Реферат: Acquaintance with geometry as one of the main goals of teaching mathematics to preschool children

The view that seems most suitable for young children is that inspired by cognitive theorists, primary among them Jean Piaget. Three types of knowledge were identified by Piaget (Kamii and Joseph, 1989), all of which are needed for understanding mathematics. The first is physical, or empirical, knowledge, which means being able to relate to the physical world. For example, before a child can count marbles by dropping them into a jar, she needs to know how to hold a marble and how it will fall downward when dropped.

The second type of knowledge is logico-mathematical, and concerns rela­tionships as created by the child. Perhaps a young child holds a large red marble in one hand and a small blue marble in the other. If she simply feels their weight and sees their colors, her knowledge is physical (or empirical). But if she notes the differences and similarities between the two, she has mentally created relationships.

The third type of knowledge is social knowledge, which is arbitrary and designed by people. For example, naming numbers one, two, and three is social knowledge because, in another society, the numbers might be ichi, ni, san or uno, dos, tres. (Keep in mind, however, that the real understanding of what these numbers mean belongs to logico-mathematical knowledge.)

Constance Kamii (Kamii and DeClark, 1985), a Piagetian researcher, has spent many years studying the mathematical learning of young children. After analyzing teaching techniques, the views of math educators, and Ameri­can math textbooks, she has concluded that our educational system often confuses these three kinds of knowledge. Educators tend to provide children with plenty of manipulatives, assuming that they will internalize mathemati­cal understanding simply from this physical experience. Or educators ignore the manipulatives and focus instead on pencil-and-paper activities aimed at teaching the names of numbers and various mathematical terms, assuming that this social knowledge will be internalized as real math learning. Some­thing is missing from both approaches, says Kamii.

Traditionally, mathematics educators have not made the distinction among the three kinds of knowledge and believe that arithmetic must be internalized from objects (as if it were physical knowledge) and people (as if it were social knowledge). They overlook the most important part of arithmetic, which is logico-mathematical knowledge.

In the Piagetian tradition, Kamii argues that "children should reinvent arith­metic." Only by constructing their own knowledge can children really under­stand mathematical concepts. When they permit children to learn in this fash­ion, adults may find that they are introducing some concepts too early while putting others off too long. Kamii's research has led her to conclude, as Su­zanne Colvin did, that first graders End subtraction too difficult. Kamii argues for saving it until later, when it can be learned quickly and easily. She also points to studies in which place value is mastered by about 50 percent of fourth graders and 23 percent of a group of second graders. Yet place value and regrouping are regularly expected of second graders!

As an example of what children can do earlier than expected, Kamii (1985) points to their discovery (or reinvention) of negative numbers, a con­cept that doesn't even appear in elementary math textbooks. Based on her experiences with young children, Kamii argues that it is important to let chil­dren think for themselves and invent their own mathematical systems. With Piaget, she believes that children will understand much more, developing a better cognitive foundation as well as self-confidence: children who are confident will learn more in the long run than those who have been taught in ways that make them distrust their own think­ing. . . . Children who are excited about explaining their own ideas will go much farther in the long run than those who can only follow some­body else's rules and respond to unfamiliar problems by saying, "I don't know how to do it because I haven't learned it in school yet."

In recent years, the National Council of Teachers of Mathematics (NCTM) has given much consideration to the international failure of Ameri­can children in mathematics, and has devised a set of standards that echo, in many ways, the Piagetian perspective of Kamii. The Curriculum and Evaluation Standards for School Mathematics (1989) prepared by the NCTM addresses the education of children from kindergarten up. Some of the more important standards are:

■ Children will be actively involved in doing mathematics. NCTM sees young chil­dren constructing their own learning by interacting with materials, other children, and their teachers. Discussion and writing help make new ideas clear. Language is at first informal, the children's own, and gradually takes on the vocabulary of more formal mathematics.

■ The curriculum will emphasize a broad range of content. Children's learning should not be confined to arithmetic, but should include other fields of mathematics such as geometry, measurement, statistics, probability, and algebra. Study in all these fields presents a more realistic view of the world in which they live and provides a foundation for more advanced study in each area. All these content areas should appear frequently and throughout the entire curriculum.

■ The curriculum will emphasize mathematics concepts. Emphasis on concepts rather than on skills leads to deeper understanding. Learning activities should build on the intuitive, informal knowledge that children bring to the classroom.

■ Problem solving and problem-solving, approaches to instruction will permeate the cur­ riculum. When children have plenty of problem-solving experiences, partic­ularly concerning situations from their own worlds, mathematics becomes more meaningful to them. They should be given opportunities to solve problems in different ways, create problems related to data they have col­lected, and make generalizations from basic information. Problem-solving experiences should lead to more self-confidence for children.

■ The curriculum will emphasize a broad approach to computation. Children will be permitted to use their own strategies when computing, not just those of­fered by adults. They should have opportunities to make informal judg­ments about their answers, leading to their own constructed understanding of what is reasonable. Calculators should be permitted as tools of explora­tion. It may be that children will compute by using thinking strategies, es­timation, and calculators before they are presented with pencils and paper (Adapted from Trafton and Bloom, 1990).

The National Association for the Education of Young Children, in its position statement regardingDevelopmental / Appropriate Practices (Bredecamp,1987), arrives at views of teaching mathematics to young children that reflect those of Constance Kamii and the NCTM. Their position regarding infants, toddlers, and preschoolers is that mathematics should be part of the day's natural activities: counting children in the class or crackers for snacks, for example. For the primary grades they are more specific, identifying what is appropriate and inappropriate practice. Table 1 summarizes their guide­lines.

Table1.APPROPRIATE MATHEMATICS IN THE PRIMARY GRADES (THE NAEYC POSITION)

APPROPRIATE PRACTICE	INAPPROPRIATE PRACTICE
Learning is through exploration, discovery, and solving meaningful problems	Noncompetitive, impromptu oral "math stumper" and number games are played for practice.
Math activities are integrated with other subjects such as science and social studies	Learning is by textbook, workbooks, practice sheets, and board work
Math skills are acquired through play, projects, and daily living	Math is taught as a separate subject at a scheduled time each day
The teacher's edition of the text is used as a guide to structure learning situations and stimulate ideas for projects	Timed tests on number facts are given and graded daily
Many manipulatives are used including board, card, and paper-and-pencil games	Teachers move sequentially through the lessons as outlined in the teacher's edition of the text
Only children who finish their math seatwork are permitted to use the few available manipulatives and games
Competition between children is used to motivate children to learn math facts.

The NCTM Standards, the NAEYC position statement, and studies with young children carried out by such researchers as Constance Kamii and Su­zanne Colvin bring us to today's best analysis of how children learn mathematics. The conclusion these researchers and theorists have reached are based not only on their work with children, but on their understanding of child de­velopment [6, pp. 426 - 436].