Abstract thinking, classified thinking, combined thinking, combination of numbers and shapes, corresponding thinking and symbolic thinking.

1. An abstract idea

This term does not appear in textbooks, but it is often mentioned in textbooks. Curriculum standards establish three basic concepts: abstraction, reasoning and model.

Conceptual explanation

Abstraction includes the abstraction of space form, the abstraction of demonstration form and the abstraction of quantity relationship, while from the perspective of primary school mathematics, abstraction mainly includes the abstraction of quantity and quantity relationship and the abstraction of graphic relationship.

Teaching suggestion

(1) Starting from the reality of life, the abstract ability is gradually improved from various angles.

(2) Make preparations for the gradual abstraction through intuitive teaching of mathematics.

2. Classification idea

Classified discussion is a common research method. There is no definition of classification in primary school textbooks, but textbooks arrange rich classification activities in different knowledge fields, and "classify these numbers" in the understanding of logarithm; In the understanding of graphics, "you classify the following graphics"; "Classify these methods" in calculation and problem solving, and "sort the data" in the study of statistical knowledge, which fully embodies the important role of the application of classification methods in concept establishment and problem solving.

Conceptual explanation

Classified thinking method is a kind of ideological and scientific classification to deal with mathematical problems on the basis of classification, and it is a basic logical way in natural science and even social science research.

Generally follow strict logical principles.

(1) Make clear the principle of variable domain and classify the set of objects, that is, variable domain, which must be clear.

(2) the principle of unified standards, the standards of each division must be unified.

(3) The classification of the confidentiality principle must be complete, and there can be no omissions.

④ The principle of non-repetition, all classifications must be mutually exclusive.

Teaching suggestion

(1) In the unit teaching of lower grade classification, we should pay attention to the idea of infiltrating classification and set.

(2) Look at the relationship between diversification and optimization of classification objectively, and gradually guide students to classify from the perspective of mathematics.

(3) Infiltrate the idea of classification in knowledge learning and problem solving in various fields.

Collect ideas

Teaching suggestion

Clarify the application of set thought in primary school mathematics. In the first grade, each number has a corresponding combination chart.

Correctly grasp the teaching requirements of set thought, guide students to understand set diagram and use it to calculate or solve problems.

Guide students to study the relationship between concepts from the perspective of structural combination. Teachers should seize the opportunity to infiltrate the set idea when learning the knowledge of numbers, the nature of numbers, the classification of triangles, the understanding of quadrangles, and the characteristics of cuboids and cubes.

4. The idea of combining numbers with shapes

The curriculum standard points out that geometric intuition mainly refers to describing and analyzing problems with figures, which also highlights that the combination of numbers and shapes is an important method and means of geometric intuition.

Conceptual explanation

The application of the thinking method of combining numbers and shapes is reflected in two aspects: one is to use shapes to assist numbers, and the other is to use numbers to solve shapes. Among them, using numbers to solve shapes is more common in middle school mathematics, and using shapes to assist numbers is more common in primary school mathematics learning.

Pupils' logical thinking ability is relatively weak, and their understanding of abstract concepts basically depends on perceptual and intuitive materials. Therefore, with the help of the intuitive means of graphics in the idea of tree combination, better teaching methods and problem-solving strategies are provided for students to learn and solve problems.

Teaching suggestion

First, study the teaching materials and grasp the starting point of the tree combination of thinking and methods as a whole.

Second, strengthen the value experience and enhance the awareness and skills of using maps.

4. Corresponding ideas

Correspondence reflects the relationship between two combined elements, and correspondence can be seen everywhere in primary school mathematics. For example, we should find the corresponding relationship and use it to solve the problem.

Teaching suggestion

Strengthen students' understanding of the corresponding relationship through intuitive teaching.

Guide students to solve problems by letters.

Conform to the idea

The curriculum standard points out that the symbol consciousness mainly refers to being able to understand and use symbols to express the quantitative relationship and changing law of numbers, knowing that using symbols can be used for operation and reasoning, and the conclusion is general.

A symbol is a concise symbol or code that abstracts a specific object. April symbol is a tool to express the relationship between spatial form and quantity, calculate and reason, and solve problems. It is the most intuitive and concise expression of people's laws of objective things, and it is the medium for exchanging and spreading mathematical ideas.

Symbol is not only a way of expression, but also related to specific contents such as mathematical concepts and propositions, which directly reflects the requirements of basic ideas such as abstract reasoning and models.

(1) can understand and use symbols to express the number-quantity relationship and variation law.

② Knowing that symbols can be used for operation and reasoning, the conclusion is average.

③ Let students understand that the use of symbols is an important form of mathematical expression and thinking.

Teaching suggestion

Mathematics learning is always dealing with mathematical symbols. In primary school, teachers should grasp the following points to infiltrate symbolic thoughts and cultivate students' symbolic consciousness.

① Understand the meaning of mathematical symbols by learning concepts, propositions and formulas.

Attach importance to the teaching of using letters to represent numbers, and initially develop students' ability to express, calculate and reason with symbols.

6. The idea of combining numbers with shapes

As a mathematical thinking method, the combination of numbers and shapes refers to the thinking method of solving problems through the relationship between numbers and shapes and their mutual transformation.

When the curriculum standard expounds geometric intuition, it points out that geometric intuition mainly refers to describing and analyzing problems with graphics, and highlights that the combination of numbers and shapes is an important method and means of geometric intuition.

Conceptual explanation

A poem in Mr. Hua's "Talking about Mathematical Problems Related to Honeycomb Structure" vividly records the relationship between number and shape. Numbers and shapes depend on each other. How can they be divided into two sides? When numbers are invisible, they are not so intuitive. When there are few shapes, it is difficult to be nuanced. When numbers and shapes are combined in various ways, everything should be separated. Don't forget that geometry and algebra are unified, always linked and never separated. The application of the combination of numbers and shapes embodies two ways, one is to help numbers with shapes, and the other is to solve shapes with numbers.

Teaching suggestion

First, study the teaching materials and grasp the starting point of the thinking method of combining numbers with shapes.

Second, strengthen the value experience of form and enhance the awareness and skills of using pictures.

7. Analogical thinking

Simple * * * existential analogy

Causal analogy

Comprehensive analogy

Teaching suggestion

Understand the learning content from the perspective of connection and development, and tap the analogical thinking in the teaching content.

In concept teaching and problem solving, experience the process of analogy and master the basic methods and steps.

8. Extreme thinking

In the process of deducing the formula of circular area, the idea of limit runs through.

The general steps of limit thought can be summarized as follows: for the unknown quantity under investigation, first try to conceive a variable related to it, confirm this variable, and the result of infinite approximation process is the unknown quantity, and finally get this result through limit calculation.

Teaching suggestion

Infiltrate and accumulate mathematical experience at any time,

Seize the opportunity, locate and limit your thoughts.

When teaching decimals, we can also seize the opportunity to infiltrate extreme thoughts with the help of mathematical stories.

9. Alternative ideas

Equivalent substitution refers to replacing its equivalent quantity with a quantity, which is a basic thinking method in mathematics and the basis of algebraic thinking method.

Conceptual explanation

Method of substitution's thought can also be understood as method of substitution, which generally means that it will be a part of a mathematical expression composed of one or several variables, and the intentional expression of variables is also conducive to solving problems.

Teaching suggestion

Equivalent substitution is a very abstract mathematical idea. Only by presenting it vividly and interestingly in a simple form that students can understand can they perceive and understand.

First, pay attention to students' interests and stimulate their desire for learning.

Second, contact life experience to guide students to explore new knowledge and understand the significance of equivalent substitution.