CHAPTER ONE
INTRODUCTION
This chapter encompasses the background of the study, statement of the
problem, purpose of the study, research questions, and significance of the
study, limitation, delimitation and organization of the study.
Background of the study
Mathematics is one of the important subjects within the foundation
subjects that constitute the core curriculum for basic (i.e. primary and
secondary) education in countries throughout the world and Ghana is no
exception.
Mathematics occupies a privileged position in the school curriculum
because the ability to cope with it improves ones social advancement. “It has
attained this position since it was made to replace classical languages like
Latin and Greek which prior to the early half of this century were used as
screening devices for entry into higher education and certain professions”
according to (http://wikieducator.org/Review of mathematics in Ghana )
.This emphasize the fact that mathematics is one of the important subjects
which is studied in our basic schools.
Again, mathematics is of a utilitarian value because it is applied in
various aspects such as farming, architecture, music etc (Agyei et al (2013).
In view of this developing pupils competency in the subject by teachers is
very much applauded.
1
Addition and subtraction are basic operations on which a lot of
mathematical concepts are built or formulated. Dexterity in these two
operations enhances pupils’ performance in other mathematical concepts.
The concept of place value will help pupils to perform addition and
subtraction of two digits numbers upward.
In an encounter with the primary six pupils of Offinso State ‘A’ on the
6th of November 2015, the researcher observed that the pupils find it difficult
to add and subtract six and seven digit numbers.
Among factors contributing to addition and subtraction difficulties among
pupils are misconceptions on the part of both pupils and teachers about
Mathematics, teachers failure to use practical oriented approaches in teaching
and lack of interest and motivation.
Also, teacher’s failure to use appropriate teaching and learning materials in
teaching is a domineering factor.
The school of practice of the researcher is located at Offinso Newtown in the
Offinso south district. The school is situated near a zongo community called
Saaboa. Offinso has farming to be predominant profession in the area. Parents
concern on their wards education is quite encouraging with respect to the
enrolment of students in Offinso State A primary but there are some hitches on
parental concern. The unsupportive nature of some parents in terms of failure to
buy Mathematics textbook for their wards is also a contributing factor to pupils’
difficulty in adding and subtracting six and seven digit numbers.
2
It is against this background that the researcher has decided to address
pupils’ difficulty in addition and subtraction of six and seven digit numbers
using the abacus and place value chart.
Statement of the Problem
An encounter with the basic six pupils of Offinso State A primary
uncovered their difficulty in adding and subtracting six and seven digit
numbers. And since addition and subtraction serve as a pre – requisite to
other mathematical concepts, pupils may lose interest in the subject as a
result of poor performance in the topic and further mathematical concepts.
Further inquiry from pupils and teachers by means of interviewing
brought to light the fact that teaching and learning materials were not
incorporated in Mathematical lessons.
Also the researcher found out that pupils’ understanding in the place value
concept is very low. A critical scrutiny into the Mathematics exercise books
of some of the pupils also brought to light that they encountered the same
difficulty in addition and subtraction of five digits numbers. This concretises
the alarming nature of pupils’ low understanding in place value concept.
Basic six pupils are expected to have an average age of eleven (11) or twelve
(12), which means they are at the concrete operational stage according to
Paiget’s theory of mental development. Some characteristics of pupils at this
stage is that they are tied to concrete materials and learn very well when made
3
to interact with concrete materials because they find it difficult to do abstract
reasoning.
Information gathered and the psychologists’ views compelled the researcher
to assist pupils to overcome their difficulty in addition and subtraction of six and
seven digit numbers with the aid of abacus and place value chart.
Purpose of the Study
The purpose of the study is to identify the causes of the difficulty in adding
and subtracting six and seven digit numbers among the basic six pupils of
Offinso State A primary and to design appropriate teaching and learning
materials (abacus and place value chart) to help the basic six pupils of
Offinso State A to overcome their difficulty in addition and subtraction of six
and seven digit numbers.
Research Questions
The following questions were formulated for the study.
1. What are the causes of basic six pupils of Offinso State A primary
difficulty in adding and subtracting six and seven digit numbers?
2. How would the understanding of place value concept help pupils of
Offinso State A primary six to overcome their difficulty in addition
and subtraction of six and seven digit numbers?
4
3. What measures have been put in place to help the basic six pupils of
Offinso State A to overcome their difficulty in adding and subtracting
six and seven digit numbers?
4. How would the use of abacus and place value chart help Offinso State
A primary six to overcome their difficulty in addition and subtraction
of six and seven digit numbers?
Significance of the Study
This study will help the basic six pupils of Offinso State A to overcome
to overcome their difficulty in adding and subtracting six (6) and seven (7)
digit numbers.
Secondly, it will help teachers institute measures to safeguard future
occurrence of such a problem by using teaching and learning materials as
well as child centred approaches in teaching concepts in Mathematics and
other subjects.
Furthermore, this study will help create awareness about the need to give
pupils difficulty in mathematics a systematic study in order to provide
scientific solutions to the problem.
Moreover, this study will assist the government and other stakeholders in
education to find a solution to the problem at stake in our basic schools.
Finally, it will also serve as a reference document to those who in
subsequent times will choose to study the same or similar problem.
5
Limitation
The study could have been done in all the schools in Offinso south
district but due to financial constraints and logistics to carry out extensively,
the researcher was compelled to focus the study on only the basic six pupils
of Offinso State A primary,
Again, the time duration at the researcher’s disposal is very limited,
because the researcher has to combine studying of the two distance courses
(i.e. Guidance and Counselling and Trends in Education) as well as the
teaching practice.
Finally, truancy on the part of some students may affect the credibility of
the research results and findings that may come out because some students in
Offinso State A primary six are habitual truants and researcher seeks to use
half the total population of the class.
Delimitation
Even though the study should have been extended to all the schools in
Offinso municipal to help all those at the basic education level, however due
to limited time as a teacher trainee with limited resources , the study was
confined to only 25 pupils in Offinso State A primary six in order to get
ample time for the research.
here are many areas of learners’ difficulty in Mathematics but this
research work concentrated on pupils’ difficulty in solving problems
involving addition and subtraction of six and seven digit numbers.
6
Finally, the researcher confined himself to using abacus and place value
chart to help pupils to overcome their difficulty in adding and subtracting six
and seven digit numbers though there are a lot of teaching and learning
materials which could have been used to help pupils to overcome their
difficulty.
Organisation of the Study
This work is made up of five chapters. The chapter one basically puts the
study into perspective and it comprises of the background of the study,
statement of the problem, purpose of the study, research questions, and
significance of the study, limitation, delimitation and organization of the
study.
The chapter two deals with the review of related literature on the study, it
talks about the views of others which are relevant and related to the study in
books, journals and other sources.
The methodology aspect of this research is captured in the third chapter. It
encompasses the research design, population, sample selection techniques
well as the research instruments used.
The fourth chapter presents the results of the study as well as discussion
of the study and analysis and data collected are expatiated in this chapter.
Finally, the fifth chapter gives an overview or summary of the study.
Findings and conclusions drawn from the study are also highlighted on and
recommendations and suggestions for further studies are presented.
7
CHAPTER TWO
LITERATURE REVIEW
Introduction
This chapter reviews relevant literature on the study. It examines the
views of authors that have relevance to the subject of study from both
theoretical and empirical perspectives. Relevant literatures were reviewed
under the following sub-headings:
a. What is Mathematics?
b. Importance of Mathematics.
c. How children learn Mathematics and their educational implications.
d. The use of teaching and learning Materials.
e. The place value concept.
What is Mathematics?
Mathematics is the abstract science of number, quantity and space either
as an abstract concept (pure mathematics) or as applied to other disciplines such
as physics and engineering (www.google.com/definition_of_mathematics
)
Oxford English Dictionary also defines Mathematics as the abstract
science which investigates deductively conclusions implicit in the
elementary conceptions of spatial and numerical relations and which includes
as its main divisions geometry, arithmetic and algebra.
8
Encyclopaedia Britannica also defines Mathematics as the science of
structure, order and relation that has evolved from elemental practices of
counting, measuring and describing shapes of objects.
Mereku (1999) opined in his speech delivered at the 6 th biennial delegates
conference of the Mathematical Association of Ghana that ‘Mathematics is a
language and provides a means of communication because it makes use of
symbolic relations which is similar across continents, abstract ideas and
concepts.’
Charles Darwin poetically describes a mathematician as a blind man in a
darkroom
looking
for
a
black
cat
which
(http://en.wikipedia.org/wiki/definitions_of_mathematics
isn’t
).
there
Darwin’s
description throws more light on the abstract nature of Mathematics.
Eugene Wenger metaphorically defines Mathematics as the skilful
science of operations which concepts and rules are invented just for this
purpose (skilful operation).
From the above definitions it can be deduced that Mathematics is purely
a science and it is well structured with sub-branches (i.e. geometry,
arithmetic and algebra).
Geometry is the visual study of shapes, size, and patterns.
Arithmetic is the branch of Mathematics which deals with the properties
of the counting numbers and fractions and the basic operations applied to
these numbers (http://en.wikipedia.org/branches_of_mathematics ).
9
Algebra is the part of mathematics in which letters and other general
symbols are used to represent numbers and quantities in formulae and
equations.
Importance of Mathematics
‘When will I use Mathematics?’ is a question often posed by students
wondering how topics like factorization and algebra will play a role in their
daily lives. However often without realising it, we use it in our day to day
activities like making purchases, tracking cell phone’s minutes and many
more.
Taylor (2013) in his book ‘How children learn Mathematics’ also asserted
that ‘Mathematics equips pupils with tools which are unique and powerful to
understand and change the world. These tools include logical reasoning,
problem solving skills and abstract thinking.
Asafo .A (2002) also stated that mathematics is a service subject applied
in physical sciences, social sciences and other related areas and it also serve
as the basis for modern technology and scientific development.
Again, mathematics is a compulsory subject for pursuing higher
education. Failure to attain a credit or higher in this subject creates the
problem of advancing into any tertiary institution for students.
According to Petty (2001) generic skills are developed through the study of
mathematics and these skills can be applied to other areas. Generic skills are
10
developed when young children are introduced to pre-number work. Some of
these generic skills are:





