1

NWOYE M.N.

PG/M.ED/02/33742

EFFECT OF GRAPHICAL-SYMBOL APPROACH ON THE PUPILS ACHIEVEMENT AND

INTEREST IN THE TEACHING OF RATIO AT UPPER PRIMARY SCHOOL IN NSUKKA

CENTRAL LOCAL GOVERNMENT AREA

Science Education

A THESIS SUBMITTED TO THE DEPARTMENT OF SCIENCE EDUCATION, FACULTY

OF EDUCATION, UNIVERSITY OF NIGERIA, NSUKKA

Webmaster Digitally Signed by Webmaster‟s Name

DN : CN = Webmaster‟s name O= University of Nigeria, Nsukka

OU = Innovation Centre

2009

UNIVERSITY OF NIGERIA

2

EFFECT OF GRAPHICAL-SYMBOL APPROACH ON THE PUPILS

ACHIEVEMENT AND INTEREST IN THE TEACHING OF RATIO

AT UPPER PRIMARY SCHOOL IN NSUKKA CENTRAL LOCAL

GOVERNMENT AREA

BY

NWOYE M.N.

REG. NO. PG/M.ED/02/33742

DEPARTMENT OF SCIENCE EDUCATION

UNIVERSITY OF NIGERIA, NSUKKA

APRIL, 2009.

3

TITLE PAGE

EFFECT OF GRAPHICAL-SYMBOL APPROACH ON THE PUPILS

ACHIEVEMENT AND INTEREST IN THE TEACHING OF RATIO

AT UPPER PRIMARY SCHOOL IN NSUKKA CENTRAL LOCAL

GOVERNMENT AREA

BY

NWOYE M.N.

REG. NO. PG/M.ED/02/33742

A PROJECT REPORT SUBMITTED IN PARTIAL FULFILLMENT OF

THE REQUIREMENTS FOR THE AWARD OF MASTERS’ DEGREE IN

MATHEMATICS EDUCATION TO THE DEPARTMENT OF SCIENCE

EDUCATION

UNIVERSITY OF NIGERIA. NSUKKA

APRIL, 2009.

4

CERTIFICATION

NWOYE M.N., a post graduate student in the department of science Education

with registration number PG/M.ED/02/33742 has satisfactorily completed the

requirements for the course and research work for the degree of MASTER IN

MATHEMATICS EDUCATION. The work embodied in this thesis is original and

has not been submitted in part or full for any other Diploma or Degree of this or

any other university.

--------------------------- ------------------------------

NWOYE M.N. Dr. K.O.Usman

Student supervisor

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DEDICATION

I dedicate this work to the Almighty God.

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ACKNOWLEDGEMENTS

I acknowledge the perception from the Almighty God who gave me the

strength and sustained me throughout the period of this work of facts findings. My

special and unreserved appreciation goes to Dr. K.O. Usman, my Project

Supervisor, who guided and directed me in carrying out this project. I thank him

mostly for this inspiration, encouragement, professional comments and correction

which made this work to be successful. I also extend my greeting to Dr Mrs C.R.

Nwagbo and Dr. Z.C. Njoku who were my content and design readers, may

Almighty God bless you. My gratitude also goes to my husband and children for

moral, financial and spiritual support which helped me to complete this work.

A special thanks goes to my dear friend, Association Prof Nkadi, Onyegegbu

for her academic advice and guidance. I also extended the thanks to my course-

mates, Mrs, Ijeoma Eze, Mrs. Euginia Onyishi, Mr. Innocent Onyeabor, Mrs.

Foluke Eze, Mrs Angy Ogboinna and Mr Uloko for the encouragement they gave

me which helped me to finish this work. May the Almighty God bless all of you

for your immense contributions in the completion of this work.

Nwoye Mercy Ngozi (Mrs)

2006.

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LIST OF TABLE

Table 1: An illustration of non-randomized control group. Pretest-posttest design --

------------------------------------------------------------------------- 39

Table 2: Table of Specification for content validation on ratio achievement

Test ------------------------------------------------------------------------- 42

Table 3: Mean and standard Deviation of pupils achievement score of subjects on

ration test. ----------------------------------------------------------------- 45

Table 4: Mean and standard deviation of pupils interest rating score of subjects in

ratio inventory. ----------------------------------------------------------- 47

Table 5: Mean achievement score and standard Deviation of male and female

pupils in ratio test. ---------------------------------------------------------- 48

Table 6: Mean and standard Devotion of male female pupils in ratio interest

inventory ----------------------------------------------------------------------- 50

Table 7: Analysis of covariance for Hypothesis 1 and 3 on pupils achievement

score in ratio test. --------------------------------------------------------------- 51

Table 8: Analysis or covariance for Hypothesis 2 and 4 on pupils gender interest

rating ----------------------------------------------------------------------------- 53

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ABSTRACT

The aim of this study is to find out the effect of graphical symbol approach in ratio

learning in upper primary school in Nsukka Central Local Government Area. Two

instruments Achievement Test in Ratio and Ratio interest inventory were used for

data collection. The data were analyzed using mean and standard deviation. And

mean hypothesis was tested by using ANCOVA. The findings showed that pupils

learn ratio better with graphical symbol approach. Graphical-symbol approach

motivates the pupils‟ interest in ratio and mathematics learning in general. Pupils

more epically females learn mathematics better with manipulative materials. On

the basis of the findings of this study, the research made some recommendations

which include: The effect of graphical-symbol approach in base-system, and with

the use of computer in the society. It was discovered that children are now more

exposed than before. Some topics, in the junior secondary schools mathematics

curriculum such as Quadratic Equation, factorization can be introduced in the

primary school mathematics curriculum to find out how it will work out.

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TABLE OF CONTENT

TITLE PAGE ----------------------------------------------------------------------------- I

APPROVAL PAGE -------------------------------------------------------------------- II

CERTIFICATION ----------------------------------------------------------------------III

ACKNOWLEDGMENTS -------------------------------------------------------------IV

TABLE OF CONTENTS -------------------------------------------------------------- V

LIST OF TABLES ---------------------------------------------------------------------VI

ABSTRACT ----------------------------------------------------------------------------VII

CHAPTER ONE

INTRODUCTION

Background of the Study --------------------------------------------------------------- 1

Statement of the Problem ---------------------------------------------------------------6

Purpose of the Study ---------------------------------------------------------------------7

Significance of the study --------------------------------------------------------------- 8

Scope of The Study ----------------------------------------------------------------------9

Research Question --------------------------------------------------------------------- 10

Hypothesis ------------------------------------------------------------------------------ 10

CHAPTER TWO

LITERATURE REVIEW

Theoretical/Conceptual Framework ------------------------------------------------- 12

Empirical Studies ---------------------------------------------------------------------- 12

Summary----------------------------------------------------------------------------------12

Models and Modeling in Primary School Mathematics Learning ----------------15

Graphical Representation in Mathematics Learning --------------------------------24

Approach to Mathematics teaching and Learning --------------------------------- 25

Empirical Studies------------------------------------------------------------------------27

