Mathematics learning is mainly to cultivate students' mathematical thinking ability. This should be based on the basic knowledge and skills of mathematics, with mathematical problems as the inducement, mathematical thinking methods as the core, mathematical activities as the main line, and follow the inherent laws of mathematics and students' thinking laws to carry out teaching.

Learning type analysis

1. modal classification

(1) Accept learning and discover learning

Definition: a learning method that presents learning content to learners in the form of conclusions.

Mode: presenting materials-explaining and analyzing-understanding and understanding-feedback and consolidation.

(2) Discovery learning

Definition: a learning method that provides learners with certain background materials and allows them to acquire knowledge independently.

Mode: Presenting materials-Hypothetical attempt-Cognitive integration-Feedback consolidation.

2. Knowledge classification 1

(1) Definition of knowledge learning: A learning activity whose main content is to understand and master the basic knowledge of mathematics. Process: Choice-Understanding-Acquisition-Consolidation

(2) Skills learning

Definition: A series of actions (internal or external) are practiced to form a skilled and automated reaction process.

Process: demonstration-imitation-practice-proficiency-automation

(3) Problem solving learning

Mathematics learning activities that focus on the problem-solving process and reflect on the problem-solving thinking process.

The process of asking questions-analyzing problems-solving problems-reflecting.

3. Knowledge classification II

(1) Learn conceptual (declarative) knowledge.

Concepts, definitions, formulas, rules, principles, laws and rules in mathematics are called conceptual knowledge.

Conceptual learning: assimilation and formation.

The way to learn related new concepts by using existing concepts is called concept assimilation; Relying on direct experience and starting from a large number of concrete examples, the way to summarize the essential attributes of new concepts is called concept formation. Concept formation is the main form for primary school students to acquire mathematical concepts.

(2) Learning skills (procedural) knowledge

Math skills in primary schools are mainly arithmetic skills. The formation of operational skills can be divided into three stages:

① Cognitive stage: "guided" trial and error. Have a preliminary understanding of the algorithm from the teacher's examples or the rules of self-study, and form the representation of the algorithm in your mind. (2) Connection stage: rule stage, that is, operating step by step according to the rules to ensure correct calculation (solving problems by using rules and providing basic operation clues by declarative knowledge)-programming stage (integrating related small rules into a complete rule system, at which time conceptual knowledge has withdrawn), which can calculate quickly and correctly. ③ Automation stage: The application of the program in the second stage is clearer and more skilled. Through more practice, the program is no longer considered, and a certain degree of program automation is realized, and the operation speed and high accuracy are obtained.

(3) problem solving (strategic knowledge) learning

By rearranging the mastered mathematical knowledge, we can find out the applicable strategies and methods to solve the current problems, so as to learn the strategies to solve the problems.

The main ways for primary school students to solve problems are: First, trial and error method (also called trial and error method), that is, correcting temporality by trying without direction.

Trial and error until the problem is solved; The second is epiphany (also called heuristic), as if the answer or method suddenly appeared, but there was one.

Based on "orientation", this is the evaluation and identification of the rules and principles on which problems are solved.

4. Task classification

(1) memory operation learning

For example, oral calculation, ruler drawing (painting), mastering basic arithmetic and accurate calculation.

(2) Understanding learning

Such as understanding and mastering the connotation of concepts, understanding mathematical principles and being able to explain or explain, understanding a mathematical proposition and deriving new propositions.

(3) Exploratory learning

If students need to explore, discover and ask questions or learning tasks by themselves, they can sum up a mathematical law or rule through their own inquiry, and gradually form new strategic knowledge through their own inquiry process.

Primary school students' mathematical cognitive learning

First, the basic characteristics of primary school students' mathematical cognitive learning

1. Common sense of life is the starting point of primary school students' mathematical cognition.

It is necessary to build a bridge between children's life common sense and mathematics knowledge, so that children can start from their own life common sense and experience, constantly try, explore and reflect, so as to realize the "mathematicization" of "common sense".

2. Pupils' mathematical cognition is a subjective mathematical activity process.

The process of mathematics cognition should become a process of "doing mathematics", so that children can discover, understand, experience and master mathematics in the process of "doing mathematics", understand the value, characteristics and laws of mathematics, learn to use mathematics, improve their self-cultivation and develop their mathematical ability.

3. Pupils' mathematical cognitive thinking is intuitive.

On the one hand, children's common sense of life is the basis of their mathematical cognition, on the other hand, children's thinking is mainly based on intuition and concrete image thinking, so intuition should be the main means for children to understand and construct mathematical cognitive structure.