Observation

Inferring

Sequencing

Comparing

Sorting etc.(www.geofpetty.com )
Furthermore, physician who has to study biological cells and bacilli need
to have knowledge of mathematics if he means to reduce the margin of error
which alone can make his diagnosis dependable.
To the mechanic and the engineer, it is a constant guide and help and
without exact knowledge of mathematics, they cannot proceed one step in
coming to grips with any complicated problem.
Also, the habit of accuracy and exactitude are developed and it prevents
man from being careless and slipshod. Mental alertness is increased as well
as sharpening the reasoning powers of man,
These outlined importance of mathematics unveil some hidden importance
of mathematics and it also emphasise on some practical experiences that
need mathematical knowledge before one can go through them.
11
How children learn Mathematics
Collins Co Build Advanced Dictionary defines learning as the process of
gaining knowledge through studying
Learning is also defined as the process of gaining knowledge through
reinforced practice.
A number of theorists have proposed ideas about how children learn
generally and these ideas can also be related to the learning of mathematics.
Piaget believes that children construct their own knowledge and
understanding through their interactions with their environment. This is
called a constructivists theory.
Vygotsky (in Atherton 2011) also known as a social constructivist
emphasised the need for a child to have guidance from ‘a more
knowledgeable other’ and also to be given opportunities to interact socially
with peers as a means of learning (Taylor 2013).
Mathematical learning is associated with mathematical understanding.
Barmby et al (2009) see this as a continuum where children ad to and refine
previous understanding. This is built on the work of Bruner (1966) who
identified the idea of spiral curriculum where children meet an idea at one
level and then later meet the idea again but are able to study it at a deeper
level and achieve better understanding of it.
Bruner again asserted that children go through three phases when learning.
The phases are enactive, iconic and symbolic. The three phases concretise
the pedagogic axiom ‘concrete to abstract’.
12
Enactive phase is when a child engages in something concrete in order to
explore and manipulate ideas; this could be related to kinaesthetic learning.
Therefore at this stage pupils are made to interact with concrete materials
when learning.
Iconic phase is where the child creates mental images to represent
concepts. This can be supported in mathematics by using models and images
so that the child eventually visualise some of them internally to assist their
thinking
The last phase which is known as the symbolic phase is where the child
uses abstract ideas as ways of representing mathematics.
Liebeck (1981) also in his book ‘How children learn Mathematics’ also
postulated that children develop their abstract thought in mathematics by first
exploring with physical objects such as toys etc. It is not long before he
recognises words to represent them. Later the child will recognise pictures of
them and written symbols are then associated with them.
Liebeck (1981) again categorise the sequence by which children learn
mathematics with an acronym ELPS which expatiated below.



