Thinking is an indirect and general reflection process of human brain on the general particularity and regularity of objective things. Mathematical thinking is an indirect reflection of the essential attributes and internal laws of mathematical objects (spatial form, quantitative relationship, structural relationship, etc.). ), and the rational activity of mathematical content is understood according to the general law of thinking.
Students' good thinking ability is the core of acquiring new knowledge, carrying out creative learning and developing intelligence. The new curriculum standard establishes the trinity of knowledge and skills, process and method, emotional attitude and values, and embodies the concept of quality education in the curriculum standard. By guiding students to actively participate, practice, think independently and explore cooperatively, we can change the learning style and cultivate students' abilities of collecting and processing information, acquiring new knowledge, analyzing and solving problems, and communicating and cooperating.
First, the meaning of mathematical thinking and mathematical thinking ability
Mathematical thinking is an indirect reflection of the essential attributes and internal laws of mathematical objects (spatial form, quantitative relationship, structural relationship, etc.). ), and the rational activity of mathematical content is understood according to the general law of thinking.
Mathematical thinking ability mainly includes four aspects:
1. Can observe, experiment, compare, guess, analyze, synthesize, abstract and generalize; 2. Reasoning by induction, deduction and analogy;
3. I will explain my thoughts and opinions logically and accurately;
4. Be able to use mathematical concepts, ideas and methods to distinguish mathematical relations and form good thinking quality. The new curriculum standard points out that the basic starting point of mathematics curriculum in compulsory education stage is to promote students' all-round, sustained and harmonious development. We should not only consider the characteristics of mathematics itself, but also follow the psychological laws of students learning mathematics. Mathematics plays a unique role in improving people's reasoning ability, abstract ability, imagination and creativity. The new curriculum standard establishes the trinity of knowledge and skills, process and method, emotional attitude and values, and embodies the concept of quality education in the curriculum standard. By guiding students to take the initiative to participate, practice, think independently and explore cooperatively, we can realize the transformation to the learning style and cultivate students' abilities of collecting and processing information, acquiring new knowledge, analyzing and solving problems, and communicating and cooperating.
The new curriculum standards focus on the objectives of mathematics curriculum, including: mathematical literacy, mathematical knowledge and skills, mathematical thinking, problem solving and emotional attitude, pay attention to the integration of students' experience, subject knowledge and social development, and emphasize that starting from students' existing life experience, students can personally experience the process of abstracting practical problems into mathematical models and explaining and applying them, so that students can gain an understanding of mathematics and make progress and development in their thinking ability, emotional attitude and values.
(4) Trial and error. In the process of understanding and applying mathematical knowledge and methods, students often make some specious mistakes for various reasons. If teachers can choose materials from them, they can create trial and error scenarios, thus providing students with time and space for trial and error. By reflecting on the causes of errors and asking critical questions, we can deepen students' understanding and mastery of knowledge and methods, improve their understanding and vigilance of errors and cultivate students' critical and rigorous thinking. This can not only stimulate students' full enthusiasm for learning, but also make them actively explore with positive attitude and vigorous energy, and also make them meditate, be infected and understand in the situation.
Strategy 2: Effective "Speaking" Mathematical Thinking Ability
Language is the shell of thinking, from the beginning of thinking, through the intermediate process, and then to the result, it must be shaped by language. In mathematics classroom teaching, in order to effectively impart mathematical knowledge to students and develop their logical thinking ability, we must attach importance to the training of students' mathematical language. Stimulating students' thinking with "saying" as the main line is one of the most effective strategies to cultivate students' mathematical thinking ability.
1, providing an opportunity to "say"
In teaching, teachers must create a better language environment, change the practice of speaking in full, leave enough time for students to express the process or result of thinking in words, encourage students to dare to think and speak, thus activating thinking factors and inducing students to carry out a series of thinking activities such as memory, imagination, analysis, judgment and synthesis.
When teaching conceptual knowledge, according to the thinking characteristics of primary school students, the concepts appearing in primary school mathematics textbooks mainly rely on intuitive demonstration methods to guide students to actively explore, and try to summarize and express in their own language, especially the key and difficult contents, chew and understand the connotation of mathematical language, and explore the ins and outs of understanding knowledge. To this end, we often design a "speaking" teaching situation: first, let students observe and compare independently, combined with the learning and experience of a certain concept knowledge, and then let students try to summarize these concepts in their own mathematical language, say them repeatedly, compare some typical examples while talking, understand the mathematical definition of concepts, especially scientifically scrutinize some words that are difficult to be accurate, so as to make the concept expression just right.