Instructional Materials on Pupils Achievement in Mathematics Learning------ 27

Models and Modeling on Pupils Achievement in primary school

mathematics ---------------------------------------------------------------------------- 30

graphic representation in primary school Mathematics learning ---------------- 33

Summary and reviewed related literature ------------------------------------------- 35

CHAPTER THREE

RESEARCH METHOD

Design of this study -------------------------------------------------------------------- 38

Area of study ----------------------------------------------------------------------------39

Population of the study -----------------------------------------------------------------40

Sample and sampling techniques -----------------------------------------------------40

Instrument for data collection ---------------------------------------------------------40

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Validation of the instrument -----------------------------------------------------------41

Reliability of instrument ---------------------------------------------------------------42

Experimental procedure --------------------------------------------------------------- 43

Lesson note ----------------------------------------------------------------------------- 43

Method of analysis -------------------------------------------------------------------- 44

CHAPTER FOUR

RESULTS

Summary--------------------------------------------------------------------------------- 54

CHAPTER FIVE

DISCUSSION, CONCLUSIONS IMPLICATION, LIMITATION,

RECOMMENDATION SUGGESTIONS FOR FURTHER RESEARCH

AND SUMMARY OF THE STUDY.

Discussion ------------------------------------------------------------------------------ 56

Conclusion ------------------------------------------------------------------------------ 57

Implication ------------------------------------------------------------------------------ 58

Limitation of the study ---------------------------------------------------------------- 59

Recommendation ----------------------------------------------------------------------- 60

Suggestions for further research ----------------------------------------------------- 61

Summary of the study. ---------------------------------------------------------------- 61

REFERENCES ----------------------------------------------------------------------- 64

APPENDICES

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12

CHAPTER ONE

INTRODUCTION

Background of the Study

Primary school is the `critical starting point for formal structured instruction.

The general and relevant objectives of the primary education had been setout in the

Natural Policy on education (Federal Republic of Nigeria (FRN, 2004). The

content of the primary education mathematics curriculum had been drawn to reflect

these policies on education. According to Betiku (2002), the committee that

worked on the primary mathematics curriculum had taken cognizance of the

limitation of time, lack of teaching aids, the influence of the environment on the

village child, levels of teachers involved and teaching methods for effective

teaching and learning of mathematics in the primary school.

Primary school is the foundation on which further learning in the discipline

of mathematics is based. Therefore, a weak mathematics learning foundation at

primary school level may lead to poor attitude, low interest, low participation and

poor achievement in mathematics at higher level of education. Therefore, effective

teaching of primary school mathematics should not be compromised. Mathematics

which is the key to all sciences has some problems in teaching and learning. Some

studies have been carried out on the causes of these problems by some earlier

researchers.

The problem of mathematics teaching and learning, according to Usman

(2002) can be classified into two. The first is called Micro-problems or problem

13

which are internal to mathematics education. These problems are related to

curriculum, teacher training, text books, use of calculator, problem solving and the

like. The second classification is called macro-problem. These are problems

associated with pressure from other sections of the society and they include

economical, political, cultural and language problems. Other researched in

mathematics education had earlier identified some other problems on related

teaching and learning of mathematics to include: those related to students, teachers,

nature of the subject (mathematics), text books, instructional materials, classroom

environment, society, the school and finally gender.

Gender has a serious effect on mathematics teaching and learning, both on

the side of the pupils and the teachers. Normally, boys performed better than girls

in mathematics learning. Some of the previous studies have proved that. Harbor

Peters (2001) in her study on influence of gender in mathematics teaching and

learning, found out that boys performed better than girls in mathematics learning

mostly before the age of eleven which is primary school age. With that gender

problem, primary school teachers who are more of females cannot teach some

difficult concepts very well because of their poor perception of mathematics.

Therefore, mathematics teaching and learning have already started having

problems from primary school.

As Science Technology and Mathematics (STM) are the bedrock of any

technological development in any economy, Nigeria is no exception to this belief.

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Therefore, STM education have been planned to commence from the primary

school level, which is the foundation of education. According to Betiku (2002) the

relevant skill acquisition ought to have been solidly laid from the primary school,

since the rest of the system is built on the primary foundation. Besides, primary

level is the key to the success or failure of the whole educational system. This

calls for concerned effort for improving teaching and learning of mathematics if

success is to be achieved.

However, a lot of efforts have been made to improve the teaching and

learning of mathematics at all levels, especially the primary and secondary schools.

Despite all these effort, mathematics has a reputation of being difficult to teach and

learn. In an attempt to reduce all these doubtful situations, bold steps had been

taken to upgrade the teaching of primary school mathematics. The main features

of those bold steps included curriculum, material production in form of syllabuses,

course outline, pupil textbooks, pupils workbooks, teachers guide and training

manual. Despite all these bold attempts and others, pupils‟ performance and

interest in mathematics have been discouraging. One of the factors responsible for

the poor performance and lack of interest is poor teaching strategies adopted by

mathematics teachers, especially at primary school level, who are mainly female

teachers. Normally, females are always poor in mathematics learning, so their

perceptions in the teaching of mathematics will also be poor. For that, primary

school mathematics teaching should be enriched with manipulative materials.

15

Moreover, based on Piaget theory of learning age and its attendant

characteristics hold that children in primary schools are concrete operators

requiring real model or illustrative objects to learn better at this level. Besides,

children between the ages of 5 to 12 years (the period of primary education for

normal children) respond positively when motivated to learn. This is possible

when source of motivation is also the object or concept being studied. The

corollary is that the provision of concrete object for teaching of primary school

mathematics had dual roles in facilitating learning by making abstract concepts real

and meaningful in the learners, and by serving as a good source of motivation.

Unfortunately, not in all concepts, can primary school mathematics teachers

be able to bring real or concrete objects to class for use as teaching aids. Again,

the difficulty is more pronounced with abstract concepts hanging which teachers

cannot handle conveniently. Notable examples of these concepts are symmetry,

place value, addition, subtraction, number system, geometry, probability and ratio.

According to Obodo (2002), the abstractness of these concepts requires so much

resources such as concrete materials in teaching the concepts.

The use of instructional materials in the teaching and learning of

mathematics cannot be overemphasized. This is because mathematics by its very

nature is abstract and extra efforts are required to bring students to understanding

concepts, principles and applications (Usman, 2002). Furthermore, many

principles and concepts in mathematics are not easily explained with common

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sense deduction or reasoning. This fact obviously adds to the difficulty pupils

encountered in the comprehension of mathematics generally.

Ratio is one of the concepts in mathematics that is very difficult to

understand. According to Betiku (2001), ratio has a wide range of application and

it cuts across so many aspects of life. The concept (ratio) is included in the

primary school curriculum for the attainment of one of the primary school

mathematics objectives. Besides, ratio is introduced only at the upper part of

primary education so as to give room for full development of a child and quick

understanding of the concept.

However, despite the pupil‟s psychological developmental considerations in

the introduction of ratio, the pupils‟ performance in the concept (ratio) still remain

very poor. According to Ale, S. O and Salau (2001), pupils‟ in primary schools

find it difficult to comprehend the concept of ratio and do perform poorly in the

concept. Factors responsible for the poor performance according to Ale et al. have

been traced to lack of appropriate instructional materials for teaching the concept.

