CHAPTER 1
INTRODUCTION
1.0
Overview
Chapter one presents the insight as well as the background to the study which
aimed at improving students’ performance in problems involving addition and subtraction
of fractions. It also includes statement of the problem, purpose of the study, objectives of
the study, research questions and significance of the research, limitations and delimitation
of the study and organization of the study.
1.1
Background to the Study
Mathematics in general is an essential subject for scientific and technological
development of any nation. It is part of life without which man cannot function (Nabie,
2002). This suggests that no nation can grow scientifically and technologically above her
mathematics status; an indication that mathematics is indispensable for science.
Mathematics is the means of sharpening the individual’s mind, shaping his reasoning
ability and developing his personality, hence its immense contribution to the general
and basic education of the people of the world (Asiedu-Addo & Yidana, 2004).
The entire world has become digital and this digital innovation is irreversible. It
has become a world culture, which is progressing at a terrific speed for good. Any person,
community or country that resists or refuses to join these forces of progress will be left
behind and abandoned to languish in ignorance, penury and backwardness forever
(Talabi, 2003).
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With this understanding, the Ghana School Curriculum made mathematics a compulsory
subject in her Basic, Second cycle and Tertiary levels of education apparently to be at par
with the developing if not the developed countries (CRDD, 2004; CRDD, 2007 & Dogbe,
Morrison and Speed, 1995). A thorough looked at the intended curriculum (syllabus) of
schools in Ghana revealed that the study of Fractions is one aspect of mathematics that
run through all levels of education of which the Colleges of Education are no exception.
At the primary level for instance, Fractions can be found in Unit 2.8 for P2, Unit 3.4 and
3.11 for P3, Unit 4.6 and 4.9 for P4, Unit 5.11 for P5 and Unit 6.2 and 6.2.7 for P6
(CRDD, 2004) with the increasing levels of scope underplay. In the Junior High schools,
Fractions can be found in Units 1.2, 1.13 and 1.14 for JHS1, Units 2.5, 2.11 and Unit 2.14
for JHS2 (CRDD, 2007). It is noted that the knowledge of Fractions is essential in
learning other concepts like Money and Taxes, Handling of Data and Probability,
Algebraic expressions, Volumes and Areas, Geometry and Trigonometry, Measurement,
Calculus just to mention but a few.
Doctors, nurses, and pharmacists need to figure out Fractions to determine the
proper dosages of medicines for unusually large or small patients. The patient may have
to take half sachet of medicine with half a glass of water two times daily. You need
Fractions to bend a straight line to turn it into a curve. Students can plan their menu and
have a balanced diet over a stipulated period of time with the idea of Fractions because
some amount of carbohydrates may be taken with some amount of protein. Mother Earth
is made up of Fractions whilst all financial operations work on Fractions in terms of
percentages in their business transactions.
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Having a solid foundation of basic Fraction concepts will make the advanced concepts
easier to learn. This is a clear indication that the importance of Teaching and Learning of
Fractions in our schools cannot be underestimated if students are to develop good
conceptual knowledge in Mathematics which will eventually enable them to perform well
in other subject areas. Most miserably however, teachers continue to skip this important
area in Mathematics without teaching the concept of Fractions. Teachers who attempt
teaching also teach without using materials that will enhance good understanding or
enable pupils to grab the concept from the onset.
There are topics that some teachers find difficult to teach. They call such topics
challenging topics (INSET Project, 2007). Some of such topics are Operations on
Fractions, Measurement of Area, Capacity, Volume and Time, Investigation with
Numbers, Shape and Space and Collecting and Handling of Data. Teachers claim that the
topics require subject teachers or specialists to teach them. However, with adequate
preparation, teaching of these topics should not be problematic. The challenging topics
are seen to be abstract in nature because they are not seen in real life situations. Besides,
there are no Teaching and Learning Materials and relevant curriculum materials to
support teachers to teach such topics. Most importantly, some teachers do not use
appropriate methodology to enable students participate fully in the lesson (INSET
Project, 2007).
In line with the above development, the educational climate in institutions
responsible for training the human resource needed for the nation for which the Colleges
of Education in Ghana are of crucial importance, is essentially driven by an overriding
interest in preparing students with the necessary Pedagogical Content Knowledge (PCK)
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to become effective and efficient teachers at the Basic schools. Notwithstanding the
government of Ghana efforts to upgrade the Pedagogical Content Knowledge of
Mathematics teachers, the subject has not undergone much change in terms of how it is
presented to students. These reflect consistently in low achievement levels in
Mathematics among students especially at the basic schools. In this regards, the Colleges
of Education formerly known as Teacher Training Colleges have undergone
transformations in many areas particularly Certification and curriculum to uplift the
standard of Content and Methodology so that trainees can handle these topics efficiently
and effectively at the basic schools even before they graduate as professional teachers.
The Mathematics Curriculum for Methods and Content were earlier integrated for
Certificate ‘A’ until 1999, but in 2005, the curriculum was changed for the diploma
programme. Between 1999 and 2005 though the Colleges were not awarding diploma, the
curriculum for Methods and Contents were not integrated. The Mathematics Content
courses for first years are now Number & Basic Algebra and Geometry & Trigonometry
for first and second semesters respectively whilst the Mathematics Content courses for
second years are Statistics & Probability and Further Algebra for first and second
semesters correspondingly and simultaneously with the Methods of Teaching Primary
School Mathematics and Methods of Teaching Junior High School Mathematics.
Mathematics is hence assigned three (3) and six (6) credit hours for Arts and
Mathematics & Science students respectively.
In spite of this, over the years students’ performance in Mathematics especially at
the basic level has not been impressive despite the huge resources and much attention
paid to the study of Mathematics. The deplorable conditions of students can be observed
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clearly from the summary of students’ performance as expressed by Anamuah-Mensah,
Mereku and Asabere-Ameyaw (2004) report on results from the Junior Secondary School
two (JSS2) students participation in Trends in International Mathematics and Science
Study (TIMSS) in 2003, that Ghana’s overall performance in Mathematics was very poor.
This performance placed:

Ghana at the 45th position out of the 46 participating countries on the overall
Mathematics achievement results table. The range of scores from 130 to 430
shows how diverse the JSS2 students were in their Mathematics abilities.

The mean percentage correct answer on all Mathematics test items for each
participating Ghanaian student was 15 and only 9% and 2% of the students
reached the low and intermediate international benchmarks respectively.

Ghana ranked 46th on the international benchmark for Mathematics.

The students’ strong content areas in Mathematics were in Number and Data
whilst the weakest areas were in Algebra, Measurement and Geometry.

In almost all the content areas, the boys achieved significantly higher scores than
the girls.
A further analysis on the results by Anamuah-Mensah and Mereku (2005) indicated that

The mean percentage of students making correct responses in Algebra,
Measurement and Geometry were 13.6%, 17.3% and 13.4% respectively.

For Number and Data, the mean percentage making correct responses were 22.6%
and 27% respectively. The Ghanaian students found the constructed response
items more difficult than the multiple-choice items.
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
The mean percentage of students who were able to provide the correct responses
to the multiple-choice items was 21.6% while that observed for the constructed
response items was 12.1%.
These performances not withstanding; Ghana’s performances in TIMSS 2007 were better
than that of 2003. According to Anamuah-Mensah, Mereku and Ghartey (2008), Ghana’s
Junior High School two (JHS2) students’ performances in Mathematics, though improved
significantly since TIMSS 2003, remains among the lowest in Africa and the world.

In Mathematics, Ghana’s score of 309 was among the lowest and was statistically
significantly lower than the TIMSS scale score average of 500. This poor
performance place Ghana second from the bottom on the overall Mathematics
results table doing slightly better than only Qatar.

Ghana’s score in Mathematics was lower than those obtained by all the
participating African countries. But the country’s performance level at TIMSS
improved from that of 2003.