 E – Experience with physical objects
 L- Spoken language that describes experience
 P- Pictures that represent the experience
 S- Written symbols that generalise the experience
Skemp (1971) described two ways of understanding mathematical ideas
that he called relational and instrumental understanding.
13
Instrumental is the shallower form of understanding. For instance we might
develop an understanding on how to add, subtract, multiply and divide using
set of procedures or algorithm by memorising steps required. However we
may not understand how the procedure worked. One major flaw of
instrumental understanding is that if our memory of procedure fails, we
would be unable to continue.
In contrast a relational understanding of these procedures would mean that
we understand how and why the procedure worked.
The work of Piaget, Bruner and Liebeck emphasise practical activity as a
starting point for learning with young pupils. Gifford (2008) also reports
neuroscientific support for this approach too.
Again, research indicates that desposition is very important in the longterm learning of mathematics (Renga & Dalla 1993). Desposition concerns
more than attitudes towards mathematics alone: persistence, risk taking,
hypothesis making and self regulation are important to a motivated
desposition (Coppley 2010).
Gayne R and Skinner B.F propounded the behaviourist theory of learning.
They hold that learning takes place through stimulus-response (S-R)
mechanism and it is the process through which the child obtains certain
desirable behaviours. They also hold the view that the child’s mind is termed
to be a tabula rasa (i.e. the child’s mind is thought to be an empty spot) and it
is the responsibility of the teacher to fill it with knowledge
14
From the above theoretical framework derived from psychologists with
different school of thoughts, the following implications were deduced the
classroom teacher in managing mathematical lessons.
Firstly, relevant teaching and learning materials should be incorporated in
lesson delivery by teachers because of young pupils’ tendency to explore and
play with familiar materials. The enactive phase and concrete operational
stage of Bruner and Piaget’s theories respectively concretise this fact.
Secondly, relational understanding on the part of pupils should be a major
concern to teachers. Pupils should be taught to understand how and why
procedures worked and do away with drill-oriented teaching. Skemp’s
relational and instrumental learning theory buttresses this point.
Again, teachers should make the classroom environment simulative enough
by giving all pupils equal chance to partake in mathematical lessons. The
stimulus-response (S-R) as asserted by the Behaviourists supports this point.
Moreover, teachers’ mastery in the subject matter content is very expedient
to a successful lesson delivery. Vygostsky assertion that children are to be
guided by ‘a more knowledgeable other’ provides the basis for this
implication. Opportunities to interact with peers as means of learning should
be taken into consideration by grouping pupils in some mathematical lessons.
Finally, motivation as means of reinforcement should be used by teachers
during mathematics lessons. Not neglecting the principle of individual
differences on the part of pupils because it’s on this basis that you can group
pupils in case of applying any type of grouping and also assessment.
15
The use of Teaching and Learning Materials
Lowe (1993) stated in his book ‘successful instructional diagrams’ that a
picture is worth a million words when used in the period of instruction. This
shows how effective lessons will be when teaching and learning materials are
used.
Lowe (1993) further explained that for instructions to be judged
successfully, the students must be able to produce the new or improved
performance as required. So, if diagrams or teaching and learning materials
are used in lessons, they help students to:


Recall the knowledge and skills they have been taught.