Such instructional materials include graph, and so many others.

The use of graph has provided a great deal of opportunity to teach some

difficult concepts in mathematics at primary school level in particular. According

to Ohuche (1990), the use of graph has been very successful for the teaching of

some concepts in mathematics that are very difficult. These concepts, Ohuche

continued, include-area, fraction and proportion. This is because, graph tend to be

17

quicker in conveying the required information. The nature of graph makes it very

easy to use for the illustration of information. For instance, a graph page is made

of several square units along several rows and columns. These features made the

use of graph for teaching concepts like area, and population very easy and

interesting (Ohuche, 1990).

Based on the above attributes, the researcher intends to use graph for the

teaching of ratio, to see if it could enhance the teaching and learning of the

concept, as well as promote pupils‟ interest in the concept. Similarly, the use of

the graph is also expected to have some effect of the pupils‟ interests in learning of

ratio.

Statement of the Problem

Primary school has been described as the foundation on which further

learning in life is built. The teaching and learning of mathematics in primary

school, marks the beginning of the subject that its usefulness cuts across all facts of

life. As primary mathematics is the bedrock for further mathematics learning. The

mathematics curriculum is arranged hierarchically so that the mathematics concept

will have links from primary school to tertiary level respectively. Besides, the

hierarchical nature of mathematics curriculum, there is a need for good

understanding of the foundation level of mathematics in primary school level

which will enable further mathematics learning to be effective. To achieve that,

the teaching of mathematics requires the use of instructional materials not only

18

because of the developmental stages of the pupils, but also for better understanding

of various concepts to be taught. However, there are some topics that some

teachers find difficult to teach without instructional materials. These topics

include area, proportion and ratio. Some researcher has successfully used graphs

to teach some concepts such as area and proportion. The use of graph on those

concepts has been reported to have a positive effect on the students.

However, there is no available study to the knowledge of the researcher on

the use of graphs in teaching ratio. Therefore, this study is designed to determined

the effect of graphical symbol approach on the pupils‟ achievement and interest in

the teaching of ratio at upper primary school.

Purpose of the Study

The main purpose of this study is to determine the effect of graphical symbol

approach on primary school pupils achievement and interest in ratio learning.

Specially, the study intends to determine.

The effect of the graphical-symbol approach on the mean achievement

scores of the primary school pupils‟ in ratio.

The effect of the graphical-symbol approach on the mean interest rating of

the primary school pupils‟ in ratio.

The influence of gender on the achievement of pupils taught using

graphical-symbol instruction.

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The influence of gender on the interest of the pupils‟ taught, using graphical

symbol instruction.

Significance of the Study

The fact that primary school mathematics is the foundation for other levels

in mathematics, emphasis are to be laid on effective teaching of mathematics from

the primary schools. Besides, the hierarchical nature of mathematics curriculum

there is a need also for calls for good understanding of the foundation nature of

mathematics at the primary school level. Therefore, this study will make a

significant contribution to the teaching and learning of mathematics at primary

school level.

It is expected that the following group of people would benefit from the

result of this study; the primary school pupils, the primary school teachers, and the

curriculum planners. The benefit for the pupils may result from the better

understanding of the concept (ratio) when it is taught by graphical-symbol

approach and discovered that there will be a positive effect on the achievement and

interest of the primary school pupils, more especially girls who do not naturally

perform well in mathematics and science related subjects. This is because there

may be no better achievement without better understanding of a concept. It has

also been revealed by various researchers that interest is directly related to

achievement in various life endeavours.

20

Similarly, teachers may also benefit from the study, if the use of graphical

symbol is of positive effect on pupils achievement and interest. This is because it

would serve as an additional approach to the teaching of mathematical concepts.

The curriculum planner may also find the study very beneficial in the sense

that the planning of primary school curriculum would be made to include the

approach that would promote better understanding of such concept like ratio. In

other words, the curriculum planners may recommend the graphical-symbol

approach for the teaching of ratio at primary school level.

Scope of the Study

The study will be limited to upper primary school pupils in Nsukka Central

Local Government Area of Enugu State. Besides, the topics to be covered would

include:

Division of ratio

Equivalent ratio

Missing addends in ratio relationship

Research Questions

The following research questions were based to guide the study:-

(1) What is the mean achievement score of the pupils taught ratio with

graphical-symbol instruction and those taught with conventional method?

(2) What is the mean interest rating of pupils taught ratio with graphical

symbol instruction and those taught with conventional method?

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(3) To what extent does gender influence the mean achievement of the pupils

taught ratio using graphical-symbol approach?

(4) To what extent does gender influence the interest rating of pupils taught

ratio using graphical symbol approach?

Hypothesis

HO1: There is no significant difference in the mean achievement score of the

pupils taught ratio with graphical-symbol approach and those taught with

conventional method.

HO2: There is no significant difference in the mean interest rating of the pupils

taught ratio with graphical-symbol approach and those taught with

conventional method.

HO3: There is no significant difference in the mean achievement score of the

male and female pupils taught ratio with graphical-symbol instruction.

HO4: There is no significant difference in the mean interest rating of the male and

female pupils.

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CHAPTER TWO

LITERATURE REVIEW

In reviewing of the related literature, the researcher organized the work

under these sub-headings:-

Theoretical/Conceptual Framework

Learning Theories on Instructional material

Models and modeling in school mathematics learning

Graphical representation and school mathematics learning

Approaches to mathematics teaching/learning

Empirical Studies

Instructional materials and pupils achievement in primary mathematics

Models and modeling on pupil‟s achievement in primary school

mathematics

Graphical representation and pupils achievement in primary school

mathematics.

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Summary

Theoretical/Conceptional Framework

Learning Theories on Instructional Materials

Mathematics Instructional materials are animated or inanimate objects that

assist the teacher in making mathematical concept skill more meaningful and

understandable to the learner (Ashforth, 1997).

Piaget (1971) as an educational psychologists and child development in

learning, in his own theory of learning stated that, “subjectivity of representation in

child development is encouraging the use of concrete models in teaching and

learning of mathematics from primary school level”. Piaget further said that

learning goes well from concrete to abstract. According to Jean proper use of

concrete model could promote the broad goals alluded in mathematics learning.

He made coherent rational for the use of concrete models in the learning of

mathematics. Primary school pupils who ware mainly at age of concrete

operational stage according Piagets need to learn more with instructional materials.

Brunner (1976) in his constructivist theory, stated that learning is an active

process in which learner construct new ideas or concepts based on their current or

past knowledge. Bruner as cited (Kearsely, 1994b), who said that, “instructional

materials are used to provide the meaning and organization to experiences and

allow the individual to go beyond the information given”. According to Bruner,

the teacher should try and encourage pupils to construct hypotheses, make

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decisions and discover principles by themselves. Bruner (1976, as cited in Kearly

1994b) stated that, a theory of instruction should address the following aspects.