In Mathematics, the 2007 score of 309 was significantly higher than the 2003
score of 276, a 33 point increase.
My experiences as a teacher also revealed that, generally, Mathematics has been one of
the subjects that most students fear to learn. As to why it is so feared, some students said
it is simply difficult and they don’t like it. Others said the way it was taught them that
made it difficult to understand. However, in my view, these situations could be attributed
to several factors. Such factors include the abstractness of Mathematical concepts, the
way the concepts are presented to the students and poor foundation among others. Today,
many Mathematics teachers barely use materials in teaching; no matter the level they
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teach, and it appears they lack the necessary pedagogical skills needed to be able to teach
the subject with competency for pupils to grasp the concept from the onset.
Like most branches of Mathematics therefore, Number and Algebra are the most
important areas where ‘Fractions’; the topic which poses problems to both students and
teachers alike can be found. The word ‘Fraction’ according to Downes and Paling (1965)
is taken from the Latin word ‘Frangere’; meaning “to break”. Fractions are a wellidentify area of difficulty for many children even adults (Pamela, 1984); and as a result,
Practicing Teachers of Dambai College of Education are no exception. It is important to
note that Fractions form an integral part of the Mathematics curriculum of every level of
the educational system. Most importantly, as implementers of the Government Policy on
Education, the Colleges of Educations’ Mathematics Curriculum lays emphasis on the
Pedagogical Content Know-how to give and enable the Teacher Trainees acquire the
requisite skills to be able to teach this all important topic at the Basic level of education.
This is evidently spelt out in the objectives of the Mathematics syllabus for Colleges of
Education as:
1. To extend the students own Mathematical ability to a level significantly beyond
that which he or she is likely to teach Mathematics in schools.
2. To give students an understanding of the Mathematical content and processes
contained in the Kindergarten, Primary and Junior High School syllabuses.
3. To provide professional skills and understanding relating to Methods of Teaching
which is appropriate for Basic Education.
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On the basis of the above, Practicing Teachers’ inability to teach Fractions using
Concrete Materials thereby enhancing pupils’ understanding of concepts better raised a
lot of concerns in the researcher’s mind. It is evident that most of the practicing teacher
trainees on teaching practice failed to teach this topic and a few who tried to teach the
topic also fumbled with the teaching. This came to light whenever the researcher went
out to supervise the students on their teaching practice. The most worrying aspect is that,
these practicing teachers’ are the implementers of the Basic school Mathematics
curriculum and as such need much more attention. More importantly, one of the
weaknesses of the 2002, 2004 and 2008 Basic Education Certificate Examination (BECE)
results listed in the Chief Examiner’s Report included candidates’ inability to answer
questions that involved Fractions. In view of this, the teaching procedures that are not in
contravention of the famous Chinese proverb that; ‘I hear and I forget, I see and I
remember, I do and I understand’ must be used. The distinction between deepen
approaches and surface approaches to learning is particularly useful for teachers who
want to understand the pupils’ learning and create learning environments which
encourage pupils to achieve desired learning outcomes. The fact that, the use of
manipulation and representation is strongly advocated by many authors such as Martin
(1994) and Apronti, Afful, Ibrahim, et al (2004), the selection of Cuisenaire rods as
effective Teaching and Learning Materials (TLM) was necessitated since it is a major
material outlined in the students’ Course to be abreast with its use and to enable pupils’
apply the three domains of learning-Cognitive (head), Affective (heart) and Psychomotor
(hand) in learning a mathematical concept. This entails the use of concrete, physical,
observable and touchable objects like Cuisenaire Rods to teach abstract concepts.
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It is upon these bases that the researcher found it most expedient to use Cuisenaire rods as
an intervention to curbing the problem which has pervaded the educational systems in
Ghana.
1.2
Statement of the Problem
Mathematics is a subject that has to do with understanding of concepts, mastery of
skills and application of these concepts in various fields of life. In the researcher’s
supervisory work as a Mathematics tutor in Dambai College of Education, it was
discovered that second year students of Dambai College of Education could not teach the
addition and subtraction of unlike Fractions using concrete materials. As a result of their
inability to teach this topic, pupils could not solve problems involving addition and
subtraction of Fractions since the concept was not well formed in their mind.
As evidence, the Chief Examiner’s Report of 2005, 2006 and 2007 for Methods of
Teaching Primary School Mathematics stated students’ difficulties in answering
questions on Fractions using Cuisenaire rods as concrete material and urged tutors to pay
particular attention to the teaching of Fractions using concrete materials.
1.3
Purpose of the Study
The main reason for carrying research in this area is to enable Practicing Teachers
of Dambai College of Education improve upon their performances in teaching addition
and subtraction of Fractions with unlike denominators by the use of Cuisenaire rods.
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1.4
Objectives of the Study
According to Cooney, Davis and Henderson (1975), objectives should be stated in
terms of observable student’s behaviour. The researcher’s objectives in this regard were
as follows:
i.
Find out the effect of Cuisenaire rods on students’ performance in solving
problems involving Fractions.
ii.
Determine the effect of Cuisenaire rods on students’ performance in teaching
problems involving Fractions.
iii.
Determine the influence of Cuisenaire rods on students’ perception in teaching
problems involving Fractions.
1.5
Research Questions
It is expected that majority of students in any learning situation will do well in a
test on a given topic if it is planned and taught well. Improving the performances of
second year students of Dambai College of Education in teaching the addition and
subtraction of fractions with unlike denominators raised the following questions in the
researcher’s mind.
1. What effect has the use of Cuisenaire rods on students’ performance in teaching
problems involving Fractions?
2. To what extent would the use of Cuisenaire rods as concrete materials sustain and
motivate students’ interest in learning of Fractions?
3. Is there any difference in students mean score performance by the use of
Cuisenaire Rods in solving problems involving Fractions?
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1.6 Research Hypotheses
The hypothesis designed to guide and direct the study is:
Null hypothesis 𝐻0 : There is no significant difference in scores between the mean pre-test
scores and the mean post- test scores of students.
Alternative hypothesis 𝐻𝑎 : There is significant difference in scores between the mean
pre-test scores and mean post- test scores of students.
1.7 Significance of the Study
Improvements of teaching methods, strategies and techniques have been the
concern of many Mathematics teachers and educators since time immemorial. These
desires have motivated mathematics teachers to carry out research work in various
aspects of the subject that interest them. These in effects serve as a guide to teaching and
learning of the subject. Since students do not only performed the various activities
involved but showed interest and asked questions, they stand a better position to explain
the concept anytime they are called to do so.
The findings of the study when implemented will help Mathematics teachers not
only to be able to teach well but also identify the usefulness of using Cuisenaire rods as
teaching and learning materials in teaching fractions. The teachers will develop less
difficulty in teaching whilst the students will develop interest and be more courageous in
solving problems involving fractions in general. The results of this study would also
serve as a guide for teachers to vary their approach and methodology to enable students
understands the concept of fractions. It would also serve as resource material for all stake
holders and others who would like to research further into this area of national interest.
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1.8
Limitations of the Study
Research of this nature will not have ended without any restraints or drawbacks to
its successful completion. However, a few of such limitations that impede the smooth
running are enumerated below:

The study would have been more representative if all the two hundred (200)
second year students of Dambai College of Education were covered. This is
because; the villages where the mentees (practicing students) practiced teaching
are distance apart from the location of the college; couple with transportation
problems. This might affect the results of the study.

There were also different unplanned programmes that distorted the organised time
set for the intervention. As a result of this, the researcher whacked time and
money to these various stations without achieving the purpose for his travels.
The conclusion will therefore be limited by these factors and as such generalizations
cannot cover all teacher trainees in Ghana.
1.9
Delimitation of the Study
Fractions are broad areas in Mathematics with so many aspects. However, the
study was restricted to the Addition and Subtraction of fractions with unlike
denominators. The study was also delimited to the use of Concrete Materials (Cuisenaire
Rods) to improving the teaching and learning of fractions. It was also confined to only
Second Year Students of Dambai College of Education in the Volta Region of Ghana;
though this problem might exist in other Colleges of Education.
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1.10
Organizational plan of the Study
The study was organised in five chapters. Chapter 1 talked about the background
to the study, statement of the problem, purpose of the study, objectives of the study,
research questions, significance of the study, delimitation, limitation and organizational
plan.
The relevant literature review was presented in chapter 2 whilst chapter 3 talked about the
methodology. Chapter 4 talked about data presentation, analysis and discussion of result
whereas chapter 5 consisted of summary of key findings, implications for practice,
conclusion, recommendations and areas for future research.
1.11 Definition of Terms
For the purpose of the study, the following definitions are implied for the terms below.
Concept: - The idea or knowledge we hold about something.
Cuisenaire Rods: - They are versatile Teaching and Learning Materials (TLMs) which
are used to teach the concepts of fractions, addition and subtraction of whole numbers
whose sum does not exceed 10. The rods are made up of 10 different colours which are
associated with numerals. Thus, 1 – white, 2 – red, 3 – light green (green), 4 – purple, 5 –
yellow, 6 – dark green, 7 – black, 8 – brown, 9 – blue, 10 – orange.
The numerals associated with the rods shows the number of white rods that can fit
exactly when laid side-by side with that rod.
Like Fraction: - They are fractions with the same denominators
Unlike Fractions: - Fractions with different denominators.
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CHAPTER 2
RELATED LITERATURE REVIEW
2.0
Overview
The review of the related literature was focused on the theoretical framework underlying
the research and related works on the study. This was based on the following themes:

Theoretical framework

Nature of mathematics

Cuisenaire rods and mathematics teaching

Meaning and concept of Fractions

Equivalent Fractions

Addition and Subtraction of Fractions

The part of language in teaching mathematics
2.1
Theoretical Framework
The theoretical framework for the study is based on the Shulman’s (1986) three
knowledge domains in teaching; grounded with the Constructivists views of teaching and
learning. These domains of Shulman (1986) encompass:

Subject Matter Content Knowledge (SMCK)

Pedagogical Content Knowledge (PCK)

