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







Executed by:

student of magistracy department

Yulia Аndreevna Dunai

(tel.: 8-029-3468595)

Scientific Supervisor:


Doctor of pedagogical science,

I.V. Zhitko

English Supervisor:

Doctor of Psychology

Associate Professor

N. G. Olovnikova

Minsk, 2009










Young children "do" math spontaneously in their lives and in their play. Mathe­matical learning for young children is much more than the traditional counting and arithmetic skills. It includes a variety of mathematical sections of among which the important place belongs to geometry. We've all seen preschoolers exploring shapes and patterns, drawing and creating geometric designs, taking joy in recognizing and naming specific shapes they see. This is geometry — an area of mathematics that is one of the most natural and fun for young children.

Geometry is the study of shapes, both flat and three dimensional, and their relationships in space.

Preschool and kindergartenchildren can learn much from playing with blocks, manipulatives (Jensen and О'Neil), different but ordinary objects ( Julie Sarama, Douglas H. Clements), boxes, snacks and meal (Ellen Booth Church). Also card games, computer games, board games, and others all help children learn geometry. 

This problem is relevant because the geometrical concepts should be formed since early childhood. Geometrical concepts help children to perceive the world. Also it will provide future success in academic achievement : as the rudiments , children learn in primary school, from the basis for further learning of geometry. Game methods help children to understand some complex phenomena in geometry. They also are necessary for the development of emotionally-positive attitudes and interest to the mathematics and geometry.


Throughout history, mathematical concepts and systems have been de­veloped in response to real-life problems. For example, the zero, which was invented by the Babylonians around 700 в.с, by the Mayans about 400 a.d., and by the Hindus about 800 a.d., was first used to fill a column of numbers in which there were none desired. For example, an 8 and a 3 next to each other is 83; but if you want the number to read 803 and you put something between the 8 and 3 (other than empty space), it is more likely to be read accurately (Baroody, 1987). When it comes to counting, tallying, or thinking about numerical quantity in general, the human physiological fact of ten fingers and ten toes has led in all mathematical cultures to some sort of decimal system.

History's early focus on applied mathematics is a viewpoint we would do well to remember today. A few hundred years ago a university student was considered educated if he could use his fingers to do simple operations of arithmetic (Baroody, 1987); now we expect the same of an elementary school child. The amount of mathematical knowledge expected of children today has become so extensive and complex that it is easy to forget that solving real-life problems is the ultimate goal of mathematical learning. The first grad­ers in Suzanne Colvin's classes demonstrated the effectiveness of lying in­struction to meaningful situations.

It’s possible to recall that more than 300 years ago, Comenius pointed out that young children might be taught to count but that it takes longer for them to understand what the numbers mean. Today, classroom research such as Su­zanne Colvin's demonstrates that young children need to be given meaning­ful situations first and then numbers that represent various components and relationships within the situations.

The influences of John Locke and Jean Jacques Rousseau are felt today as well. Locke shared a popular view of the time that the world was a fixed, mechanical system with a body of knowledge for all to learn. When he ap­plied this view to education, Locke described the teaching and learning pro­cess as writing this world of knowledge on the blank-slate mind of the child. In this century, Locke's view continues to be a popular one. It is especially popular in mathematics, where it can be more easily argued that, at least at the early levels, there is a body of knowledge for children to learn.

B. F. Skinner, who applied this view to a philosophy of behaviorism, re­ferred to mathematics as "one of the drill subjects." While Locke recommended entertaining games to teach arithmetic facts, Skinner developed teaching machines and accompanying drills, precursors to today's computerized math drills. One critic of this approach to mathematics learning has said that, while it may be useful for memorizing numbers such as those in a telephone listing, it has failed to provide a powerful explanation of more complex form: of learning and thinking, such as memorizing meaningful information or problem solving. This approach has, in particular, been unable to provide a sound description of the complexities involved in school learning, like the mean­ingful learning of the basic combinations or solving word problems (Baroody, 1987).

Rousseau's views of how children learn were quite different, reflecting his preference for natural learning in a supportive environment. During the late eighteenth century as today, this view argues for real-life, informal mathemat­ics learning. While this approach is more closely aligned to current thinking about the way children learn than is the Locke/Skinner approach, it can have the undesired effect of giving children so little guidance that they learn almost nothing at all.

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 cur­riculum. 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 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 guide­lines.



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 studiesLearning is by textbook, workbooks, practice sheets, and board work
Math skills are acquired through play, projects, and daily livingMath 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.