The most effective sequences in which to present materials

The ways in which a body of knowledge can be instructed so that is can be

most readily grasped by the learner, i.e. through instructional materials.

Gagne (2004) in his theory on instructional material was particularly

influenced in the training and the design of instructional materials in teaching and

learning. According to Gagne (1999), “a variety of learning activities should

enforce effective learning”. Gagne further said that instructional designer should

anticipate and accommodate alternate learning styles by systematically varying

teaching and assessment methods to reach every pupil to achieve Gagne‟s theory

instructional materials are to be used in mathematics teaching and learning.

A review of literature indicates that many people emphasize the importance

of instructional materials in promoting meaningful mathematics instruction. The

use of instructional materials is not new (Grossman, 2000). Grossman further

points out that because of the importance of instructional materials in teaching and

learning, educationists have been advised in 1885 to employ manipulative

materials in teaching concepts in mathematics.

Damisa (1987) stated that for a mathematics teacher to achieve his objective

in the classroom, he must find other devices for the students to see, touch, hear and

make use of these devices, but the time allocated to mathematics, may at times,

25

obstruct the seeing and using of these devices provided. Balogun (1997), observed

that failure by teachers to use appropriate instructional materials, poor teaching

methods, such as lecture and direct information dissemination method, make pupils

loose interest and thus perform poorly in mathematics.

Some psychologists have studied the effects of the manipulative materials on

achievement, retention, attitude and transfer of mathematical concepts. Their

findings led scholars, such as Jean Piaget, Jerome, Brunner, Gagne, to draw

conclusions on the effect of instructional materials on pupils achievement in

learning.

Models and Modeling in Primary School Mathematics Learning

Harbor-Peters (2001), conceptualized models for mathematics teaching and

learning as enriched devices which may be concrete or semi-concrete or abstract

for use by teachers to make mathematics concepts clearer to learners. Harbor

further stated that a model must posses a one-to-one correspondence relationship to

the mathematical concept being illustrated, using the model. It must be simple and

easy to use so that one does not use the teaching time for explaining the models.

With that, graphical illustration would also serve as model in mathematical

learning. In general, good models for teaching mathematical concepts are capable

of providing concrete and realistic experiences, which will help a learner to

discover facts or patterns. It creates curiosity and motivates the learners to explore

mathematics in a relaxed mood.

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With regard to Harbor Peters view on models as instructional materials in

mathematics learning, primary school pupils would be motivated to develop

interest in ratio learning when it is properly taught with graph sheet as a model.

Obodo (1997), viewed models as, “two or three-dimensional representatives

of objects which students learn about in the class”. The use of models according to

the Obodo, provides a mental of relating past experiences to a new situation. They

employed and provide concrete and realistic experiences from which learners can

discover facts. The minds of the students will readily accept ideas that are

illustrated by concrete example. The means that models give meaning to different

concepts and relations in mathematic by associating them directly with physical

objects. It was further explained that concept of the model is restricted to mean

only those concrete devices used by mathematics teachers and students to

demonstrate mathematical concepts. Models therefore, should represent the

natural objects, they are meant to represent. Which graphical approach can do in

ratio learning.

According to Bal, (2004), mathematics models can be considered as a

simplification or abstraction of a (complex) real world problem or situation into

mathematical problem. It was further explained that mathematics problem can be

solved using whatever known techniques to obtain a mathematical solution. This

solution is then interpreted and translated into real term.

27

Gagne and Berliner, (1992), viewed a model as a visual or picture, which

highlight the main idea of a variable in a process or system. Gagne further stated

that the use of model as learning aids have two primary benefits. Firstly, models

provide accurate and useful representations of knowledge that is needed when

solving problems in some particular domain. Secondly, a model makes their

process of understanding a domain of knowledge easier because it is visual

expression of topic. It was found out that pupils who study with models may recall

as much as 57% or more on questions concerning conceptual information than

pupils who receive instruction without advantage of seeing and discussing models.

With the view of Gagne and Berliner, graphical-symbol communication will

be a useful model in ratio learning, which will help primary school pupils to

understand ratio-relationship visually, Alesandrini, (2002), came with his own idea

similar in conclusion, when he studied different pictorial-verbal strategies in

learning from his research on the effectiveness of pictorial-verbal representation.

From his own study, the learner draw their own conclusion that the act of building

the model and running the simulation gives them a deeper understanding of the

sensitivity of the cycle outside disturbances and reinforces the concepts underlying

the model.

Richard, (2002), view model as being a representation. Richard further said

that motivation might be found by providing tools for designing interactive

stimulation. Simple interaction with model provides a far more interesting

28

exercise than observation. With his definition, Graphical-symbol model will serve

as a tool in ratio learning and bring interaction stimulation on the pupils and create

more interesting exercise than through observation and passivity.

Norman, (2002) said that one can have internal model. He viewed models as

representation of reality that people use in the environment, with others, and with

the artifacts of technology, people form internal mental models of themselves, and

explanatory power for understanding the interaction. In ratio learning, mental

model will be transformed into graphical-symbol representation, which will

interpret the mental ratio model into visual ratio representation and interpretation

Johnson-Laird (2002) viewed mental model as (“a basic structure of

cognition”). It is now plausible to suppose that mental model play a central role in

representing objects, states of affairs, sequences of events, the way the world is,

and the social and psychological action of daily life according to Holland (2002),

who suggested that mental models are basis for all reasoning processes. With

these views, graphical-symbol approach in ratio learning will create a visual

framework, which will bring visual illustration to relate mental model to visual

model.

Ryder (2004), view model as myths and metaphor that helps us to make

sense of our world. Whether it is derived from which or from serious research,

model is a means of comprehending an otherwise in comprehensible problem.

According to Ryder an instructional design model gives structure and meaning to

29

an identity problem enabling the would be designer to negotiate her design with a

resemblance of conscious understanding. Model helps us to visualize the problem,

to break it down into discrete management units. Hence in ratio relationship,

graphical-symbol model will reduce cognitive stress and increase visual idea,

which will make ratio learning more realistic and for easy understanding. Ryder

further stated that pupils who are engaged in the model building process must pull

together science content, mathematics skills and logical problem solving. Skill

manipulating materials, are also regarded (as concrete models) in mathematics

learning.

Lesh (2004) suggested that concrete models can be effectively used as an

intermediation between the real world and the mathematical world. He contended

that such used world tend to promote problem-solving ability by providing a

vehicle through which children can model real-world situations. The use of

concrete model in this manner is thought to be more abstract than the actual

situation yet less abstract than formal symbol. With Lesh‟s idea, graphical-symbol

approach will model ratio relationship visually and graphically and make it to

depart from traditional ratio learning.

Borne (2000), viewed concrete models as those objects that can be touched

and moved by pupils to introduce and reinforce a mathematical concept. With

Borne‟s view, graphical-symbol approach will reinforce mathematics learning.

30

Hartshrn (2004), suggested that manipulative materials are particularly

useful in helping pupils move from concrete to abstract level. Teachers, however,

must choose activities and concrete models carefully to support the introduction of

abstract symbols.

