AYODELE, OJO
97
Journal of Children in Science and Technology. (JOCIST) Vol. 10(1), pp 97 - 103 October, 2016.
Available online at http://www.cistng.org
ISSN 1597 – 133
EXPLORING THE CONCEPT OF ELEMENTARY GEOMETRY IN
PRIMARY SCHOOL MATHEMATICS CURRICULUM TOWARDS THE
DEVELOPMENT OF PUPILS’ LEARNING EXPERIENCE
AYODELEOJO
RISING STARS NURSERY AND PRIMARY SCHOOL, ILORIN, KWARA STATE, NIGERIA.
Abstract
The importance accorded to learning geometry in primary school mathematics curriculum
reflects the vital role it plays in developing school pupils. Thus, Geometrical notions and
properties occur in real-world problems. The primary school mathematics curriculum lays
the foundation of geometry knowledge. Pupils learn geometry notions and properties by
exploring their environment. It is therefore, very important that primary school pupils have a
good base in elementary geometry. However, this paper presents a researchon how primary
school pupils understand elementary geometry, its notions and properties as related to some
basic shapes and solids. Theresearch was conducted in a primary school within Ilorin south
L.G.A, Kwara State. The researchshowsthat, more than 50% of the pupils selected as samples
could not define correctly basic geometrical shapes neither do theyknow the correct
properties of the shapes. Also, more than 40% of the pupilscouldnotdraw the net of two and
three dimensional solidscorrectly. It is hoped this could be corrected with emphasis on the
teaching of geometry if included in the curriculum.
Keywords: Geometry, Curriculum, Primary school, Pupils, Spatial ability, Learning.
I.
Introduction
The importance accorded to learning geometry
in primary school mathematics curriculum
reflects the vital role it plays in developing
school pupils. It develops pupils’ spatial
ability,logical reasoning skills and the ability to
solve real-world problems in whichgeometrical
terminology and properties occur (Jones &
Mooney, 2003; Presmeg, 2006).
For a success in learning Geometry the
understanding of geometrical concepts is
essential. There are difficulties in understanding
geometry concepts at elementary level
(Cunningham & Roberts, 2010).
The aim of this paper is to presents a research on
how primary school pupils understand
elementary Geometry, its notions and properties
as related to some basic shapes and solids.
II.
Summary of Views about Elementary
Geometry
2.1 Elementary Geometry and Spartial
Ability
Spatial ability is a set of abilities thatincludes the
capacities to perceive the visual world
accurately. Spatial ability as a mental process is
used in perceiving, storing, recalling,creating,
Journal of Children in Science and Technology. (JOCIST)
arranging and making related spatial images.
Spatial ability could be divided intothree
elements: spatial relations, visualization, and
orientation. Spatial relations is the ability
torecognize the relations between objects in the
space. Spatial visualization is the ability to
performimagined movements of objects in twodimensional and three-dimensional space.
Spatial orientation is the ability to recognize
geometric shapes from differentpositions
identified five factors of spatial ability: spatial
perception,visualization, mental rotation, spatial
relations and spatial orientation. Spatial
perception is the abilityto recognize the
horizontal and the vertical. Mental rotation is the
ability
to
rotate
twoor
three
dimensionalfigures. Spatial visualization ability
is strongly related to the pupils’ geometry
achievement (Capraro, 2001).
2.2
Elementary Geometry and Pupils’
Mental Development
Van Hiele (1986) identified five levels of mental
development in Geometry:
Level 1 (visualization): pupils can recognize a
geometric object or concept based on a
prototypicalexample. They can name and
identify common geometric shapes.
Level 2 (analysis): pupils can recognize a
geometric shape based on its properties.
Properties are notyet ordered at this level. For
example, they might state, that a square is not a
rectangle, a rectangle isnot a parallelogram.
Level 3 (abstraction): pupils identify class
inclusion of shapes. For example, they recognize
that allsquares are rectangles, but not all
rectangles are squares. Pupils can give
definitions; they recognizehow a definition
identifies clearly a notion.
Level 4 (deduction): pupils understand the
importance of proofs and they can construct
98
geometricproofs. They are also conscious that a
proof can be done in more than one way.
Level 5 (rigor): pupils understand the relations
between geometrical concepts and they can see
themin an abstract system. At this level pupils
can study non-Euclidean geometries.
2.3 Elementary Geometry and Pupils’
misconceptions
According to Feza& Webb (2005), many
primary school pupils have problems in
recognizing different geometrical shapes in nonstandard orientation,for example, a square is not
a square if its base is not horizontal.