4. Pupils' mathematical cognition is a process of "rediscovery" and "re-creation"

Primary school students' mathematics learning is not passive acceptance learning, but active learning process of "rediscovery" and "re-creation". Let them rediscover or recreate mathematical concepts, propositions, rules, methods and principles in mathematical activities or practices.

Second, the basic law of primary school students' mathematical cognition development

1. The development of primary school students' mathematical concepts

(1) From obtaining and establishing primary concepts to gradually understanding and establishing secondary concepts.

(2) From understanding the properties of concepts to understanding the relationship between concepts.

(3) The establishment of mathematical concepts is gradually weakened by the interference of experience.

2. The development of primary school students' mathematical skills

(1) has developed from the argument orientation with perfect dependence structure to the understanding of internal meaning.

(2) From the external development of thinking to the internal compression of thinking.

(3) The gradual improvement of number sense and symbol consciousness supports the operation to develop in a flexible, simple and diverse direction.

3. The development of pupils' spatial perception ability

(1) sense of orientation is gradually established.

(2) The establishment of the concept of space has gradually developed from grasping the dominant characteristics to grasping the essential characteristics.

(3) The ability of spatial perspective is gradually enhanced.

4. The development of pupils' mathematical problem-solving ability

(1) Language expression stage (2) Understanding structure stage (3) Formation of multilevel reasoning ability (4) Symbolic operation stage.

Cultivation of Primary School Students' Mathematical Ability

First, an overview of mathematical ability

1. ability overview ability refers to the psychological characteristics of an individual who is competent for an activity.

2. Mathematical ability Mathematical ability is a personality and psychological characteristic that successfully completes mathematical activities and directly affects the efficiency of their activities.

(1) Computing power: data operation, logical operation and arithmetic operation.

(2) Spatial imagination: establish a model according to the object, restore the object according to the model, and decompose the ability to combine the model or object according to the abstract characteristics, size and position relationship of the model.

(3) Mathematical observation ability: generalization of objects, formalization of perception, perception of spatial structure and recognition of logical patterns.

(4) Mathematical memory ability: the ability to remember and reproduce generalized and formalized symbols, propositions, properties, spatial structures and logical patterns.

(5) Mathematical thinking ability: the ability to use mathematical reasoning to think about existing mathematical information.

Second, the differences of children's mathematical thinking ability

1. Reasons for differences (1) Multiple Intelligence Theory (2) Different thinking types.

2. Attitude towards differences (1) Seek common ground while reserving differences (2) Develop strengths and avoid weaknesses.

Third, the cultivation of mathematical ability.

1. Cultivate students' interest in learning mathematics

(1) Starting from students' life experience (2) Starting from setting up problem situations (3) Let students learn by doing mathematics.

2. Cultivate basic mathematical ability

(1) Hands-on operation of mathematical operation ability can not only attract students' attention, but also easily stimulate students' thinking and imagination, thus mobilizing their enthusiasm for learning, cultivating their interest in learning and enabling students to acquire knowledge actively.

In operation, students not only "play", "learn" and "think", but also improve their thinking ability, cultivate their interest in learning, and understand and digest book knowledge.

2. Mathematical language ability

In students' hands-on activities, students are also required to gradually establish a clear and profound representation of mathematical concepts through language expression, and then consciously and firmly master mathematical knowledge.

When students express mathematics, they need concise language and accurate use of mathematical terms. Strict mathematical attitude requires strict mathematical language.

3. Ability to solve problems

The ability to discover, propose, analyze and solve mathematical problems is the most important and final expression of mathematical ability.

(1) Create problem situations and cultivate problem awareness.

Create problem situations purposefully and consciously, set up obstacles to ask questions, make students feel that they have problems to think about and contradictions to solve, and put them in a situation of "eager to understand but unable to do anything".

(2) Actively explore and enhance students' subjective consciousness.

(1) make bold guesses and try to solve problems.

Guess from life experience and guess on the basis of existing knowledge and experience.

② Exchange conjectures in various forms and choose a better scheme.

(3) Expand changes and enhance students' awareness of application.

Emphasizing the application of mathematics is not only regression measurement, drawing, accounting and other teaching activities, but also a desire and way to solve problems by using mathematical knowledge and thinking methods.

(4) Apply what you have learned and solve mathematical problems.

There are many math problems in life. Guiding students to abstract the problems in life into mathematical problems in teaching can not only deepen students' understanding of what they have learned, but also help improve their ability to solve problems. For example, the painting area of the house decoration, how many bricks are used to spread the floor, the planting area and the number of trees, and why the wheels are round.

Classroom teaching process of primary school mathematics

First, the main contradiction in the process of primary school mathematics teaching

1. The contradiction between mathematics teaching and learning

Teachers are the main body and students are the main body. Students are the masters of mathematics learning, and teachers are the organizers, guides and collaborators of mathematics learning.

