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

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22

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 GOALS OF 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 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 kindergarten children 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 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.

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

APPROPRIATE PRACTICEINAPPROPRIATE PRACTICELearning is through
exploration,

discovery, and solving meaningful problemsNoncompetitive, 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
workMath skills are acquired through play, projects, and daily
livingMath is taught as a separate subject at a scheduled time each
dayThe teacher’s edition of the text is used as a guide to structure

learning situations and stimulate

ideas for projectsTimed tests on number facts are given and graded
dailyMany manipulatives are used

including board, card, and

paper-and-pencil gamesTeachers move sequentially through the lessons as
outlined in the teacher’s edition of the textOnly children who finish
their math seatwork are permitted to use the few available manipulatives
and gamesCompetition 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].

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].

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
counting 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 emphasis 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 experiences.

Today, nearly all children in the United States go to kindergarten (с 5
years). The beginning part of kindergarten focuses 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 understand 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.”

Another early goal is to teach children to understand patterns. Because
mathematics is the study of patterns, the teaching of pattern
recognition has become a part of the standard curriculum in kindergarten
through grade 3. At first, children are presented with simple patterns
to complete. Then they move on to more complex patterns of the same
kind.

These exercises are sometimes called attribute studies because such
attributes (or characteristics) as being colored or not and being square
or not are the focus of attention in addition to the attribute of
pattern. Students may use blocks or pictures to do these exercises.

At the same time, students may begin some simple work with geometry. One
goal of beginning geometry is to teach children to recognize the most
simple shapes—the square, the circle, the triangle, and the rectangle.
Teaching such basic terms simplifies classroom explanations and lays the
foundation for future work with geometry. Also, some shapes are used
when fractions are introduced.

Children respond better to three-dimensional shapes than they do to
two-dimensional pictures in books. This probably occurs because, aside
from printed materials and television, the shapes around them actually
are three-dimensional. Therefore, most educators believe that early
experiences with geometry should include such solid shapes as the cube,
sphere, cone, cylinder, and pyramid. In fact, the solid figures are
often taught first, and the two-dimensional shapes are explained in
terms of the solids. A square, for example, is one face of a cube. This
“analytic” approach to geometry is usually not tried in kindergarten,
but it may be started as early as grade 1.

Another geometric concept that is nearly always taught early is
symmetry, specifically what mathematicians call line symmetry. The
reason for including symmetry at this level is that it is useful in
design. Also, it is not difficult for young children to recognize the
difference between symmetrical objects, such as the capital letters A
and B, and an asymmetrical object, such as the letter F.

It is clear that for A and B, the dotted line separates two parts that
are identical. One part is the reflection of the other. No such line can
be drawn for F.

Another goal of early elementary education is to teach children about
measurement. This involves many skills and concepts. Often the first
notion taught is that a measurement is a number of standard units. It is
easiest to explain this idea by measuring straight lines [7, pp. 15-17].

So, а formal kindergarten math program usually starts at least six weeks
into the year and often as late as the beginning of the second term. It
generally includes counting; ordering and comparing numbers; preparation
for addition and subtraction; comparing size: preparation for telling
time; the concept of a penny; recognizing squares, triangles, and
circles; such concepts as top, bottom, front, back. in. on, between,
left, and right (which are needed to explain lessons); classifying
objects; and recognizing patterns. Most time is spent on counting and
learning to understand numbers through one-to-one matching and comparing
and ordering numbers.

Every grade has a major goal for mathematics achievement, although it is
not always formally defined. Thus, for kindergarten, the major goal is
counting [7, p. 31].

III. THE METHODS OF CHILD’S ACQUAINTANCE WITH GEOMETRIC SHAPES

Preschoolers’ intuitive knowledge of geometry frequently exceeds their
numerical skills. By building on strengths and interests that are
already present, you can foster enthusiasm for math and provide a
logical context to develop number ideas.

By age six, children often have stable yet limiting ideas about shapes.
Four-year-old Tina tells her mother, “That’s not a square. It’s too big.
A square looks like this.” Her friend Charlie adds, “Triangles have to
be this way. That’s not a triangle. It’s too upside down.” Broaden
child’s understanding by pointing out a variety of examples — squares
that are many sizes and triangles that are “long,” “skinny,” “fat,” and
turned in many directions. You can also encourage deeper thinking about
shapes not just through hands-on activities and discussions, but through
picture books such as The Greedy Triangle by Marilyn Burns [1, pp. 5-6].