Many pupils have difficulties in perceiving class
inclusions of shapes for example, they might
think, that a square is not rectangle.a square is
notrhombus,
a
rectangle
is
not
parallelogram.Somepupilscannot
recognize
geometrical solids and they cannot draw the net
of these solids(Marchis, 2008;Pittaliset.al,
2010).
III. The Basic Concept of Geometrical
Shapes and Solids
This
section
clearly
presents
thedefinitionsofsome basic geometrical shapes
and solids that the pupils were taught in their
previous geometry lesson.
i.
A parallelogram is a simple quadrilateral
with two pairs of parallel sides.
ii. A rectangle is a simple quadrilateral with
four right angles. We can also define a
rectangle based on aparallelogram: a
rectangle is a parallelogram with at least
one right angle.
iii. A rhombus is a quadrilateral with four
sides of equal length. We can define a
rhombus based on aparallelogram: a
AYODELE OJO
99
rhomb is a parallelogram in which at least
two consecutive sides are equal in length.
iv. A square is a simple quadrilateral with
four equal sides and four equal angles. A
square can also be define based on
parallelogram, rectangle or rhombus. A
square is a parallelogram with one right
angle and two adjacent equal sides. A
square is a rectangle with two adjacent
equal sides. A square is a rhombus with all
angles equal or a square is a rhombus with
at least one right angle.
v. A pyramid is a polyhedron formed by
connecting a polygonal base and a point.
If the base is a triangle we have a
tetrahedron. If the base is a square we
got a square pyramid.
vi. A prism is a polyhedron with an-sided
polygonal base, a translated copy of it to
another plane,andnother faces joining
corresponding sides of the two bases. If
the bases are triangles, the shape is called
atriangular prism. If the bases are
quadrilaterals, the shape is called
CLASS
Basic 4
Basic 5
Basic 6
TOTAL
atetragonal prism. If all the faces of a
tetragonal prism are squares, we get a
cube. Thus a cube is a particular
tetragonal prism.
Evaluation
of
Pupils’
Learning
Experience in Geometry
The remaining part of this paper focuses on the
research carried out to evaluate pupils’
understanding of elementary geometry
regardinggeometrical shapes and solids.
I.
The research was conductedat Rising Stars
Nursery and Primary School, Ilorin, Kwara
State on March, 2016. It was carried out in
form of a test.
4.1 Sample:
The sample was formed from the Upper Basic
classes of the school.
A group of 20pupils(10 males and 10 females)
were selected base on their performance in the
previous
geometry
lessons
and
their
mathematics teacher’s recommendations.
We present the samples as follows:
NO. OF STUDENTS SELECTED
5
23
6
33
9
54
20
Note that, all the pupils selected have been
taught the definitions and properties of those
geometrical shapes in their various classes. They
were also taught geometrical representations and
how to draw the net of solids. However, the
pupils were allowed torevised the above
definitions of geometrical shapes and solids with
their properties before the commencement of the
research test.
10
MALE
FEMALE
10
4.2 Procedure:
The researcher employed a set of problems
related to plane and solid geometrical shapes as
the research tool. The pupils were given those
problems in a test format.
The research problems were given to the pupils
in two parts:
Journal of Children in Science and Technology. (JOCIST)
In the first part,pupilswere asked to give the
definition of a parallelogram, rectangle,
rhombus, and square with the descriptiontheir
properties.
In the second part,pupils were asked to draw the
net of these geometrical solids: tetrahedron,
square pyramid, triangular, and tetragonal prism.
The wrong attempts made by the pupils were
observed and documented as part of the research
results.
4.3 Results:
Part One (1):
In the following we present some of the pupils’
wrong definitions of basic geometrical shapes.
First of all, let see how some of the pupils define
a rectangle.
“A rectangle is a geometrical shape, which has
four sides, two opposite sides are parallel and
equal,its angles are right angles, its diagonals are
equal and they intersect each other in the
center.” This isnot a definition; it is a list of
properties. The pupils did not know how to give
the definition.
“ Arectangle is a parallelogram whose opposite
sides are parallel and equal.” The opposite sides
of aparallelogram are parallel and equal, thus
these properties are repeated and other property
importantfor a rectangle (the angles of a
rectangle are equal) was omitted.
“A rectangle is a parallelogram whose opposite
sides are parallel and its angles are right
angles.”This definition describes a rectangle, but
some properties are repeated: the opposite sides
of aparallelogram are parallel, so we don’t need
to include this property in the definition.
As regarding to a rhombus, some pupils also
gave incorrect definition.
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“A rhombus is a parallelogram whose opposite
sides are equal.” The opposite sides of a
parallelogramare equal, so this definition repeats
two times the same property and omit others. In
this definition thepupil is aware of the fact that,
the rhombus is a particular parallelogram.