Understand the material that is being taught.
Handbook on lesson notes preparation and teaching and learning
materials in primary schools (2003) by Ghana education service states that
teachers who rely on oral presentation of lessons find the pupils frequently
unable to understand concepts effectively.
There are also visual, auditory and kinaesthetic learners. The visual
learner say ‘i see’ to mean ‘i understand’ and respond best to task involving
demonstrations or looking at illustrations and diagrams. The auditory leaner
also prefers direct instruction hence performs better when they are used
frequently
The kinaesthetic learner on the other hand prefers direct involvement
through games, role playing and interacting with materials in the classroom.
Deductions from how children learn mathematics young pupils are mostly
16
kinaesthetic learners, therefore it is very prudent for teachers to use
instructional materials in lesson delivery.
Young children are tied to concrete materials according to the Piagetian
theory and it is also backed by Bruner’s theory on how children learn
mathematics. These theorists’ postulations back the idea of young children
learning kinaesthetically and there comes in teachers’ incorporation of
teaching and learning materials in lessons.
Akinyemi also classified teaching and learning materials under six
categories.
1. Printed materials- textbooks
2. Graphic materials- posters and diagrams
3. Display materials- overhead projector, video
4. Audio materials- radio, tape recorder
5. Microfilm etc
6. Miscellaneous materials- regalia, exhibit
Lowe (1993) again stated that not all instructional diagrams found in
teaching materials are equally successful in promoting desired learning
outcomes. This is because when a diagram is poorly designed they actual
hinder than help learning.
Lowe (1993) then made an assertion that even well designed instructional
materials may be ineffective if the student is given insufficient support on how
to use them. So perhaps the adage ‘a picture is worth a million words when used
in the period of instruction’ can be rewritten as ‘a diagram is
17
worth a million words provided it has been well designed and sufficiently
supported.’
Pupils’ involvements in the collection of instructional materials are also
important because it helps them to be familiar with the environment
according to (Methods of teaching science, tutors notes).
Some examples base ten materials which are used in the teaching of
addition and subtraction of two digit numbers upwards are;
I.
Abacus
II. Dienes material (multibase block)
III. Bundles of sticks and loose ones
IV.
Place value chart etc
The place value concept
According to Swenson (1973), place value is the value of a digit in a
numeral derived from its place or position in the numeral. She illustrated that
in our modern notion, each of the (2’s) two in 2222 is a different value
depending on its position among the other digits
She again explained the place of 2 in the extreme left-hand position as
thousand times the 2 in the extreme right-hand position. She again pointed
out that the idea of place value was used in written records such as those of
ancient Babylonians and Mayons and in mechanical computing like the
Abacus.
18
Crowder and Wheeler (1968) also described the concept of place value as
the value of each symbol which is determined by its position in the numeral.
They continued to suggest that the concept of place value and positional
notation means that the symbols can be repeated within a number and the
value of each number is determined by its position its position in the
symbolised number.
Crowder and Wheeler (1968) further stated that because of place value
concept, the Hindu and Arabic decimal system needed on only ten different
symbols to represent any number. They illustrated that basic number has 2
values by saying that the numeral ‘333’ means 3 ones 3 tens 3 hundreds or in
expanded form which is (300) + (30) + (3).
According to Kramer (1971) children should be taught place value with
the use of play activities for better understanding. He again emphasised that
base ten materials like Abacus, place value chart, Dienes materials etc should
be used to help pupils to understand this concepts
From the preceding, most writers agree on a sequential approach to the
teaching of place value concept while many activities are involved.
Regrouping, grouping and conversion should be inclusive to facilitate the
skill of carrying over one value to the next step as in doing addition and
subtraction.
19
Summary
From the above reviewed literature, it is obvious that Mathematics is very
essential considering some of the importance captured. The theoretical
framework made with postulations and assertions by some psychologists
encourages the use of relevant concrete materials when teaching young
people mathematics.
Finally, pupils’ mastery in place value concept is very expedient in their
comprehension of addition and subtraction of two digit numbers upwards.
20
CHAPTER THREE
METHODOLOGY
Introduction
This chapter discusses the methods used to gather data on the topic
under study. It talks about the research design, population and sample
selection, research instruments, pre-intervention data collection/analysis,
intervention, post-intervention data collection and data analysis plan.
Research Design
The design used in the study was action research.
Action research is the study carried out in course of an activity or
occupation, typically in the field of education to improve the methods and
the approaches of those involved.
Wikipedia encyclopaedia also defines action research as the research
initiated to solve an immediate (en.wikipedia.org/wiki/Action Research).
Action research has some strength and weaknesses.
Kerry Dyke in the chapter three of his manual “Action Research” wrote
that action research helps educators to use data rather than hitches to guide
the improvement of efforts. This makes the process of action research more
scientific in nature proposing ideas and theories that can be backup by data.
Action research gives teachers something more concrete to work with
instead of just relying on the principles the teachers have used in the past. It
21
also helps to address the quality of student’s education and the progressive
growth of teachers.
In spite of the numerous advantages, action research is having several
lapses. One major disadvantage of action research is that, it is much harder to
write up because you probably can’t use a standard format to report your
findings effectively.
Moreover the cyclic nature of action research to achieve its twin
outcomes of action and research is time consuming and complex to conduct.
Population
Population is defined as a group of people that conforms to a specific
criterion and to which a researcher intends to generalise the results of the
study.
A population which constitutes the target population or the group under
study is the basic six pupils of Offinso State A primary and the teachers in
the school.
There are 850 students in the school consisting of 420 boys and 430
girls. There are also 22 permanent teachers in the school of which some
were randomly selected and interviewed on these bases:



The poor performance of pupils in mathematics.

Method suitable for teaching mathematics.

