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UNIVERSITY OF EDUCATION, WINNEBA
USING THE CUISENAIRE ROD APPROACH TO IMPROVE THE
TEACHING METHODS OF TEACHER TRAINEES’ IN TAMALE
COLLEGE OF EDUCATION-TAMALE
AHMED YAKUBU
2013
i
UNIVERSITY OF EDUCATION, WINNEBA
USING THE CUISENAIRE ROD APPROACH TO IMPROVE THE
TEACHING METHODS OF TEACHER TRAINEES’ IN TAMALE
COLLEGE OF EDUCATION-TAMALE
AHMED YAKUBU
(7100110004)
A Dissertation in the Department of MATHEMATICS EDUCATION, Faculty of
SCIENCE submitted to the School of Graduate Studies, University of Education,
Winneba in partial fulfillment of the requirements for the award of the MASTER
OF EDUCATION in Mathematics Degree
.
SEPTEMBER, 2013
DECLARATION
CANDIDATE’S DECLARATION
I, AHMED YAKUBU, declare that this thesis, with the exception of quotations and
references contained in published works which have all been identified and acknowledged,
is entirely my own original work, and it has not been submitted, either in part or whole, for
another degree elsewhere.
CANDIDATE’S SIGNATURE:………………………………
DATE:…………………………………………………………
SUPERVISOR’S DECLARATION
I hereby declare that the preparation and presentation of this thesis were supervised by me,
in accordance with the guidelines and supervision of thesis as laid down by the school of
research and graduate studies, University of Education, Winneba and recommended it for
acceptance.
SUPERVISOR: PROF. S. K. ASEIDU-ADDO
SIGNATURE:………………………………….………
DATE:……………………………………………..……
ii
ACKNOWLEDGEMENTS
A research work of this nature could only be done with the assistance of many
personalities. In undertaking this project therefore, I had tremendous advise, guidance,
encouragement and suggestion from people to whom I owe a debt of gratitude. First and
foremost, I wish to acknowledge the immense contribution of my supervisor, Prof. S. K.
Aseidu-Addo (Head of Department, Mathematics Education) for the friendly painstaking
and objective manner in which he supervised my work. To him I say a big thank you.
I am also indebted to the Principal and all the teaching and non-teaching staff of Tamale
College of Education-Tamale for being inspiration to me throughout my life.
Special thanks go to my sweet-heart (Azizatu) and my son (Abubakari Sadiq Tia) for
bearing my absence from home and their unflinching support. Finally, to God be the Glory
for seeing me through yet another hurdle and challenge.
iii
DEDICATION
To my dear wife and son (Tia) who together supported me in diverse ways to reach this
far.
iv
TABLE OF CONTENT
Content
Pages
Declaration
ii
Acknowledgements
iii
Dedication
iv
List of Tables
ix
List of Figures
x
Abstract
xi
CHAPTER 1
INTRODUCTION
1
1.0 Overview
1
1.1 Background to the Study
1
1.2 Statement of the Problem
3
1.3 Purpose of the Study
5
1.4 Research Questions
6
1.5 Significance of the Study
6
1.6 Delimitations of the Study
6
1.7 Definitions of Terms
6
v
CHAPTER 2
LITERATURE REVIEW
10
2.0 Overview
10
2.1 Inadequate Use of Teaching Learning Materials
14
2.2 Models for Fraction
15
2.3 Language Mistakes
16
2.4 Problems Encountered by Pupils
18
2.5 Problems Encountered by Teacher Trainees’
18
2.6 Overcoming Mathematics Phobia in Learning
19
2.7 Methods for Learning Mathematics
20
2.8 Mathematics Learning should be Meaningful
20
2.9 Operation with Fractions
21
2.10 Applications of Fractions
21
2.11 Conceptual Framework
22
2.12 Summary
25
CHAPTER 3
METHODOLOGY AND DESIGN
26
3.0 Overview
26
3.1 Population
26
3.2 Sampling Size and Sampling Procedure
26
3.3 Intervention Design
27
3.4 Demonstration (Folding Method)
28
3.5 Equivalent Fractions
29
vi
3.6 Teaching Addition and Subtraction of like Fraction (Same Denominators)
31
3.7 Teaching Addition and Subtraction of Unlike Fraction and Mixed Numbers
32
3.8Teaching Multiplications of Common Fractions and Mixed Fractions
and to Develop the Algorithm
38
3.9 Teaching Division of Common Fractions and Mixed Numbers
39
3.10 Method of Data Collection
42
3.11 Validity and Reliability of Instruments
42
3.12 Data Collection Instruments
43
3.13 Method of Data Analysis
43
CHAPTER 4
RESULTS/FINDINGS AND DISCUSSIONS
44
4.0 Overview
44
4.1Analysis and Data Interpretation of Questionnaire by Teacher Trainees’
44
4.1.1 Research Question 1
44
4.1.2 Research Question 2
46
4.1.3 Research Question 3
47
4.2 Analysis of Pre-Test and Post-Test Scores
49
4.3 Analysis of the Pre-Test Result
50
4.4 Analysis of the Post-Test Result
51
4.5 Analysis and Interpretation of Post-Observation Assessment
54
4.6 Comparing the Pre-Test and Post-Test Results of Trainees’
54
vii
CHAPTER 5
SUMMARY, RECOMMENDATION AND CONCLUSION
57
5.0 Overview
57
5.1 Summary of Major Findings
57
5.2 Implications for Practice
58
5.3 Conclusion
59
5.4 Recommendations
59
5.5 Implication for Further Research
60
REFERENCES
61
APPENDICES
66
viii
LIST OF TABLES
Page
Table 4.1: Responses to the use of Cuisenaire rods to improve the teaching of
Fractions
44
Table 4.2: Responses of Cuisenaire rods use in solving addition and
subtraction of equivalent fractions
46
Table 4.3: Responses on the difficulties teacher trainees’ find using Cuisenaire
rods in teaching addition and subtraction of fractions
47
Table 4.4: Scores on pre-test and post-test
49
Table 4.5: Frequency distribution table for the pre-test results
50
Table 4.6: Frequency distribution table for the post-test results
52
Table 4.7: Frequency distribution table for the pre-test and post-test results
53
Table 4.8: Paired Samples Statistics of trainees’ performance in fraction class
54
Table 4.9: Paired Sample Correlation of pre-test and post-test of the use of
Cuisenaire rods by Trainees’
55
Table 4.10: Paired Samples Test
55
ix
LIST OF FIGURES
Page
Figure 1.1: Cuisenaire Rods
8
Figure 4.1: Level of Performance of Trainees’ during the Pre-Test
51
Figure 4.2: Level of Performance of Trainees’ after the intervention process
52
Figure 4.3: Comparing the level of Performance of trainees’ after the implementation
of the intervention
53
x
ABSTRACT
The study investigates the use of “Cuisenaire rods” approach to improve the teaching of
fractions to teacher trainees’ (Sandwich - Diploma in Basic Education programme) in
Tamale College of Education – Tamale in the Northern Region.
The sample comprised thirty-six (36) teacher trainees’ in Tamale College of Education. It
involved the use of pre-test, post-test and questionnaire as the instruments for data
collection. The pre-test was conducted after which an intervention period of six weeks
which involved taking teacher trainees’ through the use of Cuisenaire rods to improve the
teaching of fractions. A post-test was conducted after the intervention. The scores obtained
from the pre-test and post-test were analysed by the use of a paired sample t-test. The
questionnaire was also analysed.
The findings revealed that the use of teaching and learning materials (Cuisenaire rod
approach) greatly improved the teaching of fractions in mathematics.
It was also realized that before teacher trainees’ could grasp the concept of addition and
subtraction of fractions, the concept of equivalent fractions should be well understood.
It was also realized that teacher trainees’ grasp a concept when they are taken through the
procedure from concrete through semi-concrete and finally to the abstract. The
implications of the findings are that teacher trainees’ should be encouraged to use teaching
and learning materials (Cuisenaire rods) to teach when introducing a concept.
The concept of Equivalent fractions should be taught first before addition and subtraction
of fractions are taught.
xi
CHAPTER 1
INTRODUCTION
1.0 Overview
This chapter provides an introduction to the research study. The introduction includes the
background to the study, statement of the problem, the purpose of the study, research
questions to guide this study. It further highlights the significant and limitation of the
study.
1.1 Background to the Study
Mathematics is one of the important subjects within the list of foundation subjects that
constitute the core curriculum for basic education in most countries throughout the world.
The subject occupies a privileged position in the school curriculum because the ability to
cope with more of it improves one’s chances of social advancement. It attained this
position since it was made to replace classical language like Latin or Greek which prior to
the early half of the twentieth century were used as screening devices for entry to higher
education and certain professions. (Howson and Wilson, 1986) as cited in Mereku (2000).
The importance of Mathematics can be seen from its application in our daily lives and its
role in technology. No other subject forms a strong binding force among various branches
of science than mathematics and without it; knowledge of science often remains superficial
(Singletary, 1997). This indicates that without a proper grasp of the underlying principles
in mathematics, the necessary skills and concepts in Science and Technology cannot be
acquired and applied by students. Ghana as a nation cannot develop fast if sustainable
strategies are not put in place to improve upon the teaching and learning of mathematics in
1
our schools. This is because these pupils or students are the future leaders of the nation. If
the educational structure cannot give them a good foundation in mathematics, then they
cannot have the requisite materials and the technical know-how needed to contribute their
quota towards the development of Ghana. McBride and Silverman (1997: page 102) said
“mathematics can enable students to achieve deeper understanding of science concepts by
providing ways to quantify and explain science relationships” .Even though mathematics is
one of the important subjects, most students shy away from it. Some students wish they
could be excluded from mathematics lessons. Even though they know it is a useful subject,
they find it difficult. That is why they develop a negative attitude towards it. Mathematics
is widely recognized as a problem in many circles. Most candidates fail to get admission
into tertiary institutions because of failures in mathematics. Every child who enters the
educational system had to study mathematics till the matriculation level (Evans 2002).