“A rhombus is a square whose opposite sides are
parallel and equal.” A square is a
parallelogram,soit’s opposite sides are parallel
and equal. Thus these properties are repeated.
As regarding to a square, some pupils also find
it difficult to give the correct definitions.
A square is a particularrhombus, thus we can’t
define a rhombus based on a square (only vice
versa).
“A square is a rectangle whose sides are equal
and parallel.” The opposite sides of a rectangle
areequal and parallel (as a rectangle is a
parallelogram), thus this information is repeated
in thisdefinition. An important property of the
square is not included: two adjacent sides are
equal.
“A square is a quadrilateral whose sides are
equal.” This definition point out one property of
asquare: all the sides have equal length. But
rhombus also has all the sides equal. Another
importantproperty is missing in order to describe
a square: at least one of the angles is right angle.
“A square is a rectangle whose sides are equal
and adjacent sides are perpendicular to each
other.”This definition describes a square but it
contains repeated information: the adjacent sides
of arectangle are perpendicular.
Part Two (2):
In the following we present some of the wrong
attempts made by the pupils while representing
AYODELE OJO
101
geometrical solids and in drawing the net of the
solids:
Figure1.Pupils’ attempt to draw pyramid; they were not aware that some of the edges cannot be seen.
Figure2. Pupils’ square pyramid
Figure 3.The pupil draw body diagonal instead of
the net of the solid.
Journal of Children in Science and Technology. (JOCIST)
102
Figure 4.Pupils’ wrong attempt; the net is not in correspondence with the given solids
Figure 5.The pupils’ wrong attempt; The net is not in correspondence with the given solid.
4.4 Observations and Discussions:
From the part one of the results,we could
observe that, some pupils were not familiar with
formulating definitions, they do not understand
thefact that, in a definition, we include the
minimum amount of properties which clearly
describe thenotion. Some of the definition, even
if they are correct, contains repeated
information. For example, in “A square is a
rectangle whose sides are equal and adjacent
sides are perpendicular to each other”, the
property that adjacent sides are perpendicular to
each other was repeated.
In other cases, the definitions given by them
were missing important properties of the defined
notion. For example, in “A square is a
quadrilateral whose sides are equal”, the fact
that, at least one angleis right angle should be
pointed out.
Some of the definitions are incorrect due to the
fact that the pupils were not aware of the class
inclusionof the shapes. For example, in “A
rhombus is a square whose opposite sides are
parallel and equal”, a rhombus was defined
based on a square.
From the part two of the results, the most
common mistake pupils made while drawing
geometrical solids wasthat,the pupils were not
aware of the fact that, some of the edges cannot
be seen (Figure 1). Some pupils gave twodimensionalrepresentation in which is not
marking the invisible sides. One pupil gave a
very interesting solidinstead of a square pyramid
AJAYI PETERET AL
103
(see Figure 2). Three pupils draw cube for
tetragonal prism, these pupilswere not aware of
the fact that, usually when we have to give some
geometrical shapes or solids wecannot choose
particular cases.
Net construction requires focusing on the
components of the solid.
Some of the pupils did not draw anything for
the net. Maybe theydidnot know the notion of
the net of a solid. One pupil drew the body
diagonal instead of the net (seeFigure 3). In
Hungarian language, the net of a solid is
“testháló”, and the body diagonal of a solid
is“testátló”, so these two notions are very
similar, this could be the origin of this
confusion.
Other pupils didnot draw the net of the solid
they were given. In Figure 4, we could observe
that, the pupil drew a cube, but the net is
composed from rectangles, so it is not the net of
a cube. In Figure 5, the solid is a square pyramid
and the net is belonging to a tetrahedron. This
lack of concordance between the solid and the
net shows that,pupils cannot imagine in their
mind the three-dimensional objects and how
they open it in order to get the net.They only try
to draw something from their memory.
IV.
Conclusion
From the research results, we could
concludethat, more than 50% of the pupils
selectedwere
deficient
in
defining,
recognizingand stating the properties of
geometrical shapes.This shows that more than
half of the pupils are only on level 2 (analysis)
on Van Hiele’slevelsof mental development in
geometry.Also, more than 40% of the pupils
have difficulties with the representations of two
and three dimensional solids. It is hoped that this
could be corrected with more emphasis on the
teaching of geometry in primary schools if
included in the curriculum.
VI.
Recommendation for Further Action
For a success in learning geometry, the
understanding of geometrical concepts is
essential.Thus,we recommend that,primary
school pupils should be motivated to learn
elementary geometry by exploringthe real world
objects within their environment.
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