The failure of teachers to use TLM’s.
22
A sample size of 25 pupils was selected from 50 students in Offinso
State A primary six. Among the 25 students were 13 girls and 12 boys
and 10 teachers were also randomly selected from the school to be
interviewed.
Sampling Procedure
In as much as the researcher wishes to work with the entire target
population and teachers, he could not do so because it could have been
extremely difficult to handle all pupils and teachers.
Simple random sampling was selected and used in the selection of
pupils from the target population i.e. basic six pupils of Offinso State A
primary. The respondents were selected to ensure fair results. Pieces of
papers with numbers written on them from 1-50 were put in a container. All
the 50 pupils who form the target population allowed to pick one slip of
paper at a time. All the pupils who picked slips of papers with even numbers
were selected, which constituted a total of 13 females and 12 males.
Research Instruments
Research instruments are testing devices for measuring a given
phenomenon, such as paper and pencil test, a questionnaire, an interview or
set of guidelines for observation. After the researcher had made a
consideration of factors that determine the appropriateness of the instruments
for research, observation, interview and tests were found appropriate to be
used to gather relevant data for the study.
Observation
23
It involves watching and listening to the subjects and recording what he
sees rather than asking questions. The researcher adopted the participant type
of observation so that he could get information which may not be accessible
anyway. But it may lose its relevance if the researcher becomes emotionally
involved and for that matter the researcher may lose its objectivity, this is the
major weakness attached to observation.
In course of teaching, the researcher observed that pupils were having
some difficulties in the addition and subtraction of six and seven digit
numbers on the part of the basic six pupils of Offinso State A primary. The
pupils were having those difficulties because thorough observation brought
to light the fact that in previous years the teachers failed to incorporate
TLM’s in their lessons. Pupils’ dexterity in place value concept was also
very low and it is also a domineering factor of the pupils’ difficulty.
Observation
Interview involves verbal questioning by the researcher to the respondents
either in face to face situation or by phone. The researcher adopted the structured
form of interview to collect relevant data to the study. Interview gives high
response rate and also gives the opportunity to observe the non verbal behaviour
of the respondents but it is very costly and time consuming.
An interview guide was prepared by the researcher for both teachers
and students and their responses were recorded (refer to appendix A and B).
Tests
24
The researcher used tests to diagnose the extent of the problem i.e. pretest and also to determine the effectiveness of his intervention procedures i.e
post-test. Each test consists of 5 questions which a correct answer is scored 2
marks and 0 for a wrong answer. (refer to appendix C and D).
Data Collection Procedure
The data collected followed a weekly planned research activity. The
researcher carried out the research in three periods thus sixty minutes each
for four weeks. The first week was used for pre-intervention and the second
and third weeks were also used for the intervention and the fourth week was
used for post-intervention.
Pre-intervention
Week one (1)
During the first week observation was made by the researcher during one
mathematics lesson. Pupils difficulty in addition and subtraction of six (6)
and seven digit (7) numbers was revealed because it was the topic slated to
be taught that very week. Pupils couldn’t solve simple problems involving
addition and subtraction of six and seven digit numbers because their
understanding of the place value concept was very low. These were some of
the ways they approached the questions given to them.
645871
879604
+ 459610
- 754821
109151481
125223
25
A pre-test was administered during the second mathematics lesson of
the week. The researcher conducted the pre-test to find out the actual
problem faced by the pupils. The pre-test consisted of 5 questions from the
pupils’ text book. The test elapsed for 45 minutes. Refer to appendix C
During the third mathematics lesson for the week, the researcher
conducted an interview using an interview guide. All the respondents
selected from the target population were interviewed. The researcher
conducted the interview to find out why the pupils were having difficulties in
the addition and subtraction of six and seven digit numbers.
The researcher again, used free periods during the first week to interview
some teachers using an interview guide prepared for the teachers. Ten
teachers were selected and interviewed .the researcher wanted to know
everything about the pupils’ difficulty that is why he tried gathering data on
the pupils problem on the first week.
Intervention
The researcher conducted the intervention using two weeks i.e. the second
and third week. The intervention was conducted in reference to pupils’
inability to add and subtract six and seven digit numbers as revealed by their
response during the pre-test and the interview held on the first week.
Week two (2)
The researcher used the abacus during the whole week. Three days
lesson was conducted during the week.
26
In the first lesson, the researcher introduced the abacus to the pupils. The
researcher explained to the pupils that an abacus is a frame containing rods
with small balls that slide alongside the rods. It is used as a tool for counting.
The pupils were asked to later describe the abacus after they have been
grouped to critically examine the abacus. Most of the pupils were able to
describe the abacus as a wooden frame with small balls used for counting.
Below is the picture of an abacus.
Figure 1.
The second day’s lesson was used by the researcher to demonstrate an
example by using three seven spiked abacuses to teach addition and
subtraction of six and seven digit numbers.
During the lesson, the researcher stressed on the collection of ones, tens,
hundreds, thousand and millions in ascending order. The process of exchanging
ones for tens, tens for hundreds, hundreds for thousands was demonstrated to the
pupils. The researcher then demonstrated to pupils how the abacus is used for
adding six and seven digit numbers . Figure 2 below shows how the researcher
demonstrated an addition operation with the abacus.
Example
5342656
+4483129
27
Figure 2
