Universal Journal of Education and General Studies (ISSN: 2277-0984) Vol. 2(3) pp. 079-083, March, 2013
Available online http://www.universalresearchjournals.org/ujegs
Copyright © 2013 Transnational Research Journals
Full Length Research Paper
Development of the concept of multiplication with
fraction
Alapa Stephen Ochefu
National Commission for Colleges of Education (NCCE), Abuja. Nigeria. E-mail : ochefu2010@yahoo.com , Phone
Number: +2348036573112
Accepted 20 March, 2013
The term concept formation describes how a person learns to form associations, where as the term
conceptual thinking refers to an individual’s subjective manipulation of those abstract associations. A
concept is a rule that may be applied to decide if a particular object falls into a certain class. Concept
learning may be understood as a two-step process: first the discovery of which attributes are relevant,
then the discovery of how they are relevant.This studyis on the development of a concrete concept of
multiplication in fraction. In the paper, practical exercises were carried out to demonstrate how and
why the result of multiplying two or more fractions brings about smaller values. The study used two
basic research questions.The analysis of the study was carried out using interview, pre- test and posttest. Thirty-five (35) pupils in primary five in primary school were interviewed and made to take the test
while forty (40) in-service teachers undertaking National Certificate in Education (NCE) were similarly
interviewed and subjected to the skill test. The pupils and the NCE students show satisfactory
performance in manipulation skills but could not logically and concisely explain what fraction means.
However, when the twenty (20) pupils out of the thirty-five and twenty (20) out the forty teachers were
taken through the modeled practical exercises designed on the concept of multiplication for three
consecutive days, the participants developed more confidence and interest in the multiplication of
fractions. Their confidence is measures by the ability to justify each step taken in the multiplication of
fraction. These lessons were taught to children of 6th grade and students in National Teachers
Institute (NTI). The students’ were tested for the retention andflexibility of these concepts based on a
retention test developed for the purpose. The findings of the study revealed that the experimental
group scored significantly higher on the various facets of attained concept like flexibility, applicability
and retention. Recommendations were made that practical demonstrations should be used at the
introductory level, practical exercises should be carried out severally to ensure proper formation of the
right concept and examples of all the exercises should be familiar activities within the learners’
environment.
Keyword: Multiplication , fraction , National Teachers Institute , Pupils and Teachers
INTRODUCTION
Education is something more than mereaccumulation of
knowledge and skills. It must beconcerned with
developing those skills and attributesthat will promote
self-discipline, responsibility, self-expression to be able to
distinguish between right andwrong, and confidence.
There should be an ability toreflect, reason and to
analyze. Basic to thedevelopment of these skills and
attitudes are the abilityand the motivation to think critically
and
communicateeffectively
with
others.
The
development of adisposition to think and to reflect on
experiencesprovides an impetus towards the more
effective use of skills and greater understanding and
appreciation ofthe knowledge and facts being acquired.
There are two fundamental types of concept – every
day and scientific concept. The dichotomy between the
two types of concept is on the bases of their formation.
Everyday concept is formed on the bases of daily
contexts and as such could be incorrectly used by
children (Vygotsk 1934 and Vygotsky, 1987).Similarly,
Schmittau (1993), observed that Vygotsky’s everyday
080. Univers. J. Edu. Gen. Stud.
concepts as concepts originated from children’ daily lives
through communication with their family, friends, or
community; and thus are closely connected to concrete
personal contexts. Children express such concepts
through their own words and use them in their thinking
without conscious awareness; hence, such concepts are
based on subjectivity. For instance, in a series of
conversation with a child in primary three in Nigeria,
when he was asked to state the meaning of “half”, he
described that half meant todivide something equally
among people. This thought would have been as a result
of his everyday concept of sharing biscuits with friends
equally without cheating.
