Teaching Strategies of Mathematical Concepts in Primary Schools (1)
1. Effective import is the basis of concept formation.
In my primary school mathematics teaching in recent years, I feel that the introduction of "familiar life cases" or "appropriate situations" can enable students to construct abstract concepts. Let me talk about the definition of volume in the lesson of volume and volume: the size of the space occupied by an object. It is difficult for them to understand this concept without combining with the reality of life.
I aroused students' interest from the story of crows drinking water, and then set the question "Why can crows drink water from bottles?" Leading to "stones occupy the space of water"; Ask the students again: "What occupies the space around us?" Through students' thinking consciousness, objects such as "schoolbags occupy classroom space" and "pencils occupy pencil box space" all occupy space. Finally, I compared a Rubik's cube with a cute little doll. "Who occupies more space?" Let students feel that objects not only occupy space, but also occupy large and small spaces.
Through these real objects in life, plus vivid examples. Students will be able to abstract the characteristics of * * * and form concepts by representing characteristics. After students know the concept, they should strengthen it in time, so that they can ask each other to say "what is its volume" by holding an object in a group or at the same table.
2. Generalization is the premise of concept formation.
Take "score recognition" as an example: by looking at the picture, use the score to represent the shaded part. From concrete concepts to abstract concepts.
(1) Divide a piece of paper into four parts, and take 1 part, which is represented by 1/4;
(2) Divide four apples into four parts, take three parts, and represent them with 3/4;
(3) Divide all butterflies into 5 groups, take 3 groups, which are represented by 3/5;
We can regard a piece of paper, four apples or five groups of butterflies as a whole, that is, the unit is "1". To sum up, divide a whole into several parts on average, and take one or several of them, which can be expressed by scores.
Mathematical concept is "abstract above abstract", and its powerful systematicness requires us to combine the age characteristics of young children, adopt appropriate teaching strategies to carry out teaching activities, and pay attention to the practical and mathematical significance of concepts, so as to improve teaching quality.
Teaching Strategies of Mathematical Concepts in Primary Schools Part II
First, provide perceptual materials to help students construct concepts.
In the process of learning the concept of geometric shapes, students should use various sensory perception concepts, listen to teachers' oral instructions, read text symbols and do practical operations, so as to understand the representation of concepts, and selectively summarize the relevant information of perceptual concepts to form representations. The thinking of primary school students is mainly intuitive thinking. In the process of understanding concepts, we can provide some perceptual materials to help students better understand concepts under various teaching guidance. Of course, when providing perceptual materials to help students understand concepts, we can adopt different teaching strategies according to different concepts.
(A) the use of intuitive teaching to help students understand concepts.
Pupils mainly think in images, and if they can use intuitive demonstrations, they will understand the essence of concepts more easily. For example, when teaching the characteristics of triangles in grade three, students can think about where you have seen "triangles" in real life. According to the students' answers, the teacher asked why the tripod of the bicycle, the beam supporting the roof, the tripod on the telephone pole and so on should be put. Are triangles instead of quadrangles? At the same time, the stability of the triangle is revealed by the intuitive demonstration of teaching AIDS. In this way, it is in line with children's cognitive law to use students' real life and some things or examples they are familiar with in real life to obtain perceptual knowledge and introduce concepts on this basis.
(B) through experimental exploration, to promote students to understand the concept.
To understand the essence of the concept of geometric shape, it is necessary to combine hands-on operation with experimental observation. Let students feel and understand concepts in the process of experimental exploration, guide students to compare similarities and differences between operating objects in time, and summarize the essential attributes of concepts. For example, when teaching the concept of "volume", students should understand the meaning of "everything occupies space" in order to understand the concept of volume. For this reason, after the story of "crow drinking water" is introduced, we put forward the question "why does water rise?" Have a preliminary understanding of "space", and then ask further, "Is the water surface rising because stones have weight or because they occupy space? Do other objects also occupy space? " Then ask students to design an experiment to prove their findings, and ask them to focus on "① How to carry out the experiment? ② What phenomenon was observed during the experiment? 3 What does this phenomenon mean? " Finally, please exchange reports. One student demonstrated, while the other students observed and thought, "What if the liquid water in the cup turned into solid sand and the stone was put into the sand?" Through group cooperation and communication, draw a conclusion. With examples, students have a deep understanding of the concept of "volume".
