### Characteristics of Young Children's Mathematical Ability Development
**(0-6 years old)**
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#### 1. Age 0～3
**Understanding Numbers**
From ages 0 to 3, children have a vague perception of quantity and can clearly distinguish between more and less. For example, if you hold 2 candies in your right hand and 3 candies in your left hand, a 2-year-old child will know to choose the hand with more candies. Under adult guidance, children around the age of 3 gradually learn to say individual number words and recite the names of numbers up to [5] by memory in a certain order. However, at this stage, children cannot truly understand the meaning of numbers; even if they can count objects one by one, they cannot state how many objects there are.
**Understanding Relationships**
Mathematics focuses on the relationships between objective things, which is an abstraction of these relationships. In mathematics, relationships are everywhere. Here, we only discuss sorting and patterns. Sorting and patterns both represent and apply relationships. Sorting refers to arranging things according to a certain logical relationship in a specific order. Patterns refer to the repetitive appearance of rule-based figures, sounds, or events. The repeating unit is the core of the pattern, and the various parts that make up this core are called elements. Based on the characteristics of the core and elements, patterns can be divided into two types: repetitive patterns, such as leaf - small stone - pencil, leaf - small stone - pencil... and growth patterns, such as □○, □○○, □○○○...
Children aged 0-3 have no sorting ability, but they can identify the largest and smallest objects.
**Understanding Quantity**
Children’s understanding of the quantity of objects mainly relies on perception, which has significant limitations and lacks differentiation and precision.
Children aged 0-3 generally do not recognize differences in the quantity of objects.
**Understanding Shapes**
Children aged 0-3 have limited understanding of shapes.
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#### 2. Age 3～5
**Understanding Quantity**
Children aged 3-5 can generally correctly distinguish the size differences of objects and use some simple words to express corresponding quantities, such as ["I am holding a big doll"]. Their accuracy in perceiving object sizes improves, allowing them to judge the largest or smallest object in a group where the differences are not too obvious. However, children at this age cannot yet recognize other differences in quantity or express them with words. They often describe height, thickness, length, width, etc., vaguely as [big] or [small].
**Understanding Numbers**
Children aged 3-5 can state the total after counting physical objects and can retrieve a corresponding number of objects based on the number said by an adult. At this stage, children gradually understand the relationships between numbers, such as 2 being smaller than 3, 3 being larger than 2, and 3 consisting of 2 and 1. If provided with manipulative materials, children can perform simple addition and subtraction. The **Whole Brain Mathematics** curriculum [http://www.djjcw.cn/product_detail-0-118.aspx] can provide teaching guidance for children during this period. All of this indicates that children have formed a basic concept of numbers at a relatively low level.
**Understanding Shapes**
Children aged 3-5 can correctly recognize and differentiate circles, squares, and triangles, and have a certain matching ability for ellipses, rectangles, and semicircles. They can find similar shapes based on examples provided by adults. At this stage, children can correctly recognize basic shapes and gradually understand the fundamental characteristics of plane figures. They can achieve shape constancy, recognizing shapes regardless of their size or orientation. They can also compare similar plane figures and understand simple relationships between shapes.
**Understanding Relationships**
Children aged 3-5 still have low sorting abilities and tend to arrange objects randomly.
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#### 3. Age 5～6
**Understanding Numbers**
Children aged 5-6 realize that numbers do not change with changes in physical objects, forming the concept of "number conservation," though this understanding is limited to numbers within 10. They can perform abstract calculations of small numbers without relying on physical objects and can count up to 100. At this stage, children have formed a relatively advanced concept of numbers and begin transitioning from concrete imagery to abstract numerical operations.
**Understanding Quantity**
Children aged 5-6 significantly improve their precision in perceiving quantities. They can distinguish and arrange objects of different sizes and more accurately recognize and differentiate height, thickness, length, and width, learning to use appropriate words to describe these attributes. Although they can judge equal quantities, they still lack an understanding of "quantity conservation." For instance, if equal amounts of water are poured into a tall, thin glass and a short, wide glass, they may struggle to determine whether the two glasses contain the same amount of water. Additionally, children at this age can correctly recognize and describe various characteristics of object quantities and better understand the relativity of quantities. They gradually learn to understand the reversibility and transitivity of quantities based on logic.
**Understanding Relationships**
Children aged 5-6 can sort objects by length or size, though this is not done systematically. Sorting tasks are often completed through trial and error. Children at this age find it easier to master repetitive patterns compared to growth patterns.
**Understanding Shapes**
Children aged 5-6 can basically understand the typical features of shapes and form a "standard model" of shapes in their minds, allowing them to make correct judgments. They can further understand complex relationships between shapes. Research abroad shows that children aged 5-6 can generalize relationships between shapes at a certain level of abstraction. At this stage, children can also recognize basic three-dimensional shapes such as spheres, cubes, and cylinders, and name them correctly.