Curricular Knowledge (CK)
To teach all students according to today’s standards, teachers need to understand subject
matter deeply and flexibly so they can help students create useful cognitive maps, relate
one idea to another, and address misconceptions. Teachers need to see how ideas connect
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across fields and to everyday life. This kind of understanding provides a foundation for
Pedagogical Content Knowledge that enables teachers to make ideas accessible to
students (Shulman, 1987).
Teaching is not a matter of knowing something. It is far more than mere transmitting of
concepts and ideas to learners. It involves bringing out the accumulated ideas and
experiences that students come to class with and working on those ideas and experiences
together with the students by way of refining, reorganizing, co-constructing and repairing
these ideas and experiences into meaningful and compressible form for students to
assimilate (Shulman, 2000). This forms the foundation on which teaching mathematics
through problem solving leans on.
According to Shulman (2000), teaching is about making the internal and external
capabilities of an individual and can only be achieved if teachers engage students in the
classroom discourse. It is only when students are engaged in an interactive classroom
environment that their ideas, conceptions and experiences are made bare to the teacher to
put them on truck.
2.1.1 Subject Matter Content Knowledge
Shulman (1986) defined Subject Matter Content Knowledge as the amount and
organization of knowledge intrinsically in the mind of the teacher. He argues that
teachers’ subject matter content knowledge should not be limited to knowledge of facts
and procedures; but also an understanding of both the substantive and syntactic structures
of the subject matter.
The substantive structures are the various ways in which the basic concepts and
principles of the discipline are organized to incorporate its facts. Teachers will therefore
15
be able to use appropriate materials to teach mathematics well only when they
comprehend the network of fundamental concepts and principles of problem solving in
holistic manner (Shulman, 1986).
The syntactic structure of a discipline is the set of ways in which truth or falsehood,
validity or invalidity are established (Shulman, 1986). The syntactic structure is used to
establish the most appropriate claims about a particular phenomenon. Teachers’
knowledge must therefore go beyond mere definitions of accepted truths in the subject
matter domain.
In sum, to provide for effective teaching and learning of mathematics, mathematics
teachers’ Content Knowledge of concepts cannot be underplayed. The question that arises
is ‘how can an individual handle a subject matter competently if the content knowledge is
weak? The researcher believes that teacher’s knowledge of mathematics is essential to
their ability to teach effectively as Brophy (1991) cited in Asiedu-Addo and Yidana
(2004) indicates
“where (teachers) knowledge is more explicit, better connected and more
integrated, they will tend to teach the subject more dynamically, represent
it in more varied ways, encourage and respond fully to students comments
and questions. Where their knowledge is limited, they will tend to depend
on the text for content, de-emphasise interactive discourse in favour of
seatwork assignments and in general portray the subject as a collection of
static, factual knowledge.”
This suggests that the teacher uses mainly non thought provocative questions, and
often selects only what he/she thinks can teach.
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2.1.2 Pedagogical Content Knowledge
Pedagogical knowledge includes generic knowledge about how students learn,
teaching approaches, methods of assessment and knowledge of different theories about
learning (Harris, Mishra & Koehler, 2009; Shulman, 1986). This knowledge alone though
necessary; is insufficient for teaching purposes. According to Shulman (1986),
pedagogical content knowledge is knowledge about how to combine pedagogy and
content effectively. It includes, knowing what approaches fit the content, knowing how
elements of content can be arranged for better teaching. It also involves knowledge of
teaching strategies that incorporate appropriate conceptual representations to address
learner difficulties and misconceptions and foster meaningful understanding; and
knowledge of what the students bring to the learning situation; knowledge that might be
either facilitative or dysfunctional for the particular learning task at hand. Shulman
(1986) lay emphasis on the pedagogical content knowledge as the combination of the
most regular taught topics, the most useful forms of representations of those ideas, the
most powerful analogies, examples, illustrations, explanations and demonstrations in the
art of teaching. Pedagogical Content Knowledge also includes the ways of representing
and formulating the subject matter that makes it comprehensible to students with diverse
views and understandings. In teaching Mathematics through activity oriented base and
problem solving techniques, teachers need to design and present the lesson using
appropriate teaching learning materials (TLMs) that can enable the students construct
their own knowledge of the concept. They need to know the pedagogical strategies and
techniques most appropriate for reorganizing the understanding of learners who might
appear before them as blank slates (Shulman, 2000); hence the knowledge of subject
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matter in the training of a mathematics teacher in particular and the classroom teacher in
general is as important as the methodology aspect of it, and that the course outline in the
Teacher Training Institutions should be reviewed in a more pragmatic approach by
encouraging students to appreciate the need for both methodology and content courses
(Asiedu-Addo & Yidana, 2004).
2.1.3
Curriculum Knowledge
The word ‘curriculum’ comes from a Latin root which originally meant ‘a course
to be run’, that is, a course in the sense of ‘race-course’ (Mereku & Agbemaka, 2009).
Curriculum has numerous definitions which can be slightly confusing; especially meeting
it the first time. It refers to all the courses offered at a school; it is the prescribed course
of studies which students must fulfil in order to pass a certain level of education.
Curriculum is really more than just what is taught in the classroom. The term was once
used to refer only to the content of educational provision. It was therefore barely
distinguishable from terms like ‘syllabus’ or even timetable (Mereku, 2004). It is
anything and everything that teaches a lesson planned or otherwise. Humans are born
learning, thus the learned curriculum actually encompasses a combination of the hidden,
null, written, political and societal etc. Since students learn all the times through exposure
and modelled behaviours, it means that they learn important social and emotional lessons
from everyone who inhabits the school.
According to Tanner and Tanner (1975) cited in Mereku and Agbemaka (2009),
“Curriculum is the planned and guided learning experiences
and intended outcomes, formulated through the systematic
reconstruction of knowledge and experience under the auspices
of the school, for the learner’s continuous and wilful growth in
personal-social competence”
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This definition according to Mereku and Agbemaka (2009) highlights the fact that the
curriculum must take into account not only established knowledge but also emergent
knowledge. This is because curriculum while transmitting the cumulative tradition of
knowledge also concerns with the systematic reconstruction of knowledge in relation to
the life experience, growth and development of the learner.
An interesting interpretation of the term ‘curriculum’ by Costa and Liebmann (1997)
cited in Mereku and Agbemaka (2009) is given below:
“Curriculum is the pulse of the school; it is the currency through
which educators exchange thoughts and ideas with students and
the school community. It is the passion that binds the
organization together. Curriculum, in the broader sense, is
everything that influences the learning of students both overtly
and covertly, inside and outside the school.”
Whereas, Young (1998) cited in Mereku and Agbemaka (2009) looks at the ‘curriculum’
as ‘socially organized knowledge’; and said
“….. academic curricula are as much the products of people’s
actions in history as any other form of social organization. They
are not given, nor, in today’s language, do they represent an
unchanging gold standard. They can therefore be transformed.
The issue is one of purposes and the extent to which the existing
curriculum represents a future society that we can endorse or a
past society that we want to change”,
From the definitions above, it is possible to state that a curriculum has the following
characteristics:

It comprises the experiences of children for which the school is
responsible.

It has content.

It is planned.

It is a series of courses to be taken by students.
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In addition, a curriculum considers the learners and their interaction with each
other, the teacher and the materials. The output and outcomes of a curriculum are
evaluated. Bringing all these points together, the curriculum is viewed as a composite
whole including the learner, the teacher, teaching and learning methodologies, anticipated
and unanticipated experiences, outputs and outcomes possible within a learning
institution (Mereku & Agbemaka, 2009). The Mathematics Curriculum therefore is
represented by a full range of programmes designed for the teaching of mathematics
topics at a given grade level. It covers a wide variety of instructional materials available
in relation to the subject matter to be handled and the set of characteristics that guides the
use of particular curriculum materials in particular circumstances (Shulman, 1986).
Teachers need to think hard about students mathematical ideas analyze textbooks
presentations and judge the relative value of two different representations in the face of a
particular mathematical issue (Ball & Bass, 2000). Mathematics teachers need to have
thorough understanding of the curricular resources available for instruction so as to make
them available to students when teaching mathematics for students to make their own
meaning of concepts.
2.1.4
Constructivist Idea of Learning
Constructivism is a philosophy of learning founded on the premise that by
reflecting on our experiences, we construct our own understanding of the world we live
in. I believe in constructivism because I view an individual as a knowledge constructor.
Constructivists believe that mathematics does not grow through a number of indubitable
established theorems, but through the incessant improvement of guesses by speculation
and criticism (Fletcher, 2005). Constructivism can be traced at least to the eighteenth
20
century and the work of the Neapolitan philosopher Giambattista Vico, who held that
humans can only clearly understand what they have themselves constructed
(http://www.sedl.org/scimath/compass/v01n03/2.html). Each of us generates our own
“rules” and “mental model”, which we use to make sense of our experiences. Learning
therefore, is simply the process of adjusting our mental models to accommodate new
experiences. The constructivists claim that learning is an active process and that
knowledge is constructed rather than innate, or passively absorbed. Knowledge is
invented not discovered and that all knowledge is personal, distinctive and socially
constructed. To the constructivist, learning is essentially a process of making sense of the
world and requires meaningful, open-ended, challenging problems for the learner to
solve. However, social constructivist thesis is that mathematics is a social construction, a
cultural product, fallible like any other branch of mathematics. They claim that
knowledge is not passively received but actively built up by the cognizing subject and the
function of cognition is adaptive and serves the organization of the experiential world
(vonGlasersfeld, 1989).
Many others worked with these ideas, but the first major contemporaries to
develop a clear idea of constructivism as applied to classrooms and childhood
development were Jean Piaget and John Dewey
http://www.sedl.org/scimath/compass/v01n03/2.html).
For Dewey education depends on action. Knowledge and ideas emerged only from a
situation in which learners had to draw them out of experiences that had meaning and
importance to them (Dewey, 1966). These situations had to occur in a social context,
such as a classroom, where students joined in manipulating materials and, thus, created a
21
community of learners who built their knowledge together. Piaget's constructivism is
based on his view of the psychological development of children. In a short summation of
his educational thoughts, Piaget called for teachers to understand the steps in the
development of the child's mind (Piaget, 1973). The fundamental basis of learning, he
believed, was discovery: "To understand is to discover, or reconstruct by rediscovery”
(http://www.sedl.org/scimath/compass/v01n03/2.html); which suggest that a student who
understands a concept can explain it a variety of ways anytime without following a
rigidly procedure.
The philosophical view on how we come to understand or know; is characterized by three
(3) propositions:

Understanding is in our interactions with the environment
This is the core concept of constructivism because we cannot talk about what is learned
separately from how it is learned as if a variety of experiences all lead to the same
understanding. Rather, what we understand is a function of the content, the context, the
activity of the learner, and perhaps most importantly, the goals of the learner.
Understanding is an individual construction; as such cannot be shared but rather, we can
test the degree to which our understandings are compatible. An implication of this
proposition is not just within the individual but rather it is a part of the entire context
(Gaffney and Anderson, 1991).

Cognitive conflict or puzzlement is the stimulus for learning and determines
the organization and nature of what is learned
When we are learning in an environment, there is some stimulus or goal for learning. The
learner has a purpose for being there. That goal is not only the stimulus for learning, but
22
it is a primary factor in determining what the learner attends to, what prior experience the
learner brings to bear in constructing an understanding, and, basically, what
understanding is eventually constructed. In Dewey’s terms, it is “problematic” that leads
to and is the organizer for learning (Dewey, 1938; Rochelle, 1992), but for Piaget, it is
the need for accommodation when current experience cannot be assimilated in existing
schema (Piaget, 1977; vonGlaserfeld, 1989).