(C) strengthen the concept of variant to help students understand the concept
Variation refers to the change of irrelevant characteristics of the positive example of the concept. Variants are used to illustrate a group of examples of the same concept with the same essential characteristics but different non-essential characteristics. In the teaching of geometric concept, we can make full use of variants to help students understand the concept more deeply. For example, when learning the concept of "vertical", students are often used to understanding vertically, crossing a point outside a straight line into a vertical line and drawing horizontally. When the direction and position of a straight line change, it will be influenced by fixed thinking and make mistakes, which will lead to the wrong place and height in the triangle (parallelogram and trapezoid) with changed position or shape, which will affect the correct calculation of area. The reason is that the concept of "vertical" failed to provide enough variant materials for students in the formation stage, and students failed to abstract the essential meaning of "perpendicular to each other" of "two straight lines intersecting at right angles". When we know and draw the height of triangle (parallelogram, trapezoid), we should also do it in variant graphics. Then guide students to analyze and compare and find out their similarities and differences, so as to help students understand the essential characteristics of "triangle height" from different aspects.
Second, build a network system of concepts and deepen the essence of concepts.
When teaching concepts, you can't teach concepts in isolation. Before preparing to teach a new concept, students should be provided with a framework in which the concept can be put. If you study concepts in isolation, the level of learning is limited. Therefore, in teaching, teachers should take some appropriate ways to understand students, find the connection point between old and new knowledge, text knowledge and life, and start teaching, so that students can learn new concepts from the perspective of contact, promote active construction and form a network system of concepts.
(A) compare the similarities and differences of concepts, and promote the understanding of concepts.
By comparing similar things, it is helpful for students to find similarities and essential characteristics of similar concepts. In the process of learning, similar concepts often appear. For example, when teaching the concepts of "acute triangle", "right triangle" and "obtuse triangle", we provide students with a large number of examples, and classify triangles according to their angles on the basis of measurement, so as to guide students to discuss why they are divided like this and what are the characteristics of triangles in a group. Finally, the teacher gave three concepts. Three different types of triangles are presented. Comparatively speaking, generalization is more refined, which further clarifies the essential characteristics of these concepts.
(2) Reveal the relationship between concepts and deepen the understanding of concepts.
The understanding of new knowledge depends on what is already in the mind. In concept teaching, seeking suitable knowledge in students' original cognitive structure is an important basis for understanding new concepts. For example, in the study of "knowing parallelogram", parallelogram is learned on the basis of learning squares, rectangles and other graphics. It can be said that the knowledge of rectangle and square is to learn the upper knowledge of parallelogram, master the knowledge background of students, review the characteristics and inquiry methods of rectangle and square in view of students' recent development fields, establish representations, and let students abstract the characteristics of parallelogram through guessing, operation and verification. Then, students are required to explore the relationship between these three kinds of graphics through comparison, observation and hands-on operation, find out their similarities and differences, connect scattered graphics in series and dynamically, build a cognitive structure, and go through a process from part to whole, further enrich the extension of the concept and clarify the essence of the concept.
(C) the use of schema to establish a structure to promote the internalization of concepts
Schema refers to an organized, repeatable and generalized thing, and it is an individual's way of perception, understanding and thinking about the external world. When we help students learn concepts, we should purposefully guide them to classify, sort out and summarize related concepts, express them with charts, establish concept structure and promote concept internalization. For example, when teaching triangle classification, students can further clarify the essential characteristics of various triangles with the help of Wayne diagram. For another example, in the review process of plane graphics, we can guide students to draw the plane graphic structure diagram of primary school step by step through comparison, generalization and classification, so as to further understand the essence of various concepts and clarify the connections and differences between concepts.
In a word, promoting the development of students' spatial thinking is the highest realm of geometric figure concept teaching. Only according to the essence of concepts, starting from students' cognitive characteristics and realistic starting point, teachers can use various effective teaching strategies, carry out teaching with a developmental perspective, teach concepts in the concept system, establish the relationship between concepts, closely follow the essence of concepts, and help students deeply analyze and understand the essence of concepts in observation, exploration, experience and practice, can we realize effective teaching of geometric concepts.