When, reasonable.
In the practice of test questions, teachers can first conduct sufficient listening and speaking training to form a good learning environment for reading, examining and analyzing the meaning of the questions, so that students can read the questions, talk about the places that easily lead to our calculation errors, and talk about the steps of answering questions. In the long run, students will gradually overcome their thinking inertia, optimize their thinking quality and improve their thinking ability.
When solving problems, the best way is to integrate mathematics knowledge into the most basic listening and speaking activities that every student can do. Teachers can use illustrations, objects or line drawings in teaching materials for oral training, so that students can tell the observed appearance, and talk about the operation process while doing it in students' hands-on operation, so that the external operation process can be closely combined with the internal intellectual activities.
2. Guide the norms of "saying"
Accurate and standardized use of mathematical language to express mathematical thinking process smoothly and logically describe mathematical laws or discoveries is not only the concrete embodiment of students' profound thinking and strict logic, but also the deep demand of learning methods advocated by the new curriculum.
(1) Pay attention to the conversion between students' life language and mathematics language, and gradually form an accurate mathematics language. The language of life is free and loose, with no fixed constraints. Different mathematical languages, influenced by the essence of mathematics, have the characteristics of rigor, accuracy and strong logic. One task of refining life mathematics is to guide students to transform their life language into mathematics language. For example, the price of each commodity in mathematics is called unit price, the number of pieces bought is called quantity, and the total amount of money is called total price. Of course, in the teaching process, we should not only pay attention to the transformation from life language to mathematics language, but also guide students to learn how to use mathematics language in life, explain life and embody the idea that mathematics serves life.
2. Attention should be paid to guiding students to use accurate mathematical language in their daily study. Accurate language cannot be formed overnight. Need repeated training. Listening and speaking activities are the key to the accuracy of mathematical language. In daily life teaching, language training teachers should be targeted at students, guide and persuade students with language difficulties, let them practice, try and train repeatedly, and they will also speak a standard mathematical language. In addition, teachers' teaching language must be accurate and structured, and set an example for students with standardized mathematical language and become an example for students to learn.
3. Experience the process of "speaking"
(1) Observation: Intuitive images and vivid demonstrations are the ways for primary school students to gain perceptual knowledge, from which they can get inspiration and get materials for language expression. Teachers should be good at guiding students to observe the demonstration process of drawings, objects and teaching AIDS, and make the mathematical knowledge contained in drawings, objects and demonstration process clear and complete in mathematical language. For example, when teaching the characteristics of geometric shapes and the derivation of calculation formulas, students should be guided to collect relevant physical objects first, and with the help of physical observation, operation demonstration and teaching AIDS, students should be encouraged to express the characteristics of shapes and the derivation process of formulas in language, so as to cultivate the development of students' thinking from images to abstract thinking in speaking. At the same time, teachers should encourage and guide students to understand mathematical concepts on the basis of perceptual materials or make simple judgments and inferences through quantitative relations, so as to master the most basic knowledge.
(2) guessing: guessing also requires students to express their intentions in their own mathematical language, why they think so, and the basis for guessing. For example, when teaching the classification of triangles, the two corners of the triangle are blocked by paper. Please think about what triangle it may be according to the exposed angle, and tell your thoughts and reasons. Let students experience listening and speaking activities, so that learning is always in an exciting state, so as to truly experience the fun of learning, which will help students form real mathematical ability and establish mathematical ideas.
(3) Reasoning: Reasoning is often accompanied by reasoning and explanation. Reasoning can be divided into oral reasoning and written reasoning, and oral reasoning is widely used in teaching. "Reasoning" in mathematics teaching is a learning process to explore mathematical problems. Frequent reasoning activities are conducive to improving students' logical reasoning ability. Of course, oral reasoning is more difficult than written reasoning, because the accuracy of language use and the consistency of reasoning process should also be considered. But it is helpful for students to learn the essence of mathematics and improve their quality to make oral reasoning and often experience the reasoning process.
(4) Group cooperation and communication: Mathematics learning should be carried out in an atmosphere of cooperation and communication, and all organs should communicate in cooperation, including careful listening, clear and well-organized explanation and scientific and rigorous reasoning. Through cooperation and communication, students can speak, discuss and listen boldly in their study, so that their senses are all open. This whole-hearted learning is the real experiential learning. (5) The combination of modern information technology and mathematics teaching also promotes students' experiential learning. Multimedia teaching software integrates various activities such as watching, listening, speaking and hands-on. This multi-organ experience, like multimedia, allows students to enter the kingdom of mathematics and experience the excitement brought by mathematics. No one can forget this kind of learning.