Hedden (1986), divided the transition iconic level (the level between

concrete and abstract) abstract levels in the following way:-

The semi concerted level is a representation of a real

situation; pictures of the real items are used rather than

the items themselves. The semi abstract level involves a

symbolic representation of concrete item but the pictures

do not look like the objects for which they stand.

Howden (1986) placed specific concrete models as those manipulative ranks

from the concrete to the abstract. In place value, for example (going from concrete

to abstract), they include pebbles, bundled straws, based ten blocks, chip trading

and the abacus. Howden cautions that building the bridge between the concrete

and abstract level requires careful attention. Graphical-symbol approach will

transform ratio learning from visual to abstract level.

Suydam and Higgins (1977) said, “Mathematics achievement increased

when manipulative materials were used”. Graphical-symbol approach will

increase achievement in ratio.

Penner and Rich Lehren (1998), said that,

31

When students have space and geometry, measure, and

data at their disposal, as well as the more traditional

forms of numbers sense, the transition to mathematical

modeling of natural phenomena becomes feasible and

powerful, even in the physical models of elbows can lead

in turn, to graphical and functional descriptions of the

relationships between the position of a load and the point

of attachment of the tendon. Thus elbows can be

modeled as third class levers, an idea we explored with

third grade pupils.

Umoren, Ukaha, Chukwu (1984), said; that “mathematical modeling has

recently become a prominent term among mathematicians and the users of

mathematics, particularly in science and technology”. Graphical-symbol will help

the young users of mathematics to process ratio relationship graphically.

Allen (1981), view mathematics modeling as the total process involved in

the steps we go through when we use mathematic to deal with any real situation

that we have Graphical-symbol communication will be used to process ratio

relationships visually.

Lassa (1981), view mathematic modeling as a unifying theme for all

application of mathematics, but key steps in any activity in technology and indeed

in most forms of the modeling process. Lassa further said that mathematic

32

modeling is one particularly powerful way of representing reality. It is usually

better in technology to have some mathematical model to be accurate. He said that

the teaching of mathematics modeling is one new approach to the teaching of

mathematics because of the need to make mathematics more relevant to everybody

life. Graphical-symbols approach will serve as a new approach in ratio learning in

primary school level.

Ang. Keng Cheng (2001), viewed mathematical modeling as a process of

representing real world problems in mathematical term in an attempt to find

solutions to the problems. Graphical-symbol communication will serve as

modeling in ratio learning, it will be a process of real world problem in ratio

relation and help the solution to graphical problem.

Mathematics Modeling Our World is founded on the principles that

mathematics is a necessary tool for understanding the physical and social worlds in

which we live. Mathematics Modeling Our World is a grade 9 – 12 curriculum,

which included the primary six mathematics curriculum, in which pupils not only

learn mathematics, they also learn to use mathematics, in solving their problems.

MMOW support that pupils are taught to use a variety of resources to solve

problems, and they learn to choose resources that meet the need of a particular

solution.

33

With the view of Mathematic Modeling Our World, graphical-symbol

communication will serve a variety resource to solve ratio problem that will meet

the need of ratio solution.

(1) English and Halford (1995), said that, modeling involves the

establishment of links among representations of a mathematical concept and its

relationship to other concepts. More importantly a model needs to externalize the

links to the learner in ways that would help him or her to visualize them.

Graphical symbol model will externalize the ratio relationship to the ways that

would help them to visualize the relationship. Furthermore, English and Halford

explained that modeling involves the deputation of the relations that are embedded

in a scheme both graphical or concretely. The modeling process could also

contribute to expansion of Networks of schemes that are associated with

mathematical concepts resulting in deeper understanding. Modeling activities

must also have an inbuilt flexibility to help children externalize constituents of a

model. These activities need to be grounded within the experiences of children,

including observation of concerts in real life contexts. Explanation would also

reveal conjecture about other situations and solved problems.

California (1989) symposium, pargarmon, view modeling as the use of a

formal language (symbolic or grammatic) to present some knowledge short.

34

Graphical-symbol communication will be used as a language to communicate ratio

relationship ideas graphically and visually.

Graphical Representation in Mathematics Learning

Mega mathematics (2004), said that when mathematics talk about graphs

they are mostly likely to be thinking of collection of dots and lines that you see in

the illustration of this section. Sometimes, graphs are called networks, and a

glance at pictures of them will show you why. Mega mathematics further added

that graph could be seen as one of the mathematical objects, which make

mathematics learning more transparent. Graphical-symbol will make ratio learning

transparent.

Graph game, (2004), show that a number is also a mathematical object that

is probably the most familiar to everyone. Some other mathematical objects are

knots maps and infinite state machine, linear graph and picture representation will

make ratio learning more transparent. The idea of graphical-symbol approach will

help pupils to make ratio relationship more transparent.

National Council of Teachers of Mathematics of American (NCTM) gave a

recommendation which includes in their goals that pupils in grades 3 – 6 should be

able to represent data using tables and graphs such as lines and pots bars graphs

and line graphs, pie chat stem graphs (standard 5, p. 178), graphical-symbol

approach will serve positively in ratio learning and the idea of linear graph should

be introduced to make a very important point about this goal of (NTCM).

35

Van de Walle (2004), suggested that teachers should not “get overly anxious

about the tedious details of graph construction”. According to Walle, teachers

should take one or two approaches to graph construction. Pupils should either be

encouraged to do their best when creating their own graphs or that student should

use technology and computer to generate exact graphs. Some types of graphs that

should be taught include bar, stem and leaf plots and continuous data graphs. Van

de Wall has been encouraging the use of graph in primary school mathematics

learning. As he had been mentioned the other types of graphs used in primary

school. Graphical-symbol approach in ratio learning will bring about the idea of

line graph in primary school mathematics learning.

Approach to Mathematics Teaching and Learning

Burner in Obodo (2004) said that after some years at least 10 years of

experience, teacher will find it difficult in helping their students to discover

mathematical ideas for themselves without involving them in activity-method.

Obodo (2002) said that approaches in mathematics teaching and learning

enable the pupils acquire mathematics skills. Some of the approaches are:

Area Title Approach: It is the type of approach whereby Tiles are used in

teaching of area to the pupils. The tiles represent the square units. Collins (1998),

„Encourage the Area Tile Models‟ Approach in teaching of area in primary school.

It will bring area learning home and realistic to children because tiles are what they

36

can see in homes and schools. It makes a mathematics learning begins from known

to unknown-from real to abstract. Which graphical-symbol method will also do in

ratio learning.

Discovery Approach: It is the type of approach whereby a pupils are guided to

build mathematics ideas, to think more to discover things by themselves. Ale in

Imoke (2005) said that discovery approach as a pupils-dominated and activity-

oriented method of teaching and learning of mathematics. With graphical-symbol

approach, discovery will be more encouraged because the pupils will be allowed

the view the graphical-representation and discover the given ratio relationship

vividly.