Knowledge evolves through social negotiation and through the evaluation of
the viability of individual understandings
The social environment is critical to the development of our individual understanding as
well as to the development of the body of propositions we call knowledge.
At the individual level, other individuals are a primary mechanism for testing our
understanding. Collaborative groups are important because we can test our own
understanding and examine the understanding of others as a mechanism for enriching,
interweaving and expanding our understanding of particular issues or phenomena.
vonGlaserfeld (1989) noted that, other people are the greatest source of alternative views
to challenge our current views and hence serve as the source of puzzlement that
stimulates new learning.
2.1.5 Classroom Implication of Constructivism to the Teacher
In constructivism, teachers and pupils are viewed as active meaning makers who
continually give contextually based meanings to each others' words and actions as
they interact. From this perspective, mathematical structures are not perceived, intuited
or taken in but are constructed by reflectively abstracting from and re-organising
23
Sensori-Motor and conceptual activity. Thus the mathematical structures that the teacher
'sees' are considered to be the product of his or her own conceptual activity and could be
different from those of the pupils. (VonGlasserfeld, 1989). Consequently, the teacher
cannot be said to be a transmitter of such structures nor can he or she build any structures
for pupils. The teacher's role here is viewed as that of a facilitator in the learning process.
Indeed if pupils are to be empowered and given greater control over their own
lives, then as Fletcher (1997) points out, they should be encouraged to choose
their own areas of study in mathematics and should also be encouraged to work
in groups and generate mathematical problems. In the classroom, the teacher’s view of
learning must point to a number of different teaching practices. It means encouraging
students to use active techniques to create more knowledge and then to reflect on and talk
about what they are doing and how their understanding is changing. The teacher makes
sure he/she understands the students' pre-existing conceptions, and guides the activity to
address them and then build on them. Constructivist teacher encourages students to
constantly assess how the activity is helping them gain understanding. By questioning
themselves and their strategies, students in the constructivist classroom ideally become
"expert learners." This gives them ever-broadening tools to keep learning. With a wellplanned classroom environment, the students learn how to learn. When they continuously
reflect on their experiences, students find their ideas gaining in complexity and power,
and they develop increasingly strong abilities to integrate new information. The teacher's
main role is to facilitate and encourage this learning and reflection processes.
24
2.2 The Nature of Mathematics
Mathematics is not only a list of facts and techniques which children memorise
but is made up of a number of processes which together form a way of thinking. These
processes are; problem solving and investigation, generalizing, abstracting, specializing,
classifying, conjecturing, communicating mathematically, justifying, forming and testing
hypothesis, applying, comparing and ordering among others (Martin, 1994; Apronti, et al
2004). Views held on the nature of mathematics according to Mereku (2004) can be
described in terms of the constituents or elements of knowledge embodied in the subject.
The constituents of mathematical knowledge or things that have to be learned to possess
mathematical knowledge are usually expressed by rules, definitions, methods and
conventions. Ernest (1985) and Van Dormolen (1986) cited in Mereku (2004) referred to
this constituent as ‘Objects’ and ‘Kernels’ respectively. These constituents have
theoretical, communicative and methodological implications. Children find it difficult to
understand abstract concepts. Besides, the difficulties intrinsic in mathematics itself are
termed subject-based factors. They arise from the nature of mathematics, its symbolism
and language. The mathematical concepts are very many; and are represented using
mathematical symbols, which by their nature are very abstract right from the concept one;
that even the mathematical concepts we teach in the primary one are far removed from
reality (Nabie, 2002). Similarly, mathematical symbols are seldom experienced in real
life situations that have meaning to children. If they experience them at all, they have no
real value to them until they start with symbolic work in school; hence this abstract
nature and structure of mathematics make abstraction, generalisation, deduction and
recall of concepts and principles difficult for learners.
25
2.3
Cuisenaire Rods and Mathematics Teaching
Cuisenaire rods are versatile manipulative materials for teaching concepts in
mathematics and one of such very important concepts is Fractions. These rods were
invented over 75 years ago by George Cuisenaire – a Belgian Mathematics teacher
(Kurumeh & Achor, 2008). However the use of Cuisenaire rod is still prominent in the
intended curriculum of Colleges of Education today due to the important role it plays in
teaching mathematical concepts especially Fractions. These materials were invented to
help students grasp abstract concepts in mathematics using coloured cardboards strips of
varying lengths called Cuisenaire rods. The original pack of Cuisenaire rods consist of 74
rectangular rods in 10 different lengths and 10 different colours as shown below:
W
R
e
d
G
r
e
e
n
P
u
r
p
l
e
Y
e
l
l
o
w
D
a
r
k
g
r
e
e
n
26
B
l
a
c
k
B
r
o
w
n
B
l
u
e
O
r
a
n
g
e
Each colour corresponds to a different length. The content of the pack is thus: 22 white
rods of 1cm each, 12 red rods of 2cm each, 10 light green rods of 3cm each, 6 purple rods
of 4cm each, 4 yellow rods of 5cm each, 4 dark green rods of 6cm each, 4 black rods of
7cm each brown rods of 8cm each, 4 blue rods of 9cm each and 4 orange rods of 10cm
each. These rods could be used as manipulative and symbolic concrete representations in
teaching concepts in mathematics. Learners explore whole numbers, Fractions,
measurements, ratio, area and perimeter etc using Cuisenaire rods (Thompson, 1994) and
develop a link between ordinal and cardinal numbers and counting (Martin, 1994).
The use of Cuisenaire rods’ approach is a hands-on and minds-on manipulative
activity filed approach for teaching abstract concepts in mathematics and sciences. It is a
valuable educational tool for modelling relationships between what is taught in school
and their everyday life activities (Elia, Gagatsis & Demetrico, 2007) thereby enabling
students to work independently and in groups on meaningful mathematics while the
teacher provides individual attention to other students. Because Cuisenaire rods are
ready-made tools, its approach minimizes preparation and set up time both for the teacher
and the students. This approach helps to develop key skills such as classification, critical
thinking, problem solving and logical mathematics and spatial reasoning (Rule &
Hllagan, 2006). However, Taylor-Chapman (1967) said there are several advantages for
colouring the rods. Some of these may be doubtful but seem beyond argument:
1. You can use the rods for sorting and matching by colour; eg ‘all yellows together’
or ‘match each green with a black’.
2. The children quickly learn to pick out any length very quickly.
27
3. We can call a rod by its colour and make it our unit. Thus if a brown is the unit,
purple will be found to be half, white (natural) will be an eighth and if light-green
is the unit, then dark-green is two; and natural is one-third. These are great
advantages when learning about relationships.
2.4
Meaning and Concept of Fractions
The word Fraction is taking from the Latin word ‘Frangere’ which means ‘to
break’ (Downes & Paling, 1965; Apronti et al, 2004). This suggests that, a Fraction may
be described as a part of a whole where the whole could be ‘a unit’ or a set of objects. In
a related development, Martin, et al (1994) pointed out the importance to realise that the
pairs of numbers ‘12’, ‘23’, etc and the phrases ‘one third’, ‘two fifths’ etc are not Fractions
but merely symbols and words representing the concept of particular Fractions. They are
of the view that; to understand what a Fraction is, then we must first look at how they
arise. ‘A half’ is what we get when we share something equally into two parts. They
noted that what ‘a half’ is depends upon what we started with. This suggests that ‘a half’
of Mr. A may not be the same as ‘a half’ of Mr. B. It is not possible therefore to show
any single object and say ‘this is what a half is’. They illustrated these in a diagram as all
representing a half.
28
From the above, Fractions are not objects but actions (Martin, et al 1994, Kusi-Appau,
1997) and that; we perform an action to halve something. It is only when we learn to
represent these actions that these symbols can be treated as objects.
However, Hilton and Pedersen (1983) said the words ‘a half’, ‘a quarter’, ‘a third’, and
‘three-quarter’ etc are used frequently in everyday speech and their meaning is clear to
the reader; suggesting that one can say ‘a fourth’ instead of ‘a quarter’. The phrase ‘half’
can be used in subtly different ways. For instance, would you like a piece of cake? We
may reply ‘please just a half’. The host or hostess may then cut the piece of cake into two
equal pieces and give him one of those pieces. He has received ‘a half’ of the original
piece of cake. On the other hand, a realtor showing us two possible lots for purchase may
say ‘Lots A is more attractively situated but there is only half as much land as on lots B’.
There is no suggestion that lots A has been created by cutting up lot B; the realtor only
means that the amount of land on lot A is the same as the amount of land we would get
by taking half of lot B. This means that when we say that 10 Ghana pesewa is a tenth of
GH¢ 1.0, we certainly do not mean that 10 Gp is obtained by cutting or breaking GH¢ 1.0
into ten equal pieces and taking one of the pieces; rather, we mean that 10 Gp is worth a
tenth of GH¢ 1.0, that we would require ten 10 Gp to purchase what we can purchase for
GH¢ 1.0.
Fractions, Decimals and Percentages are number ideas that are not whole
numbers. These three concepts are closely related to each other for the fact that one can
move from one domain to the other. In line with this, the teacher must help the child to
see these relations and how to move from one form to the other (Kusi-Appau, 1997). In
developing the concept of Fractions, the teacher must be able to use activities in real life
29
situations (Apronti et al 2004). According to Barnette and Ted (2000), Kusi-Appau
(1997) and Apronti et al (2004); Fractions can be considered in three ways:
i. Part-whole model (Sharing)
ii. Part-group model
iii. Ratio model

Part-whole model (sharing)
This is the case where children share a number of items like oranges, among themselves.
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 to share; and as
they do this they make use of Fractions to denote part of a whole. Very often, teachers
1 1 2
make mistake of telling children that the statements like 2, 3, 3 etc which are symbols and
words representing the concept of Fractions, are Fractions. When teaching Fractions in
schools, the emphasis is often on situations where the object can easily be cut, folded,
split or coloured in equal parts. Although there is some need for this sort of activity,
children should be exposed to a wide variety of situations, some where such folding or
splitting strategies will not be successful. In experiencing a variety of situations where
Fractions can be found, learners will have the opportunity to reflect and abstract critical
relations in different contextual situations. In other words, children must see a whole in
all its representational forms. To overcome such misconception, teachers must let
children see that Fractions are not objects but are actions of dividing objects into equal
parts and taking some parts. In developing a sound understanding of the part-whole
concept of Fractions, it is necessary for teachers to present situations of fair sharing,
where the child is expected to reason out the consequences of different actions
30
http://nrich.maths.org/2550. Children must therefore know how Fractions arise.
Confusion among children is from what we take as a whole. Very often teachers use a
unit object as a whole and therefore when children come to meet groups of objects, they
become confused when we take a Fraction from it. Considering the figures below:
(i)
(ii)
(iii)
The child may see the part shaded in (i) as half. However, in other situation when you
have an object of the shape in (ii) and divide it into two as in (iii), the child finds it
difficult to understand that the shaded portion is half. This creates misconception and
confusion in the child’s mind. To overcome this confusion, the teacher must help the
child understand that the whole could be one unit, a group or part of a unit or anything we
are taking part could be our whole (Kusi-Appau, 1997). Fractions taught as a part-whole
concept, can ensure that children have a sound foundation for conceptualizing other
concepts in Fractions. However, it must be noted that despite the wealth of possible
examples, an approach to Fractions based solely on "part-whole" is too restricted yielding proper Fractions only. Therefore other concepts of Fractions need to be explored
if children are to have a fuller and better understanding of rational numbers
http://nrich.maths.org/2550.