Teaching Strategies of Mathematical Concepts in Primary Schools Part III
First, the importance of teaching mathematical concepts
Mathematical concept is the most basic knowledge and an important part of mathematical knowledge. The first is relative independence. The concept reflects the essential attributes of a class of objects, that is, the intrinsic and inherent attributes of such objects, abandons the concrete material attributes and concrete relations of such phenomena, and abstractly summarizes the quantitative relations and formal structures. Therefore, to some extent, it is relatively independent of the specific content of the original. Secondly, it is the unity of abstraction and concreteness. Mathematical concepts reflect the essential attributes of a class of objects. Take the concept of "rectangle" for example. In the real world, you can't see abstract rectangles, only various concrete rectangles. In this sense, mathematical concepts are "divorced" from reality. Because of the use of formal and symbolic languages in mathematics, mathematical concepts are farther away from reality and have a higher degree of abstraction. It is precisely because of the high degree of abstraction and the weak connection with the real things that the application of mathematical concepts is more extensive. No matter how abstract, high-level concepts always take low-level concepts as concrete content, and mathematical concepts are the basic parts of mathematical propositions and mathematical reasoning. As far as the whole mathematical system is concerned, the concept is true. Therefore, it is both abstract and concrete. Third, it is also logically related. Most concepts in mathematics are formed on the basis of the original concepts, and are fixed in the form of language or symbols with the help of logical definitions, so they have rich connotations and strict logical connections. In the process of learning mathematical concepts, primary school students often don't know the connotation and extension of the concept, which is easy to have a vague understanding of the concept and affects their ability to analyze, solve and process information. Therefore, a correct understanding of mathematical concepts is the premise of mastering the basic knowledge of mathematics, and concept teaching is the key to the whole mathematics teaching. Teachers should strengthen concept teaching, strive to make students thoroughly understand concepts, firmly grasp concepts, flexibly use concepts, and strive to cultivate students' thinking ability and problem-solving ability, thus improving teaching quality.
In the process of primary school mathematics teaching, the cultivation of students' mathematical ability and the solution of mathematical problems are actually the process of judging and reasoning by using concepts. Among the three thinking forms of concept, judgment and reasoning, concept, as the "cell" of thinking, is the premise of judgment and reasoning. Without a correct concept, there can be no correct judgment and reasoning, let alone the cultivation of logical thinking ability. Therefore, learning concepts well is the most important part of learning mathematics well. Judging from the reality of mathematics concept teaching in primary schools, students have two attitudes towards concepts: one thinks that basic concepts are boring, does not attach importance to them, and does not seek good solutions, which leads to vague understanding and understanding of concepts. The other is to attach importance to the basic concepts but just memorize them, but can't really understand them thoroughly, which will inevitably seriously affect students' mastery and application of basic mathematics knowledge and skills. Only when students really master the basic concepts in mathematics can they master the knowledge system of mathematics, and they can correctly, reasonably and quickly carry out operations, argumentation and spatial imagination. In a sense, the key to the level of mathematics lies in the differences in understanding, application and transformation of mathematical concepts. ; Therefore, grasping concept teaching is a basic link to cultivate mathematical ability.
There are many factors that affect the teaching of mathematical concepts in primary schools. On the one hand, teachers' emphasis on concept teaching is the main external factor that affects teaching. In concept teaching, teachers often deliberately pay attention to the "accuracy" of concept expression, while ignoring its essence and practical background; Emphasize the careful consideration of definitions and theorems, but ignore the process of their occurrence and development and the basic facts and phenomena reflected; Excessive pursuit of logical rigor and formalization of the system ignores the thinking image of students at a certain age. On the other hand, it is pointed out in the curriculum standard of primary school mathematics that the concepts in the basic knowledge of primary school mathematics mainly include number, set diagram, four operations, measurement, ratio and proportion, formula and so on. These concepts are abstract and general, which brings difficulties to concept teaching.
As far as primary school students are concerned, these factors will affect the effectiveness of mathematics concept teaching in primary schools because of their young age, lack of sufficient perceptual materials and real life experience, and poor abstract logical thinking ability and language understanding ability.