Laboratory Approach: This is the type of approach whereby the pupils are led to

find the importance of the mathematics tools and instructional material, which they

have in learning mathematics. Usman (2002) emphasized that the pupils are made

to carry out some measurements often using simples equipment such as rulers and

compass, collect data by experimentation or discovery, making drawing and

models, make computational devices, and perform experiments with materials. The

graphical-symbol approach will also encourage this approach more because some

of the materials such as graph sheets, ruler, and pencils are to be used in plotting of

the graphic in ratio relationship.

37

Target-task Approach: In this approach the pupils will first of all give some

hander topic to solve which they may or may not be able to solve. After that an

easier problem will be solved with help of the teacher‟s guidance.

With graphical-symbol approach, some difficult ratio problems which have

been given to the pupils to solve can now be solved graphically and easier with a

simpler understanding.

Small-group Approach: This approach is the type of approach whereby a

mathematics teacher group pupils in smaller groups during mathematics learning.

For graphical symbol approach, small group approach will be more effective

because the teacher will group the pupils and give them some ratio problems to

plot graphically.

Empirical Studies

Instructional Materials on Pupils Achievement in Mathematics Learning

Okorie, Onuoha, Anayanwu and Ugochukwu (1989) researched on, “the

extent of use of visual aids in the teaching of mathematics in Okigwe and Owerri

Educational Zone in Imo State. The research was a descriptive survey research.

The research was carried out in all the secondary schools in the two above-

mentioned educational zones. 55 mathematics teachers were chosen in the schools

and were used. 850 students were randomly selected from the secondary schools.

The main instrument used for the data collection was questionnaire, seven research

questions were tested, 5 hypotheses were made.

38

Two sets of questionnaire were administered to the two groups of

respondents by the researchers themselves. Pearson‟s moment correlation

technique was used to compare the response. A correlation co-efficient of 0.81

was obtained. This was considered high enough. Chi-square was employed in

data analysis. From the findings, it was concluded that visional aids are of

immense important in teaching and learning of mathematics.

They spaced the lessons and spread interest hence reducing boredom that is

often necessitated due to the abstract nature of mathematics. Instructional material

help to reduce verbalism and give a concrete touch to the teaching of mathematics.

It is often said that seeing is believing. With instructional materials students are

convincingly taught mathematical facts without necessarily imposing facts on them

and hence forcing them to cram these facts. Learning thus becomes natural and

active participation of every member of the class is ensured. A combined effect of

hearing seeing and doing will be enabling and ensuring retention.

Emmanuel C. (1977), in worked on the, “Impact of the audio-visual aids in

the teaching of mathematics”. The research was deceptive survey. It was carried

out in the secondary schools in Otukpo in Benue state. 6 schools were used out of

10 schools in the area. Interview was used to collect data from mathematics

teachers and students from the six schools.

One hypothesis was tested, which stated, schools which employ more audio-

visual aids in mathematics get better results in mathematics than those which

39

employ fewer”. Tables of percentages were used respectively for the presentation

and analysis of the data obtained during the study. From findings, it implies that

mean calculated was approximately 0.9. This implies that there is a strong positive

linear relationship between dependent variable (x) and the independent variable (y)

tested. This means that x increases, and also y increases. This means that when

more audio visual aids are employed, in teaching and learning of mathematics,

more students will pass therefore the hypothesis which employ more instructional

materials in mathematics teaching, get better results holds.

Piaget, and Diene (1976), stated that, in order to create a meaningful mode.

After validating their model through, they used their model to test the effectiveness

of model on pupils achievement in learning. This hands-on approach to

constructing knowledge about a system results in laboratory type setting and has

been attributed to the words. According to them, proper and use of concrete models

could be used to promote the broad goals alluded, in mathematics learning. Each

of these men has made coherent rational for the use of concrete models in the

learning of mathematics concepts. Bruner et al (1967) wrote “seeing is a decision

making process”.

Post on his studies on the effect of concrete model and operation in pupils‟

achievement in a mathematics. His theoretical study was base on the works of

Piaget, Brunner and Diene. The learning theories were based on the effect of

concrete model and operation in pupils‟ achievement in mathematics. The men

40

discussed based on their intellectual views on pupils‟ achievement mainly on

concrete operations effect in mathematics learning. Based on that, concrete models

were encouraged for pupils achievement.

On the research of the role of concrete models in the learning of

mathematical concepts. Given share number of studies Post (2004), cited some

undertaken, it is perplexing to note that more is not known about the precise way in

which concrete model effect the development mathematical concepts. Perhaps that

largest contribution factor to this has been the lack of coordinated research efforts

that have mapped out a priori and have designed individual investigation that

would have provided coordinated answers to sets of related question rather, the

past pattern of research has been that of large numbers individually conducted

investigations and then postersion attempts to relate them in some fashion. This

has not been particularly fruitful and has left many unanswered questions and

hungers gaps in our knowledge.

Models and Modeling on Pupils Achievement in Primary School Mathematics

One obvious fact is that the study of models is a crucial need in the area of

primary mathematics learning. Due to some positive effects of these models on

primary school mathematics learning some researchers has worked on its uses and

effects. The most recent and comprehensive review of research on the use of

concrete models was complied at the mathematics and science information

references center at Ohio State university (ERIC) by Suydam and Higgins (1976).

41

The result generally concluded that concrete models are effective in promoting

pupils achievement but emphasizes the need for additional research.

Lowrie (2001), did a study in presentation and mathematics learning. The

research was carried out to identify the effect of representation of mathematics

concept by using models. Again Lowrie (2001) identified three categories of

problem solving approaches to include visualizes, verbalizes, and both user and we

as the role of imagery in problem solving. Lowrie (2001) found out that 42% of

the participants solving the mathematical participants solved the mathematical

problems using the visual technique.

Umeron Ukaha, Chukwu (1984), in his study on mathematics modeling

approaching to the task in teaching algebra among students. The research was

carried out with quasi-experimental design to investigate the effectiveness of using

geometric models in teaching algebraic expansions and factorization to junior

secondary. A total number of sixty – (60) students were selected randomly and

grouped into mathematics modeling approach (MMA) group and traditional

method (TM) group for the study. TM was used as control over MMA, which the

researcher set out test.

Each group containing 30 students with equal males and females. The study

was carried out in a mix school. The instrument used for the data collection was

test. The test was organized in form of pre-test and post-test. The data were

42

analyzed by use of (ANCOVA). Four research questions were tested and analyzed.

Their scores were organized under MMA and TM and analyzed.

The analysis of the pre-test revealed that there was equivalence in the

mathematical abilities of the experimental and control groups at the beginning of

the experimental.

It was also found that there were significant differences between their

performances

The mean performance of students taught using MMA and that of those

taught using TM.

The mean performance of boys taught using MMA and girls taught using

MMA.

The mean performers of girls taught using MMA and that of those taught using

TM.

From the analysis the mean score of the students in the experimental

group was 23.43 while that of those in the control group was 15.93. And since the

mean score of those in experimental group was 15 – 93 and the mean score of

those in experimental group was higher, the claim was that the Mathematics

Modeling Approach (MMA) was better methods of teaching algebraic expansion

and factorization than traditional or control method (TM or CM).