Part-group model
In everyday activities of children, it is often becomes necessary for them to consider part
of a set in relation to the major set. This is illustrated in the diagram below:
31
1
3
Part of a group

Ratio model
According to Apronti et al (2004) the ratio model shows the relationship between objects
or quantities of the same kind. It is a way of comparing the objects and this ends up in the
form of a Fraction; that is to say if there are 30 boys and 50 girls in a class then ratio of
30
3
boys to girls is 30: 50 = 50 = 5 (Kusi-Appau, 1997). Using Cuisenaire rods to compare
the lengths of two rods side by side for instance, it takes 5 white rods to equal 1 yellow
rod. Hence the length of the white rod is
1
5
of the length of the yellow rod. This ratio of
the length of the white rod to yellow rod is 1:5 as illustrated diagrammatically as:
W
Yellow
According to Pamela (1984), Fractions are well identified area of difficulty for many
children and even adults. There are two obstacles to understanding of Fractions.
1. Fractions cannot be taught of as separate, independent entities. They have
meaning only in relation to the whole to which they apply. To recognise Fraction
of something, you need a concept of the whole. It is relatively easy to imagine the
whole apple of which you have a quarter, but it is not easy to imagine the whole
kilogram of which you have a quarter, or the whole hour of which a quarter has
passed.
32
2. Complicated notations by which Fractions are symbolized. The numeral at the
bottom of a Fraction (denominator) has an entirely different function from the
numeral at the top (numerator). For instance, the denominator of the Fraction
2
3
tells us that the ‘whole’ has been divided into three equal shares. The numerator
tells us that two of those shares are under consideration. The word ‘denominator’
means ‘the thing that names’. The denominator of the Fraction
2
3
gives the
Fraction its name; ‘third’. The word ‘numerator’ means; ‘the thing that numbers’.
Hence the numerator of the
2
3
tells us the number of thirds to be considered. The
denominator and numerator for Fractions also make it possible to denote the same
2
Fraction in infinitely many ways. For instance
4 6
, ,
8
,
10 14
,
6 9 12 15 21
3
is the same as
2
𝑒𝑡𝑐. is called equivalent Fractions of 3. This idea takes a long time
to sink in, and can prove another obstacle to understanding.
To overcome the first obstacle, we should always in the early stages refer to the whole to
which any Fraction applies. We should not talk about a ‘quarter’ but ‘a quarter of an
apple’, ‘a quarter of a metre’ or ‘a quarter of twelve’ etc.
2.5
Equivalent Fractions
Apronti et al (2004) explained equivalent Fractions as Fractions of the same value
but different names. They are Fractions which represent that same number but have
different names (Kusi-Appau, 1997: 97). For instance 12, 24, 36,
4
8
are equivalent Fractions.
According to Apronti et al (2004) equivalent Fractions can be introduced using:
33
i.
Paper folding and shading
ii.
Fractional boards
iii.
Cuisenaire rods
Paper Folding and Shading
They illustrated ½ by folding vertically, a sheet of paper strip into two equal parts and
shaded one part shown below:
1
2
To have its equivalence, they again fold the strip of paper horizontally thereby having
four equal parts with two parts shaded.
2
4
By folding the same paper again you will have eight equal parts with four of the parts
shaded
4
8
It is noted from the above that the portion shaded for the first time, no other part had been
shaded again. It is the same portion that has been named differently.
34
Fractional Boards
Using Fractional boards, Apronti et al (2004) to identical strips of cards or paper, fold
one into two equal parts and another into four equal parts and so on as shown below:
Whole
1
2
1
2
1
4
1
8
1
4
1
8
1
8
1
4
1
8
1
8
1
4
1
8
1
8
1
8
From the illustrations, 2 halves make 1 whole, 2 one-fourths make one-half, 4 one1
2
4
8
eighths make one-half and so on. This implies that 2 = 4 = 8 = 16 etc.
Using Cuisenaire Rod
According to Kusi-Appau, (1997) and Apronti et al (2004), one can choose any rod or set
of rods like Orange and Dark green to be the whole. You can then make up as many rows
using rods of one colour only as shown below.
Orange
Dark green
Brown
Purple
Red
Red
W W
W
Brown
Purple
Red
W W
Red
W W
Purple
Red
W W
Purple
Red
W W
Red
W W
It can be seen from the diagram that:

Two browns make the orange and dark green whole

Two purples make one brown
35
Red
W W W

Four purples make two browns

Two reds make one purple

Four reds make one brown

Eights reds make the orange and dark green whole

Two whites make one red

Four whites make one purple

Eight whites make one brown sixteen whites make the orange and dark green
whole
These colour observation can then be turned into Fractional statements as a brown is onehalf of the orange and dark green whole, a purple is one-fourth of the orange and dark
green whole, a red is one-eighth of the orange and dark green whole and a white is onesixteenth of the orange and dark green whole.
Pupils can the progress from concrete objects to diagrams, to words and eventually to
3
4
symbols and deduce that 12 = 24 = 36 = 48 etc and 14 = 28 = 12
= 16
and so on.
2.6
Addition and Subtraction of Fractions
Dolan (2000) observes that apart from whole-number computations, no topic in
elementary mathematics curriculum demands more time than the study of Fractions. To
him, for students to understand, the 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 must be developed
before formal work with Fractions.
36
According to Owusu and Manu, (2007) before pupils are introduced to addition and
subtraction of Fractions, they must be able to rename Fractions using their equivalence to
confirm their readiness for operations on rational numbers. However, children often think
that whenever two Fractions are added, the result is less than 1 (Owusu & Manu, 2007).
This is because their exposure to addition of Fractions is always less than 1. This means
that they need early exposure to problems where the sum is greater than 1 to erase such
misconception.
Teaching addition and subtraction of Fractions for better understanding, it is
expected that we use concrete materials. However, usually the first step is to learn to add
and subtract Fractions with the same denominator which is fairly straightforward and
activities using concrete materials are easy to devise (Martin et al, 1994). To Apronti et al
(2004), paper folding and shading, number line and Cuisenaire rods could be used to
teach addition and subtraction of like Fractions.
i.
Using Paper folding and Shading
3
8
2
8
5
8
This shows that
3
8
2
5
+8=8
and
3
8
2
1
− 8 = 8 respectively.
37
1
8
ii.
Using Number line
0
2
8
1
8
3
8
3
8
2
8
3
8
3
8
iii.
6
8
5
8
7
8
8
8
6
8
7
8
+ 28 = 58
1
8
0
4
8
4
8
5
8
8
8
− 28 = 18
Using Cuisenaire rods
Pupils pick the brown rod as the whole.
W
W
W
W
W
W
W
W
Brown
Pupils to take two whites and three whites to represent two-eighths and three-eighths
respectively.
2
8
W
3
8
W
W
W
W
They have to join the white rods end-to-end and then compare to the whole as:
5
W W W W W
8
Brown
2 3 5
+ =
8 8 8
For subtraction, the two whites are subtracted from the three whites to have one white.
One white rod is then compared to the whole which is the brown rod as shown below.
1
8
W
Brown
38
2.7
The Part of Language in Teaching Mathematics
The language used by the teacher in teaching any topic is very important if he or
she is to make positive impact on his or her learners. Language plays an important role in
the teaching and learning process. Mathematics language should be carefully and
accurately used from the beginning of the child’s learning experiences (William, 1986).
Torbe (1982) cited in Mereku and Cofie (2008) pointed out that “without language,
without the telling and listening, the reading and the writing which fills every school day,
there could be no communication and no educational process; it is language which makes
the whole educational process possible”. One of the main reasons why children
experience difficulty in mathematics is in the understanding the nuance of mathematical
language (Warrant, 2006) and since mathematics, as a language makes use of symbolic
notation as such requires using and interpreting this symbolic notation and grasping the
abstract ideas and concepts which underlie it (Mereku & Cofie, 2008). They laid
emphasis on the fact that the child’s inability to use language in mathematics will not
only hinder his understanding of the subject but will also prevent the teacher from having
a deeper insight into the child’s grasp of mathematics. The appropriate use of
mathematical language and symbols can also help children develop mathematical
concepts. An understanding of the mathematical symbols and examples will enhance
children’s mathematical ability (Nabie, 2002, 2009). This is more so if they are combined
jointly manipulated and linked to their everyday life situations. In line with this, Skemp
(1986) cited in Martin (1994) and Apronti et al (2004) proposed that:

“concepts of a higher order than those a person already has cannot
be communicated to that person by a definition. Only by arranging
for the person to encounter a suitable collection of examples can
such concept be communicated”.
39

“Since in mathematics examples are almost invariably other
concepts, the concepts used in the examples must already be formed
in the mind of the learner”.
Beginning a lesson with children’s previous knowledge, of which the mother tongue
forms a part, gives them a perpetual momentum to forge ahead in the learning process.
New ideas are easily developed and understood if they are linked with already existing
ones. Hence children mathematical concepts can easily be developed if incoming
concepts are correctly linked with what the child knows already in an understandable
language.
40
CHAPTER 3
METHODOLOGY
3.0
Overview
This aspect of the research dealt with the methods and procedures used to obtain data for
the research work. This was done under the following themes:
3.1