Pupils usually learn mathematical concepts in two ways: assimilation and formation. The assimilation of concepts requires students to retrieve concepts related to new concepts from existing cognitive structures, and prompt the essential attributes of new concepts through interaction. Students' intelligence is different. Even students of the same age or grade, due to different levels of intellectual development, the speed of reaching the corresponding learning level is also different. The main reason is the difference of students' cognitive strategies and metacognition level. The formation of concepts mainly depends on students' direct experience, abstracting and summarizing from a large number of perceptual materials, prompting the essential attributes of concepts, thus forming concepts. The concept teaching of primary school mathematics has obvious cognitive intuition and needs concrete experience to support it. Therefore, the clarity and stability of concepts in students' original cognitive structure, students' original life experience and the richness of perceptual materials will play an important role in concept teaching.
Students' abstract generalization ability and language expression ability are both internal factors affecting the effect of concept teaching, which deserve attention. In the process of concept formation, students find various attributes of things by observing objective things, and then abstract the essential attributes from them. After mastering the content of the concept, these essential attributes can be extended to similar things, so as to have a general understanding of the similar things reflected by the concept, which is to understand the concept. For example, when teaching the concept of rectangle, students should first observe all kinds of objects with rectangles, guide them to find out what their sides and corners have in common, and then abstract the graphics and summarize the characteristics of rectangles. Without the necessary abstract generalization ability, the connotation and extension of the concept will be expanded or reduced unilaterally. Students' language expression ability is also very important to the teaching of mathematical concepts. If the expression ability of mathematical language is poor, the expression of concepts will be inaccurate, which will affect the understanding, consolidation and application of concepts. For example, the accurate definition of "radius" should be: "The line segment connecting the center of the circle to any point on the circle is called the radius of the circle." If students describe it as the distance from the center of the circle to the circle, there will undoubtedly be deviations in practical application.
Second, the strategy of mathematical concept optimization
The teaching of mathematical concepts in primary schools generally involves the introduction, establishment, consolidation and deepening of concepts. This is a complicated thinking process, which is not only a process of recreating knowledge and gradually understanding concepts, but also a process of improving students' thinking quality, developing students' thinking ability and cultivating students' innovative consciousness and creativity.
1, concept introduction
The introduction of concepts is the first step in mathematics concept teaching, which is directly related to students' understanding and mastery of concepts.
An intuitive introduction. It is an active and complicated cognitive process for primary school students to master concepts, and their abstract thinking is directly related to perceptual experience. Therefore, first of all, we should provide rich and typical perceptual materials so that they can gradually abstract and internalize concepts through intuitive images. Intuitively introduce concepts, that is, ask questions and introduce concepts through familiar life cases and vivid metaphors of primary school students; Or use teaching AIDS, models, charts, projection demonstrations and hands-on operations to increase students' perceptual knowledge, and then gradually abstract and introduce concepts. In this process, we should attach importance to the role of life cases in introducing concepts. Mathematics comes from real life, and it is everywhere in life. Combining with real life, the introduction of concepts conforms to the psychological characteristics and cognitive laws of primary school students. For example, when teaching the characteristics of triangles, students can think: where is the "triangle" used in real life? Why the tripod of bicycle, the beam supporting the roof, the tripod on the telephone pole, etc. All made into tripods instead of quadrangles? Through examples in life, it is pointed out that triangles have the characteristics of stability. Let some students get perceptual knowledge by using things or examples they are familiar with in real life, and introduce concepts on this basis. Modern psychology believes that practical operation is the source of children's intellectual activities. Introducing concepts through students' practical operations can concretize abstract concepts. Homework activities greatly promote the development of students' thinking ability. In teaching, students can do it themselves, measure, divide, calculate and set it up, and get first-hand perceptual materials from it, which lays the foundation for abstract generalization of new concepts. For example, when teaching the concept of "pi", students can make several circles with different diameters, roll them on a ruler or measure the circumference of the circle with a rope to calculate how many times the circumference is. Let the students find out for themselves that although the size of a circle is different, its circumference is always more than three times its diameter. At this time, the teacher introduced a concept that the circumference of a circle is more than three times the diameter of the same circle, which is a fixed number called "pi".
Introduce from the original concept. The relationship between mathematical concepts is very close, which can be derived from students' existing conceptual knowledge and directly deduce new concepts. This not only consolidates old knowledge, but also learns new concepts, strengthens the internal connection between old and new knowledge, helps students to establish a systematic and complete concept system, and fully mobilizes the enthusiasm and initiative of learning. For example, the concepts of "divisor" and "multiple" are based on the concept of "divisibility"; Derive "common divisor" and "greatest common divisor" from "divisor"; Derive "common multiple" from "multiple" and then derive "minimum common multiple". For another example, in geometric knowledge, the area formulas of squares, parallelograms, triangles and trapeziums can be derived from the area of rectangles.