According to Institute of Education, University of Cape Coast (2007) Chief Examiner’s
Report of Diploma in Basic Education Teachers (Sandwich) indicates that many students
disliked questions that involve fractions. Students had difficulties with addition of fractions
with different denominators. They could not make use of equivalent fractions. Besides,
questions that involve division and multiplication of fractions were not well answered.
Students could not state the vital steps needed before writing the final answer. They
therefore developed phobia towards questions involving fractions. The fraction concept in
its broader sense starts from the day the child enters the world. The child experience with
fractions is through everyday life. For instance, the components of breast milk that the
child starts to suck are expressed in fractional forms, as specified by the Ghana Health
Service advocacy for exclusive breast feeding for the first six months of the child’s live the
2
breast milk consists of seventy-five percent (75%) water and twenty-five percent (25%)
milk. Burton and Ted (2000) reported that the common form of fraction has the advantage
of being intuitively understandable when we want to talk about parts of a whole. Working
with fractions poses serious problems to most candidates especially “Diploma Teachers”.
Candidates had difficulties in solving questions that involve fractions. Important steps of
working are often left out by candidates (Institute of Education, University of Cape Coast
(2007) Chief Examiner’s Report). A distinguished American Professor of Mathematics, Dr
Whiney said “for several decades we have seen increasing failure in Mathematics in spite
of intensive efforts in many directions to improve matters. It would be reasonable also to
suspect that the causes are fundamental in the ways in which they are taught, how children
learn and the teaching situations” (Skemp 1991 page 72). The most effective means of
learning a concept or skill is through active participation.
1.2
Statement of the Problem
One of the weaknesses the 2002, 2004 and 2008 Basic Education Certificate Examination
(BECE) listed in the Chief Examiner’s Report of the West African Examination Council
included candidates inability to answer questions that involved fractions. Further, the 2007
Chief Examiner’s report of the Institute of Education, University of Cape Coast, also
stressed that most of the Untrained Teachers pursuing ‘Untrained Teachers Diploma in
Basic Education’ (UTDBE) also could not answer questions involving fractions during
their end of semester examination on Methods of Teaching Primary school Mathematics.
This is a clear indication that the teachers on the Untrained Teachers Diploma in Basic
Education (UTDBE) Programme, who are service teachers on the field, have passed on
some of their difficulties onto the pupils and must be checked. In addition, teacher
3
trainees’ were also not able to answer questions that involve addition, subtraction,
multiplication and division of fractions. The researcher is not surprised that some
candidates of the BECE could not effectively answer questions that involved fractions,
simply because most of these service teachers do not understand the concept of fraction so
well that, they are therefore unable to effectively teach the concept of fraction.
More so, the performances of the Diploma in Basic Education (DBE) teacher trainees’
during their On Campus teaching Practice (OCTP), in the process of teaching of addition
and subtraction of fractions were not encouraging. They find it difficult in making their
lessons on fractions look practical. All they do is to use the Least Common Multiple
(LCM) to add or subtract but the practical meaning is not taught. Those who attempt to use
Teaching Learning Materials (TLM’s) are not able to demonstrate the operations on unlike
fractions such as + , or - . Pre-service teachers, who are expected to be conversant with
the use of TLMs’ such as the Cuisenaire rods to illustrate fractions, are themselves not able
to do so. Some cannot even associate the colors to the lengths of the rods. The mere
hearing of the name Cuisenaire rods scare most of them. The few who are comfortable
with Cuisenaire rods are able to use them in teaching the operations of whole numbers but
are not able to extend the ideas to fractions.The study therefore sought to address some of
the difficulties teacher trainees’ face in teaching fractions.
To say that, the quality of mathematics teaching depends on teachers’ Mathematics
knowledge of teaching (MKT) the content should not be a surprise. Perhaps curriculum
programmers do not take this into account, they only concentrate on providing logistics
which they feel will promote effective teaching and learning. We should not lose sight of
4
the fact that little improvement is possible without direct attention to the practice of
teaching. Unsurprisingly, many Ghanaian teachers lack sound mathematical understanding
and skill (Akyeampong, 2003). Meanwhile, the kind of importance given to mathematics
makes it more imperative that every child be taught to get the understanding and the flair
for it.
Strong standards and quality curriculum are important, but no curriculum teaches itself. To
implement standards and curriculum effectively, school systems depend upon the work of
skilled teachers who understand the subject matter. Since pupils and their teachers have a
peculiar difficulty in solving addition and subtraction of fractions, it is therefore important
that, mathematics educators adopt good mathematics knowledge of teaching (MKT) to
build the capacity to use instructional materials wisely, to assess students’ progress, and to
make sound judgments about presentation, emphasis and sequencing.
1.3 Purpose of the Study
The study is to find out;
i.
Whether the use of manipulatives (Cuisenaire rods) would improve the teaching of
fraction.
ii.
Whether the use of Cuisenaire rods could be used in solving addition, subtraction,
division and multiplication of fraction.
iii.
The difficulties Diploma teacher trainees’ encountered in using the Cuisenaire rods
in teaching adding, subtracting, dividing and multiplying fractions.
5
1.4 Research Questions
The following research questions will guide the conduct of the study;
1. To what extent would the use of Cuisenaire rods improve the teaching of fractions?
2. To what extent would Cuisenaire rods be used in solving addition and subtraction
of equivalent fraction?
3. What are the difficulties Diploma teacher trainees’ encountered in using Cuisenaire
rods in teaching addition and subtraction of fractions?
1.5 Significance of the Study
The findings of the research will serve as a guide to teacher trainees’ in Colleges of
Education as to how they can improve upon the teaching of fractions using the Cuisenaire
rod approach. Again, it will guide policy makers and other stake holders in teacher
education to make it a priority in providing teaching learning materials and especially
Cuisenaire rods to Colleges of Education in Ghana.
1.6 Delimitations of the Study
The study is made up of teacher trainees’ at Tamale College of Education – Tamale.
Besides, it will be confined to only second year students. The study is delimited to the use
of manipulatives (Cuisenaire rods) to improve the teaching of fractions. It is therefore not
possible to generalize the results of this study.
1.7 Definitions of Terms
For the purpose of the study, the following definitions are implied for the terms below;
Colleges of Education: They are institutions where students are trained to become
6
professional teachers, at first they were called Teacher Training Colleges but all the
colleges are being upgraded to diploma status hence the name.
Concept: It is the knowledge or the idea we have about something.
Cuisenaire Rods: They are teaching learning materials which are use to teach fractions,
addition and subtraction of whole numbers whose sum is less than 10. The rods are made
up of 10 different colours which are associated with numbers 1 - White, 2 - Red, 3 - Light
green, 4 - Purple, 5 - Yellow, 6 - Dark green, 7- Black, 8 - Brown, 9 - Blue and 10-Orange.
7
Purple
Light Green
Red
White
Figure 1.1: Cuisenaire Rods
8
Black
Dark Green
Yellow
Brown
Orange
Blue
Cuisenaire rods
Like Fraction: They are fractions with the same denominators.
Teacher Trainees’: They are the students of the colleges of Education
Unlike Fractions: They are fractions with different denominators.
Sandwich Courses: A course of study which includes periods of study and periods of
working.
Cert ‘A’ Teachers: Certificate ‘A’ teachers are teachers who are being trained to obtain
Diploma Certificates.
9
CHAPTER 2
LITERATURE REVIEW
2.0 Overview
This chapter deals with the review of relevant and related literature. It is to acknowledge
works of other authorities so as to avoid plagiarism. Literature review is necessary to avoid
the risk of duplicating previous studies, using unproductive techniques, and therefore not
contributing much to the advancement of human knowledge. The researcher tries to
evaluate previous studies, observations, opinions and comments related to this intended
research.
The word fraction is taken from the Latin word “frangere” which means to break. This
suggests that fraction may be described as a part of a whole where the whole could be a
unit or a set of objects.
Paling (1996) states that children in primary and secondary schools spend many hours each
year work involving fractions. They learn how to add and subtract two fractions, to
multiply one fraction by another and to divide one fraction by another. They are then
introduced to quick methods and to phrases such as “turn upside down and multiply”. Yet,
at the end of all their computations with fractions, students on teaching practice are often
asked to repeat work that the children had done before, but did not appear to understand.
Paling (1996) reveals that children who go to secondary schools or other forms of further
education there is the need to be able to use operations (addition, subtraction,
multiplication and division) with common fractions. This is particularly true in
mathematics and science, especially when letters are used to represent numbers for
10
example, in solving equations and in changing formulae, algebraic fractions often occur
and an understanding of the methods used for the operations (addition, subtraction,
multiplication and division) are essential.
Dolan, (2000) observes that apart from whole - number computation, no topic in the
elementary mathematics curriculum demands more time than the study of fractions. Yet,
despite the years of study, many students enter high school with a poor concept of fractions
and an even poorer understanding of operations with fractions. When asked about their
memories of fractions, adults will often reply’, “Yours is not to reason why, just invert and
multiply.”