Research design

Population and Sample

Sampling Techniques

Instrument

Intervention

Method of data collection

Data analysis procedure
Research Design
The design for this study is an Action Research in the form of pre-test, intervention
and post-test structured to examine how the performance of second year students in
teaching fractions is improved using Cuisenaire rods at Dambai College of Education.
Action research according to Cannae (2004) involves the application of scientific
methods to solve classroom problems. It uses pre-test and post-test data from the teaching
of two instructional units to identify student teacher controlled factors which promote or
inhibit pupils’ academic achievement (Bill, 1986).
Action research is attractive to educational researchers because it seeks to identify
peculiar problem in the educational field especially in the classroom and suggest possible
41
rectification to the problem by offering suitable intervention and recommendations for
use by other educators to also apply such intervention. Put simply by O’Brien, (1998),
action research is “learning by doing” - a group of people identify a problem, do
something to resolve it, see how successful their efforts were, and if not satisfied, try
again http://www.web.net/~robrien/papers/arfinal.html.
This design was chosen in order to find possible solution to the problem identified and
that teachers will be able to have command on the teaching of fractions in Ghanaian
schools.
3.2
Population and Sample
According to Anamoah-Mensah et al (2004), the quality and integrity of any
study depend on the validity and the efficiency of the samples used in the study. In this
regard, the sample was carefully selected. Out of the target population of two hundred
(200) Second Year Students of Dambai College of Education in the Krachi-East District
of the Volta Region for which 165 are males and 35 females, a sample size of 50 students
comprising 41 males and 9 females were selected for the study.
3.3
Sampling Techniques
The sampling technique used was the stratified sampling alongside with the
random sampling techniques to select the samples for the study. Stratified sampling
according to Awanta and Asiedu-Addo (2008) is the process of selecting a sample in such
a way that identified subgroups in the population are represented in the sample the same
proportion that they exist in the population. The percentage representation of male and
female students was calculated to be 82% and 18% respectively. Therefore to have a
42
good representation, these percentages were used on the sample size of 50 students. The
sample was therefore selected by writing “YES” and “NO” on paper and fold for both the
males and the female to pick at random. For the female group, 9 “YES” and 26 “NO”
was written on paper for the 35 female to pick randomly. All those with the “YES” were
selected for the study. The same process was repeated for the male students to have 41 of
them on the study.
3.4
Research Instruments
The instruments used for the collection of data were tests. The tests were used in two
folds, that is, pre-test and post-test.
3.5
Validity and Reliability
3.5.1 Validity
Validity of an instrument according to Taale and Ngman (2003) refers to whether
the instrument truthfully does what it is constructed to do. In other words, when the
instrument measures what it is intended to measure, then it is valid.
To ensure the validity of the test items, the researcher consulted the curriculum for
methodology and some prescribed mathematics textbooks for teacher trainees. The
purpose was to gain insight into what learners were expected to learn in order to develop
the instrument accordingly. The researcher made sure that the content of the test was
based on what the research questions were set to find out. Thus,, only questions on
teacher trainees Pedagogical Content Knowledge were asked.
After constructing the test items, the researcher approached other tutors in the
Mathematics Department to cross check the appropriateness of the test items. Durrheim
43
(1999) suggests that the researcher approach others in the academic community to check
the appropriateness of his or her measurement tools. Colleague tutors responses indicated
that the contents examined in this study reflected the prescribed Content for students’
Pedagogical Content Knowledge
3.5.2
Reliability
Reliability on the other hand refers to how well the instrument provides a
consistent set of results across similar test situation, time periods and examiners (Taale &
Ngman-Wara, 2003). It means the degree of dependability of a measuring instrument. It
is worth mentioning that it is possible to have an instrument which is reliable because the
responses are consistent, but may be invalid because it fails to measure the concept it
intends to measure (Fraenkel & Wallen, 2000).
In this study, the split-half method was used to check the reliability of the instrument
because it is a “more efficient way of testing reliability” and was less time consuming
(Durrheim, 1999). The split-half method requires the construction of a single test
consisting of a number of items. These items are then divided or split into two parallel
halves (usually, making use of the even-odd item criterion). Students’ scores from these
halves were then correlated using the Spearman-Brown formula used in reliability testing.
The value of the coefficient was 0.72. This value indicates a good degree of reliability of
the instrument as asserted by (Fraekel & Wallen, 2000).
3.6 Data Collection Procedure
All the fifty (50) students sampled for the study responded to the pre-test
administered to determine their previous thoughts on teaching Fractions using concrete
44
materials. The pre-test was conducted on 5th February, 2011. In the 2nd week of March
2011, the implementation for the intervention begun. After four (4) weeks of lessons and
activities on the teaching and learning of Fractions, the students were again tested (Posttest) which involved similar concepts but different set of questions as compared to that of
the pre-test to determine the amount of knowledge the students have gained from the
intervention activities. The Pre-test and the Post-test were marked and the scores by the
students are shown in appendices C and D.
3.7
Intervention
The intervention the researcher employed in the study is the use of Cuisenaire
rods as a Concrete Material to help teacher trainees in teaching and learning of fractions.
The intervention spanned four (4) weeks and lessons were conducted three times a week
for 60 minutes per meeting. Students were introduced to the concept of Fractions,
comparing Fractions (equivalent fractions) and addition and subtraction of fractions using
materials including Cuisenaire rods, paper folding, number lines and fractional boards.
The researcher demonstrated the concepts using the materials in range of activities to
help the teacher trainees overcome their difficulties in teaching the concept of Fractions.
The lesson was taken out of their syllabuses and other sources of information that
researcher deem vital to use. Different methods, techniques and strategies were employed
to enable students’ involvement in the lesson by relating the object of learning to the
needs of the learner, their involvement in the learning process become increasingly
significant. The students were allowed to work in groups and in pairs as they manipulate
these materials while the teacher serve as a facilitator providing help when needed and
asking thought provoking questions to stimulate critical analytical and complex thinking
45
in order to help them construct their own meaning as the study focused on promoting
constructivist approach of learning.
3.8
Intervention Implementation
Students were taken through series of activities using concrete materials but concentrated
on the use of Cuisenaire rods in solving and teaching problems involving Fractions.
3.8.1
Concept of Fractions
The researcher demonstrated the concept of fractions by folding a sheet of paper
equally for the students to see. The researcher discussed with students to identify one part
as one-half because a whole has been divided into two halves.
1
2
After demonstration the students were asked to do likewise.
Students were put in groups and the Cuisenaire rods were given to them. The researcher
instructed them to choose a rod say (orange) and try to make up as many rows as they can
using rods of one colour only as shown below.
W
W
W
W
W
W
Yellow
Red
W
W
W
W
Yellow
Red
Red
Red
Red
Orange
From the diagram above, five red rods make an orange rod. In fraction statement a red is
1
one-fifth of the orange whole which is written symbolically as 5. In the same vein, two
46
yellow rods make an orange rod. In a fraction form, a yellow is one-half of the orange rod
1
1
written as 2 and a white is one-tenth written as 10.
3.8.2
Equivalent Fractions
The idea of equivalence occurs and every opportunity should be taken during
discussion 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 the various