The calculation method is introduced. Refers to finding problems through calculation and drawing out concepts through calculation. Some concepts are inconvenient to introduce with examples and have little connection with existing concepts. Through the observation and analysis of the operation, we can find the essential attributes contained in it and achieve the purpose of introducing concepts. For example, when teaching the understanding of "reciprocal", we can first give several formulas with the product of two numbers as 1, so that students can calculate the results, then observe and analyze them, find out the rules from them, and lead to the definition of "reciprocal".
2, the establishment of the concept
The establishment of concept is the central link of concept teaching. Perception and experience are only the orientation of entry, and the revelation of the essential attributes of concepts can be the basis for judgment.
Use variants. The so-called change means that the examples or materials provided constantly change the presentation form, change the non-essential attributes and make the essential attributes "unchanged", which can help students form concepts accurately. The expression of perceptual materials has an important influence on the learning and mastering of mathematical concepts. If the perceptual materials provided to students are "standard" objects or graphics, students' understanding of concepts will inevitably be one-sided. Through the use of variants, students can see the essence through phenomena and truly master concepts.
Distinguish by comparison. When establishing concepts, we should compare and analyze some adjacent and confusing mathematical concepts in time to find out their connections and differences. Such as the greatest common divisor and the least common multiple; Separable; A proportional, inverse, disproportionate amount, etc. This can not only consolidate the concept, but also make the new concept clear, which is helpful to the gradual formation of students' concept system.
Set it off with the opposite side. Negative contrast is an effective method of concept teaching, which can be directly illustrated by negative examples or analyzed from both positive and negative aspects. Students can further deepen their understanding of concepts by contacting these positive and negative examples related to concepts.
Establish a multi-level and phased concept system. Understanding the concept is not achieved overnight, and it needs a long-term and repeated understanding process. Similarly, the establishment of a complete conceptual system should also be carried out at multiple levels and in stages. For example, when teaching "the preliminary understanding of fractions", it can be divided into three levels: the first is to highlight the "average score" of a fraction, and then "take a share"; The second is to solve the relationship between "number of copies" and "whole"; Third, it is clear that the unit "1" can be an object or a collection of objects. Through such repeated concept teaching, students can not only master the basic concepts of scores well, but also lay a good foundation for continuing to learn the essential attributes of scores.
3. Consolidation and deepening of concepts
From the process of cognition, the formation of concept is a process from perceptual knowledge to rational knowledge. That is to say, the general law is summed up from individual cases, and consolidating concepts is a process of memorizing and maintaining concepts, a process of deepening understanding and flexible use of concepts, that is, a process from general to individual. It is impossible for primary school students to master mathematical concepts overnight, and they must deepen their understanding of concepts through timely consolidation.
Generally, concepts are consolidated by memorizing, applying and establishing a concept system. Rote memorization requires students to master the concept definition through repeated perception and memory on the basis of understanding. Application means that students consolidate concepts in applying concepts, and its main form is practice. For example, after teaching the meaning of fractional multiplication, let the students talk about 3? 4? 5,5? 3? 4,2? 3? 3? The meaning of 4, etc. For another example, after learning "Understanding the Circle", let the students judge which line segment in the picture is the radius of the circle and which line segment is the diameter of the circle.
Students' understanding is a development process from shallow to deep, from concrete to abstract. Students' mathematical knowledge is carried out in stages, and concept teaching is also arranged in stages. Therefore, concept teaching should not only pay attention to the stages of concepts, but also pay attention to the continuity of concept development, develop the meaning of concepts in a planned way, and develop students' abstract generalization ability in stages. Through the application, students can deepen their understanding of concepts, make them find out the vertical and horizontal connections between concepts, form a systematic cognitive structure, and achieve the purpose of deepening concepts.
In a word, all stages of mathematics concept teaching in primary schools are closely linked. After the concept is introduced, it should be established immediately, consolidated in time, deepened in understanding and prepared for the development of the concept. In concept teaching, teachers should combine the characteristics of concepts with students' reality, flexibly design different links and adopt various teaching strategies, so that students can master mathematical concepts and improve their mathematical ability.
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