CHAPTER ONE
INTRODUCTION
1.1. Background to the Study
Common Fraction has been one of the oldest topics in the basic school syllabus for
mathematics. In our daily life activities, common fraction is one important factor that
enables us to share a whole with a number of people in our community, school and other
places.
There were some groups of teachers who boldly say that they can teach effectively
without the use of teaching learning materials. Others are also of the view that a teacher
can not deliver his/her lesson to the fullest understanding of the learner without teaching
learning material.
During my attachment programme at Patriensa District Assembly Junior Secondary
School, I found out that, the exercises given by the mentor under common fraction, the
pupils performed poorly. I contacted the subject teacher on why he taught that topic
(common fraction) without teaching learning materials and the response was that pupil
performance has nothing to do with teaching learning material rather the pupils are low
minded.
On realizing this, I then decided to prove the mentor wrong therefore find ways and
means of eradicating this problem from Patriensa District Assembly Junior Secondary
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School (J.S.S. 2) for delivering common fraction with the use of teaching learning material
called Fraction Board.
It is against this background that the project work wishes to direct its attention to a
topic in Mathematics “Common Fraction” as a representation of the rest of the topic.
1. 2. Statement of the Problem
Common Fraction has been a useful topic in our everyday life. Both literate and
illiterate alike use it in one way or the other.
Common Fraction is applicable in sharing a whole into parts and taking parts of the
whole. It can also be used in grouping and percentages. Thus everyone needs to have a
knowledge of this Mathematical concept so as to apply it now or then. But it seems that
most of pupils or students have problem with this topic due to poor foundation acquired at
the basic level.
Some students tend to dislike this important mathematical concept.
Especially, in operation of like and unlike fraction, finding of least common denominator
etc. Students normally get confused.
If this anomalous behaviour of future leaders are not rectify at the basic level, they
will grow to become cheats or being cheated in terms of sharing etc. Also using teaching
material like Fraction Board will make the teaching of Mathematics interesting and pupils
will like the subject even to the higher level of education.
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1. 3. Purpose of the Study
It was my main aim to use Fraction Board – a teaching aid to help pupils at
Patriensa District Assembly Junior Secondary School to acquire the knowledge or concept
of Common Fraction especially addition of unlike fraction.
1. 4. Research Questions
The study attempted to address the following research questions
i.
Do teachers use Teaching Learning Material when presenting Mathematics
concept such as Fraction to Junior Secondary School pupils?
ii.
How do the pupils respond when teaching fraction with and without teaching
aids?
iii.
How do the pupils add unlike Fractions?
1. 5. Significance of the Study
The results of this study will provide among other information which will help into
expose any discrepancy that might affect the teaching learning of common fractions so that
the education authorities could address it.
Moreover, this study will help to eradicate from pupils mind thus their perception
about fraction especially their way of adding two or more unlike fraction.
1. 6. Limitation
To make excellent results for the study there exist some limitations. The population
for this study was particularly Junior Secondary School pupils. Since the researcher can not
3
reach all these students in the entire nation for the study, a cross section of the target group
had to be considered to represent the population.
1. 7. Delimitation
Although the study seeks to appraise the whole nation, it is delimited on Patriensa
District Assembly J.S.S. Two. Production of the Teaching Learning Material “Fraction
Board” was another delimitation as only few were made due to lack of finance.
1. 8. Preliminary Definition of Terms and Acronyms
For the purpose of the study, the following definition and acronyms were used.
Fraction – Is part of a whole or part of a group.
Equivalent Fraction – different fractions of the same value. E.g. ½ and 2/4
Like Fraction – they are fractions with the same denominators. E.g. 1/3 and 2/3
Numerator – if a number is expressed as a fraction, the top number or digit is the
numerator. The bottom number or digit is the denominator.
Unlike Fraction – they are fractions with different denominators. E.g. ½ and 1/3
Fraction Board – is a teaching learning material used in the teaching of fractions.
D/A - District Assembly
J. S. S. - Junior Secondary School
4
CHAPTER TWO
LITERATURE REVIEW
2. 1. Overview
In this chapter, the related literature on this research is discussed
2. 2. Definition of Fraction
According to Baffour A. Asafo-Adjei, (Mathematics Methodology for Teachers Training
Colleges -1992), a fraction is the result of dividing something into number of parts and each
part is termed as a fraction. He further said in simple terms that ‘a fraction is part of a
whole’.
A fraction contains two terms, the numerator and the denominator. The number
below the line is the denominator and it represents the number of equal parts in which a
whole unit has been divided. The number above the line (top number) is the numerator and
represents that number of equal parts of a whole. The line separating the numerator and the
denominator is also called a bar and indicates division. For example 7/8 may be read as
“seven – eighths”: seven divided by eight: seven equal parts out of eight equal parts.
(Berston Fisher, 4th Edition in Collegiate Business Mathematics)
5
2. 2. 1. Types of Common Fractions
Robert J. Hughes (1993) I Business Mathematics Essentials defined the three
different types of fractions as:

Proper Fractions: a fraction whose top number (numerator) is smaller than its
bottom number (denominator) for example, ½, 4/5, 7/12 etc.