Dolan, (2000) further states that the fraction
can be viewed as the solution to the
problem of dividing. 3 dollars among 4 people: 3÷4. For students to understand this
connection between whole number and fractions, teaching about fractions and their
operations must be grounded in concrete models. A firm foundation for number sense
involving fractions, and a deeper understanding of the algorithms for operations with
fractions must be developed and preceded formal work with fractions.
Brunner as cited in Martin et al (1994) observes that when a child or an adult is learning a
new concept, one must go through three stages. These stages are used simultaneously for
acquisition of new skills throughout man’s lifetime. They also revealed that it was the duty
of the teacher to use concrete materials to introduce the topic first and gradually proceed to
picture and diagrams, that is from concrete to semi-concrete and finally to the abstract
stage. Hemstock and Costelpe (1975) explain the concept “fraction”. The two authors
11
explain vulgar fractions as the same as simple fraction for example
they explained that
the lower number that is the 4 is the denominator and 1 is the numerator. The golden rule
sets that before fraction could be added or subtracted, they must have the same
denominators that is and could be added to get
however and cannot be added until
they changed into fractions of the same denominators. They also explained that if a fraction
is multiplied by a whole number, thee denominator remains the same. For instance
3x
, moreover when a fraction is multiplied by a fraction, multiply the
numerator together and the denominators also together, for example, for example
The rule also says that divisions of fractional numbers averts the divisor and multiply the
numerator and the denominator for instance
Obeng (2005) stated that addition or subtraction with different denominator can be solve
using the fractions as illustrated in the examples below:
Equivalent fractions of are
And those of are
12
In the same way;
Equivalent fractions of are
And of are
a)
Elizabeth (1988) maintains that “diagrams can be used to make some classifications which
young children have done, which facilitates knowledge building. Elizabeth (1988) further
described the usefulness of diagrams as teaching and learning materials when one has to
solve mathematics problem. From the above assertion it is evident that diagrams enable the
problem - solver to obtain an insight of the problem at hand, enhance formulation of
various strategies that will help one solve the problem and aid one to construct proofs of
several theorems.
Dirkes (1980) emphasizes that when children use drawings to evaluate their attempts to
show an idea of fractions, they soon use space to help them produce answer to
computational examples through a problem - solving process. Dirkes goes further to state
that pictures and concrete objects, for that matter models serve to communicate
mathematics in a way that words and symbols cannot do. This implies that concepts are
well understood only when they are illustrated through objects diagrams or pictures.
13
According to Stanley and Peck (1981), it is good to start fraction by sharing. That is giving
each child an empty egg carton and have them place the egg cartons bottom side upon their
desks in front of them. Ask them to place a piece of yarn across the carton to show where
they cut it, if they were to share a carton with two people (including themselves as one of
the two), three people, four people and so on.
The “3” is written as an instruction to perform the sharing act and is read “share with 3”,
“4” share with four and so on, becomes an instruction. Similarly, other fractions are
conceived in terms of instructions for sharing and covering. For example
is read “Share
with three, cover up two shares”. This simple approach to fractions removes many of the
difficulties children frequently have with fractions. Working with physical materials that
can be seen and felt seems to be helpful. Children like to do the physical sharing and
manipulating of the objects. A wide variety of materials to share increases understanding
and skillful questioning sustains interest.
2.1 Inadequate Use of Teaching Learning Materials
In Ghana, most schools learn without adequate teaching learning materials. Dotse (2000)
states that ‘instructional aids and games when properly employed make learning
simultaneous, effortless and quite enjoyable. Rather unfortunately, the use of these
materials is on the decline’. Cuisenaire rods can be used to teach fractions.
14
2.2 Models for Fraction
Three concepts of fractions illustrated by Bernett and Ted (2000) are stated below;
a) Sharing (Part - Whole Model)
This is the case where some children share a number of items like oranges or sweets. In a
situation where the number of items being shared is not enough for the children, it becomes
necessary for them to cut or break the items up into bits and share. This is the use of a
fraction to denote part of a whole. In the fraction
the bottom number “b” indicates the
number of equal parts in a whole and the top number “a” also indicates the number of parts
(s) taken out from the whole.
b) Part Group Model
In the day to day activities of children, it often becomes necessary for them to consider part
of a set in relation to the major set. For instance in a primary school set up, Basic School 1
(B. S. 1) is part of the school which implies that it is a subset of the school.
c) Ratio Model
This shows the relationship between objects of the same kind. It is a way of comparing the
objects and this ends up in the form of a fraction. For instance in using the Cuisenaire rods
to compare the lengths of two rods, it takes 3 red rods to equal the length of 1 dark green
rod in fig (1) below. Hence the length of the red rod is of the length of dark green rod.
15
The ratio of the length of the red rod to the dark green rod is 1:3
Red
Dark green
2.3 Language Mistakes
The language used by the teacher is important if he or she is to make positive impact on his
or her learners. William, (1986: page 21) states that “more importantly mathematical
languages should be carefully and accurately used from the beginning”. They further
observed that “mathematical vocabulary needs to be taught and should be taught in the
context of practical experience” for instance, the fraction should be described in words as
two-seventh, with emphasis on the two. A phrase such as two over seven should be
avoided.
Watson (1980) gives a concise description of the errors many pupils make in the operations
on fractions. The author includes the common mistakes such as the following
2 x1 = x =
Watson explains that this is the result of the applications of the rule for addition of
fractions with the same denominator.
According to Apronti (1987) identified fraction as a challenging topic for both teachers and
pupils. The author outlined the following strategies for teaching the concept of fraction.
16
i. Let pupils see fraction as part of a whole object or a group of objects.
ii. Note that the concept of fraction is based on equal division.
iii. Build up the concept of fraction gradually with the concept of one fourth, one third
and so on.
iv.
Use paper folding and sharing, the chart or fractional board to compare fractions
and find equivalent fractions.
According to Amissah (1998) in a paper delivered at the 26th National General Workshop
of the mathematical Association of Ghana (MAG), August, 24 - 29, 1998, at St. James
Seminary School, Sunyani, the analyses presented revealed that mathematics performance
at the primary level gives much cause for concern. Primary Education Program (PREP)
conducted a study on selected teacher trainees using the same set of “Criterion Referenced Test” (CRT) test items. The study revealed that some of the teacher trainees
faulted in the very areas that the pupils had performed poorly. Amissah further stated that
he was privileged to serve on a panel for selecting the National Best Teacher’s award
winners. The candidates who appeared before the Panel were those adjudged the best
Regional award winners. One of the questions a panelist asked at the primary level was
‘how do you teach +
to primary four?” Two of the ten award winners gave this method
“Add the numerators and the denominators”.
Hence
.
17
2.4 Problems Encountered by Pupils
Pupils find it difficult to deal with fractions. The reason might be that they do not grasp the
concept well. The understanding depends upon the ability of pupils to relate symbols with
words. Multiplying fractions by a whole number was a problem, but adding fraction is a
deeper crisis. For instance 2 x was given as
(treated as a vector). Below are samples of
pupils’ work presented to the researcher.
i. -7 + 12 = -31+62 = 31. The pupils’ explanation was that we multiplied 4x7 and
added 3 and also multiplied 5 by 12 and added 2 to obtain -31 and 62
respectively which were summed up to 31.
ii.
, the pupil explained that she added the numerators and denominators to
obtain that is (2+ 4) and (3 + 5). In analyzing the computation errors made in
the fraction exercise, it is evident that mush of the difficulty lies in a lack of
conceptual understanding as emphasize by Carpenter (2001:36). It can be
concluded that children should be exposed to variety of experiences with
grouping before being introduced to fractions.
2.5 Problems Encountered by Teacher Trainees’
Teacher trainees’ find it difficult to teach fractions effectively. Some teacher trainees’ also
use oranges to introduce fractions to their pupils which are not the appropriate teaching
learning materials for those concepts. Below are samples of teacher trainees’ work when
the researcher conducted quiz to find out their level of understanding the concept fraction.
18
i.
. Most of the teacher trainees’ subtracted the numerators that is 5-1
and the denominators that is 4 – 4 and their difference were 4 and 0 respectively.
They finally subtracted 0 from 4 which resulted 4.
ii.
+
=
= 3 + 12 = 15. Most of the teacher trainees’ added the numerators and
denominators and again added the results of both the numerators and denominators
to get their final answer 15.
iii.
. Most of the teacher trainees’ divided both the numerators and the
denominators and further divide them that is 2 ÷ 2 = 1 and 9 ÷ 3 =3. They finally
got . For multiplication they multiply both the numerators and denominators to get
their results. It can be concluded that teacher trainees’ have not grasp the concept of
fraction and are unable to impact the knowledge to their pupils.
2.6 Overcoming Mathematics Phobia in Learning
Mathematics is widely recognized as a problem subject many parents find it difficult to
help their children with mathematics at home.
At a mathematics clinic in Hagpur, the parents brought their children to know why they
were not coping with formal mathematics to their level of expectations. During
unstructured dialogue with parents/facilitators and learners, it was discovered that fractions
and rations accounted for the largest number of troubles for the children.
19
More than half of the parents were of the opinion that most of what the children studied as
part of the mathematics curriculum has no practical use. At the clinic, special card games
were designed and used to teach concepts such as equivalent fractions. This was to help the
children to learn while playing.
It also became clear that one needs to integrate formal school mathematics within the home
and community and make efforts to better the perception of mathematics. Civil (2003).