The last day’s lesson was also used by the researcher to teach subtraction
of six and seven digit numbers using an abacus. The pupils were taught that
the difference between two numbers is found by subtracting a smaller
number (subtrahend) from a larger number (minuends).
Example:
7 8 5 6, 9 7 4
- 2 4 3 4, 5 2 3
28
Figure 3
All the three days lessons taught in the week was based on a detailed lesson
plan prepared by the researcher. Refer to appendix E
Week three (3)
During the third week of intervention the pupils were taken through
addition and subtraction of six and seven digit numbers using the place value
chart.
The three days lessons used for the intervention lasted for sixty
minutes and a detailed lesson plan was again prepared to guide the researcher
in his intervention.
In day one’s lesson, the researcher introduced the pupils to the place
value chart where it was explained to pupils how it is used. The pupils were
told that materials including cardboards, pencils, pens, felt pens, erasers and
ruler are used in preparing the place value chart.
29
The researcher further explained that with the use of a ruler and felt pen
the positions on the place value chart were marked and drawn from positions
representing ones to millions.
Example of the amount in the chart (i.e 5 milliinton,793 thousand,6
hundred and 12; five million, seven hundred and ninety three thousand, six
hundred and twelve.
Millions
1,000,000
5
Hundred
Ten
Thousands Hundreds
thousand
thousand
100,000
10,000
1,000
100
7
9
3
6
Tens
Ones
10
1
1
During the second day’s lesson, the researcher used the place value chart
to teach addition of six and seven digit numbers. The researcher involved the
pupils in an addition game involving (0-20), before introducing them into the
use of place value chart to add six and seven digit numbers. The game is
played by two pupils at a time. One starts from zero and add 1,2 or 3 . The
addition alternates between the players up to the one who gets 20 first and he
or she is declared the winner. The game is aimed at helping pupils to develop
problem-solving strategies.
The pupils to solve the question below using the place value chart.
Example: Add 4,967,547 and 3,387,686
30
2
Figure 5
Place value chart
M
H-Th T-Th
Th
H T
4
9
7
5
4
7
3
6
O
3
8
7
6
8
6
8
13
15
15
12 13
13
8
3
5
5
2
3
3
During the last days lesson of the third week the researcher then
guided the pupils to solve questions on subtraction of six and seven digit
numbers using the place value chart. It was explained to pupils that
subtraction of numbers can be illustrated by the removal of a number
from a group of objects.
Example :
733526
- 375863
The above question can be solved with the place value chart as shown in
figure 6.
31
Figure 6.
Place Value Chart
H-th
T-th
Th
H
T
O
7
3
3
5
2
6
3
7
5
8
6
3
3
5
7
6
6
3
_
Post Intervention
The fourth week was used to administer a post-test to find out the
effectiveness of the researchers intervention procedure. The post- test
consisted of 5 questions which was scored 2 marks each. Forty minutes
was used to conduct the test and twenty minutes was used for making
corrections and marking of the test.
Data Analysis Procedure
The research findings were based on the analysis of the pre-test and
post-test as well as answers the teachers and pupils provided to the
interview guide .the findings were analysed using tables and descriptive
statistics such as frequencies and percentages together with charts
32
CHAPTER FOUR
RESULTS, FINDINGS AND DISCUSSIONS
Introduction
This chapter presents the results of the study and discussion of findings.
The results were the outcome of pupils’ performance in the pre-test and posttest. It further deals with the analysis of findings made through interviews
administered to some selected teachers and students. The researcher used
tables and bar charts to analyse the pre-test and post-test findings and only
tables for the findings from the interview.
Summary 1
Table 1 shows the results of the pre-test scores
Table 1
Marks
Frequency
Percentage
∑fx
0
6
24
0
2
9
36
18
4
8
32
32
6
2
8
16
8
0
0
0
10
0
0
0
Total
∑f=25
100
∑fx=66
33
Mean=∑∑ =
Mean= 2.64
The data is further explained with the bar chart below
A bar chart showing the pre-test scores
of pupils
10
frequency
8
6
frequenc
4
2
0
0
2
4
6
8
10
marks
Figure 4.1 A bar graph showing the pre-test scores of pupils
Data analysis of pre-test scores
From table 1 and figure 4.1 i.e. the bar graph, it was observed that out of
the twenty five students (25) selected for the study, six (6) of the
representing 24% scored zero (0). Nine (9) students representing 36% scored
two (2) marks, eight (8) pupils representing 32% scored four marks each and
two pupils (2) representing 8% scored six (6) marks each.
The pre-test results again produced a mean of 2.64. this shows that
averagely their performance was very low and shows pupils disability in
adding and subtracting six and seven digit numbers.
34
Summary 2
The table below shows pupils post-test scores
Table 2
Marks
Frequency
Percentage
∑fx
0
0
0
0
2
0
0
0
4
0
0
0
6
8
32
48
8
10
40
80
10
7
28
70
Total
∑f=25
100
∑fx=198
Mean=∑∑ =
Mean= 7.92
The data is further explained by the bar graph below
35
A bar chart showing the post-test
scores of pupils
12
frequency
10
8
6
frequenc
4
2
0
0
2
4
6
8
10
marks
Figure 4.2. A bar graph showing the post-test results
Data analysis of post-test results
From table 2 and figure 4.2, it can be observed that out of the twenty-five
(25) students, eight pupils (8) representing 32% scored six (6) marks, ten
(10) students eight (8) marks representing 40 % and seven (7) students
scored ten (10) marks representing 28 %.
The post-test results also produced an average of 7.9 as compared to 2.64
of the pre-test scores. This implies that intervention in the form of abacus
and place value chart very successful hence contributing to the massive
improvement of pupils results.
36
Interview
The data below are the results of the structured interview administered by
the researcher. The researcher used an interview guide for both teachers and
pupils (refer to Appendix A and B respectively). Though the responses were
open ended but the researcher made efforts to categorise them.
Table 3
Table 3 shows the responses teachers gave to the interview question one
meant for teachers (refer to appendix A).
Why do pupils perform poorly in mathematics?
Responses
Respondents
Percentage
Bad perception about mathematics
2
20
Lack of access to maths books
3
30
Pupils background
3
30
Poor teaching methods
2
20
Total
10
100
Table 3
The table above displays information from some selected teachers on their
responses to the interview question, why do pupils perform poorly in