Improper fraction: a fraction whose numerator is equal to or bigger than its bottom
number (denominator). Examples are 2/2, 5/3, 7/2, 12/9 etc.

Mixed numbers (fraction): this is a number that contains a whole number and a
proper fraction. For example: 12/3, 51/2, 83/6, 104/5 etc.
From Johnson et al (1988) – Essential Algebra 5th Edition, “every fraction” has three
signs associated with it: the sign of the numerator e.g.
2
/5, -4/2, etc, the sign of the
denominator e.g. 2/3m 4/-2, etc and the sign of the entire fraction e.g. 2/3, -2/3 etc. If any
two of the three signs of a fraction are changed, the value of the fraction is unchanged.
They (Johnson et al 1988) also defined equivalent fraction as fractions that
have the same value. According to them, if the numerator and the denominator of a
fraction are multiplied by or divided by the same non-zero number, the new fraction is
equivalent to the original one.
D. Paling (1991) – Teaching Mathematics in Primary Schools, children are
more confused by operations with common fractions and decimal fractions than by any
other topic in Primary School Mathematics. Unfortunately, this confusion often stays
with children right through the Senior Secondary School. Once they get confused, it
seems almost impossible to find a remedy. He said that the causes of this confusion are
not difficult to find. They see to be three fold.
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The introduction of rules is too early. The children cannot understand them
and apply them incorrectly. For example:
(a)
Turn upside down and multiply or invert and multiply: children can use this
parrot – like for a division such as 1/4 ÷ 2/3 but often cannot deal with 2/3 ÷
2. (For this, they are sometimes told to change the ‘2’ to 2/1. Yet another
rule).

