1, the formation of the concept of child number
Mathematics is a highly structured abstract symbol system with internal logical relations. Meaningful mathematics learning involves the active construction of understanding these relationships, which is based on children's existing development level and knowledge and experience. It takes several years to form the concept of the number of children. Research shows that although 2-year-old children can't count yet, they already have the ability to visually check the numbers. Visual inspection means that you can quickly and accurately tell the number of small sets without counting. Adults and children have this ability. 2-year-old children can accurately observe 1-3 objects, and the visual range of 3-year-old children is improved to 4, and the maximum visual number of 4-5-year-old children is 5. Children's innate ability to observe small numbers may provide a basis for the formation of their acquired concept of numbers.
To a great extent, the development of children's number concept is an acquired learning and construction, which depends on a lot of perceptual experience and reflection on their own operation experience. At first, children unconsciously learn to sing Arabic numerals with adults, but they don't really understand their quantitative significance; Then they have to learn the skills of counting procedures under the demonstration of adults, and try to correspond oral counting, finger counting and counted objects one by one. At first, this kind of correspondence is not easy for children, because it involves the cooperation of several skills, so their counting is often wrong, sometimes they miss counting, and sometimes they miss pointing objects. Children in our country generally form the concept of cardinal number between the ages of three and a half and four. In other words, many children often count before the age of 3-4, but they don't really understand the concept of cardinality. Children's understanding of cardinality is gradually developed from their counting practice. It is counting, especially the action of counting and the reflection on action, which plays a bridge role and helps children establish the connection between concrete objects and abstract number concepts. Children construct a preliminary understanding of logarithm with the help of objects and operations. With the improvement of children's understanding of logarithm, the abstraction of the entity they want to use is also improving. For example, at first they have to solve numbers on the object level, then they can use their fingers as substitutes for objects, and then they can solve numbers on the representation level. Finally, they can fully understand the meaning of a number with only symbols, without relying on any specific media.
2. Children's understanding of addition and subtraction and the development of calculation ability.
The development of the concept of children's cardinality is a necessary prerequisite for addition and subtraction. The development of children's addition and subtraction ability has gone through the process from physical operation to representation to concept, from concrete to abstract. Children often encounter practical problems related to the increase or decrease of food or toys in their daily life, so they have actually accumulated rich experience in addition and subtraction at the physical level. For children, addition and subtraction seem to be an extension of counting. Addition is to count the objects of two sets together, that is, to use the physical operation method of "full number". With the accumulation of experience, when children use the method of "full number" to complete the addition operation, they may find it unnecessary to count all the objects one by one, so they can take one of the known object sets as a given condition and join the other object set. For example, three objects plus two objects, starting from three, plus four plus five. This method is called "from one addend to another". After children learn how to count from one addend to another, they may soon find it easier to count from a big addend. This method of "adding from one number to the next" can be done not only at the physical operation level, but also at the representation and symbol level.
The addition and subtraction operation at the representation level means that children can complete the addition and subtraction operation by imagining physical objects and using oral calculation. It is considered that this is a transitional period from physical operation to symbolic operation, not everyone has to go through this stage, and the development of addition and subtraction operation ability of representation level varies from person to person. When children have been able to skillfully add and subtract with counting instead of objects or representations, their understanding of logarithm has completed the transformation from physical operation to concept. In the end, children can completely get rid of physical or digital help. Conceptually, if they only use mental arithmetic to add and subtract, that is, they use the knowledge of decomposition and combination of numbers within 10 or the knowledge of formulas, such as knowing that 2+2=4, 5+5= 10, 2+8= 10 and 3+7 =.
The development of preschool children's addition and subtraction ability has a clear development track from concrete to abstract, but the stages of development are not uniform, and there will be cross before and after, that is, different levels of operation strategies may be used at the same stage. Children are spontaneously constructing their own understanding of addition and subtraction and strategies to solve the problems of addition and subtraction, whether in the physical, representation (counting) or conceptual development stage. For example, in the first two stages, some children will spontaneously switch from "all numbers" to a more effective operation method of "counting from an addend"; In the process of using physical objects or counting methods, some children will spontaneously construct a more effective strategy of "counting from big numbers"; In the development stage of mental arithmetic, children will actively construct methods of calculating with different numbers.
3. The development of children's concepts of shape and space.
Perception and abstraction of object shape. Children's understanding of geometric shapes depends on spatial perception, and it takes a development process from the perception of geometric shapes to the ability to express them with corresponding words, which is manifested in three levels: (1) being able to match plane geometric figures with names; Graphics can be identified according to their names; Can name plane graphics (Lin Jiasui, Li,, 1999). Children's understanding of geometric shapes is plane first, then solid. The research shows that the correct rate of preschool children's recognition of several common plane geometric shapes is circle, triangle, square and rectangle from high to low. Preschool children's recognition accuracy of other shapes is semicircle, trapezoid, diamond and polygon from high to low (Chang Hong, 2009). 3-4-year-old children can have good plane graphics matching ability; 4-5 years old may be an important period for children to understand shapes. Most children aged 4-5 can recognize ellipse, semicircle, trapezoid, sector and so on besides several common shapes. They have begun to understand the simple relationship between plane graphics, that is, they can make a preliminary transformation of the graphics they know, such as dividing, combining, disassembling and spelling. A considerable number of 4-5-year-old children show the ability of graph conservation, that is, they can correctly identify and name graphs regardless of their size, color and placement. Children have the ability to recognize basic plane geometric figures since they were five years old, and their ability to combine figures has also developed significantly.
Spatial cognitive ability mainly refers to the cognition of the spatial relationship of the object and the position of the subject itself in space, which reflects the individual's perception, understanding and application of spatial information (Dong Qi, Tao Sha, 2002). In the concept of space, children only master some basic concepts and words of orientation, such as up and down, front and back, inside and outside, left and right, etc. Because these concepts of orientation are generally relative, and because the foothold of judging orientation will change, the orientation relationship will also change. This relativity of spatial position and spatial relationship may be one of the reasons for the slow development of children's spatial cognitive ability. The research shows that children's understanding of the basic orientation of space is first up and down, then back and forth, and finally left and right. At the age of 3, the concept of up-and-down orientation has been established. At the age of 4, the concept of up and down began to develop at the age of 5. Children can first determine the left and right with themselves as the center, and then develop to determine the left and right with objects as the center. Children can completely and correctly distinguish up, down, front and back directions when they are about 6 years old, but their self-centered ability to distinguish left and right directions has not been developed (Li Dan, 1992).
The development of spatial representation. The research shows that the development of children's spatial representation shows three reference systems: egocentric reference system, fixed reference system and coordinated reference system. In early childhood, spatial reasoning basically uses a self-centered reference system, but in a familiar environment, we can sometimes judge spatial relations with the help of external environment.
4. The development of children's cognitive ability.
This model is to discover or create regular auditory, visual and physical movements (rosalind Charlesworth, 2007). Common patterns include simple patterns, such as arranging two objects or patterns according to certain rules, such as colors or shapes; Digital mode, such as arranging numbers according to certain rules; Patterns in nature, such as cobwebs and shells. The development of children's pattern cognitive ability is generally manifested in this feature. After the age of 3, children show the ability to distinguish simple patterns in the surrounding living environment, including the basic characteristics and arrangement rules of things in the patterns. Subsequently, children can make some predictions on the arrangement of patterns on the basis of pattern copying, which is embodied in that they can complete the arrangement of existing patterns with the same elements on the basis of the arrangement of existing patterns; Finally, children can create their own patterns without any pattern demonstration, or further, they can express patterns with different forms of abstract symbols.