mathematics? Out of the ten teachers selected to be interviewed, two (2)
respondents representing 20% pointed out that the bad perception pupils have
about mathematics contributes to their poor performance in mathematics. Three
(3) respondents representing 30% also talked about pupil’s background being a
domineering factor in their low performance in mathematics. Another
37
three (3) representing 30% also said lack of access to mathematics books
contributes to their poor performance in mathematics.
Lastly, two (2) teachers representing 20% also responded that poor
teaching methods on the part of teachers also contribute to their abysmal
performance in mathematics.
Table 4
The table below shows teachers responses to the interview question two
for teachers (refer to appendix A).
What method is suitable for the teaching of mathematics?
Responses
Respondents
Percentage
Activity method
7
20
Demonstration
3
20
Total
10
100
Table 4
From table 4 seven respondents representing 70% pointed out that Activity
method is the most suitable method for the teaching of mathematics. Three
(3) others representing 30% also said demonstration is suitable for the
teaching of mathematics
Table 5
The table 5 below shows the responses teachers gave to the interview
question three (refer to appendix A).
38
Why do teachers fail to use TLM’s?
Responses
Respondents
Percentage
Not necessary
1
10
Lack of support from the government
9
90
Total
10
100
Table 5
From table 5 only one respondent representing 10% said that it is not
necessary to use TLM’s in lesson delivery, nine respondents representing
90% also said lack of support from the government has resulted to the failure
of teachers to use TLM‘s.
Table 6
This table shows the responses students gave to the first interview
question for students (refer to appendix B).
Do you have access to mathematics books both at home and in school?
Responses
Respondents
Percentage
Sometimes
5
20
Not at all
15
60
Very often
5
20
Total
25
100
Table 6
From table 6 five (5) students representing 20% said that they sometimes
get access to mathematic books at home and in school. Fifteen (15) others
representing 60% responded that they don’t get access to mathematics books.
39
Five students representing 20% also said they always get access to
mathematics books.
Table 7
This table shows the responses students gave to the interview question
two meant for students (refer to appendix B).
Are you given the opportunity to interact with the TLM’s teachers bring to
the class?
Responses
Respondents
Percentage
Sometimes
1
4
Not at all
3
12
Very often
21
84
Total
25
100
Table 4.7
From table 7 it can be deduced that pupils are not always allowed to
interact with the TLM’s teachers bring to class. Because one (1) student
representing 4% said he always have access to the materials teachers bring to
class for instruction, three other representing 12% also responded that they
are sometimes made to interact with the materials teachers bring to class for
instruction. Twenty-one (21) pupils representing 84% also said they don’t get
the chance to interact with the instructional materials teachers bring to class.
40
Conclusion
From the above tables, findings and discussions, it is noted that a whole
lot of problems was associated to the inability of the basic six pupils of
Offinso State A primary to add and subtract six and seven digit numbers.
The researcher took into consideration all these problems and designed a
suitable intervention in a form place value chart and abacus and used them. It
is worth knowing that the study to a large extent successful and the post-test
results of the basic six pupils of Offinso State A is also evident to the success
of the intervention
Hence, there have been remarkable improvements in the pupils’
performance.
41
CHAPTER FIVE
SUMMARY, CONCLUSIONS AND RECOMMENDATIONS
Introduction
This chapter concludes the study by presenting a summary of the research
findings, conclusions, recommendations and suggestions for further research
Summary
The research unveiled and examined the causes of the basic six pupils of
Offinso State A primary difficulty in adding and subtracting six and seven
digit numbers. After unveiling the causes, how to improve upon pupils’
performance was a key goal. Finally the study was to find out whether the
abacus and place value chart used as intervention to teach the basic six pupils
of Offinso State A primary school yielded any gains in pupils performance in
the post- test.
The study consist of twenty-five (25) pupils selected from the fifty (50)
pupils in basic six of Offinso State A primary school.
The instruments used were tests, observations and structured interview.
Responses of the twenty-five (25) pupils and ten (10) ten teachers from the
interview were analysed with tables and percentages
The results of the pre-test showed that pupils indeed faced problem with
addition and subtraction of six and seven digit numbers based on the scores
they had.
42
However analysis of numerous data collected revealed the following as
the causes of pupils’ difficulty.
Firstly, it was found that the pupils were having bad perception about
mathematics and this impeded their understanding of basic mathematical
concepts
Secondly, most of the pupils were not having access to mathematics
books. This implied that they don’t have any textbook to dwell on for
practice after lessons.
Thirdly, it was revealed that pupils’ background was also expedient in
their poor performance in mathematics. Most of the parents of the pupils
were farmers since the school is situated at an area which has farming to be a
predominant profession. Their interest on their wards education wasn’t
encouraging; sometimes they even take their wards to farm during schooling
days which made their wards to miss some mathematics lessons.
Again, poor teaching methods on the part of some teachers were also
noticed. Though most teachers accepted the fact that activity method was the
most suitable method for the teaching of mathematics yet very few of them
applied it. Lack of teaching aids for mathematics lesson also resulted to
pupils’ abysmal performance in the pre-test and their performance before
intervention as a whole. Most teachers attributed it to lack of support from
the government. The very few teachers that made efforts to send teaching
aids to class also denied the pupils from interacting with them.
43
After the causes were obtained a painstaking effort was made to curb the
situation. Therefore the researcher designed an abacus and place value chart
as intervention to curb the situation.
The intervention procedures put in place brought light the following
findings