The use of words and phrases which have little meaning
to the children.
For example:
(b)
“Cancel” this word is alright if its meaning is understood. Often, however, it
is not understood. This again shows itself at the J. S. S. and the S. S. S. level
in incorrect canceling such as;
To many children, canceling means crossing out anything at the top of a
fraction which looks like something at the bottom of the fraction. That is all
they know about it.
(c)
Find the L. C. M. (lowest common multiple). Why they have to find it in the
addition and subtraction of fractions is often not clear to children. Some
children do not even know what the L. C. M. is.
7
Some teachers themselves do not fully understand operations with fractions.
All they can do is to introduce a quick as possible ‘rules’ which they
themselves learned at school.
2. 3. Summary
C. B. Duedu and A. Asare-Inkoom, both lecturers at University of Cape Coast in
their book, Mathematics Course Book for Diploma in Basic Education for Distance
Learning say that “the language of fraction is used in our everyday conversation to simply
mean part of a whole, a set or a measured quantity.
Common fraction is of three types: Proper fraction, Improper fraction and Mixed
Fraction or number. Equivalent fraction is said to be different fractions with the same value.
It is of these facts and other principles especially dealing with the operation on common
fractions using teaching-learning materials that this study into preparation of fraction Board
to teach operation in common fraction at J. S. S. is being researched into.
8
CHAPTER THREE
METHODOLOGY
3. 1. Overview
This chapter deals with the research design, the population and the sample s well as
the sampling method. In addition, the procedure, the instrumentation, the validity and
reliability of the instruments are discussed.
3. 2. Research Design.
The research design used was classroom experimentation. That was appropriate
since the study was attempting to find out the effectiveness of using teaching learning
materials such as fraction board to teach mathematics topic like fraction. The experiment
was to involve the students in teaching and learning process of operation on common
fraction in general.
3. 3. Population and Sampling
A population is the group to which the researcher would like the results of a study
be generalized (Gray, 1979).
The targeted population of this study is J. S. S. two students in Patriensa in AsanteAkim District. Since the whole class could not be used, simple random sampling was used
to select twenty students in the class. That was made up of ten (10) males and ten (10)
females.
9
3. 4. Procedure
In the first week, the researcher taught the operation of unlike fractions (i. e.
addition and subtraction) without using teaching learning materials. At the end of the week,
a test was conducted for the whole class. The selected students were marked and their
results were recorded.
In the second week, the teaching aid (thus fraction Board) was used to teach the
same concept again. At the end of that week, the same questions used in the first week were
used to administer another test to the students.
Their scripts were collected and the selected students scripts were marked and
recorded. The two records were later analyzed and compared. All the twenty students
selected took part in both tests and that accounted for hundred percent (100%) of the total
sample taken.
3. 5. Instrumentation
The instrumentation used in this study was detailed lesson notes and fraction board
to teach the concepts. There were three lesson notes to teach addition and subtraction.
Before the operation of common fractions, equivalent fractions and like fractions were
discussed. Exercises on the two operations were conducted after each lesson. The test was
administered after teaching all the two operations. Each question carries equal marks of ten.
It was marked out of hundred percent. The questions comprises of all the two operations in
different forms.
10
The lesson notes used for the study is presented in Appendix A, diagram of fraction
board in Appendix B, Sample questions in Appendix C and sample of students scripts in
Appendix D. the results of students work are analyzed in chapter four.
3. 6. The Validity of the Instrument
The validity of the instrument used is assured as it has been used by D. Parling
(1991) in his book, ‘Teaching Mathematics in Primary School’, C. B. Duedu et al in their
book , ‘Methods of teaching Primary School Mathematics’ and finally by J. L. Martins et al
in their book, ‘Mathematics for Teacher Training Colleges’.
3. 7. Reliability of the Instrument
The reliability of the instrument is not a problem due to the fact that the
developmentalists of learning such as the American Psychologist, Jerome Bruner in a
number of publications suggests that children and adults alike can be taught effectively in
some intellectually honest form at any stage at any three stages.
The implication of his theory was that learners learn any concept better when first of
all using concrete materials with the gradual introduction if abstract symbols via the use of
pictures and diagrams. Thus using fraction chart to teach operation of fractions is more
reliable than just using the symbols representing fractions.
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3. 8. The Mode of Analysis of the Data
The descriptive statistics was used to analyze the data collected. The procedure
adopted was by using frequency table to find the numbers and percentages of students who
scored specific groups of marks. The two frequency tables for before and after using the
materials were analyzed and compared.
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CHAPTER FOUR
RESULTS AND DISCUSSION
4. 1. Overview
This chapter discusses the results of the intervention as well as the statistical data of
the results.
4. 2. Results
Table 1: Frequency table for the Pre-test Results
Marks
Tally
0 – 10
Frequency
Percentages (%)
13
65
11 – 20
////
4
20
21 – 30
//
2
10
31 – 40
/
1
5
41 – 50
0
0
51 – 60
0
0
61 – 70
0
0
71 – 80
0
0
81 – 90
0
0
91 – 100
0
0
Total
20
100
13
Figure 1: Bar graph showing Pre-test results
4. 3. Data Analysis of Pre-test results
From both frequency table and bar graph, it was observed that out of twenty (20)
pupils selected, thirteen (13) of them scored a range of to ten (0 -10)marks. That
represented sixty-five percent (65%) of the total number selected. Four (4) pupils
representing twenty percent (20%) had eleven to twenty marks (11-20). Two pupils (ten
percent) scored twenty one to thirty marks (21-30) and one pupil had thirty one to fourty
marks (31-40) representing five percent (5%).
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That means that from the results, the highest range of marks scored was from thirty
one to fourty. This was scored by only one pupil or five percent of the population. Most of
them scored thirty marks and below. None of them scored above fourty marks. Most of
them (sixty-five percent) had ten percent and below. From those results, it was realized that
the pupils did not do well at all. Their performances were below average or poor.
Based on the pupils’ performances of the pre-test, an intervention was done to arrest
the situation. A post-test was therefore conducted. The post-test result is as shown below.
Table 2: Frequency Table for Post-test Results
Marks
Tally
Frequency
Percentages (%)
0 – 10
0
0
11 – 20
0
0
21 – 30
0
0
31 – 40
1
5
41 – 50
/
1
5
51 – 60
/
1
5
61 – 70
///
3
15
71 – 80
////
4
20
6
30
91 – 100
5
25
Total
20
100
81 – 90
/
15
Figure 2: Bar graph showing Post-test results
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4. 4. Analysis of Post-test Results
From the frequency table as well as the bar chart representation of the post test
results, it was observed tat one pupil each representing five percent (5%) scored marks
ranging form fourty one to fifty (41-50) an fifty one to sixty (51-60) respectively. Fifteen
percent and twenty percent had mark intervals of sixty one to seventy (61-70) and seventy
one to eighty (71-80) respectively. Thirty percent scored a range of marks of ninety to
hundred (90 -100) which represent six pupils out of the twenty pupils.
The results shows that ninety percent (which represents eighteen out of twenty
pupils ) had marks which is fifty one (51) and above.
The lowest range of marks scored was fourty one to fifty (41 -50) while the highest
mark interval was ninety to hundred (90 -100). None of them scored below fourty one mark
(i. e. Zero percent).
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Comparison Between pre-test and Post-test Results
Marks
Tally
0 – 10
Frequency
///
%
Tally
Frequency
(%)
13
65
0
0
11 – 20
////
4
20
0
0
21 – 30
//
2
10
0
0
31 – 40
/
1
5
0
0
41 – 50
0
0
/
1
5
51 – 60
0
0
/
1
5
61 – 70
0
0
///
3
155
71 – 80
0
0
////
4
20
81 – 90
0
0
6
30
91 – 100
0
00
5
25
Total
20
100
20
100
/
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4. 5. Analysis of Pre-test and Post-test Results
From the pre-test results, it was observed that, out of twenty pupils only one pupil
got the highest mark which was as thirty one to fourty (31-40), though not satisfactory.
In the post-test, the highest mark scored was ninety one to hundred (91-100)
percent and five out of the twenty pupils scored that mark. None of the pupils scored below
fourty one to fifty marks.
The above information shows that the pupil’s performance at the post-test was not
discouraging.
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CHAPTER FIVE
SUMMARY, CONCLUSION AND RECOMMENDATION
5. 1. Overview
In this chapter, the researcher will discuss the research problems of the
methodology, summary of the findings and conclusion. Some suggestions would also be
given for future research.
5. 2. Summary
Although the research was successful, there were some research problems
encountered by the researcher.
Due to time factor, only on school was selected. Only one class and a sample of
only twenty students randomly picked for the project.
Financial constraints also prevented the researcher from providing enough teaching
and learning materials for even each of the selected students. The general school activities
such as sports and games, cultural festivals and public holidays were other research
problems.
During those periods, there were no classes and that affected the period of the
research. Because of that, some of the research activities such as attending to individual
20
student during the intervention became difficult. These and other problems hindered my
research.
In the pre – test, the general performance of the students were not encouraging. All
of them scored below fifty out of hundred. That gave the impression that they did not
actually understand the concept. Their poor performance called for an intervention.
Teaching – learning materials called Fraction Board was then used to teach the concept for
two weeks. It was observed from the post – test results that after the intervention, their
performance was far better than the pre – test results. The lowest mark obtained in the post
– test was within 41 – 50 and that was scored by only one person (that was ten percent of
the sample selected). The two results finding indicates that the material used really helped
the students to understand the concept.
The findings revealed that the use of teaching – learning materials is more effective
and efficient in isolation or in abstract.
5. 3. Conclusion
In general terms, it can be said that the research was successful but there was one or
two limitations in terms of internal validity, external validity, measurement issues and
statistical problems. Internally, the design used, that is classroom experimentation was
perfect and valid but it was time wasting since each child had to perform an activity to find
solution to a given problem. After all, the tedious activities, the symbolic methodology
which the learners need to know was used later.
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Externally, it can be generalized that the findings from the twenty selected students
was applied to the entire general. But some of the students selected still find the concept
difficult.
The instrument used – detailed lesson notes and Fraction Board was very reliable
and valid. It was valid in the sense that some experts or author like Pioget and Brunner
have shown that learners learn better when materials are used to present a concept to them.
Hence the use of Fraction Board is valid.
It was reliable because that is the methodology and teaching strategy all
professional teachers have been using. In the absence of the researcher, every professional
teacher can use to present the concept to the pupils.
There was no particular statistical problem that the variables used was even so was
the class interval. The only problem was the zero frequencies which gave large interval
between two bars in the bar chart.
5. 4. Recommendation
Though the researcher was successful, there were some limitations. Therefore
suggests that in the near future, the researchers of the same issue should take the following
suggestions into consideration

