In recent years, the National Council of Teachers of Mathematics (NCTM) has given much consideration to the international failure of Ameri-can 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 chil-dren 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, partic-ularly 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 col-lected, 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 of-fered by adults. They should have opportunities to make informal judg-ments about their answers, leading to their own constructed understanding of what is reasonable. Calculators should be permitted as tools of explora-tion. It may be that children will compute by using thinking strategies, es-timation, 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 regarding Developmental / 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 guide-lines.

Table 1. APPROPRIATE MATHEMATICS IN THE PRIMARY GRADES (THE NAEYC POSITION)

The NCTM Standards, the NAEYC position statement, and studies with young children carried out by such researchers as Constance Kamii and Su-zanne 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 de-velopment [6, pp. 426 - 436].

II. THE PURPOSES AND THE CONTENT OF MODERN MATHEMATICAL EDUCATION IN PRIMARY SCHOOL

Often children question the importance of learn-ing mathematics. Now that handheld calculators and home computers are commonly available, questions about the relevance of learning math have become louder. Nevertheless, educators continue to make math the second most time-consuming subject in elementary school (after reading). The reasons for teaching math are many, and the goals of general education require that math be a major part of the curriculum.

The goals of math education change slowly from grade to grade. Most children require all the time from preschool through the end of grade 6 just to learn the meaning of whole numbers, fractions, and decimals and how to perform operations with them (Of course, a number of other mathematical ideas are also taught along the way). Although actual computations can of-ten be done with a calculator, answers are of no use without an understanding of basic math processes.

Businesspeople who are involved in setting prices find that elementary algebra is helpful. Geometry is more than useful in planning many sewing projects. Scien-tists of all kinds, including biologists and social scien-tists, need calculus to solve problems and do research.

As a result, high-school math courses are largely designed to provide the basics that are needed in such situations and to prepare students for college. Some colleges require all students to take mathematics, but many have math requirements only for students of sci-ence, engineering, and advanced planning for business.

Nearly everyone starts learning mathematics before going to school. When television first became popular in the 1950's, some people joked that children were coming to kindergarten already able to count at least as high as the numbers on the channel selector. But the joke turned serious when people realized that very young children really were learning to count from TV, especially if they watched educational shows such as today's "Sesame Street." The tots also learned colors, shapes, and directions--subjects that usually form a large part of the kindergarten mathematics program [7, p. 13].

Mathematics learning is sequential--one idea builds on another. Consequently mathematics is taught in nearly the same sequence in almost every school in the United States [7, p. 29].

In preschool (с 2 till 5 years) the children gain informal practice with count-ing and shapes. So, one of the first goals of a kindergarten mathematics program is to present numbers and counting in ways that show how words, meanings, and the symbols that represent them are related. The symbols, such as the numerals 1 through 10 are especially important because many children can count correctly before they are able to get any meaning from the symbols [7, p. 14].

They also learn the meaning of words such as top, in, and left. Preschools put much empha-sis on games and activities that use simple counting. Reading and writing numerals are almost never taught.

Not all children go to preschool. All of the topics covered at that level are taught again in kindergarten and grade I. The schools cannot assume that all children will have had the same early math experi-ences.

Today, nearly all children in the United States go to kindergarten (с 5 years). The beginning part of kindergarten fo-cuses on informal experiences similar to those in preschool. Later in the year, more formal experiences start. Sometimes books or kits are used to organize mathematics learning, but many kindergarten teachers believe that it is too early to ask children to work with books or even with specific mathematics materials. Some classrooms may have a computer with math-related software to help teach early math concepts.

Children learn two ways to compare numbers. Thus, even before they learn the order of the numbers, children can un-derstand that some numbers are larger than others and that some numbers are smaller [7, p. 30].

Another important early skill is writing numerals. This skill is essential because it enables children to communicate on paper with their teachers and with others in later life.

Although understanding the meaning of numbers is the main goal of beginning mathematics education, it is not the only goal. There are numerous subgoals. In kindergarten or grade 1, the first subgoal may be to teach such basic concepts as top, left, and before. These ideas have many important uses both inside and outside the classroom. For example, a teacher will lose much time explaining if children do not understand a simple direction such as "Look at the picture at the top of the page."

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