4. Encourage the freshness of "speaking"
In the classroom, teachers sometimes ask students to repeat their own language expressions only according to the correct answers of individual students in order to make students proceed smoothly according to their own designed procedures, which makes the development of thinking confined to a narrow space. Therefore, teachers should encourage students to speak creatively, be good at tapping the potential of students' thinking, and peep into the vast thinking space through students' unique opinions, which is helpful to cultivate students' flexible thinking ability. If students are encouraged to associate more and talk more, it is to induce students to associate and speak other conditions or characteristics related to them through one condition or characteristic, thus cultivating the divergence of students' thinking. When reviewing the relationship between fractional application problems and ratio, the scores in key sentences can often be expressed in the form of component number and ratio at the same time. For example, according to the condition that the number of boys in a class is 3/5 of the number of girls, students can be inspired to think that the number of girls is 5/3 of the number of boys; Boys are 5-3/5 less than girls; The number of girls is 5-3/3 more than that of boys; The ratio of girls to boys is 5: 3; The number of boys is 3/3+5 of the whole class; The number of girls in the class is 5/3+5 and so on.
The language expression process of students reflects the thinking process of students. Strengthening language training can improve the logic, flexibility and accuracy of students' thinking. However, if we want to really improve our thinking ability through language training, we should not only strive to make students "speak more" purposefully, but also let teachers give correct guidance in time, and more importantly, teachers should persevere.
Strategy 3: Effectively "organize" the environment of mathematical thinking
Teachers help students to sort out the thinking context and pay attention to the starting point and turning point in the thinking process, which is the focus of cultivating thinking ability in primary school mathematics teaching.
In teaching, for each problem, we should consider both its original knowledge base and its related knowledge content, guide students to derive new knowledge from existing knowledge, and at the same time compare and analyze it with old knowledge, distinguish similarities and differences, and cultivate students' systematic and grounded thinking. Only in this way can we better stimulate students' thinking and gradually form the context of knowledge. The key to our teaching is to make students' thinking context clear, and the key to clearing the thinking context is to seize the starting point and turning point of thinking.
1, guide students to grasp the starting point of thinking. The context of mathematical knowledge is connected and closely linked, and always constitutes the knowledge system of each unit according to the natural law of occurrence-development-extension. The same is true of students' thinking process of acquiring knowledge, either starting from existing experience or introducing old knowledge. This is the beginning of thinking, starting from the starting point of students' thinking, grasping all levels of thinking development, and gradually deepening until the end. If this beginning does not conform to the students' knowledge level or thinking characteristics, students will feel the solution to the problem.
If we don't start from scratch, our thinking will not develop in an orderly way.
Of course, different knowledge and students have different starting points of thinking, but regardless of the starting point, thinking training in mathematics teaching must start from the "occurrence point" of thinking, rely on old knowledge, and make students' thinking process clear, organized and logical through "migration" and "transformation".
2. Guide students to grasp the turning point of thinking. Students' thinking sometimes gets stuck, which is the obstacle point of thinking. At this point, the teaching should be timely guidance and guidance, to promote the change of students' thinking, and take this opportunity to promote the development of students' thinking.
For example, both parties process a batch of parts at the same time, and the number of parts that Party A plans to process is 2/5 of that of Party B. In fact, A has processed 34 pieces more than planned, which is exactly 7/9 of that of B. How many parts are there in this batch? When students think about this problem, although they can accurately judge that the scores of 2/5 and 7/9 are based on the number of parts processed by B, the values of these two standard quantities are not equal, so students' thinking is hindered. Teachers should seize this opportunity in time to guide students to start thinking: "The number of parts processed by A is 2/5 of that of B", indicating how many parts A and B plan to process? "Exactly 7/9 of the number of parts processed by B" also shows how many parts actually processed by A and B? In this way, the fractional relationship of standard quantity B will be transformed into the fractional relationship between standard quantity and total quantity until this problem is solved. In this process, the process of teachers guiding students to associate scores with ratios is actually the process of students' thinking turning. Grasping this turning point is conducive to overcoming students' thinking obstacles and cultivating divergent thinking.
In a word, mathematics is a logical, abstract and systematic subject. How to cultivate primary school students' basic mathematical thinking ability will be our long-term conscious teaching goal. In teaching, improving students' learning ability, cultivating students' thinking consciousness, giving students opportunities to think in many ways and cultivating students' thinking quality will surely become the direction of our mathematics teachers' efforts.