ideas which they have acquired. Using paper folding, students were taken through the
following activities.
3.8.3 Paper Folding and Shading
The students were guided to fold vertically, a sheet of paper strip into two equal parts and
shade one part to represent one-half as shown below:
1
2
The activities continued by guiding them again to fold the strip of paper horizontally
thereby having four equal parts with two parts shaded.
2
4
47
By folding the same paper again you will have eight equal parts with four of the parts
shaded
4
8
It was noted from the above that the portion shaded for the first time, no other part had
been shaded again. It is the same portion that has been named differently.
3.8.4
Using Cuisenaire Rods
The researcher guided the teacher trainees to choose any rod or set of rods to be
the ‘whole’ for instance the orange and dark green rods joined end-to-end to make up as
many rows as possible using rods of the same colour only making sure that each row is of
the same length as the original ‘whole’ chosen and write down their observations in
words.
W W
W
Red
Red
W W
W W
Red
Purple
W W
Red
Red
Purple
W W
W W
Red
Red
Purple
Brown
W W W
Red
Purple
Brown
Orange
Dark green
From the diagram students were guided to identify the following:
i.
Two browns make the orange and dark green rods put end-to-end as ‘whole’.
ii.
Two purples make one brown.
iii.
Four purples make two browns.
iv.
Four purples make orange and dark green whole put end-to end.
48
v.
Two reds make one purple.
vi.
Four reds make one brown.
vii.
Eight reds make the orange and dark green whole put end-to-end.
viii.
Two whites make one red.
ix.
Four whites make one purple.
x.
Eight whites make one brown.
xi.
Sixteen whites make the orange and dark green whole. etc
These colour observations were then turned into fractional statements as
1. A brown is one-half of the orange and dark green rods put end-to-end.
2. A purple is one-fourth of the orange and dark green rods wholes put end-to-end.
3. A red is one-eighth of the orange and dark green whole.
4. A white is one-sixteenth of the orange and dark green whole.
A critical look at the diagram shows a pattern as
1
2
=
2
4
4
=
8
=
8
16
and also
1
4
=
2
8
=
4
16
etc. Equivalent fractions are fractions of the same value but different names illustrated by
the diagram above i.e.
1
2
=
2
4
=
4
8
=
8
16
1
2
etc and 4 =
8
=
4
16
etc. It was therefore deduce
that when the top (numerator) and bottom (denominator) of a Fraction is multiply by the
same counting number, the value of the fraction remains unchanged though with different
names.
3.8.5 Comparing Fractions
With the idea of equivalent fractions, students were also guided to compare both
like fractions and unlike fractions using paper folding. I guided the students to compare
firstly like fractions. For instance comparing
49
1
4
2
and 4. Students were guided to take two
strips of paper of the same size and fold them such that each one is divided equally into
four parts and shade the corresponding parts. Students were asked to compare the shaded
parts by putting the strip of papers side by side as shown below. It was clear from the
2
1
diagram that 4 is greater than 4 since they all the same denominator.
𝟏
𝟒
2
4
Activities concerning unlike Fractions were also carried out using the paper folding. For
instance using
2
3
1
and 2 , the students were asked to take two strips of paper with the same
2
size, fold first one into three equal parts and shade two of the equal parts as 3. The second
1
strip of paper is also folded into two equal parts and shade one part of it to have 2. The
strips of papers were placed side by side for students to identify which of the shaded area
2
1
is larger as shown below. Students did see clearly that 3 is greater than 2.
2
3
𝟏
𝟐
50
3.8.6
Addition of like Fractions
Cuisenaire rods were used extensively here to teach and develop algorithm of
Fraction. Specific examples were used e.g. addition of
1
5
2
+ 5. Students were guided to
choose a rod (whole) that can be split into five exactly of other rods. Students were able
to pick orange and yellow since orange can be split into five reds and yellow into five
whites a rod of each represents one-fifth respectively. I did not restrict the students of
which rods to work with, rather some used the orange rod whiles others used the yellow
rods as shown below.
W
W
W
W
W
Yellow
Students were guided to pick one and two of the white rods to represents
respectively. Putting these rods end-to-end, we have
W
W
1
5
and
2
5
W
Comparing the three rods put end-to-end with the whole (i.e. yellow rod), we have this
diagram below.
W
W
W
Yellow
Thus
1
5
+
2
5
=
3
5
.
On the other hand students who picked orange as the whole were also guided to pick one
red rod and two red rod to represent
1
5
and
2
5
respectively. They joined the rods and then
compared to the whole which is the orange rod and got
Red
Red
Red
Orange
51
Red
3
5
.
Red
Red
Red
Red
Orange
Thus
1
5
+
2
5
=
3
5
.
1
2
In the same way, adding 3 and 3, students were guided to pick a rod that can be split into
three equal rods. Students pick light green and dark green as the whole and worked with
them as follows.
Red
Red
Red
Dark green
2
1
The red rods are each one-third so two red rods are taken as 3 whiles a red rod is taken as 3.
Thus two reds and one red joined end-to-end and compared to the whole fit exactly onto
1
the whole as above. Thus 3 +
2
3
3
= 3 = 1.
3.8.7 Subtraction of like Fractions
Students were guided to understand that the algorithm of subtraction is done in
3
the same way as in addition. In solving 5 −
1
5
for instance, students were again guided to
pick rod or a train of rods that can be split into five of other rod. Students were able to
pick yellow and orange rods. The yellow can be split into five whites whiles the orange
rod can also be split into five red rods. It suggests that any of the rods could be worked
with. Using the yellow rod as shown bellow, each of the white rods represent one-fifth.
3
Students were guided to take three whites rods and compare to the whole represent 5 and
out of the 3 whites rods take one from it and the result is 2 whites.
52
W
W
W
W
W
Yellow
3
5
3
1
Thus 5 −
3.8.8
5
W
W
W
1
5
Yellow
W
Yellow
2
= 5.
Addition of unlike Fractions
Through discussion, I explained to the understanding of students that Fractions with
unlike denominators can be classified under the following categories:
i. one denominator as a multiple of the other
ii. a common factor
iii. denominators being prime numbers
There was a thorough discussion on fractions with one denominator as a multiple of the
other the same before they add them. For instance
1
3
=
2
6
was obtained by multiplying
the numerator and denominator by 2. In adding fractions with different denominators for
instance
1
+
2
1
3
we listed down sets of Fractions that are equivalent to both
shown below.
1
2
1
3
=
=
2
4
2
6
=
=
3
6
3
9
=
=
4
8
4
8
=
=
5
10
5
15
=
=
6
12
6
18
7
= 14 =
=
7
21
=
8
16
8
24
53
etc.
etc.
1
2
and
1
3
as
For the set of equivalent Fractions shown above, we pick those with the same name to
1
1
3
1
2
represent the original fractions 2 and 3. This shows that 6 is equivalent to 2 whiles6 is also
1
1
equivalent to3. This shows that 2 +
1
3
=6 +
3
2
6
5
= 6.
3
2
Using Cuisenaire rods as a concrete material to solve for example 5 +
, it is clear that
10
10 is a multiple of 5 hence we need to choose a rod which is the whole such that the rod
can be split into ten which is the orange rod.
W
W
W
Red
W
W
Red
W
W
Red
W
W
Red
W
Red
Orange
1
1
From the diagram one red represents 5 and one white represents 10 of the whole which is
3
2
the orange rod. Therefore in the question 5 +
3
Red
whiles 3 red rods represents 5
2
, 2 white rods represent 10
10
Red
W
W
Red
It is clear that we cannot combine two different rods and have meaningful explanation
6
hence we change 3 red rods for 6 whites rods to have its equivalent as 10.
W
W
W
Red
Now the two white rods
2
10
W
W
W
Red
W
W
6
10
3
5
Red
and six white rods
W
W
W
W
W
are joined end-to-end and then compared side-by-side to the whole which is the orange
rod.
8
10
W
W
W
W
W
W
Orange
Therefore
3
5
+
2
10
6
becomes 10 +
2
10
=
8
10
.
54
W
W
W
6
10
1
For the question 3 +
1
, you need a rod that can be split into 2 and 3 respectively and that
2
rod is dark green rod. 1 red rod represents
1
3
and 1 light green rod represents
1
2
of the
whole below.
W
W
W
Red
W
Red
Light green
W
W
1
3
Red
Light green
1
2
Dark green
Students were guided to exchange 1 light green rod for 3 whites and 1 red rod for 2
whites.
i.e.
1
2
=
Light green
3
6
W
W
W and
1
3
Putting the exchanged white rods end-to-end we have
=
Red
W
W
2
6
W
Comparing the 5 white rods to the whole which is Dark green rod we have
1
Hence 3 +
1
2
=
2
6
+
3
6
=
W
W
5
6
W
W
.
5
.
6
Also there are situations in which one has to choose a set of rods as the whole.
2
For instance in questions like 3 +
1
, one need to choose a rod that could be split into 3
4
and 4. No single rod could be split in this way hence any of the following could be
chosen as the whole

