledge and improve their ability to solve problems, but also train their mathematical thinking and develop their intelligence. At the same time, homework design is targeted, hierarchical, comprehensive and creative. It is necessary to combine the teaching content with the students' reality, and train all kinds of students in a targeted manner to achieve the goal of "reaching the standard at the same starting point and at different levels".
Second, we should teach students mathematical thinking methods.
Confucius said that "learning without thinking is useless, thinking without learning is dangerous", which properly explained the relationship between learning and thinking. In order to make students think actively in mathematics learning, it is necessary to teach students the basic methods of analyzing problems, which is conducive to cultivating students' correct mathematical thinking mode. To be good at thinking, students must attach importance to the study of basic knowledge and skills. Without a solid double foundation, mathematical thinking ability cannot be improved. We should adhere to heuristic teaching and cultivate students' thinking ability of drawing rules.
Mathematics teaching is to inspire students' thinking. In the teaching process, teachers should guide students to observe, discover, summarize and master the laws. Mastering the law is an effective way of learning, which can overcome interference and improve students' cognition, thus reaching a new realm of thinking. The formation process of concepts and laws should be regarded as an important teaching link in the example class. Students should not only know how to do it, but also know why to do it, and what makes them think so. This formation process can be completed by teachers guiding students, or by teachers telling their own exploration process.
For example, learning "the invariance of quotient". First of all, by preparing questions, let students make it clear that "a number multiplied by several times can be said to be a number multiplied by several times, and a number multiplied by several times is a number multiplication"; Dividing a number by a few can be said to reduce a number by several times, and reducing a number by several times is divided by several times. Secondly, guide students to observe and compare the invariance of quotient. Step 1: Observe the following set of formulas, compare the dividend and divisor first, and then find the quotient to see what changes have taken place.
① 12÷3=②24÷6=③ 120÷30= ④240÷60=
(1) Compare Formula 2344 with Formula ①, and ask: What's the change? What hasn't changed?
(2) In ② ③ ④, how can the dividend and divisor change to keep the quotient unchanged? Let the students draw the following conclusions:
Divider divisor
Zoom in twice, zoom in twice.
Zoom in 10 times, zoom in 10 times.
Expand 20 times, expand 20 times
(3) Can you give another example like this? Seeing that the quotient remains unchanged strengthens the "simultaneity" and "sameness".
(4) What laws can be found through this top-down observation? Based on the above reasons, the following achievements have been made here. The students successfully concluded that in division, the dividend and divisor are expanded by the same multiple at the same time, and the quotient remains unchanged.
(5) Through the comparison between Formula 123 and Formula 4, it is concluded that the dividend and divisor in division are reduced by the same multiple at the same time, and the quotient remains unchanged. Step 2: Try to strengthen the two laws summarized above. Step 3: Summarize the nature. Q: Through the observation and comparison of the students just now, we have come to two laws of constant quotient. Who can sum up these two laws and talk about them? Through the previous laws and exploration process, students can sum up the invariance of quotient.
In mathematics practice, we should carefully examine the questions and observe them carefully. The key to solving problems is to have the ability to dig hidden conditions and learn the positive and negative analysis methods from conditions to conclusions or from conclusions to conditions. For a mathematical problem, we must first judge which interval it belongs to and which concepts, laws or calculation formulas are involved. Try to learn the use of mathematical language and symbols in the process of solving problems.
There are many ways to cultivate students' thinking ability. To make students' thinking active in mathematics, the most fundamental thing is to arouse students' enthusiasm for learning mathematics. Teachers should be good at enlightening, guiding, guiding and dispelling doubts, so that students can turn learning into thinking. Of course, good mathematical thinking quality is not formed overnight, but as long as we persist in various means according to the actual situation of students, we will certainly achieve certain results.