Geometry is the study of shapes, both flat and three dimensional, and
their relationships in space. Geometry enters infants’ lives from birth
as they attempt to make sense of the shapes in their environment: crib
bars, stuffed animals, mother’s breast and face, the door to the
bedroom. Geometric shapes become some of the first intentional scribbles
made in young children’s drawings, and they delight in their awareness
of the shapes around them.

Such natural interest deserves encouragement and informal teaching
intervention. In the 1950s, two Dutch educators developed a theory of
stage development in geometric understanding. The van Hiele theory has
gained acceptance in the United States in recent years, and applies to
children from the early years through high school. An important tenet of
the theory is that children do not grow through the stages automatically
but, with teacher assistance, will do so competently (Teppo, 1991). What
children are exposed to in the early years sets the stage for learning
in geometry throughout their entire school experience. Through the
primary grades, children are at the earliest, visual, stage in which
they explore their environment to learn to identify the shapes within
it. Activities such as describing, modeling, drawing, and classifying
help them develop a spatial sense [6, p. 457 ].

Douglas H. Clements offers such reception for formation of knowledge of
children about shapes. In a preschool classroom you might see a scene
like this: A teacher challenges Michelle and Debbie to use their bodies
to make a shape together. The girls sit down facing each other and
stretch their legs apart. With feet touching, they create a diamond.
Another child takes a look, sees the diamond shape, and says: “If we put
someone inside, we can make two triangles.” Immediately, they ask Ray to
scrunch in and lie across the middle.

It works. A diamond can be divided into two triangles. Michelle notes
that there’s a shape that has six sides, and she wants to try making one
of those. Another child may even know that the shape is called a
hexagon. After a brief discussion, Michelle gets five other children
together. Then, under her direction, they all lie down on the floor and
create a six-sided shape [1, p. 5].

Ellen Booth Church ( a former professor of early childhood ) suggests
parents to apply following receptions in the course of acquaintance of
children with shapes:

· Sort the house: Collect a variety of household objects, like bottle
caps and envelopes, and invite your child to sort them into different
piles — one for circles, rectangles, and so on. Invite her to go on a
treasure hunt around the house to find “one more thing” for each pile.

· Cut shape sandwiches: Use cookie cutters to make dainty tea sandwiches
for a Shape Tea Party! Cut the bread with the cutters, then spread the
shapes with cream cheese. Try filling the bread slices with foods of
different shapes, like a round tomato slice on square bread or a
triangle of cheese on round bread.

· Make puzzles: Use large, colored file cards, folders, or poster board
to make shape puzzles together. Cut out big basic shapes, like circles
or triangles, and then cut each into two or three pieces. Your child can
decorate the shape puzzles with crayons and then put the puzzles back
together.

· Explore letters: Hunt for shapes within the letters of the alphabet.
Write out the alphabet in capital letters, and have your child find
which shapes make up each. You can also show her how shapes form as you
write; invite her to experiment by drawing large shapes of her own and
turning them into letters.

· Paste a picture: Provide your child with paste and a variety of
cut-out paper shapes in different sizes and colors for creating
pictures. Encourage him to combine shapes to create designs or familiar
images. For example, he might make a shape person with a circle head, a
square body, and rectangle legs.

· Search through stories: Many children’s picture book
author-illustrators boldly use basic shapes in their illustrations
(books by Tana Hoban, Eric Carle, and Leo Lionni are particularly good
places to start). As you read with your child, ask her to point out the
shapes she finds in each picture.

· Take a walk: Move shape play outside by taking a walk with cardboard
cut-outs of a triangle, circle, square, and rectangle. Your child can
match them to objects like signs, plants, doors, and car tires. Take
along your digital camera to snap photos of the shapes. Then print them
and make a family shape book [2, p. 12].

Ellen Booth Church considers that boxes are the raw materials of
creative thinking. Exercises: «Big and Little Sorting», «Boxes and Lids
Matching», «Serial Boxes», « Fill ‘Em Up», «Make a Shape Feely-Box»,
«Treasure Boxes», «Shadow Box Collages», «Play Store», «You-in-the-Box»
and others [3, pp. 9-10].