43

Graphic Representation in Primary School Mathematics Learning:

Researchers on graphical representation of bivariate and multivariate data by

grades 4, 5 and 6 students revealed the level of pupils-generated graphical-

representation of bivariate, and multivariate association (Moritz, 2000). Regarding

the tools of Moritz‟s study, it was reported that a survey item was prepared

including two parts, namely part (B) items related to bivariate association and part

(M), items related to multivariate association. During the data collection

procedure, one item was asked from the related area and the other three items were

asked from unrelated ones. The levels of response were categorized as level 1

unsuccessful bivariate association, level 2; partial bivariate association, level 3,

complete bivariate solution and the case of multivariate representation. The

numerical (symbol) analysis revealed that there was an association of the response

level to part (M) according to the response level of part (B).

Outhred and Sardonic (1997) a research on representation of numerical

(symbolic) situation through problem-solving at Kindergarten (kg) level.

Moreover, it had been concluded that students understood that a cube

represents unit for representing a number. Similarly, it was reported that students

developed the ability to writing equation in the end of five-month problem-solving

sessions. The research was a result of those data revealed through a regular KG

classroom. A problem sheet was prepared comprising of “addition” (combine),

44

“subtraction” (combine and separated), “Multiplication” (equal groups), “division”

(portative) and “fraction” (one half) related problems.

The data collection procedure was during the problem-solving session

focusing on student-generate problem-solving strategies, for example, “crossing

out and petitioning sets for subtraction”, separation for subtraction and addition”,

“drawing” ( Outhred & Sardelich, 1997, p. 380). The pupils-generated drawings,

produced during the problem-solving session, were kept as a basis for analysis.

Outhred & Sardelich‟s research was an extension of the previous research

carried out by Carpenter et al (1993) as they concluded that Kindergarten pupils

had solved a variety of difficult problems in the end of eight months.

Diezmann (1999) reported that in order to develop the pupil‟s ability of

using diagram as cognitive tools, teachers need to assess the quality of diagrams

and provide them with the necessary support. The theoretical framework of

Diesmans, (1999) research on “Assessing the research on representation of

mathematical problems by using diagrams. Furthermore, matrices networks and

hierarchies and range of that represent part-whole characteristics. Diezmann,

Francis, Horley & Novice, (1999) were taken into account of this study the aim of

Diezmann‟s study was to explore how the quality of diagrams can be assessed

using theoretical prototype, and specifically, how prototypes can be used to

identify the different levels of performance.

45

Diezmann‟s (1999) research was a case study of 12-year-old five pupils who

were both from high and low achievers in mathematics as well as high and low

performers in visual methods of solution.

The instruction of twelve half-hour lesson addressed the four general

purpose of diagrams generation and its use in novel problem-solving tens.

Interviews were conducted before and after the instruction and along with the task

related to the five “isomorphic problem”. The levels of study-generated diagrams

were criteria; level 0 was assigned when the diagram, level 1 was categorized for

the plausible diagram but lacking on assigning the appropriate component of the

structure, level 2 was assigned for the diagram which represented at least one but

not one components of the structure, and level 3 was labeled for such diagram that

represented all the components of the structure appropriately.

Summary of Reviewed Related Literature

The review of the related literatures in mathematics learning was carried out

in three areas. Namely:-

(1) Instructional materials and learning theories in mathematics learning;

(2) Model and modeling approach in mathematics learning and;

(3) Graphical representation in mathematics learning.

(4) Approaches to Mathematics teaching and learning

From the review, it was observed that instructional materials enhanced

the teaching and learning of mathematics more especially in the primary school

46

level. In other hands, the educational psychologists in their learning theories

encouraged the use of concrete objects in teaching and learning of mathematics

mostly at early stage of learning. When the children are at concrete operational

stage, they learn more by seeing, doing and hearing.

On the other hands, the review shows that models are more valuable and

effective in mathematics teaching and learning. It makes mathematics learning to

be real instead of abstract. It gives mathematics learning visual understanding.

Literatures also revealed that modeling process in mathematics teaching, enhances

children to learn mathematics, easier and faster. It makes mathematics learning

clearer.

Literatures showed that graphical representation has also been used in some

areas like pie chart, histogram, and bar chart, to represent mathematics

information. It was observed that the graphical approach gives mathematics

learning visual understanding. Instructional materials had been found to arouse the

pupil‟s interest, motivation and curiosity in some areas of mathematics learning.

Literature revealed that most of problems of poor performance of pupils in

mathematics could be solved by proper use of adequate instructional materials in

teaching and learning of some areas in mathematics from primary school level.

In the empirical studies, it was reviewed that pupils who were taught

mathematics with manipulative materials, performs better than those taught

without manipulative materials. It has been observed also that some studies

47

were conducted in other areas such as fraction, percentage, algebra and geometry.

Therefore there is a need to carry such study on ratio, which is one of the abstract

concepts in primary school mathematics contents, to verify if the result will remain

the same with the other areas. However, from the review of few studies carried out

by some researchers Suydam and Higgins (1977), Post (2004) and Umeron Ukaha,

Chukwu (1984), it appears that models as manipulative materials enhance pupils

achievement in mathematics. Literatures did not show studies on ratio. This then

created a gap that needs to be filled. Therefore, there is a need to see how effect

the graphical-symbol approach will have in pupils‟ achievement in ratio. The

present study is therefore a step in the right direction with the hope that graphical-

symbol approach will motivate primary school pupils in mathematics learning.

48

CHAPTER THREE

RESEARCH METHOD

This chapter is organized to cover the following areas: design of the study,

area of the study, population of the study, sample and sampling techniques,

instrument for data collection, validity and reliability of instruments,

administration, experimental procedures and method of data analysis.

Design of the study:

The design of this study is the Quasi-experimental research design.

Specifically, it is the pretest, posttest nonequivalent control group design. The non-

randomized control group was chosen because according to Ali (1999) “Non-

randomized control group is used when the researcher cannot randomly sample and

assign his subjects”. It controls the internal validity threats of the initial group

equivalence and researcher‟s selection bias, since there were no randomizations of

subjects into groups. The classes were already in existence, so intact classes were

used in order not to disrupt the normal class setting, in terms of seating

arrangement and classroom schedules.

49

Table 1: An illustration of non-randomized control group. Pretest- posttest

design

S Grouping Pre-test Treatment Pos-test

–

–

Experimental

Control

O1

O1

X1

control X2

O2

O2

Where;

S – denotes sample

X1 – denotes treatment 1 (graphical approach)

X2 – denotes the control group 2 (conventional/traditional approach)

O2 - denote pre-test

O1 - denote post-test

Area of study

The study was carried out in Nsukka Central Local Government Area. This

area was chosen because no known research had been carried out in primary

schools in this area on graphical symbol approach. The area is tickly populated and

the researcher felt that the findings will give a good generalization of what is

happening in other areas.

50

Nsukka Central Local Government Area is made up of four towns which

are: Nsukka town, Lejja, Obukpa and Obimo with a total of 46 primary schools.

(Appendix A).