Scientific concept on the other hand is conceived on
the bases of a system that has developed in human
history and therefore lacks concrete contexts. Kozulin
(1990) opined that Vygotsky’s scientific concepts are
based on “formal, logical, and decontextualized
structures”. Blackwell (1990) compared Vygotsky’s
scientific concept with mathematical concepts which are
based on a system and as such have logic and
objectivity. Mathematical concepts are expressed in
mathematical language and introduced to children in a
formal and highly organized education.
Zack (1999) discussed extensively the relationships
between every day and scientific concepts and concluded
that the two concepts are mutually dependent in the
process of developing children’s concepts in daily lives
and in school.
While students may have some facility in using
fractions, many of them appear not tohave fully
developed an understanding that fractions are numbers
(e.g., Kerslake, 1986, Domoney, 2002 and Hannula,
(2003). Kerslake (1986)) emphasizes the need for
students to understand fractions at least as an extension
of the number system. Her report presents some of the
difficulties 12 to 14 year old students have in connection
with fractions such as the pupils’ inability to recognize
that fraction are numbers that can be used in
calculations. In order to develop a conceptualknowledge
of rational numbers, students should be able to both
differentiate andintegrate whole numbers and fractions. It
seems important to use several models for eachconcept,
but two or more related concepts, whenever possible,
should be representedtogether so that their relationship
becomes clear. The concrete method that can ensure
that learners form a clear and unambiguous concept of
fraction is through practical exercises.The concept of
fractions is an important area of study that is applicable to
daily life situations and alsoa prerequisite topic to virtually
all other mathematical concepts. In the light of this, the
Nigerian basic curriculum has addressed this concept
right from primary one.
Fraction has applications in every field of study be it
science or arts. It is in recognition of the importance of
fraction that makes it necessary for learners (especially
children) to form a clear and comprehensive concept that
will be useful through their life’s span.
This paper is designed on practical approach to the
teaching of the skill of multiplication of fraction by another
fraction which has been identified by the author as a
serious challenge in the teaching and learning of fraction
in both primary and secondary schools in Nigeria. It has
been observed from the result of this research that if
learners understand the basic concept and principles
behind any theory. The learners will be creative,
innovative in their thinkingand further expand their
understanding of fraction.
When teaching children how to multiply fractions, it is
important to make the process meaningful. This may be
done best by using a six-step process that helps children
to visualize fraction multiplication, understand fraction
multiplication, be able to do fraction multiplication, be
confident when multiplying fractions and identify that the
product of two or more fractions is actually smaller than
any of the fractions involved in the multiplication.
In the first step of this instructional process, children
use a model to find answers to some fractionmultiplication examples. In the second step(which really
happens concurrently with step one) the children keep a
record of the results from step one.After enough
examples have been completed, the children move to the
third step by looking for a pattern that suggests how to do
the multiplication without the model. In the fourth stepthe
children hypothesize how to do the multiplication without
the model.The fifth step is to complete examples using
the hypothesized procedure and then redo those
examples with the model to check the correctness of the
procedure and the sixth step is to observe the size of the
product and compare it with the sizes representing each
of the fractions involved in the multiplication.
Barge (2012) thought of concepts as a complex and
true act of thinking that cannot be mastered through
memorization. Concept is an act of generalization which
lies in the transition from one structure of generalization
to another and the process of generalization is completed
only at the formation of true concept.
The partition concept is an activity-based exercise
whereby the facilitator and the learners are involved in
dividing a whole paper into equal parts, selecting and
shading some of the equal parts out of the total equal
parts in the whole. For learners to use partition concept
effectively, the facilitation of the concept should come
after the learners have gone through the exercises on
how to divide a unit into various (2, 3 4, 5…) equal parts.
This suggests that the facilitation of the partition concept
should take place after theability to measure lengths and
share the lengths into equal parts. In most cases, the
precision and fluency in the executionof the skills are the
requisite
vehicles
to
convey
the
conceptual
understanding.
