MINISTRY OF EDUCATION OF THE REPUBLIC OF BELARUS
EDUCATIONAL ESTABLISHMENT “ BELARUSIAN STATE
PEDAGOGICAL UNIVERSITY NAMED AFTER MAXIM TANK”
FOREIGN LANGUAGES DEPARTMENT
ACQUAINTANCE WITH GEOMETRY AS ONE OF THE MAIN GOALSOF TEACHING MATHEMATICS TO
PRESCHOOL CHILDREN
Executed by:
student of magistracy department
Yulia Аndreevna Dunai
(tel.: 8-029-3468595)
Scientific Supervisor:
Professor
Doctor of pedagogical science,
I.V. Zhitko
English Supervisor:
Doctor of Psychology
Associate Professor
N. G. Olovnikova
Minsk, 2009
CONTENTS
INTRODUCTION
I. HISTORICAL PATTERNS AND PERSPECTIVES OF TEACHING MATHEMATICS IN PRIMARY SCHOOL
II. THE PURPOSES AND CONTENT OF MODERN MATHEMATICAL EDUCATION IN PRIMARY SCHOOL
III. THE METHODS OF CHILD’S ACQUAINTANCE WITH GEOMETRIC SHAPES
CONCLUSION
CONTENTS
REFERENCES
INTRODUCTION
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 sectionsof 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.
I. HISTORICAL PATTERNS AND PERSPECTIVES OF TEACHING MATHEMATICS IN PRIMARY SCHOOL
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 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 isused to motivate children to learnmath 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].
II. THE PURPOSES AND THE CONTENT OF MODERN MATHEMATICAL EDUCATION IN PRIMARY SCHOOL
Often children question the importance of learning 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 often 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. Scientists of all kinds, including biologists and social scientists, 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 science, 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].