Teaching and learning materials make lessons interesting and
practical


Pupils understand concepts better when made to interact with
TLM’s and this affirms Piaget’s assertion that children are
tied to concrete materials


Activity oriented lessons ensures the maximum participation
of all pupils


There
was
a
considerable
improvement
in
pupils’
performance in the post-test. The post-test produced an
average an average of 7.92 as compared to 2.64 of the pre-test
Conclusion
The research based on the totality of action research concludes that the
use of TLM’s i.e. abacus and place value chart has helped the basic six
pupils of Offinso State A primary school to overcome their difficulty in
adding and subtracting six and seven digit numbers
To add to the above strategies, teachers should adopt different approaches
and techniques during lesson delivery. Therefore parents, teachers,
44
stakeholders and beneficiaries of education should help find appropriate
means to help pupils improve in mathematics since it is an indispensible
subject in the curriculum.
Recommendations
Despite efforts made by the researcher to achieve the purpose of the
study, there is still the demand for recommendation.
Firstly, the researcher recommends that child centred methods should be
adopted and used by all teachers in the teaching of mathematics and other
subjects in the curriculum. This will ensure effective participation of pupils
in the lesson.
Secondly, the Ministry of Education and Ghana Education Service should
provide enough mathematics textbooks so that pupils will also have them as
reference materials after lessons in the classroom. This will improve their
practice and further improve their performance.
Again, the Ministry of Education and other stakeholders should provide
some basic teaching and learning aids used in teaching mathematics so as to
help make lesson delivery easier for teachers and also aid easy understanding
on the part of pupils.
Moreover, parents should assist their wards at home and also buy
mathematics textbook for them so as to improve their performance.
45
Finally, the Ministry of Education, school authorities as well as NGO’s
should institute special packages for pupils who excel in mathematics as a
means of motivation. It is hoped that it will raise the morale and interest of
pupils in mathematics.
Suggestion for further research
In every human activity, there is no perfection since we are bound to
make mistakes. It is based on this that the researcher wishes to suggest the
following modifications for further researchers who will undertake similar
research in order to enhance efficiency.
The researcher suggests that other researchers should try different
intervention procedures such as using multi base block, number tracks,
bundles of sticks and loose ones to teach addition and subtraction of six and
seven digit numbers
Also, it is suggested that the researcher should look at multiplication and
division of numbers to make teaching and learning of mathematics worthwhile
46
REFERENCE
Asafo Adjei (2002); Teaching Basic Mathematics for Training Colleges
(Methodology), USA; Global Journals Inc.
Barmby P, Lynn B, Tony H & Higgins S (1990); Primary Mathematics
Teaching for Understanding: maidenhead; Open Head press.
Brunner J (1966) ; Towards the theory of Instruction; Cambridge mass ;
Belknap press Harvard University.
Collins CoBuild Advanced dictionary (2009); Henle cengage Learning.
Coppley J.V (2010); our young Children and Mathematics; Virginia:
National Association of Education of Young children
Crowder & Wheeler (1968), Teaching Elementary School Mathematics:
George overhead limited’
Chen J & Weiland L (2007), Helping young children learn Mathematics,
(digital paper), Available on children exchange website:
www.children exchange.com
Encyclopedia Britanica (2008), 15th edition; Encyclopedia Britanica Inc
(official website).
En.wikipedia,org/Action Research.
http://wikieducator.org/ review of mathematics in Ghana.
http://en.wikipedia.org/wiki/definition of mathematics.
http://en.wikipedia.org/wiki/branches of mathematics.
Kramer (1971), Teaching elementary school Mathematics,
Ohio; Claire parkyns publication.
47
Kerry Dyke (2012), Action research The Manual; a journal to be a guide and
Understanding of Action research (available online), www.mun.ca/ chapter 3
Kilpatrick, J., Swafford, J., Findell, B., & National Academy of Sciences
National Research Council, W. N. (2001). Adding it up: Helping children
learn mathematics; Washington DC :national academy press
Lowe R (1993), Successful instructional diagram, London:
Kopan page limited
Liebeck P, (1981), How children learn Mathematics a guide to
Parents and teachers
Mereku K.D (1999), School Matthematics in Ghana 1960 -2000, a paper
Delivered at the 6th biennial delegates conference of mathematical
association of Ghana (MAG)
Petty R (2001), Teaching generic skills, a journal of using assessment to raise
achievement in mathematics, available online at www.geofpetty.com
Renga R and Delta C (1993), Effects of Mathematics: London, pinguin books
Skemp R (1971), Psychology of learning Mathematics, New York;
pinguin books
Swenson T (1973), Teaching Mathematics to children ,New York :
Oxford university press
Vygostsky L.S (1986), Thought and language , Cambridge MA; MIT press
Zoltan D (1971), 4th edition, Building up Mathematics, London UK;
Hutchinson educational limited
48
APPENDICES
APPENDIX A
Interview guide for teachers
1. Why do pupils perform poorly in mathematics?
2. What method is suitable for teaching mathematics?
3. Why do teachers fail to use TLM’s?
49
APPENDIX B
Interview guide for pupils
1. Do you get access to mathematics books both at home and in school?
2. Are you given the opportunity to interact with the TLM’s teachers
bring to the classroom?
50
APPENDIX C
Pre-test items
Solve the following questions
1.
243,936
2.
+186,432
2,147,865
3. 3,466,609
+ 1,489,347
_ 2,386,677
4. 4,149,903
5.
891,142
_
+
364,686
1,387,646
51
APPENDIX D
Post-test items
Solve the following questions using the abacus or place value chart
1.
879,604
2. 645,871
3. 4,103,827
_
754,821
+459,610
+2,398.758
4. 6,786,785
5. 2,461,289
_ 247,898
+4,287,920
52
APPENDIX E
LESSON PLAN
SCHOOL: OFFINSO STATE A PRIMARY
AVERAGE AGE: 11+ YRS
SUBJECT: MATHEMATICS
REFERENCE(s): MATHEMATICS SYLLABUS PG.
TEACHER’S GUIDE PG.
CLASS: BS 6
PUPILS’ TEXTBOOK 2, PG.
WEEK ENDING: 29-02-2016
DAY/
TOPIC/SUB
TOPIC
OBJECTIVE
(S)
ASPECT
RPK
T L M ‘s / T L A ‘s
CORE POINTS
DURATION
REMARKS
DAY
TOPIC
OBJECTIVES
TLM: Abacus
Monday
ADDITION
AND
SUBTRACTION
OF SIX
AND
SEVEN DIGIT
NUMBERS
By the end of
the lesson, the
pupil will be
able to:
INTRODUCTION
DATE
EVALUATION
i.
Use abacus to
add six and
seven digit
Introduce the lesson by asking
pupils to add and subtract four
and five digit numbers. E.g.
5415 65471
+3457 _45360
25-02-2016
53
Abacus is
a frame
containing rods with
small balls that slide
alongside the rods
EXERCISE
Solve
following:
1.243936
+ 186423
the
2.2147865
+1489347
numbers
SUB-TOPIC
DURATION
60minutes
ADDITION
AND
SUBTRACTION
OF SIX
AND
SEVEN
DIGIT
NUMBERS
USING
THE
ABACUS
ACTIVITIES
RPK
Pupils can add
and
subtract
four
and five
digit numbers
1. Introduce the abacus to pupils by
guiding them to know that it
consist of column and beads.
The first column from right
represents ones, second column
represents tens, third column
represents hundreds up to the
seventh
column
which
represents millions
3. 3466609
- 22386677
4.4149903
ii. Demonstrate to pupils how
beads on each column is
exchanged with the other.
E.g, ten beads on the ones
column is equivalent to one
bead on the tens column
iii. Guide pupils to solve 422345 +
344321 using the abacus by
guiding them to first represent
422345 on the abacus
iv. Ask pupils to represent 344321
on the abacus
v. Ask pupils again to combine all
the beads represented on each
column and count them..
vi. Pupils will find out that
422345 + 344321
54
- 1387646
5.891142
+364686
766566
vii. Lead pupils to use the idea they
used in addition of six and
seven digit numbers using the
subtraction operation
viii. Solve more examples
with pupils
55
REMARKS
abacus to perform the
APPENDIX F
LESSON PLAN
SCHOOL: OFFINSO STATE A PRIMARY
SUBJECT: MATHEMATICS
AVERAGE AGE: 11+ YRS
REFERENCE(s): MATHEMATICS SYLLABUS PG.
TEACHER’S GUIDE PG.
CLASS: BS 6
PUPILS’ TEXTBOOK 2, PG.
WEEK ENDING: 29-02-2016
DAY/
DURA
TION
TOPIC/SU
B TOPIC
OBJECT
IVE (S)
ASPECT
RPK
DAY
TOPIC
Monday
ADDITIO
N
AND
SUBTRAC
TION
OF
SIX AND
SEVEN
DIGIT
DATE
OBJECT
IVES
By
the
end
of
the
lesson,
the pupil
will
be
T L M ‘s / T L A ‘s
CORE POINTS
EVALUA
TION
REMARK
S
TLM: Place value chart
Place value chart is made with a cardboard
, felt pens and ruler
EXERCIS
E
solve the
following
INTRODUCTION
Revise pupils knowledge on the addition
and subtraction of six and seven digit
numbers using the abacus
56
1. 243936
+ 186423
25-022016
NUMBER
S
able to:
Place value chart
M
DURA
TION
60minut
es
ix. Use place
value
chart to
SUBadd six
TOPIC
and
seven
ADDITIO
digit
N
AND
numbers
SUBTRAC
TION
OF
SIX AND
SEVEN
DIGIT
RPK
NUMBER
Pupils
S USING
THE
can add
PLACE
and
VALUE
subtract
CHART
six and
seven
digit
numbers
using the
abacus
HTh
T Th
-
H
T
O
2.2147865
+ 1489347
T
ACTIVITIES
h
1. introduce the place value chart to pupils and
demonstrate how it is designed with
chalkboard illustrations
3. 3466609
- 22386677
x. Guide pupils to solve 4967547 + 3387686
using the place value chart
4.4149903
Place value chart
- 1387646
M
HTh
TTh
Th
H
T
O
4 9
6
7
5
4 7
3
8
7
6
8 6
8 13
15
15
12
13
13
8 3
5
5
2
3
3
3
5.891142
+364686
57
xi. Lead pupils to perform subtraction
operation using the place value chart
xii. Lead pupils to solve more examples.
REMARK
S
58