Student’s
re-occurring
poor
performance
in
mathematics in secondary schools in Nigerian has
Stephen
become an educational cancer. This worries every
parent, guardians and stakeholders in Nigeria. Adeniji
(1998) and Amoo (2001) expressed tales of woes about
low achievement in mathematics in Nigeria secondary
schools. Formation of proper concepts is an effective
activity in the teaching and learning of Mathematics in
order to demystify the abstract nature of Mathematics
and one of the sure methods of the demystification of
abstract concept is through practical exercises that can
engage learners to perceive and observe the results of
their activities. Experience has shown that teachers rarely
use practical procedures in presenting the concept of
multiplication in fraction which leaves learners with the
option of memorizing only the techniques of manipulation.
The research is designed for learners to form a proper
concept of fraction, multiplication involving fractions and
to justify why the product of two or more fractions is
smaller than any of the fractions involved in the
multiplication.
Purpose of the Study
The purpose of the study is to develop the concept of
fraction, multiplication of fraction by whole numbers and
fraction by fraction and observe reasons for the product
of two or more fraction to be smaller than any of the
fractions involved in the multiplication.
Research Questions
bridged with real life situations and the two groups were
once again subjected to the similar questions. It was
observed that the group that carried out the practical
exercises performed very well in both the skills and the
interpretation of the word problems in the multiplication of
fractions.
The research was conducted for pupils in primary six at
an average age of ten (10) years) and NTI study centre in
Kwali Area Council of Federal Capital Territory, Abuja.
Seventy-five (75) learners were involved in the
exercise, thirty-five (35) from primary five and forty (40)
students from NTI study center in Kwali Study center.
The thirty-five (35)leaners were randomly selected from
two classes of primary six and the forty (40) NTI students
were selected from a class containing sixty (60) students
to make up the sample for the research.
(a)
Survey research design was adopted for the work due to
the nature of the study. Two groups were identified in
each of the schools that have been used for the research.
Questions were set to test the learners’ skill in
multiplication of fractions and some word problems
involving fraction. The result of the test shows that 70%
of the pupils correctly carried out the multiplication skills
but only 20% could correctly interpret the word problems.
However, when the word problems were interpreted into
figures, the learners successfully carried out the
multiplication. This demonstrates that the concept of
fraction was not properly formed prior to the conceptual
facilitation. One of the groups was taken through the
practical exercises in which the results gotten from
theoretical skills were practically demonstrated and been
A fraction multiplied by a whole number
The concept of multiplication of whole numbers was
orally discussed with learners as entry behaviour such as
5 times 3. (5 x 3) means 5 + 5 + 5 (3 times). i.e.
+
+
= 5 x 3 = 15.
Then, let a whole strip of paper be represented by,
Therefore,
The following research questions were used for the
study.
1
What is fraction?
2
Why is it that the product of two or more fractions
smaller than any of the fractions involved in the
multiplication?
METHODOLOGY
081
, hence x 7 =
When the shaded areas are rearranged to complete
whole strips of paper, we obtain
=
,i .e. 5 whole strips and one out of
four equal parts of a strip.
082. Univers. J. Edu. Gen. Stud.
(b)
A fraction multiplied by another fraction
The principle of subdividing the two sides of a two–
dimensional shape into equal strips was used to explain
the concept behind the multiplication of one fraction by
another fraction. For example,
could be carried out
on just one rectangle, such as
i.
Divide the breadth into four equal parts and
shade three out of the equal parts
ii.
Divide the length into three equal parts and
shade two out of the equal parts.
iii.
Count the number of equal parts that belongs to i
and ii above
iv.
Count the total number of equal parts that are in
the rectangle.
The division of the breadth into four equal parts
produced the fraction of the shaded parts as:
Hence,
x
=
equal parts inside the rectangle.
Rearranging the area covered by the product of the
fractions, it was found out that the rectangle was divided
into two equal areas. i.e.
=
This shows that
=
The division of the length into three equal parts produced
the fraction of the shaded parts as:
, i.e. six equal parts out of the total
=
=
=
.