2.7 Methods for Learning Mathematics
Piaget, Bruner and Dienes each suggests that learning proceeds from the concrete to the
abstract. It is believed that children should be actively informed in the learning process and
opportunities for talking about their ideas should be provided. It is also believed that
symbols and formal representation of mathematical ideas follow naturally from the
concrete level, but only after conceptualization and understanding hence taken place
(Rober: 1998).
2.8 Mathematics Learning should be Meaningful
The notion of meaningful learning advanced by William Brownell in the first half of the
twentieth century provides the cornerstone for a quality school mathematics program.
Brownell conceived of mathematics as a closely knit system of ideas, principles and
processes a structure that teaching should not only take advantage of, but emphasize, so the
‘arithmetic is less a challenge to that pupils’ memory and more a challenge to his or her
intelligence’ (Marilyn and Suydan: 1998).
20
2.9 Operation with Fractions
The key to a meaningful presentation of the operations with fractions is to establish a firm
background in fractions, especially equivalent fractions and modeling fractions. Then,
problem situations that involve fractions and operations should be presented. Children also
gain a better understanding of operations with fractions if they can estimate answers by
using whole numbers and fractions such as one-half or one-tenth. For example, before
actually computing the answer to three whole numbers two on three plus four whole
number five on six, they should realize that the answer is more than 7. Infact, since and
are each more than
the answer is more than 8. Developing these types of number and
operation sense will make it easier to establish what reasonable answers to problems are.
2.10 Applications of Fractions
i. The concept of equivalent fractions is needed to mix two or more ingredients in the
kitchen when cooking.
ii. The concept of fractions is needed to take the right amount of drugs when one is
sick. The student may have to take half sachet of medicine mixed with half a glass
of water two times daily.
iii. Students can best plan their menu and have balanced diet over a stipulated period of
time with the idea of fractions. This is because some amount of Carbohydrates may
be taken with some amount of protein.
iv. Until recently, the mineral components in milk were considered to be a little value.
Their use in various applications was considered excessive and could adversely
21
affect the taste and texture of the finished product. For these reasons, the mixing of
milk components with other ingredients to produce industrial blends is a way of
better responding to the particular requirements of end users. This offers different
components of high-performing milk and fruit beverages and the idea of rations is
the reason for the different components of milk in the market.
2.11 Conceptual Framework
Manipulatives are concrete objects that can be viewed and physically handled by students
in order to demonstrate or model abstract mathematical concepts. They include a host of
colourful shapes and objects including tangrams, cubes, and Cuisenaire rods. The use of
multipulative materials to teach mathematics has a long history. In the 19 th century,
influential Swiss educator Johann Pestalozzi (1746-1827) advocated the use of various
manipulatives, such as blocks, to help children acquire abstract concepts, such as number
sense, through concrete means (Saettler, 1990).
In the early part of the 20th century, Montessori (1870 – 1952) founded schools and
acquired a host of followers that believed in and stressed the importance of concrete,
authentic learning experiences. It was her belief that children actualized their innate desire
to learn through self –directed exploration of developmentally appropriate manipulatives
(Ward, 1971).
In the 1960s and 1970s, the appearance of manipulative in the elementary classroom
increased rapidly following the publication of Zoltan Dienes’ theoretical justification for
their use. Since Dienes’ work, a number of studies investigating the effectiveness of
manipulative have been conducted. The results have been somewhat mixed. However,
22
there is general agreement among educators today that an effective mathematics
curriculum in the elementary grades must include liberal use of manipulative materials.
Such criteria understandings as number, place value operations, fractions, decimals,
geometry and algebra all can be effectively taught the use of manipulatives.
Much research into the use of manipulative materials and their benefits to children has
been conducted over all the years. Although some researchers have questioned the value of
manipulatives in teaching mathematics (Raphael and Wahlstrom, 1989), research has
demonstrated that manipulative generally have a positive effect on student achievement
and mathematical learning compared to more traditional instructional methods that place a
heavy emphasis on the use of worksheets and computational fluency (Bisio, 1971;
Fennema, 1972; Suyadan & Hiiggins, 1977; Driscoll, 1981; Parham, 1983; Sowell, 1989;
Cramer et al., 2002).
Bisio (1971) analyzed the effectiveness of the three different methods for teaching the
addition and subtraction of fractions with like denominators. In the first method, teachers
and students did not use manipulative materials; in the second method, the teachers used
manipulatives only to demonstrate concepts for students; in the third approach, both
teachers and students used manipulatives. Although Bisio found that the passive use of
manipulatives by teacher (method 2) was just as effective as the active use by students
(method 3), both approaches were much more effective the instructional approach in which
manipulatives were not used at all.
23
Fennema (1972, P, 635-640) summarized research on the use of Cuisenaire rods to teach
arithmetic compared to more traditional approaches. She found that research generally
supported the use of manipulative for first-graders, but that the value of the rods for
second- and third – graders was less conclusive. Fennema (1972, P. 635) concluded,
“There is some indication that children learn better when the learning environment
includes a predominance of experiences with models suited to the children’s level of
cognitive development.” Her recommendations were that teachers use manipulative to
teach mathematics in the early grades and gradually decrease their use as students are able
to gasp concepts more symbolically.
Suydam and Higgins (1977) performed a meta – analysis of 40 research studies into the use
and effectiveness of manipulatives on student achievement in math. 60% of the studies
indicated that manipulatives had a positive effect on student learning; 30% showed no
effect on achievement; and 10% showed significant differences favouring the use of more
traditional (non – manipulative) instructional approaches. In similar work, Sowell (1989)
performed a meta –analysis of 60 additional studies into the effectiveness of the various
types of manipulatives with kindergarten through post secondary students. On the basis of
this research, she concluded that achievement in mathematics could be increased through
the long-term use of manipulatives.
Parham (1983) analyzed 64 research studies into the use of mathematics manipulatives at
the elementary level. Generally, she found that there were significant positive differences
in achievement of students who had used manipulative as part of their math instruction
compared to those who had not.
24
In a recent study of 1,600 fourth – and fifth graders, Cramer et al. (2002), compared the
achievement of students using.
2.12 Summary
Below are the summary of the literature review.
i.
Studies show that the teaching of fractions and their operations should be grounded
by concrete models (e.g. Dolan 2008; Martins 1994).
ii.
Other studies (e.g. Obeng 2005; Hemstock of Costelpe, 1975; Paling, 1996) have
shown that in teaching fractions pupils or students should be taught in the use of
rules in solving problems relating to fractions.
iii.
Other authorities are of the view that the language of the teacher is important if he or
she is to make positive impact on his or her learners. Mathematical language should
be carefully and accurately used from the beginning. For instance, the fraction
should be described in words as three-fourths. The phrase such as three over four
should be avoided (E.g. William 1986).
Even though authorities cited in this piece of work have come out with their findings, the
researcher is of the view that the use of teaching and learning materials enriches learners’
visualization and solid foundation for better understanding and therefore must be
encouraged.
25
CHAPTER 3
METHODOLOGY AND DESIGN
3.0 Overview
This chapter deals with the methodology employed in this study, and includes the research
design (research problem, sample strategy, participant selection), and fieldwork strategies.
This study employed both qualitative and quantitative research methods. The purpose of
this study was to improve the ability of pre-service teachers’ in the teaching of fraction
using Cuisenaire rods.
3.1 Population
A population is a group of elements or variables. Be it human, objects or events that can
form to specific criteria and which are interested to a researcher for intending to
generalization of result. The population of the research consists of Diploma students in
Tamale College of Education - Tamale. In all they were 200 students. It was selected
because of proximity of the area of study of the researcher. The choice of this was based
purely on convenience.
3.2 Sampling Size and Sampling Procedure
Probability sampling techniques was used to select second year students of Tamale College
of Education, because the researcher happened to be the Mathematics teacher assigned to
teach (Methods of teaching mathematics in basic schools).
The researcher selected 36 teacher trainees’ comprising 18 men and 18 women. All the
men were grouped and the researcher wrote 18 “yes” and the rest “No” on a sheet of paper
26
and folded them. So those who picked “yes” were selected. The women went through
similar procedure and the 18 were also selected. The average age of the students selected
were 20+ and the percentage of students selected is seven point two percent (18%).
3.3 Intervention Design
The structured table below served as a guide for the intervention strategy.
Date
Activity/Exercise
6th – 10th August, 2012
Class exercises were given to students.
13th August, 2012
Pre-test was conducted
15th August, 2012
Intervention began
20th – 24nd August 2012
Continuation of the intervention
27th – 31th August, 2012
Class exercise on equivalent fractions and
addition and
3rd – 7th September, 2012
Continuation of the intervention
10th – 14th September, 2012
Exercise on multiplication and division of
fractions coupled with interesting activities
17th September, 2012
Post –test was conducted
The researcher decided to use activities and demonstrations to help the teacher trainees’ to
overcome the difficulties they face in teaching fractions.
The demonstrations helped the teacher trainees’ to develop the Concept of fractions when
an idea is explained; the researcher had to demonstrate it for clarity on the part of the
learners. Activities were used and this made teacher trainees’ to participate meaningfully in
the lessons. Below was how interventions were implemented.
27
i. Demonstration: Real objects and Cuisenaire rods were used to demonstrate the
concept of fraction.
ii. Equivalent fractions.
iii. Teaching Addition and Subtraction of like fractions
iv. Teaching Addition and Subtraction of Unlike fractions and mixed fractions.
v. Teaching multiplication of Common fractions and Mixed fractions.
vi. Teaching division of Common fractions and Mixed fraction.