Also Ellen Booth Church offers to use snacks and mealtime to teach big
ideas with taste and ease. The kitchen is filled with many wonderful
foods and cooking tools in a variety of colors, sizes, and shapes. It is
the perfect laboratory for exploring some of the first topics children
learn in school: color, shape, and size. Understanding these concepts is
important because your child uses them in observing, comparing, and
discussing all she sees and encounters. The ability to notice, use, and
voice similarities and differences are at the heart of beginning math,
science, and reading skills. For example, reception for acquaintance
with the shape. Eat a square meal. We have all heard of the importance
of eating a square meal of healthy foods, but why not have a really
“square” meal? Serve waffles (big and little squares) with a side dish
of pineapple chunks for breakfast. Have a snack of square cheese slices
on square crackers placed on a square napkin. As you are preparing and
enjoying your meals, ask your child to notice the similarities and
differences between the different squares. Help her notice that all the
squares have four sides, but can be various sizes. For a fun challenge,
give your child a slice of pre-wrapped American cheese. As she unwraps
it, ask her how she can fold her cheese square into a triangle (point to
point). One way to focus on a particular color is to have a meal all in
the same color. This will help your child to not only focus on learning
the name of a particular color, but also it will help her see the many
different shades of a particular color. For example, not all oranges are
the exact same shade and so on [4, p. 3].

Julie Sarama, PhD, and Douglas H. Clements, PhD offer are some
activities parents can try at home to support math learning:

1. Play with many different but ordinary objects. Children stretch their
imaginations when they play with ordinary objects. Many of these have
interesting geometric properties. For example, some cylindrical objects,
such as paper towel rolls and toilet paper rolls, can be looked through,
rolled, and used to represent objects such as towers in a castle. All
these activities develop the foundation of understanding
three-dimensional shapes.

2. Play with the same objects in different ways. Creativity and thinking
are enhanced if children play with the same object in different ways.
When the box is a container, then a house, then stairs, then cut apart
into a track for a car, children see the relationships between shapes,
real-world objects, and the functions they serve.

3. Play with the same toys again and again. Some materials are so
beneficial that all children should play with them again and again
throughout their early years. Blocks for all ages, Duplos, and Legos, at
the right age, can encourage children to build structures, learn about
shapes and combine them, compare sizes, and count. They also learn to
build mental images, plan, reason, and connect ideas. Sand and water
play are invaluable for learning the foundations of measurement
concepts. Creating patterns and designs with stringing beads, blocks,
and construction paper develop geometric and patterning ideas. Puzzles
develop spatial thinking and shape composition.

4. Everyday objects can be fun, as is playing with constructive toys
again and again. However, buying too many different types of commercial
toys can decrease children’s mathematical thinking and creativity. Less
can be more! Rotating toys keeps children interested.

5. Count your playful actions. Many games and playful activities just
call out for counting. How many times can you bounce a balloon in the
air before it touches the ground? How many times can you skip rope?

6. Play games. Card games, computer games, board games, and others all
help children learn mathematics. They count dots on cards and spaces to
move. Counting helps them connect one representation of numbers to
another. They learn to instantly recognize patterns of numbers, such as
the dot patterns on dice or dominoes. Some games involve using a timer.
Concentration and Bingo involve matching. Checkers and Candy Land
involve spaces and locations.

7. Play active games. Beanbag tossing, hopscotch, bowling, and similar
games involve moving and distances. Most of the games involve numbers
and counting for score keeping, too. Games such as “Mother May I?”
involve categories of movement. “Follow the Leader” can be played using
math concepts, such as announcing you will take five large steps
backward, then two small steps to the side.

8. Discuss math playfully. Math will emerge when you help your children
see the math in their play. Talk about numbers, shapes, symmetry,
distances, sorting, and so forth. Do so in a playful way, commenting on
what you see in the children’s constructive play.

9. Provide ample time, materials, and teacher support for children to
engage in play, a context in which they explore and manipulate
mathematical ideas with keen interest [5, pp. 10-11].

