Considering the importance of order, environment and accuracy, Dr. Montessori emphasized: "It is necessary to prepare in advance, that is, sensory education must be carried out before mathematics education."
(A) based on sensory education
In real life, there are countless things around children, but for many things, the same attributes (size, shape, color, weight, etc. ) are limited. First of all, we explain and demonstrate things to children by summarizing various attributes of things, fully mobilize children's various sensory organs, let children concentrate on their own sensory organs, achieve full attention, and let children firmly grasp those abstract elements and relationships.
Then, based on the three operating modes of sensory teaching AIDS (see below), help children analyze and synthesize, cultivate their logical thinking ability, and make them master the essence of things.
Understanding these * * * same properties of all things in nature is the basis of children's mathematics education, because they are not quantified before entering the cognitive quantity in the future. Dr. Montessori summarized and classified these attributes and infiltrated them into colorful sensory teaching AIDS. By repeatedly operating these teaching AIDS, children's sensory organs are stimulated, children's attention is attracted, and children gradually feel and understand these abstract elements and their relationships.
There are three basic ways to operate the sensory teaching aid mentioned above (see the book Sensory Education and Teaching Aid Operation Manual in this series of tutorials for details), namely, the three elements of teaching aid operation, namely:
1. Pairing-Corresponding Pairing (P)
The operation of pairing in pairs. For example, in a cylindrical teaching aid, find a type with the same size (or the same height and thickness), or find the same color on the color drawing board, so that children can understand the concept of "equality" first, and then further learn and accept the concept of number.
2. Sequence sorting and grading (G)
Hierarchical sorting and sequential operation. The following operations include sequential relations: cylinders have size order, red bars have length order and thickness order. Numbers themselves have size and order, and natural numbers are arranged in the order of 1, 2, 3 and 4. If you want children to understand the relationship between inequality and greater than less than, you must fully understand and master the order. Only by mastering the order can children make a correct judgment.
3. Classification and sorting
The operation of distinguishing categories and classifying is the operation of collecting things with the same attributes and classifying them. When children do addition and subtraction, it is important to understand the relationship between the whole and the part. Children often forget the part when considering the whole, and forget the whole when considering the part. When using "number" to do classification operation, we must know the concepts of whole and part and the coherent relationship between them.
The ability of "pairing", "grading" and "classification" is the basis of one's logical thinking. Operations such as "matching", "grading" and "classification" can help children improve their abstract thinking ability of mathematical analysis, induction and synthesis and cultivate their keen observation ability.
In this regard, Dr. Montessori has the following discussion: "Feeling is the entrance of the spirit, and all cognition is first obtained from feeling. The basis of external psychological cognition is the basis of sensory cognition, that is, sensory cognition. Collecting and comparing various experiences is the first step to form a spirit. " (Quoted from "The Secret of Children")
She also said, "What we call sensory exercise is to cultivate children's ability to distinguish or classify." In this way, through the sensory organs and the three basic operations of sensory teaching AIDS, children's ability to think logically about things should be cultivated. First, children should be trained in comprehensive analysis and thinking, and the structure of things or phenomena should be clearly grasped.
Therefore, feeling practice and feeling education are necessary preparatory courses for mathematics education, and learning mathematics must be based on feeling.
(B) the concretization of abstract things
Facts have proved that language is helpful to thinking and is a link that cannot be ignored in teaching. In mathematics education, especially for the concretization of abstract things, we need the guidance of language.
In the process of implementing mathematics education, whether it is educational tips or the use of teaching AIDS, we need to combine specific examples to carry out "naming exercises" to guide children to be familiar with quantity in life and help them master basic mathematical concepts.
In order to help children better understand teaching AIDS, and at the same time let teachers know how much children absorb and understand teaching AIDS, "three-stage teaching method" can be adopted in the teaching process of name practice, which runs through every link of teaching. This can not only achieve the purpose of heuristic teaching, but also expand children's vocabulary.
When children operate, they need to show them different objects and compare them. For example: big-small, big-bigger-largest, rough-smooth, light-heavy, uppercase-lowercase, small-smaller-smallest, heavy-light, hard-soft, etc.
"Three-stage teaching method" refers to:
1. The first stage-be worthy of the name.
"This is 1" "This is big" and so on.
2. The second stage-find the object with the same name.
"Which is 1?" "Which is bigger?" Wait a minute.
3. The third stage-remember names and objects
"How much is this?" "Which is the biggest?" Wait a minute.
For the detailed knowledge of Montessori's "three-stage teaching method", please refer to this series of tutorials "Montessori Daily Life Education and Teaching Aid Operation Manual".
(C) a complete curriculum system
Another feature of Montessori mathematics education is that it has a complete curriculum system, and the curriculum design and teaching activities follow the scientific logical process. Dr. Montessori used appropriate teaching AIDS in Mathematics Education, focusing on the decimal method, and then entered the four operations of using memory, closely combining "quantity", "number" and "numeral". In the implementation of teaching, we should start with the concept of "quantity" and give systematic guidance repeatedly and carefully according to the teaching system. In the teaching process, adults should choose appropriate teaching AIDS according to the different characteristics of children and make a learning plan for teaching AIDS purposefully.
In the second chapter of this book, "Implementation of Mathematics Education", we will describe how to make a learning plan for teaching AIDS and how to arrange mathematics teaching AIDS according to certain scientific categories. For example, when should teaching AIDS be presented throughout the course? How much does the child know about mathematics? To what extent, etc.
It all depends on the progress of the course. In short, on the one hand, teachers should teach children according to the learning plan of teaching AIDS; On the other hand, children should be guided to practice math teaching AIDS step by step according to their own understanding.
(D) The characteristics of Montessori mathematics teaching AIDS
In the field of abstract mathematics, Montessori mathematics teaching AIDS play a unique and important role, which has the following distinctive characteristics.
1. Take sensory education as the basis of mathematics education.
2. Start with real quantitative understanding. For example: counting rod, spindle box, etc.
3. Pay attention to the relationship between quantity, number and numeral. For example, count sticks, numbers and chips.
4. Use Arabic numerals and unify fonts. For example, the same Arabic numeral font is used for sandpaper numeral boards, numerals and chips, headstock, etc.
5. Pay attention to the concept of "0" and decimal calculation. For example: headstock and decimal.
6. Generally, the operation standard of synthetic decomposition is "10" instead of "5". For example, several strokes.
7. In order to indicate the orientation or calculate the quantity, the principle of "color classification" should be applied to the quantity and quantity of teaching AIDS. For example, the unit, tenth and hundredth digits of stamp games are green, blue and red respectively; Color beading is also the number corresponding to color, usually red beading 1, green beading 2, etc.
8. In the "error correction" project in mathematics education, the function of prompting is realized in the form of calculation or error control panel. For example: 100 board, addition table, etc.
9. The design and teaching of Montessori mathematics teaching AIDS follow certain logical principles, so when operating teaching AIDS, we should first master the basic concepts of numbers, and then proceed in the direction of "combination" and "decomposition".