Orange and Red

Blue and Light Rod

Brown and Purple

Black and Yellow

Dark green and Dark green
55
Any of the above combination could be used as a whole for the question and discussion
was done for all for students to realize that they all arrive at the same answer. Using the
orange and Red rods as the whole, I guided the students to put the two rods end-to-end
and look for a rod that can go into the whole 3 and 4 respectively as shown below.
Light green
Light green
Purple
Light green
Purple
Light green
1
4
Purple
1
3
Orange
Red
From the above diagram 1 purple represents one-third (1/3) and 1 light green represent
one-fourth (1/4).
Purple
2
3
Purple
1
4
Light green
Students now change the two purple rods for eight (8) white rods and one light green rod
for three white rods, join end-to-end and compared to the whole.
11
12
W
W
W
W
W
W
W
W
W
W
W
Orange
It implies that
2
3
+
1
4
=
8
12
+
3
12
=
Red
11
.
12
3.8.9 Subtraction of unlike Fractions
The researcher guided the students to understand that subtraction and addition of
Fractions follow the same procedure or algorithm. The only different thing you have to
do is where you need to take away instead of addition. Using
1
2
1
− 8, students were
guided to choose a rod which could be split into eight equal parts. Since 2 is a factor of 8
56
or 8 is a multiple of 2 the rod chosen can be split into 2 too. The appropriate rod is the
brown rod. As shown in the diagram below, 1 purple rod represents
1
2
and 1 white rod
1
represents 8 of the whole (which is the brown rod).
W
W
W
W
W
W
Purple
W
W
Purple
Brown
1
2
Purple
1
8
W
Students change 1 purple for 4 white rods and compare with the whole and can now take
away1 white rod from 4 white rods to have 3 white rods. Students now compare 3 white
rods with the whole to have 3/8.
1
2
4
8
1
Hence 2 −
1
8
=
Purple
W
4
8
+
W
1
8
W
W
=
4
8
W
W
-
1
8
W
=
3
= 8.
57
W
3
8
W
W
W
W
CHAPTER 4
DATA PRESENTATION, ANALYSIS AND DISCUSSION
4.0
Overview
This chapter dealt with the presentation of data, analysis of scores collected from
the pre-test and the post-test and discussions based on the results of the study.
The data collected was analyzed quantitatively and qualitatively. The results from the
pre-test and post-test raw scores were analyzed using both the descriptive and inferential
statistics employed on the Statistical Package of Social Sciences (SPSS). The descriptive
statistic used to analyze the data projected the sample size, minimum and maximum
scores, the mean scores and standard deviation for both the pre-test and post-test. The
data was further analyzed using inferential statistics to project the p-values and t-values
from the paired sample T-test.
4.1 Data Presentation and Analysis
Table 4.1 shows the frequency distribution of the raw scores of the pre-test conducted for
fifty (50) students. (See Appendix C).
Table 4.1: Frequency distribution of Pre-test Scores by percentage
Scores
Frequency
Percentage (%)
1 – 10
10
20
11 - 20
27
54
21 – 30
10
20
31 – 40
3
6
41 – 50
-
0
Total
50
100
58
Table 4.1 shows that 10 students representing 20% of the total number of students scored
between 1 and 10 inclusive, 27 students representing 54% of the total number of students
scored marks ranging from 11 to 20. Again, 10 students representing 20% of the total
number of students involved in the study scored marks ranging from 21 to 30 whiles only
3 students representing 6% the number scored marks between 31 and 40 inclusive. No
student scored marks ranging from 41 to 50. It is obvious from the marks that out of the
50 students who took the test, 42 students representing 84% obtained marks less than half
of the total marks and only 8 students representing about 16% of the total number of
students scored half or more of the total marks indicating poor performance of students in
teaching problems involving addition and subtraction of Fractions.
Table 4.2: Frequency distribution of post-test results for fifty (50) students by
percentage
Scores
Frequency
1 – 10
-
0
11 - 20
2
4
21 – 30
22
44
31 – 40
19
38
41 – 50
7
14
Total
50
100
Percentage (%)
From the post-test results in Table 4.2, it can be seen that no student obtained
marks between 1 and 10 inclusive. Two (2) students representing 4% of the total number
of students got marks from 11 to 20 whiles 22 students representing 44% of the total
59
number of students scored marks ranging from 21 to 30. Again, 19 students which
represent 38% of the total number scored marks between 31 and 40 inclusive whiles 7
students representing 14% of the students’ total number scored marks from 41 to 50. It
was realized from the post-test results that, 44 students representing 88% of the total
number of students who took the test obtained half or more of the total marks. These
improvements in students’ performance indicate the effect of the use of Cuisenaire Rods
with questioning skills in teaching students on problems involving Fractions. However, 6
students representing 12% of the total number of students scored marks less than half of
the total marks which indicate that some students still have little problems in solving
Fraction related problems using concrete materials.
Table 4.3: Descriptive Statistics of Pre-test and Post-test Scores
N
Minimum
Maximum
Mean
Std. Deviation
Pretest
50
5
37
16.94
7.175
Posttest
50
20
48
32.12
6.915
From table 4.3, the mean of pre-test score was 16.94 and that of post-test score
was 32.12. Thus, the Gain score which is 32.12 − 16.94 is 15.18 when compared to the
pre-test mean score of 16.94 show a significant improvement in students’ performance in
solving problems involving Fractions by the use of Cuisenaire rods.
A comparison of standard deviations of the pre-test score which was 7.175 and post-test
score which was 6.915 revealed that the standard deviation of the post-test was less than
that of the pre-test which indicates that the scores in the post-test were more spread
around the mean mark which is 32.12 than it was in the pre-test scores. The minimum
and maximum marks of pre-test scores and post-test scores are respectively 5, 37 and 20,
60
48. It is clear that both the minimum and the maximum marks of post-test scores are by
far larger than that of the pre-test scores. Also, the range which is the difference in the
maximum and minimum marks of pre-test and post-test is 32 and 28 respectively. It
could be realised that, the range for pre-test is larger than the range for post-test which
buttresses the interpretation for standard deviation; the fact that the data for the pre-test
are less spread around the mean compared to that of the post-test.
Testing of the Hypothesis
Null hypothesis 𝐻0 : There is no significant difference in scores between the mean pre-test
scores and the mean post- test scores of students.
Alternative hypothesis 𝐻𝑎 : There is significant difference in scores between the mean
pre-test scores and mean post- test scores of students.
Table 4.4: Paired Sample Test for Pre-test and Post-test scores
𝑃𝑟𝑒𝑡𝑒𝑠𝑡
𝑵
𝑴𝒆𝒂𝒏
𝑺𝒕𝒅 𝑫𝒆𝒗𝒊𝒂𝒕𝒊𝒐𝒏
𝒕
𝒅𝒇
𝑷 − 𝒗𝒂𝒍𝒖𝒆
50
−15.180
8.395
−12.786
49
0.000
− 𝑃𝑜𝑠𝑡𝑡𝑒𝑠𝑡
From the Table 4.4 above, analyzing the data with paired sample test produced
the P-value (0.000) which is less than the level of significance (0.05). Hence we reject the
null hypothesis 𝐻0 and accept the alternative hypothesis 𝐻𝑎 (Asiedu-Addo et al 2004 ).
Accordingly, it is concluded that there is a significant difference in mean scores between
students in the Pre-test (𝜇 =16.94) and the post-test (𝜇 = 32.12).
61
4.2
Discussions of Results
Considering the pre-test scores, 84% of the students obtained marks less than the
total mark. This is an indication that students had difficulty in working with fraction
related problems. From the post-test results, 88% of the students scored half or more of
the total mark. This is an improvement in students’ performance compared to their
performance in the pre-test. The mean pre-test score of 16.94 and the mean post-test
score of 32.12 with a gain score of 15.18 indicated that students’ performance was about
twice better than when the intervention was not administered. In answering the research
question, it is clear that after the intervention, the evidence gathered suggest that
incorporating the intervention tool (Cuisenaire Rods) into mathematics classroom
teaching of fractions improved the achievement scores of students.
Also in the paired sample t-test, the P-value of (0.000) is far less than the significance
level of (0.05) which means that the null hypothesis must be rejected in order to accept
the alternative hypothesis indicating that there is significant difference in the mean scores
between students in the pre-test and post-test scores.
4.3
Summary of Findings from Research Questions
4.3.1 Research Question 1
What effect has the use of Cuisenaire rods on students’ performance in teaching
problems involving Fractions?
In answering the first research question, the initial result (Pre-test results) from
Table 4.1 and appendix C suggest that the overall performance of students in teaching
was very poor which reflected in the performance of pupils as well. It was also realised
from Table 4.1 and appendix C that about 84% of the students obtained marks less than
62
half of the total mark. This is a clear indication that students had difficulty in teaching
Fraction. After the intervention, the evidence gathered from the results (Post-test results)
suggested that incorporating the intervention tool (Cuisenaire Rods) into Mathematics
classroom teaching improved the achievement scores of students, since about 88% of the
students scored half or more marks of the total score. This is shown in the mean value of
32.12 over the mean value of 16.94 by the pre-test scores with a gain score of 15.18
indicating that students’ performance was twice better than when the intervention tool
was not used. These evidences showed that the use of Cuisenaire rods on students have
positive effect since students’ performance improved tremendously as a result of the use
of the intervention tool.
4.3.2 Research Question 2
To what extent would the use of Cuisenaire rods as concrete materials sustain and
motivate students’ interest in teaching and learning of Fractions?
Table 4.3 shows descriptive statistics of pre-test and post-test scores. The
minimum and maximum marks of pre-test and post-test scores are respectively 5, 37 and
20, 48. Comparing the standard deviations of pre-test scores which was 7.175 and posttest scores 6.915 (Table 4.3) revealed that the standard deviation of post-test was less
than that of the pre-test indicating that the scores in the post-test (ie by the use of
intervention tool) were more spread around the mean mark of 32.12 than it was in the
pre-test indicating significant improvement in students performance. This improvement is
as a result of the motivation derived from the use of the intervention tool (Cuisenaire
Rods) thereby sustaining their interest in problems involving Fractions.
63
4.3.3
Research Question 3
Is there any difference in students mean score performance by the use of Cuisenaire Rods
in solving problems involving Fractions?
From Table 4.4, the Paired Sample Test analysis of the data yielded the P-value of
(𝑃 = 0.000) which is less that the level of significance of 0.05. Hence, we reject the null
hypothesis (𝐻𝑜 ) which states “there is no significant difference in students means score
performance by the use of Cuisenaire rods in solving problems involving Fractions”,
accordingly accept the alternative hypothesis(𝐻𝑎 ). We conclude from the results by this
result that there is a significant difference in the mean scores of students in the pre-test
(16.94) and the post-test (32.12).
64
CHAPTER 5
SUMMARY, CONCLUSION AND RECOMMENDATION
5.0 Overview
This chapter summarises the research findings, conclusion and gives recommendation
and suggestions for further research and for curriculum development.
5.1
Summary
The research was conducted to improve second year students of Dambai College
of Education in solving problems involving Fractions using Cuisenaire rods. The data
collected from the pre-test scores and post-test scores were first subjected to descriptive
statistics using Statistical Package for Social Sciences (SPSS). It was realised that before
the intervention, only 16% of the students were able to solve problems involving
fractions satisfactorily. But after teaching them by the use of Cuisenaire Rods through
activities, about 88% of the students were able to solve problems involving fractions
satisfactorily. The main research question the researcher asked was to find out whether
“there is any difference in students’ performance in using Cuisenaire rods in solving
problems involving Fractions”
The process of intervention revealed that there is a significant difference in the mean
achievement scores of students using the Cuisenaire rods to their mean achievement
scores than when Cuisenaire rods were not used. The statistical difference showed that
the intervention tool (Cuisenaire rods) used improved students knowledge in problems
involving fractions. The students now develop more positive attitudes towards fractions
and mathematics as a whole because they were excited as they could easily answer
65
thought provoking problems and reach conclusions once they can manipulate the
materials.
The findings have serious implications for mathematics teaching and learning.
The question about teachers’ pedagogical and content knowledge requirements cannot be
over-emphasised for effective classroom mathematics teaching and learning. This is
because effective lesson presentation requires expert execution of a set of decisions and
actions in the pre-instructional, interactive and post instructional phases of teaching that
depend on the knowledge base of the teacher. At the pre-instructional phase, decisions
about what content to include in lesson presentation and organising the content in a
logical and meaningful manner require extensive content knowledge base with a
repertoire of pedagogical strategies; thereby enabling the students to construct ideas and
make meaning of on their own. The teachers’ pedagogical content knowledge cannot
exclude the issue of language since it is only through language of a kind that any form of
teaching can be possible.
5.2
Conclusion
It is evident from the findings of the study that using Cuisenaire Rods improved
immensely on students’ achievement in Mathematics and Fractions in particular. There
was a common feeling of confidence among students using the Cuisenaire rods. There is
therefore growing evidence in the research conducted by Kurumeh and Achor (2008)
who have found Cuisenaire rods effective in the teaching and learning of Fractions and
other topics in Mathematics.
66
Since Cuisenaire rods are materials recommended in the curriculum of the Colleges of
Education, a deep insight into its use is required so that students can follow the step by
step procedure in its use to teach the topics in Mathematics at the Basic schools.
5.3
Recommendation
The importance of Mathematics cannot be over-emphasised. From the study, it
was useful in helping students through the use of Cuisenaire rods develop a meaningful
understanding in problems involving Fractions. Based on the results gathered from this
study, the researcher has this recommendations and suggestion to make.

Since Cuisenaire rod is one important material that students are required to be
abreast with its usage in teaching mathematics concepts, the researcher
recommends that tutors teaching mathematics in the Colleges of Education must
be familiar with its usage so that they can incorporate these materials in their
teaching processes.

This study as well as the study conducted by Kurumeh and Achor (2008) in
Nigeria suggests that there is significantly positive effect of using Cuisenaire rods
in teaching mathematics. It is therefore recommended that more emphasis should
be laid on its use especially at the Colleges of Education.
67
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74
APPENDICES
APPENDIX A
PRE-TEST ITEMS
1
1
1. Show, step by step, how you would use concrete materials to solve 4 + 2 = ?
2. Describe one way in which you would guide pupils in primary class 4 to
3
6
4
8
determine for themselves that and are equivalent Fractions, using concrete
materials.
2
1
3. Describe an activity you would use to guide pupils in primary class 4 to find 3 + 6
3
4. Describe an activity you would use to guide pupils in primary class 4 to find 5 −
2
.
10
5. Describe briefly how you would use a named concrete material to introduce the
3
Fraction 8 to pupils in primary class 3.
75
APPENDIX B
POST-TEST ITEMS
1
1. Describe an activity you would use to guide pupils in primary class 4 to find 3 +
1
.
5
7
3
2. How will you explain to an upper primary pupil that 8 is greater than 4 using
concrete material.
1
1
2
3. A primary class four (4) pupil does the following 2 + 3 = 5
i. State the child’s mistakes.
ii. Describe how you would use concrete materials to help the pupil to overcome
his/her problem.
2
1
4. Show, step by step, how you would use concrete materials to solve 3 − 4.
3
1
5. Describe an activity you would use to guide pupils in primary class 4 to find 5 − 4
76
APPENDIX C
PRE-TEST SCORES
12
23
11
9
13
16
18
12
19
20
11
16
21
14
17
18
16
14
19
17
25
18
31
28
33
10
15
8
21
9
19
7
37
23
25
12
18
10
14
15
11
20
15
9
26
8
30
5
21
8
77
APPENDIX D
POST-TEST SCORES
26
27
26
29
30
31
30
36
48
29
38
27
31
46
25
43
41
32
40
20
46
34
43
36
37
22
36
20
33
38
30
31
32
25
29
33
30
28
26
32
23
24
33
27
24
30
44
36
39
30
78