Preschool and kindergarten children can learn much from playing with
blocks (Jensen and О’Neil, 1982). They can

? compare and seriate shapes. Start with single shapes and have children
compare two pieces according to size. Later, more pieces can be added
until several items of one shape can be seriated;

? classify and name shapes. Provide two shapes at first then, after much
practice, add a third and fourth. Make large loops of yarn on the floor
or provide boxes to place the blocks in. A good game to play for naming
is Pass the Block. Children sit in a circle and, as music plays, they
pass blocks in one direction. When the music stops, each child names the
shape he or she holds;

? trace and feel shapes. Children trace around a block with pencil, then
color it in if desired. With a little practice, they can superimpose
different shapes on one paper, coloring in some of the sections. Blocks
can also be used as items in a “mystery bag.” Two or three familiar
shapes are placed in a bag and children take turns reaching in, feeling
a block, and identifying its shape.

Plane figures can be explored through active interaction. Colored tape
is laid on the floor in geometric shapes large enough for children to
walk on. Ask the children to jump, walk, crawl, and so on across
specific shapes. They might count the number of children who can fit in
one triangle or the number of steps it takes to walk the perimeter of a
square.

Kindergarten and primary children continue to learn best from working
with manipulatives and may find the illustrations in math textbooks
confusing. Some materials that are appropriate are tiles, pattern
blocks, attribute blocks, geoblocks, geometric solids, colored cubes,
and tangrams. Computer games in which geometric shapes appear from
different angles help children overcome their misunderstandings of book
illustrations, which may show a shape from only one or two viewpoints.
Some appropriate activities are building structures with various types
of blocks to enhance spatial visualization. Folding and cutting
activities such as origami or snowflake making, exploring the indoor and
outdoor environments to identify shapes and angles made by people and
nature, reading maps, making graphs, playing Tic-Tac-Toe, Battleship,
and other games that use grid systems [6, pp. 458- 459].

All day long there are opportunities for children to increase their
awareness of mathematics in the world around them. Mathematical
phenomena may not always be as obvious as those of language, however.
Thus, the teacher must take extra care to include mathematical learning
whenever natural learning situations arise. Often this means recognizing
opportunities to incorporate mathematics into other areas of the
curriculum.

CONCLUSION

Mathematical learning for young children is much more than the
traditional counting and arithmetic skills; it includes a variety of
mathematical concepts (classification, ordering, counting, addition and
subtraction, measuring, geometry).

Children may begin some simple work with geometry in primary school.
Main goal of beginning geometry is to teach children to recognize the
most simple shapes—the square, the circle, the triangle, and the
rectangle. Teaching such basic terms simplifies classroom explanations
and lays the foundation for future work with geometry.

By age six, children often have stable yet limiting ideas about shapes.
It’s possible to broaden child’s understanding by pointing out a variety
of examples — squares that are many sizes and triangles that are “long,”
“skinny,” “fat,” and turned in many directions.

Thus teachers must keep in mind that children learn geometry most
effectively through active engagement with toys, blocks, puzzles,
manipulatives, drawings, computers and teachers!

It’s possible to develop deeper thinking about shapes not just through
hands-on activities and discussions, picture books but through playing.
In primary school playing is used as the main method of teaching.

Huge experience of using games and playing exercises during the
children’s studying of mathematics (and geometry) is accumulated in
practice of working of preschool organizations.

REFERENCES

1. Douglas Clements. Ready for Geometry! From an early age, children
make sense of the shapes they see in the world around them //
International Journal of Mathematical Education. Science and Technology.
– 2006. – № 2, pp. 5-6.

2. Ellen Booth Church. Exploring simple shapes sets the stage for
creative thinking // International Journal of Mathematical Education.
Science and Technology. – 2007. – № 11, pp. 12-13.

3. Ellen Booth Church. Boxes are the raw materials of creative thinking!
// International Journal of Mathematical Education. Science and
Technology. – 2006. – № 10, pp. 9-10

4. Ellen Booth Church. Color, Shape, and Size. Use snacks and mealtime
to teach big ideas with taste and ease // International Journal of
Mathematical Education. Science and Technology. – 2007. – № 8, pp. 2-5.

5. Julie Sarama, Douglas H. Clements. Some activities teachers can try
to support math learning// International Journal of Mathematical
Education. Science and Technology. – 2005. – № 1, pp. 10-11.

6. Suzanne Lowell Krogh. Educating Young Children. Infancy to Grade
Three. New York.: McGraw-Hill, Inc., 1994. – 605 p.

7. The World Book of Math Power. Volume 1. Learning Math. – Chicago.:
World Book, Inc., 1995. – 420 p.

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