Similar activities were carried out with the learners using
various examples of fractions for three days. The result of
the multiplication exercise showed clearly that the area
covered by the product of fractions is smaller in size than
the area covered by each of the fraction involved in the
multiplication.
DISCUSSION AND RESULTS
There are two basic concepts of fraction that a child
needs to understand in order to form a clear concept of
fraction during the first encounter in learning fraction.
These are concepts of partition fraction and quantity
fraction. The partition concept requires that a child should
be able to divide a given whole into equal parts of equal
sizes out of which a certain number of equal parts are
selected. For example, requires that the whole be
Therefore, combining the equal division of breadth and
length to form one rectangle to obtain:
divided into five equal parts out of which three of the
equal parts are selected. On the other hand, the
prevailing meaning of quantity fraction is the type of
fraction that has a universal unit. Hence the most
characteristic of quantity fraction is the unique unit-whole
which is independent of any situations. For instance,
meter, where the unit-whole corresponds to one (1)
meter. Mathematicians are divided over which of the
concepts should be used at the introductory class for
pupils who are to take the topic on fraction for the first
time. Kaori (2004) opined that using partition approach at
the introductory point could hamper the pupils
understanding of a wider concept of fraction, especially
quantity fraction while others prefer the use of partition
Stephen
concept to introduce fraction in order to ensure activitybased teaching. This paper used partition concept to
facilitate multiplication of fractions in primary six and it
was observed that this improves pupils’ understanding of
multiplication of fraction. In a co-educational primary
school in Kwali Area Council in Abuja, pre-lesson
questions on fraction were given to pupils in primary six
and students of National Teachers’ Institute (NTI) in order
to ascertain their prior opinions on fraction.
It was observed that very few respondents gave clear
and precise explanation for the questions that require
interpretation of word problems. Only 70% of the pupils in
primary school gave accurate answers to questionson the
skills of working out the product of fractions while only
20% of the learners could accurately interpret word
problems in fraction. However, it was observed from oral
interview after the test, that none of those whogave
accurate answers could confidently explain why they got
their answers. This suggests that they must have come in
contact with the process of working out fractions without
the proper concept of it.
When a group of seventy-five(75) learners was taken
through the activities of multiplication of fraction using the
concept of partition, they overcame their challenges and
also gave reasons for the product of multiplying two or
more fractions and justify why the product of two or more
fractions should be smaller than any of the fractions
involved in the multiplication. The activities above were
severally performed with the learners for three days
before the terminal assessment was conducted.
CONCLUSION
Students’ understanding of fractions as an extension to
the number system appears to benefit from the use of
partitioning concept in the teaching of the concept of
multiplication in fraction. This is due to the fact that
learners could notice and see the result in practical form
rather than memorizing it as skills to be remembered.
Learners could also justify why the product of multiplying
fractions is smaller than any of the fractions involved in
the multiplication process as against what they
experienced in the multiplication of whole numbers
whereby the product of multiplying two or more whole
numbers is bigger than any of the numbers involved in
the multiplication process Hannula (2003).
083
RECOMMENDATIONS
In order to motivate the interest and visa-vice the
attitudes of teachers and learners towards fractions and
all Mathematics topics in general, I wish to make the
following recommendations:i. Practical demonstrations should be used at the
introductory level.
ii. Practical exercises should be carried out severally to
ensure proper formation of the right concept.
iii. Examples of all the exercises should be familiar
activities within the learners’ environment.
iv. The use of ready-made formulae should be avoided
at the introductory stage; learners should be guided to
discover formulae themselves.
v. Simple and familiar words to the learners should be
used.
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andNumerical Constructs, in J. Winter and S. Pope (eds.),
Research in Mathematics Education
Volume 4, Papers of the British Society for Research into
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Hannula MS (2003). Locating Fraction on a Number Line,
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to 3-24, Honolulu,
Hawai’i.
John DB (2012). MATHEMATICS. GRADE 5. UNIT 4: Adding,
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