3.4 Demonstration (Folding Method)
The researcher introduced the concept of fraction by folding a sheet of paper equally
among two students. The researcher further discussed with students that one part is called
one-half because a whole has been divided into two equal halves that is why one part is
considered as one-half.
When a whole is divided into seven equal parts and three parts are taken away, the fraction
taken away is three-seventh and the other fraction not taken away is four-seventh. In
groups of threes, the teacher trainees’ were given Cuisenaire rods. The researcher
instructed them to choose the orange rod as a whole. They should try and make up as many
rows as they can use rods of one colour only. Each row must be of the same length as the
rod you first chose (or orange rod) as illustrated below.
Orange
Red
W
W
Red
W
W
Red
W
W
Red
W
W
28
Red
W
W
W = white
From the diagram above, five red rods make an orange rod. In fraction statement, a red is
one- fifth of the orange whole which is written as
. Also, ten white rods make an orange.
In fraction statement, a white is one-tenth of the orange which is written as
3.5 Equivalent Fractions
The idea of equivalence occurs and every opportunity should be taken during discussing
with teacher trainees’. The idea should grow out of the teacher trainees’ experience rather
than being taught as a separate topic. It is helpful to draw together the various ideas which
they have acquired.
For example long strips
Whole
Using Cuisenaire rods to teach equivalent fractions
The research instructed trainees’ to choose any rod or set of rods to be the “whole” for
example Orange and Dark green make up as many rows as possible using rods of the same
colour only for instance all red or all brown. Each row must be of the same length as the
original whole chosen.
29
Orange and Dark green
Brown
Brown
Purple
Red
W
W
Purple
Red
W
W
Red
W
W
Purple
Red
W
Red
W
W
W
Purple
Red
W
W
Red
W
W
Red
W
W
Guide trainees to write down their observations in words. For instance from the table
above, the following observation can be identified.
i. Two brown rods make the orange and dark green whole.
ii. Two purple rods make one brown rod.
iii. Four purple rods make two brown rods.
iv. Two red rods make one purple rod.
v. Four red rods make one brown rod.
vi. Eight red rods make the orange and dark green whole.
vii. Two red rods make one purple rod.
viii. Four white rods make one purple rod.
ix. Eight white rods make one brown rod.
x. Sixteen white rods make the orange and dark green whole.
These colour observations are then turned into fractional statement such as:
i. A brown is one half of the orange and dark green whole.
ii. A purple is one-fourth of the orange and dark green whole.
iii. A red is one-eighth of the orange and dark green whole
iv. A white is one-sixteenth of the orange and dark green a hole.
30
From table 3, we have the pattern as
and also
Equivalent
fractions are fractions of the same value but have different names or structure for instance
and .
3.6 Teaching Addition and Subtraction of like Fraction (Same Denominators)
Example 1: Using the Cuisenaire rods to teach addition of like fraction such as
and
to
develop the algorithm for addition of like fractions. Chose the yellow rod as a whole and
the white rods each represent one-fifth. The reason why the yellow rod was chosen is that
the yellow rod can be split into five equal rods (yellow).
Yellow
W
W
So we take
W
W
W
and
W
W
W = white
W to represent one-fifth i.e.
and two-fifth i.e.
1 white
2 white
respectively. Putting these white rods together, we will have 3 whites that are three-fifths.
W
W
W
Yellow
we want to solve the question
by using the Cuisenaire rods.
Taking the dark green as the whole, then the red rods are each one-third of the whole.
is taken as 2 reds. is also taken as 1 red.
31
Dark green
Red
Red
Red
We now have 2 red rods and 1 red rod which are 3 red rods. This is the same as the whole
dark green. Hence
.
Example III: To solve
using the Cuisenaire rods. Choose a rod or a train of rods to be
your whole. Use the orange rod as a whole, and then the red rods are each one-fifth.
Orange
Red
Red
Hence
Red
Red
Red
.
3.7 Teaching Addition and Subtraction of unlike Fraction and Mixed Numbers
Example 1: Using the Cuisenaire rods solve the following fraction
and develop the
algorithm for addition of unlike fraction. 10 is a multiple of 5 and as such we need to
choose a whole rod which can be split into ten. Therefore, the orange is the appropriate rod
needed.
32
Orange
Red
W
Red
W
W
Red
W
W
Red
W
W
W
Red
W
1 red rod represents and 1 white rod also represents
rods represent
W
W
of the whole rod (orange) 2 white
W
Change 3 red rods for 6 white rods, that is has changed to
W
W
W
W
W
(equivalent fractions).
W
Now putting all the white rods together
W
W
W
Therefore
W
W
W
W
W
becomes
Example II: Solve the following fraction
. 6 is a multiple of 2 and 3. You need to
choose a rod which can be split into six. Therefore the dark green is the appropriate rod.
Dark Green
Green
Green
Red
W
Red
W
W
W
Red
W
W
1 green rod represents , 1 red rod represents
whole that is dark green.
33
and 1 white rod also represent
of the
Change 1 green rod for 3 white rods, that is has change to (equivalent fraction).
Change 1 red rod for 2 white rods, that is has change to (equivalent fraction).
Now putting all the white rods together.
W
W
Hence
W
W
W
becomes
Example III: Solve the following fraction
. Now, 8 is a multiple of 2 and as such you
need to choose a whole rod which can be split into eight equal parts. The appropriate rod is
the brown.
Brown
Purple
W
W
Purple
W
W
W
W
W
1 purple rod represents and 1 white rod also represents of the whole rod (brown).
1 purple rod represents
w
Change 1 purple rod for 4 white rods i.e.
W
W
W
has changed to
W
34
(equivalent fractions).
W
Therefore
W
-
W
has now become
W
W
Brown
W
W W W
Brown
Brown
Out of the 4 white rods take one from it. You will realize that there will be three (3) white
rods left.
Hence
-
=
-
=
Example IV: Solve the following fraction
-
using Cuisenaire rods. 20 is a multiple of 5
and 4. So you need to choose rod(s) which can be split into twenty. The appropriate rods
are two oranges.
Orange and Orange
Yellow
Yellow
Yellow
Purple
Purple
Purple
Yellow
Purple
Purple
W W W W W W W W W W W W W W W W W W W W
1 purple rod represents
and 1 yellow rod also represents
of the whole that is orange
and orange.
Change 3 purple rods which represents
for white rods and that will be
fractions).
35
(equivalent
W
W
W
W
W
W
W
W
W
Change 1 yellow rod which represents
fractions)
W W W W W
Therefore
-
has now become
-
W
W
W
for white rods and that will be
(equivalent
out of the 12 white rods take away 5 white rods
from it and 7 white rods would be left.
Orange and Orange
Orange and Orange
W W W W W W W W W W W W
W W W W W
Orange and Orange
=
W W W W W W W
Hence
-
=
-
=
Example V: Solve the following mixed fraction 1 + 2
This can be done either by converting each mixed fraction into an improper fraction as
+
1 +
or by adding the whole numbers together and the fractions together as
= 3+ +
36
Using the Cuisenaire rods to choose a whole which can be split into fifths and tenths. The
orange rod is appropriate.
Orange
Red
W
W
Red
Red
Red
Red
W W W W W W W W
For take 2 red and 1 white rods for
W W
W
Change each red for 2 whites.
W W W W
And
Joining together we have 5 white rods.
W W W W W
And exchanging for 1 yellow
Yellow
Comparing the yellow to the original whole we have
Hence 1 + 2
=3+
+
=3+
=3
=3+ +
=3
37
3.8 Teaching Multiplications of Common Fractions and Mixed Fractions and to
Develop the Algorithm
Example 1: Multiplication of a whole number by a fraction such as 5 x
multiplication is
thought as a repeated addition. Therefore, 5 x can be interpreted as 5 lots of or 5 groups
of
Using the rods, we need to identify a whole which can be divided into thirds. The light
green is the appropriate rod. Take two thirds five times.
Light Green
W
W
W
For two thirds we take 2 white. Therefore for 5 we take two whites 5 times. This gives 10
white rods. Comparing this to our original whole of light green we have
=3
Light Green
Light Green
W
W
Light Green
W
W
W
Light Green
W
W
W
Light Green
W
W
The diagram above shows 3 holes and one third that is 5 x
W
=
W
38
W
W
=3
Example II: Multiplication of a fraction by another fraction such as
.
W
x
is thought as of
We choose a suitable whole rod which divides exactly fifths. An orange rod is appropriate.
We then find which colour of rods represents one-fifth. An orange is the same as 5 reds.
We take three of these rods to represents
Orange
Red
Red
Red
Red
Red
Our whole now is of which we need to find two thirds. The new whole is three reds
Red
Red
Red
which divides the original whole.
Red
Red
Hence
x =
of
= of orange whole
3.9 Teaching Division of Common Fractions and Mixed Numbers
Using the Cuisenaire rods to develop the algorithms.
The operation of division applied to fraction is extremely difficult for students to
understand for instance 4 ÷ 1 means how many one-thirds are in four.
Example 1: Dividing a whole number by a fraction such as 4÷
39
Choose two rods one of which is a third of the other. Dark green and red rods are the
appropriate. 4 wholes are taken as 4 dark green.
Dark Green
Dark Green Dark Green Dark Green
4 Wholes
R
R
R
R
R
R
R
R
One red indicates
R
R
R
R
R
R
R
we have 12 reds and this gives twelve-thirds that is 4 ÷ = 12
Example II: Dividing a fraction by another fraction
Solve ÷
Take a rod that can be divided into three and two. The dark green is the most appropriate
rod. One light greed rod will be equal to
of the dark green whole and two red rods equal
to of the dark green whole.