2. The contradiction between pupils' cognitive characteristics and mathematics knowledge.

There will be contradictions between the abstraction of mathematics and the concrete visualization of primary school students' cognition, between the rigor of mathematics and the simplification and intuition of primary school students' cognition, and between the universality of mathematics application and the narrow knowledge and little contact with real life of primary school students.

3. The development level of pupils' cognitive structure and teachers' teaching.

Contradiction between mathematical knowledge First of all, there is a contradiction between the teaching of mathematical knowledge by teachers and the understanding and mastery of mathematical knowledge by students. Second, the contradiction between teachers' mathematical language expression and students' understanding of it. Third, the contradiction between the new knowledge mastered by primary school students and the old knowledge.

Second, the primary school mathematics teaching process

1. The process of primary school mathematics teaching is a process of communication and interaction between teachers and students.

The basic attributes of communication are interactivity and reciprocity, and the basic ways of communication are dialogue and participation. For primary school students, communication provides space for their open mind, prominent subjectivity and creative liberation; For teachers, classroom communication is to share their understanding of mathematics with students and feel the joy of learning. The teaching process of primary school mathematicians is a process of equal dialogue and communication between teachers and students. The contents of this dialogue and exchange include information on mathematical knowledge and skills, as well as information on emotions, attitudes and attitude values. It is through this kind of dialogue and communication that teachers and students realize the interaction between teachers and students in the classroom.

Effective interaction should pay attention to the following two aspects:

(1) We should fully mobilize the initiative and enthusiasm of primary school students.

Mathematics teaching process is a learning process of exploring, practicing and thinking about mathematics content, and students are the main body of learning activities. Only by guiding students to observe, calculate, compare, guess, reason, communicate and other activities can teachers urge students to construct their own understanding of mathematics, master mathematical knowledge and skills, gradually learn to observe things from a mathematical perspective, think about problems, and generate interest and desire to learn mathematics.

(2) Realizing the change of teachers' roles.

The leading role of teachers can be reflected in the following activities.

① Arouse students' learning enthusiasm, stimulate students' learning motivation and guide students to actively participate in learning activities. (2) Understand students' ideas, give targeted guidance and help students solve their learning difficulties; At the same time, encourage different views, participate in students' discussions, evaluate their learning and make adjustments. ③ Create a good classroom environment and spiritual atmosphere for students' study, and guide students to carry out active mathematics activities.

2. The process of primary school mathematics teaching is a process in which teachers guide students to carry out mathematics activities.

(1) Organize and guide students to experience the process of "mathematization"

Students' mathematics learning should be a "mathematical" process. That is to say, students start from specific situations, and after thinking activities such as induction, abstraction and generalization, they look for mathematical models and draw mathematical conclusions. Teachers should be good at guiding students to upgrade their life experience into mathematical knowledge and methods.

(2) The process of generating and constructing mathematical knowledge by both teachers and students.

Under the background of school study, teachers play an important role in guiding students to construct mathematical knowledge. In the process of mathematics classroom teaching, teachers should pay attention to guiding students to effectively construct mathematics knowledge and "generate" knowledge and methods. This "generation" process is completed through the interaction between teachers and students and the external factors of teachers.

(3) Experiencing mathematics in activities and obtaining the process of mathematics development.

The process of mathematics teaching in primary schools should be an activity process in which teachers and students participate together. In this process, teachers design and provide meaningful situations for students, and organize students to operate, communicate and think together. It is necessary to provide students with relatively sufficient time and space, so that students can have the opportunity to explore and practice independently, learn mathematical knowledge from practical problems, raise mathematical questions from related disciplines and existing knowledge, and explore and study the inherent laws and principles of mathematics.

3. The process of primary school mathematics teaching is the process of common development of teachers and students.

(1) Promoting students' development The basic purpose of primary school mathematics teaching is to promote students' development and lay the foundation for the lifelong development of primary school students. Students should develop in four aspects: mathematical knowledge and skills, mathematical thinking, problem solving and emotional attitude and values. These four aspects should be intertwined, infiltrated and inseparable, forming a whole.

(2) Promoting teachers' professional growth Excellent teachers all grow up in teaching practice. Good knowledge structure, ability structure, professional guidance, peer communication and constant self-reflection are the key factors for the growth of excellent teachers. Teachers' professional abilities include teaching design, teaching implementation and teaching reflection. The teaching process must follow the law of education and the law of children's physical and mental development, and teachers should be able to creatively solve the conflicts of cognition, emotion and values between teachers and students and form a unique teaching style. Teaching is a personalized creative process.