First of all, it outlines the depth and breadth of thinking.
(A) the meaning of the depth and breadth of thinking
People's thinking is that in life, when encountering difficulties or problems, they will think with their brains. The process of thinking is a process of analysis, comparison, simulation, synthesis and summary, that is, a process of proposing solutions to difficulties through one's own cognition and understanding. The cultivation of thinking ability is one of the important tasks in primary school mathematics teaching. Students' study, play and life are inseparable from thinking activities, and thinking ability is the basis for students to understand things.
By consulting relevant literature, the author has a preliminary understanding of the depth and breadth of thinking. It is believed that the breadth of thinking is a high-content way of thinking, which is mainly reflected in being good at thinking about this problem from multiple angles and in all directions according to the whole problem, that is to say, when solving the problem, we should pay attention to analyzing the essence of things and fully consider the specific details. Suppose a mathematical problem is placed in a three-dimensional space and analyzed from all angles and directions. Some people call it "three-dimensional thinking" For example, the simple math problem 475÷25 can be solved by (500-25)÷25=500÷25-25÷25 or (400+75) ÷ 25 = 400 ÷ 25. This is the breadth of thinking. The depth of thinking means that when students think about a problem, they put aside superficial phenomena and grasp the core of the problem, that is, from far and near, from the outside to the inside, step by step and step by step.
(B) the importance of thinking depth and breadth in mathematics teaching
People must be different from the moment they are born. Coupled with the influence of acquired family education, environment and other external factors, the depth and breadth of primary school students' thinking are also different. It is precisely because of this difference that we should pay more attention to cultivating the depth and breadth of primary school students' thinking in primary school mathematics teaching. In addition, more importantly, in the teaching process, teachers should not only pay attention to imparting knowledge to students, but also pay attention to improving students' quality in many ways, especially the ability of mathematical thinking to penetrate into knowledge. If teachers neglect to expand the depth and breadth of students' thinking in the teaching process, students will not be able to better digest the knowledge taught by teachers, and they will form the bad habit of only "listening".
The ancients said, "Learning without thinking is useless, and thinking without learning is dangerous." This sentence explains the subtle relationship between thinking and learning. In the teaching process, teachers should clarify the relationship between thinking and learning, pay attention to activating students' thinking, and let students learn knowledge better. In this regard, students put forward higher requirements in understanding and analyzing problems.
Second, in the teaching process to improve the depth and breadth of students' thinking suggestions
(A) focus on a variety of solutions
As mentioned above, there are many solutions to a math problem. In the process of solving and thinking, teachers should support students to think independently, solve problems through their own way and understanding, and support students to exchange ideas. In this teaching process, students can answer questions through independent thinking, improve their autonomous learning ability and inquiry ability, and deepen their thinking. While exchanging ideas with each other, students compare, discuss and study various solutions to the same problem, and integrate new solutions into their own thinking, which effectively cultivates students' all-round thinking ability and expands the depth and breadth of thinking.
(B) pay attention to the variability of the problem
The so-called variability of problems refers to changing the conditions of problems in the teaching process. In the process of students thinking about a math problem, the conditions of the problem have changed, and the direction, angle and way of students thinking will also change. We should look at this problem from many aspects and find the correct answer in a new way. For example, "It is known that every inner angle of a polygon is equal to 135. What is the degree of this polygon? " This math problem can be changed to "The sum of the internal angles of a known polygon is equal to 1080". What is the degree of this polygon? " It can also be translated into "The number of sides of a known polygon is 8. What is the sum of the internal angles of this polygon? " . On this same problem, let students analyze the problem from many aspects, solve the problem through different ways, break through the mindset and greatly improve the breadth of students' thinking.
(3) Pay attention to cultivating the habit of asking questions.
The subject of mathematics puts forward high requirements for students' logicality, requiring students to constantly think about problems and be good at asking questions. Only in this way can they master the law. Although the traditional teaching concept has always been based on the teacher's "speaking", it is also very popular to let students boldly put forward their own opinions. The ancients said, "You can always ask the Eight Immortals Temple for advice, and you can't be wrong if you are stupid." In this passage, we can see that the ancients attached great importance to asking questions in the learning process. In many speeches, Mr. Li Zhengdao emphasized that the teaching process should focus on "learning" rather than "learning to answer". In addition to rote learning, we need to master the basic concepts, theorems and formulas of mathematics. It is necessary to understand the connotation and extension of the basic concepts, theorems and formulas of mathematics, as well as the necessity of introducing them and the connection with other knowledge. Only by cultivating students' habit of asking questions can students' thinking penetrate the surface and shallow level of knowledge, deeply understand the internal essence of knowledge and improve the depth of thinking.
(D) focus on combining relevant knowledge points
There is a certain correlation between mathematical knowledge, including the vertical connection between each part of knowledge in their respective development process and the horizontal connection between each part. Being good at finding the relationship between them is helpful for students to think about the problem from a systematic perspective and grasp the essence of the problem. For example, when teaching the relationship between circles, teachers compare the learned knowledge points, the relationship between points and circles, and the relationship between straight lines and circles to help students find the relationship between circles. In this way, combining the relevant knowledge points learned will help students accept new knowledge points and penetrate into the inherent nature of new knowledge points. The most important thing is to expand the depth of thinking in the process of classifying, combing, synthesizing and finding the law of knowledge. Mathematics is the science of thinking, and thinking ability is the core of mathematics discipline ability. It is found that the thinking quality of mathematics is based on profundity and extensiveness. Therefore, in order to improve students' thinking depth, teachers must use mathematical knowledge in the process of optimizing teaching, create opportunities to improve students' thinking ability and open the door to students' wisdom.
(5) Cultivate students' thinking methods of guessing before testifying.
Conjecture plays an important role in the process of discovery, and teachers should take this as the basis to expand the depth and breadth of students' thinking. In this process, teachers should provide students with space and opportunities to guess, so that students can understand that reasonable guessing must be based on careful use of induction and analogy until "argumentation reasoning" is completed. In the teaching process, whether learning new knowledge or reviewing old knowledge, we should analyze the specific content. In view of different knowledge points in each section, teachers should ask relevant questions for students to think independently, and indirectly guide and help students to recall different knowledge in each section, and conduct in-depth analysis, understanding and reasoning, so as to reach the final correct conclusion. Finally, and most importantly, we must summarize all the contents of each chapter.
Mathematics teaching is closely related to the depth and breadth of thinking, and mathematical ability is different from general ability, so cultivating mathematical thinking ability is an important task of mathematics teaching. In the process of cultivating students' mathematical thinking ability, we should not only consider the general requirements of ability, but also deeply study the characteristics of mathematical science, mathematical activities and mathematical thinking, seek the laws of mathematical activities and expand the depth and breadth of students' mathematical thinking. The purpose of primary school mathematics teaching is not only to impart knowledge, so that students can learn, understand and master mathematics knowledge, but also to pay attention to teaching students learning methods and cultivating students' thinking ability and good thinking quality, which is the need to improve students' quality in an all-round way.