One Whole
Dark Green
Dark Green
Red
Light Green
Light Green
R
R
W
W
of original whole
Whole
Light Green
W
Red
W
40
When we lay the light green rod along the two red rods we have one light green plus one
white rod. 1 white rod is equal to
of light green rod. The indicate that halve in
will be
1 .
Hence
÷ =
=1
Example III: Dividing a mixed fraction by a fraction using the Cuisenaire rods such as
2 ÷
. You need a rod that can be divided into 2 and 3. Dark green rod will do. We take
three dark green rods to represent three wholes, one light green rod will be equivalent to
of a dark green whole and 1 red rod will be equal to of a dark green whole.
Dark
DarkGreen
Green
Light
Green
RR
Light
Green
RR
Light
Green
R
Dark Green
RR
Light
Green
R
R
Light
Green
R
Light
Green
R
Light
Green
R
Dark Green
R
Light
Green
R
R
Light
Green
R
Light
Green
R
Light
Green
R
R
2 of the original
whole
W W W W W W W W W W W W W W W
of two reds
41
When we lay the red rods along the 2 new whole, we have 7 red rods plus one white rod.
One white rod is equal to of the red rod. This indicates that one thirds in 2 will be 7 .
Hence 2 ÷
the mixed fraction could be first change to improper fraction that is 2 ÷
becomes ÷ =
=7
3.10 Method of Data Collection
The researcher used six weeks for the collection of the data. The first week was used to
conduct the pre-test. Four weeks used for the intervention and the last week used for the
post-test.
3.11 Validity and Reliability of Instruments
Statistical analysis is about processing data for the purpose of extracting relevant
information. Basic to any successful statistical analysis, therefore, is the collection of data.
Gordor and Howard (2006) contend that the validity and reliability drawn from a statistical
analysis largely depends on the nature, relevance and accuracy of the data. Validity of an
instrument is the extent to which the items in an instrument measure what they are set out
to measure. Reliability on the other hand is the extent to which an instrument measures the
idea that consistent results will be given by a measurement instrument when subjects is
measured repeatedly under near identical conditions or in the same setting. For the sake of
reliability of the instrument, the researcher administered the same questions and the
intervention strategies to all students under the study in a number of times (at least 2 times)
to ensure consistency of responses and scores. Again, triangulation approach was
42
employed to collect the data, which is, combining several methods to collect the data for
the study. Here the researcher used pre-test, post-test and questionnaire to actually get the
‘holistic’ understanding of the situation. During the pre-test and the post-test, the
researcher did supervise the students.
3.12 Data Collection Instruments
Two kinds of instruments were used – a questionnaire and tests (i.e. pre-test and posttest).To answer the research question on teacher trainees’ knowledge on the use of the
Cuisenaire rods approach to teaching fraction. The questionnaire was intended to
investigate why teachers-trainees’ perceptions about the nature of fractions and the use of
the Cuisenaire rods method of teaching fractions.
The pre-test was designed to assess the teacher-trainees’ ability to work with fractions. The
contents of the test were computation of addition and subtraction of fraction. The post-test
was designed to assess the teacher-trainees’ ability on fractions after the intervention. The
questions came from the Diploma in Basic Education textbooks and institute course
outline. See Appendices B to E for the pre-test and suggested response to pre-test and posttest and suggested response to post-test respectively.
3.13 Method of Data Analysis
The researcher based on the pre-test and post-test scores for the analysis. The Paired
Sample t-test was used to analyze the data. .
43
CHAPTER 4
RESULTS/FINDINGS AND DISCUSSIONS
4.0 Overview
In this chapter, the data collected from the teacher trainees’ of Tamale College of
Education- Tamale were analyzed and interpreted by the researcher. The data information
includes those obtained from the responses to questionnaire, pre-test and post-test of the
teacher trainees’.
4.1Analysis and Data Interpretation of Questionnaire by Teacher Trainees’
4.1.1 Research Question 1: To what extent would the use of Cuisenaire rods improve
the teaching of fractions?
Table 4.1: Responses to the use of Cuisenaire rods to improve the teaching of
fractions
Note: SA represents Strongly Agreed, A represents Agreed,
SD represents Strongly
Disagreed, D represents Disagreed, T represents the number of teacher trainees’ who
responded to the questionnaire.
No.
1.
2.
Statement
The use of Cuisenaire rods as a
teaching and learning material
greatly improve the teaching of
fractions in Mathematics.
Concepts of fractions were
better understood when
Cuisenaire rods were used as
teaching and learning materials.
SA
A
(T) %
(T) %
(26)72.2
(7) 19.4
(2)5.6
(1)2.8
(23)63.9
(7)19.4
(0)0
(6)16.7
44
SD
(T)%
D
(T)%
Item 1 in the table 4.1 indicates that twenty six (26) students representing seventy two
point two percent (72.2%) of the sample and seven (7) representing nineteen point four
percent (19.4%) strongly agreed and agreed respectively that the use of Cuisenaire rods as
a teaching learning material would greatly improve the teaching of fractions in
mathematics.
This finding supports that of Bernett et al (2000) when they described ratio model and
stressed the use of Cuisenaire rods as helping to improve the understanding of fractions in
mathematics. The current finding revealed that the use of Cuisenaire rods as a teaching and
learning materials greatly improve the understanding of the concepts of fractions.
Item 2 in table 4.1 which indicate the trainees’ understanding of the concept of fractions
when Cuisenaire rods are used as teaching and learning materials. It indicated that twentythree (23) representing sixty-three point nine percent (63.9%) and seven (7) representing
nineteen point four percent (19.4%) of the respondents strongly agreed and agreed
respectively to the notion that trainees’ understood the concept of fractions better when
Cuisenaire rods were used as teaching and learning materials. Whiles six (6) representing
sixteen point seven percent (16.7%) disagreed to the notion. This finding is in agreement to
Martin et al (1994) in citing Brunner (1989) who are on the opinion that a new concept
must be learnt with concrete teaching learning material (such as Cuisenaire rods) as this
would help learners to understand concept of fraction better. The use of Cuisenaire rods
has a great influence on the understanding of students.
45
4.1.2 Research question 2: To what extent would Cuisenaire rods be used in solving
addition and subtraction of equivalent fractions?
Table 4.2: Responses of Cuisenaire rods use in solving addition
and subtraction of equivalent fractions
No.
Statement
SA
A
SD
D
(T) %
(T) %
(T) %
(T) %
1.
Students were able to
solve addition of
equivalent fractions using
Cuisenaire rods.
(28) 77.8
(6)16.7
(1) 2.8
(1) 2.8
2.
Students were able to
solve subtraction of
equivalent fractions using
Cuisenaire rods.
(28) 77.8
(8) 22.2
(0)0
(0)0
Item 1 in the table 4.2 indicating the Cuisenaire rods been used to solve addition of
equivalent fractions was analyzed and it was realized that thirty (28) respondents
representing seventy-seven point eight percent (77.8%) and six (6) respondents
representing sixteen point seven percent (16.7%) strongly agreed and agreed respectively
to the notion that Cuisenaire rods can be used to solve addition of equivalent fractions.
Whiles one (1) respondent representing two point eight percent (2.8%) and one (1)
respondent also representing two point eight percent (2.8%) strongly disagreed and
disagreed respectively to the notion.
The above finding is in line with Obeng (2005) who was of the opinion that Cuisenaire
rods can be used to teach addition of equivalent fractions. The findings revealed that the
46
teaching of addition of fractions is made easier when pupil’s relevant previous knowledge
in equivalent fractions is no doubt.
Item 2 in table 4.2 indicating Cuisenaire rods which was used in solving subtraction of
equivalent fractions was analyzed. Twenty-eight (28) representing seventy-seven point
eight percent (77.8%) of the respondents strongly agreed to the notion that the Cuisenaire
rods can be used to solve subtraction of equivalent fractions. Whiles eight (8) representing
twenty-two point two percent (22.2%) agreed to the notion and zero percent however
disagreed to the above mentioned notion. This finding is also in agreement to Obeng
(2005) when he mentioned that Cuisenaire rods can be used in the teaching of subtractions
of equivalent fractions. The finding revealed that before subtractions in equivalent
fractions are taught pupils should be conversant with the use of Cuisenaire rods.
4.1.3 Research question 3: What are the difficulties teacher trainees’
encountered using Cuisenaire rods in teaching addition
and subtraction of fractions?
Table 4.3: Responses on the difficulties teacher trainees’ find using Cuisenaire rods in
teaching addition and subtraction of fractions
No. Statement
SA
A
SD
D
(T) %
(T) %
(T) %
(T) %
1.
Students can add fractions without
much difficulty when taken through
the Cuisenaire rod approach.
(0) 0
(22) 61.1
(7) 19.4
(7) 19.4
2.
Trainees can subtract fractions
without much difficulty when taken
through the Cuisenaire rod approach.
(0) 0
(22) 61.1
(8) 22.2
(6) 16.7
47
Item 1 in the table 4.3 which deals with the notion that trainees’ can add fractions when
taken through the Cuisenaire rods approach indicates that twenty-two (22) respondents
representing sixty-one point one (61.1%) percent agreed to the above notion that trainees’
can add fractions when taken through the Cuisenaire rods approach. Whiles seven (7) and
another seven (7) respondents representing nineteen point four percent (19.4%) in each
instance strongly disagreed and disagreed in both instances in the discussed notion.