At least, two schools should be used instead of one for time reflection of the
instrument used.

Classroom teachers should always use teaching materials to present the concept to
their learners. Also, more exercises should be given to pupils to practice.
22

Teachers should try as much as possible to provide manipulative for their pupils to
play with.
Also, the researcher suggests that Ghana Education Service, NGO’s, District Assembly
and other stakeholders of education should try to support the future researchers .in
education research findings in cash and in kind.
Last but not least, the period for research work in the colleges should be at least a year
duration.
23
REFERENCES
1. Osafo – Affum et al (1986). Ghana Mathematics series: Pupils Book Four. Pg. 116
– 121.
2. B. Kwakye Addo et al (1988). Ghana Mathematics Series: Teacher’s Handbook 4
Pg. 133 -139.
3. J.L. Martin (1994) . Mathematics for Teacher Training in Ghana.
Pg. 98 - 103
4. Robert Asafo-Adjei (2000). Teaching Basic Mathematics
Pg. 72 – 81.
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APPENDIX A
Fraction Board
25
APPENDIX B
Pre – test and Post – Test
1. 3/6 + 2/4
6. 3/6 – 2/3
2. 1/6 + 2/4
7. 4/6 – 2/5
3. 1/3 + 1/4
8. 6/12 – 2/4
4. 4/6 + 2/2
9. 4/6 – 3/12
5. 2/7 + 1/14
10. 5/4 – 7/12
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APPENDIX C
Pre-intervention Results
27
Post-intervention Results
28
29