This finding is in line with Elizabeth (1988) when this author maintained that solving to
obtain an insight of the problem at hand enhance formulation of various strategies that will
help the one solving the problem and aid one to construct proof of several theorems.
The finding indicated that sixty-one point one percent (61.1%) of the students can add
fractions when taking through the Cuisenaire rods approach.
Item two (2) in table 4.3 indicated that twenty-two (22) respondents representing sixty-one
point one percent (61.1%) agreed that they could subtract fractions without much difficulty
when taken through the Cuisenaire rods approach. Eight (8) respondents representing
twenty-two point two percent (22.2%) and six (6) respondents also representing sixteen
point seven percent (16.7%) strongly disagreed and disagreed respectively to the notion.
This finding confirms the notion Elizabeth (1988) made when this author mentioned that
the use of teaching and learning materials (Cuisenaire rods) in subtraction of fractions can
make some classifications which young children have done, which facilitate knowledge
building and enhance formulation of various strategies that will help in solving problems
and aid one to construct proofs of several theorems. Sixty-one point one percent of the
48
trainees’ indicated that they could subtract fractions without much difficulty when taken
through the Cuisenaire rods approach.
4.2 Analysis of Pre-Test and Post-Test Scores
The table below shows the marks of teacher trainees’ in pre-test and post-test after they
have been given ten questions to answer in each situation which was marked over 20. Both
tests were administered under normal examination conditions to ensure that the students do
not copy from each other. The table below shows the scores obtained by the teacher
trainees’ for the pre-test and post-test.
Table 4.4: Scores on pre-test and post-test
Student
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
Pre-Test
2
4
4
8
10
8
6
0
12
4
6
8
10
4
6
6
0
8
Post-Test
10
10
14
16
20
16
10
12
16
16
12
20
20
12
14
16
4
18
Student
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
49
Pre-Test
6
8
10
0
12
10
4
6
0
4
6
8
6
0
4
6
8
4
Post-Test
10
20
20
12
20
20
12
16
12
14
12
16
16
4
12
14
16
10
4.3 Analysis of the Pre-Test Result
Looking at the raw scores of the pre-test, it was realized that no teacher trainee had all the
ten (10) questions correct, only two (2) teacher trainees’ representing five point six percent
(5.6%) out of thirty-six (36) trainees’ scored twelve (12) marks out of the ten questions.
Four (4) trainees’ representing eleven point one percent (11.1%) had half of the marks that
is ten (10) and twenty-five (25) trainees’ representing sixty-nine point four percent (69.4%)
had eight (8) up to two (2) and five (5) trainees’ representing thirteen point nine percent
(13.9%) had zero (0). A critical look at the pre-test scores showed that the overall
performance was not encouraging. Below is the frequency distribution table for the Pre-test
results.
Table 4.5: Frequency distribution table for the pre-test results
Marks
Frequency
Percentages
Below 1
5
13.89
1-3
1
2.78
4-6
17
47.22
7-9
7
19.44
10-12
6
16.67
Total
36
100
50
Level of Performance of Trainees at Pre-Test
Figure 4.1: Level of Performance of Trainees’ during the Pre-Test
4.4 Analysis of the Post-Test Result
An observation of the post-test scores showed that the general performance was
encouraging. Seven (7) trainees’ representing (19.4%) were able to score all questions
given them. Twenty- two (22) trainees’ scored twelve (12) and above representing (61.1%)
while only seven (7) trainees’ out of the thirty-six (36) scored ten (10) and below
representing (19.4)%. Since the trainees’ scored higher marks in the post-test than in the
pre-test, it means there has been an improvement in their performance as a result of the
intervention put in place by the researcher. Below is the frequency distribution table for the
Post-test results.
51
Table 4.6: Frequency distribution table for the post-test results
Marks
Frequency
Percentages
Below 1
1-3
4-6
7-9
10-12
13-15
16-18
19-21
Total
0
0
2
0
13
4
10
7
36
0
0
5.56
0
36.11
11.11
27.78
19.44
100
Bar chart of post-test
Figure 4.2: Level of Performance of Trainees’ after the intervention process
52
Table 4.7: Frequency distribution table for the pre-test and post-test results
Marks
Below 1
1-3
4-6
7-9
10-12
13-15
16-18
19-21
Total
Pre-test
5
1
17
7
6
0
0
0
36
Post-test
0
0
2
0
13
4
10
7
36
Comparison of Pre- and Post-Test Results of students
Figure 4.3: Comparing the level of Performance of trainees’ after the implementation
of the intervention
53
4.5 Analysis and Interpretation of Post-Observation Assessment
After the researcher intervened by structuring and teaching addition, subtraction, division
and multiplication of fraction for the six weeks intervention period, he assessed the
trainees’ performance in addition, subtraction, division and multiplication of fraction once
again by test to find out if the intervention mechanism has worked to perfection.
4.6 Comparing the Pre-test and Post-Test Results of Trainees’
The researcher coded the data and inputted in the statistical package SPSS to find out the
relationship between the use of Cuisenaire rods and the understanding of the trainees’.
Table 4.8 below shows the results and analysis of both the Pre- and Post-test of the
trainees’ performance in fractions.
Table 4.8: Paired Samples Statistics of trainees’ performance in
fraction class
Variable
Mean
Std.
Deviation
N
Pair
PRE-TEST
POSTTEST
Std. Error
Mean
5.7778
36
3.33904
.55651
14.2222
36
4.18918
.69820
The descriptive Table 4.8 above displays the mean, sample size, standard deviation and
standard error for the pre-test and post-test performances of the college students. From
table 4.8 above, it is seen that the mean score of students before the intervention was 5.78
which is smaller than that of the Post-test (14.22). This displays a great improvement in
performance of students after the intervention.
Similarly, there is a difference of (0.85) in standard deviation from (4.19) to (3.34). This
shows a better performance of students in the post-test. It suggests that the use of
54
Cuisenaire rods in teaching fractions changed the performance of the students for the
better.
Table 4.9 below; indicate the correlation between the use of Cuisenaire rods and the
competency level of college student in addition, subtraction, division and multiplication of
fractions.
Paired Sample Correlation of pre-test and post-test of the use of Cuisenaire rods by
Trainees’
Table 4.9: Paired Samples Correlations
Variable
N
Pair PRE-TEST & POSTTEST
Correlation
36
.796
Sig.
.000
From Table 4.9 above, there is a strong positive correlation between the post-test and pretest. This also indicates that there is a relationship between the use of Cuisenaire rods and
achievement in students’ performance.
Table 4.10: Paired Samples Test
Paired Differences
Variable
Mean
PRE-TEST –
8.4444
POST-TEST
Std.
Deviation
2.53484
95% Confidence
Interval of the
Difference
Std. Error
Mean
.42247
55
Lower
Upper
t
9.30211
7.58678 19.988
Sig. (2tailed)
Df
35
.000
Table 4.10 above contained the paired differences between the pre-test, where college
students learn the use of Cuisenaire rods and the post-test, where college students learn
fractions after mastering the use of Cuisenaire rods. From the table the 95% confident
interval of the difference provides an estimation of the boundaries between which the true
mean difference lies in 95% of all possible intervention for the pre-test and post –test
results.
For the pre-test and post-test scores, there is a lower score of 9.302 and an upper score of
7.587. The t static is obtained by dividing the mean difference by its standard error. The t
static for the pre-test and post-test scores was 19.988. The sig. (2-tailed) column displays
the probability of obtaining a t static whose absolute value is equal to or greater than the
absolute t static. In table 4.10 above, we see that the significant value for change in
performance by college students in addition, subtraction, division and multiplication of
fraction is less than 0.005. Base on the above findings and analysis, the researcher
concluded that the perceived poor performance of the college students in fractions is not
due to chance and can be attributed to lack of using appropriate teaching and learning
materials such as Cuisenaire rods.
56
CHAPTER 5
SUMMARY, RECOMMENDATION AND CONCLUSION
5.0 Overview
The study is about using Cuisenaire rods to improve the teaching of fractions to teacher
trainees’ in Tamale College of Education, Tamale in the Northern Region. The study was
guided by the following research questions:
1. To what extent would the use of Cuisenaire rods improve the teaching of fractions?
2. To what extent would the Cuisenaire rods be used in solving addition and
subtraction of equivalent fraction?
3. What are the difficulties teacher trainees encountered using the Cuisenaire rods in
teaching addition and subtraction of fractions?
The sample comprised thirty-six (36) teacher trainees’ in Tamale College of Education. It
involved the use of pre-test, post-test and questionnaire as the instruments for data
collection. The pre-test was conducted after which an intervention period of six weeks
which involved taking teacher trainees’ through the use of Cuisenaire rods to improve the
teaching of fractions. A post-test was conducted after the intervention. The scores obtained
from the pre-test and post- test were analyzed by the use of a paired sample test. The
questionnaire data were also analyzed by using likert-type scale.
5.1 Summary of Major Findings
Below are the major findings of the research work.
 The findings revealed that the use of Cuisenaire rods greatly improve the teaching
of fractions in mathematics.
57
 The findings also revealed that teacher trainees’ lack the understanding of the
subject matter as related to everyday life.
 Finally it was also realized that teacher trainees’ grasp the concept of fraction well
when they are taken through the Cuisenaire rods approach and finally to the
abstract.
5.2 Implications for Practice
From the analysis and discussion of the data, the findings of the study have the following
important educational implications;
 Teacher trainees should be encouraged to use teaching and learning materials to
teach when introducing a concept like fractions.
 The above implication is an agreement to Martin et al (1994) in citing Brunner
(1989) when they revealed that it is the duty of the teacher to use concrete
materials to introduce fraction first and gradually produced to pictures and
diagrams, that is from concrete to semi-concrete and finally to the abstract stage.
 Cuisenaire rods approach should be used in teaching addition and subtraction of
fractions first. This implication is in agreement to Obeng (2005) when the auther
stated that addition or subtraction with different denominators are first express as
equivalent fractions by the use of Cuisenaire rods before solving them.
 The language used should be very simple and be at the level of learners. This
implication is in agreement to Williams (1986) when this author viewed the
language of the teacher as very important if he or she is to make positive impact on
his or her learners. This author stated that mathematics language should be
carefully and accurately used from the beginning. For instance the fraction 3/4
58
should be described in words as three fourths. The phrase such as three over four
should be avoided.
5.3 Conclusion
The concepts of fractions should be well explained to students by using the appropriate
teaching and learning materials.
 Teacher trainees should work in pairs and present ideas in different ways and solve
real life problems using fractions.
 The language used in communicating to learners is important if teachers want to
make a lasting impact on the learners. Expressions such as ‘four over six’ and ‘two
over seven’ should be avoided.
In conclusion, there is no guaranteed way of making sure students learn, but if they are
well motivated and the information is presented in a visible structure using interesting and
the appropriate teaching and learning materials then students will perform better.
5.4 Recommendations
Based on the findings of this study, the following recommendations are proposed for
consideration.

The use of Cuisenaire rods should be used to introduce lessons to enhance the
concept of topic being taught in fractions.

Teacher trainees should work in pairs or groups when necessary. This will enable
them to exchange ideas freely in class.
59

Great emphasis should be laid down on the systematic approach of taking teacher
trainees’ through fractions and more exercises must be given to teacher trainees’ to
practice since practice makes man perfect.

Teaching methods should be varied.

Workshops, seminars and conferences must be organized frequently to enable
teacher trainees’ to be abreast with new innovations of the subject matter.
5.5 Implication for Further Research
In designing future studies, the following suggestions may be considered.
1. The Cuisenaire rods approach should be supplemented by use of jargons and songs
accompanied with beats.
2. The sample size should be increased. A much larger size would enhance the
validity of the findings.
3. Different environmental settings are suggested for future study at the same time.
60
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mathematics Connection. Published by the Mathematical Association of Ghana.
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Hoffinmann Press Inc.
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Burton, R. & Musser. G (2000). Mathematics in Life, Society and the World. A. NJ. USA:
Vacom Company.
Carpenter, T. (2001). Fractions: (Results And Implications From National Assessment)
The Arithmetic Teacher, Vol. 28, Number 8.
Civil, M. (2003). Mathematical Thinking and Emotions, New York, USA: McGraw - Hill
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Costelpe, P. J & Hemstock H. F. (1975). Modern Business Arithmetic. Canada: Thomas
Nelson and Sons Ltd.
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Cramer, K. Post, T., & DelMas R. (2002). “Initial Fraction Learning by Fourth- and Fifthgrade Students: A Comparison of the Effects of using Commercial Curricula with
the effects of using the Rational Number Project Curriculum”. Journal for
Research in Mathematics Education, 33:111-144.
Dirkes, M. A. (1980). Mathematics Pursuits One; The Macmillan Company of Canada
Limited.
Dolan, D. (2000). Activities for Elementary Mathematics Teachers. 4th Edition.’ London
Dotse, D. (2000) Mathematics and the threshold of the new millennium. Journal
Mathematics Connection, Vol.1 no.1.
Driscoll, M. (1981). Research within Reach: Elementary School Mathematics. Reston,
VA: National Council of Teachers of Mathematics.
Elizabeth, W. (1988). Primary Mathematics for the age of the Calculator, 3d Edition,
printed by Longmans Group, UK Limited.
Evans, K. (2002). Teaching of Mathematics in Senior Secondary Schools in Ghana.
Unpublished Monograph.
Fennema, E. (1971). “Models in Mathematics Arithmetics Teacher, P. 635-640.
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Hawson, G. & Wilson, B. (1986). School Mathematics in the 1990s Cambridge University
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Institute of Education U. C. C. (2007). Chief Examiners’ Report of UTDBE Cape Coast
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Manu, A. M, Apronti, D. O., Afful, J. A., Ibrahim, M. & Oduro, E. 0. (2004). Methods of
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Marilyn, N. & Suydan N. (1998). Multidigit Numbers Sense Framework, Journal for
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65
APPENDIX A
QUESTIONNAIRE FOR TEACHER TRAINEES’
Research Question One (1)
To what extent would the use of manipulatives (Cuisenaire rods) improve the
teaching of fractions?
1. The use of Cuisenaire rods as teaching learning material would greatly improve the
teaching of fractions in mathematics.
Strongly agreed [
]
Agreed [
] Strongly Disagreed [
]
Disagreed [
]
2. Trainees understood the concept of fractions better when Cuisenaire rods were used as
teaching learning materials.
Strongly agreed [
]
Agreed [
]
Strongly Disagreed [
]
Disagreed [
]
Research Question Two (2)
To what extent would Cuisenaire rods be used in solving addition and
subtraction of equivalent fraction?
3.
Cuisenaire
rods can be
Strongly agreed [
]
used to solve
Agreed [
]
addition of
Strongly Disagreed [
equivalent
]
fractions
Disagreed [
]
4. Cuisenaire rods can be used to solve subtraction of equivalent fraction.
Strongly agreed [
]
Agreed [
]
Strongly Disagreed [
66
]
Disagreed [
]
Research Question Three (3)
What are the difficulties teacher trainees encountered in using the Cuisenaire rod
approach in teaching addition and subtraction of fractions?
5. Students can add fractions without much difficulty when taken through the Cuisenaire
rod approach.
Strongly agreed [
]
Agreed [
]
Strongly Disagreed [
]
Disagreed [
]
6. Students can subtract fractions without much difficulty when taken through the
Cuisenaire rod approach
Strongly agreed [
]
Agreed [
]
Strongly Disagreed [
]
Disagreed [
]
7. Without the use of Cuisenaire rods students do find questions on fractions easy.
Strongly agreed [
]
Agreed [
]
Strongly Disagreed [
67
]
Disagreed [
]
APPENDIX B
PRE-TEST QUESTIONS
Q1. Write the fractional form for the unshaded parts of the diagram below
Q2. Describe one way in which you would guide pupils in primary class 4 to determine for
themselves that and are equivalent fractions, using the Cuisenaire rods.
Simplify the following fractions
Q3.
Q4.
Q5.
Q6.
Q7.
Q8. 2
Q9. 5 ÷
Q10.
68
APPENDIX C
SUGGESTED RESPONSE TO PRE-TEST QUESTIONS
Q1. The response will be because the diagram has been divided into seven equal parts for
four part have not been shaded.
Q2.
-
Guide the children to choose any rod or set of rods to be their ‘whole’ for example
the Brown rod.
-
Red
W
Guide the children to make up as many rows using rods of one colour only.
W
Red
W
Brown
Red
W
W
W
Red
W
W
The whole i.e. Brown rod is divided into four equal red rods so when 3 of the rods are
under consideration is represents
Again, the whole i.e. Brown rod is divided into eight equal white rods so when 6 of the
white rods are under consideration, it represents . Guide the children to compare and
and they will realize that, they are the same in value that is equivalent fractions.
Q3.
Q4.
69
Q5.
Q6.
Q7.
=3+
Q9. 5 ÷ = 15
Q10.
=
=1
70
APPENDIX D
POST –TEST QUESTIONS
Q1. Draw and shade a box that will represent the given fraction
Q2. Write two equivalent fractions for each of the fractions in (a) and (b)
Q3. Using the example
, describe, step by step, how you would use Cuisenaire rods to
develop the algorithm for the addition of common fractions with primary class 4 pupils.
Q4. Simplify fraction below
4
Q5. Simplify the fraction below
Q6. Show how you would teach a child to find the Cuisenaire rods
using the Cuisenaire rods
Q7. Simplify the fraction below
Q8. Simplify the fraction below
Q9. 4
Q10.
71
APPENDIX E
SUGGESTED RESPONSE TO POST-TEST QUESTIONS
Q1.
Q2. The equivalent fractions are
a.
b.
Q3. The dark green and purple rods are chosen as that whole.
Dark green
Dark green
Purple
Purple
Purple
Light green
Light green
Light green
Light green
Red
Red
Red
Red
Red
Red
W
W
W
W
W
W
W
W
W
W
W
W
1 purple rod represent of the whole
1 light greed rod also represents of the whole
Change 1 purple road for white rods which is 4 white rods i.e.
Change 1 light green rod for white rods which is 3 white rods i.e.
Summing up the 4 white rods and the 3 white rods give 7 white rods. Hence
72
Q4. 3
= 8+
= 8 +1 = 9
Hence 3
Q5.
Q6.
is thought of as
Choose a suitable whole rod which divides exactly in 4. A brown rod is an appropriate.
Then find which colour of rods represents one fourth.
red
Brown
red
Red
Red
Take three of these red rods to represent
Red
Red
red
new whole
73
The new whole of which we have to find three fifths. The three red will now be divided
exactly into three of 3 reds is 1 red remaining 4 of them.
Comparing the 1 red remaining 4 ‘red’ to the original whole gives
Therefore
Q7.
Q8.
Q9. 4 ÷
Q10.
74
of the original whole.
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