Population of the study:

The population of the study consists of 1,067 primary six pupils in the 46

primary schools in Nsukka Central Local Government Area.

Sample and sampling techniques:

The sample for this study was 120 primary six pupils. The schools were

randomly drawn from each town, making a total of four school with 120 primary

six pupils. In the schools selected, one was used as control group, where one

school serves as the experimental group. Simple balloting (with replacement) was

used to assign those schools to both experimental and control groups. Only primary

six pupils were used for the study because that is the class that are mature enough

to handle graphs.

Instrument for data collection

Two instruments were used for data collection. They are: The Achievement

Test Ratio (ATR) and Ratio Interest Inventory (RII). These instruments were

developed by the researcher. The curriculum and table of specification served as a

guide 24 questions were developed considering the three concepts of ratio outlined

51

in the curriculum: Division of ratio, Equivalent ratio, Missing addends. The

Achievement Test Ratio (ATR) is on appendix B, C & D while the Ratio Interest

Inventory (RII) is on appendix E.I

In developing the Ratio Interest Inventory (RII), the items were based on 4

point scale rated as Strongly Agree (SA), Agree (A), Disagree (D) and Strongly

Disagree (SD). These were put in figures as SA = 1, A = 2, D = 3 and D = 4. See

appendix E.

Validation of the instrument:

The instrument were subjected to face and content validation. Two experts

from mathematics education and one from measurement and evaluation unit

validated the instruments. The validation were required to assess the

appropriateness of the test items, sequencing and coverage of ratio concepts. The

experts certified that the instruments are valid for the purpose of the present

investigation. For the Ratio Interest Inventory, the comments of the validators

helped intriming down the checklist to 12 items as shown in appendix B, C and D

for the content validity, see the table of specification below. And the letter for

request for face validation of achievement test see Appendix L. Application for

permission to carry out experiment see Appendix M.

Table 2: Table of Specification for Content Validation on Ratio Achievement

Test

52

Ratio

Problem

Ability Process Dimension

Topics % Low Cognitive

Process (60%)

Higher

Process (40%)

Division of

ratio

30 4 3 7

Equivalent

ratio

30 4 3 7

Missing

addends

40 6 4 10

Total 100 24 16 24

Reliability of instrument

53

The researcher carried out a trial testing of the Achievement Test Ratio to

estimate the internal consistency and reliability coefficient of the instrument. She

administered the instrument to 20 primary six pupils in a school that is not part of

the schools selected for the study. The essence of the trail testing was to:

(a) Identify the extraneous variable that could disrupt the study.

(b) To find out if time allotted for the test was enough.

(c) To find out low easy or hard students will find the items.

Crookback Alpha (α) was used to estimate the internal consistency and

reliability coefficient 0.70 was got for the Achievement Test Ratio (ATR).

and 0.65 was got for the Ratio Interest Inventory (RII).

Experimental procedure

Classes in each school were randomly assigned to both experimental and

control group. Those in the experimental group were taught ratio following the

graphical approach while those in the control group were taught following the

conventional/traditional approach.

Two graduate teachers of mathematics education that have been teaching in

the primary school for at least three years were co-ordinated and used for the study.

The lesson notes for both experimental and control groups were used for the co-

ordination.

54

The researchers with the help of the research assistants visited the schools

and administered the tests to the pupils. The time allocated for the test was 1 hour.

The test were scored following the marking scheme on Appendix I.1, I.2, I.3, I.4,

J.1, J.2, J.3, J.4, K.1, K.2, K.3, K.4.

Lesson Notes

Six lesson notes were prepared and used for the study. Three lesson notes are for

the control group following the traditional approach, while the other three are for

the experimental group following the graphing symbol approach. See appendix F1,

F2, F3, G1, G2, G3, H1, H2, H3.

Method of data analysis

Means score and standard deviation was used in answering the research

questions. Research hypothesis were tested using the analysis of covariate

(ANCOVA) at P<.05. The pretest scores were used as covariate to the post test

scores.

55

CHAPTER FOUR

RESULTS

This chapter presents the results of the study. This is presented according to

research questions and hypothesis.

Research Question 1

What is the mean achievement score of the pupils taught ratio with graphical

– symbol approach and those taught with conventional approach?

TABLE 3: Mean and Standard Deviation of pupils achievement score of subjects

on ratio test.

PRE TEST POST TEST

GROUP MEAN STANDARD

DEVIATION

MEAN STANDARD

DEVIATION

MEAN

GAIN

EXPERIMENTAL

(NO = 60)

75.71 16.98

79.03 16.59 3.32

CONTROL

(NO = 60)

53.82 14.57

55.57 13.60 1.75

DIFFERENCE IN

MEAN

21.89 23.46

56

Table 3: The result of the pupils achievement was presented in table 3 and it

was observed that for the pretest administered on RAT. The experimental group

taught using graphical-symbol approach had a mean of 75.71 with a standard

deviation 16.98. . While the control group taught with conventional approach had a

mean of 53.82 and standard deviation of 14.57. In the posttest, administered on

RAT, the experimental group taught using graphical- symbol approach had a mean

of 79.03 and standard deviation of 16.59 while control group had a mean of 55.57

with the standard deviation of 13.60.

The experimental group had a mean gain of 3.32 while the control group had

a mean gain of 1.75. The mean difference of pre and post tests were 21.89 and

23.46 respectively.

Table 3 shows that there is a great difference in the mean achievement of pupils

taught using graphical-symbol approach, and those taught using convention

approach as their difference recorded 21.89 and 23.46 respectively.

Research Question 2

What is the mean interest rating of pupils taught ratio with graphical symbol

approach and those taught with conventional approach?

TABLE 4: Mean and Standard Deviation of pupils interest rating score of subjects

in ratio inventory.

57

PRE TEST POST TEST

GROUP MEAN STANDARD

DEVIATION

MEAN STANDARD

DEVIATION

MEAN

GAIN

EXPERIMENTAL

(NO = 60)

17.77 4.48 42.38 3.17 24.61

CONTROL

(NO = 60)

17.23 4.41 30.73 7.48 13.50

DIFFERENCE IN

MEAN

00.54 11.65

Table 4: The result of the pupils interest rating is presented in table 4 and it

was observed that for the pre test administered on R11. The experimental group

taught using graphical-symbol approach had a mean of 17.77 and standard

deviation of 4.48 where as their counter parts taught using conventional approach

had a mean of 17.23 and standard deviation 4.41. For the posttest, experimental

group taught using graphical symbol had a mean of 42.38 with the standard

deviation of 3.17 while the control group taught using conational approach had a

mean 30.73 with standard deviation of 7.48. The mean gain of both the experiential

and control groups is recorded 24.61 and 13.50 respectively.

The mean differences of pre and post test of the groups is recorded 0.54 and

11.65, which indicates that there is a great difference mostly in the posttest,

58

In the mean interest rating of pupils taught ratio with graphical-symbol approach

and those taught with commotional method

Research Question 3

To what extent does gender influence the mean achievement score of the

pupils taught ratio using graphical – symbol approach?

TABLE 5: Mean achievement score and Standard Deviation of males and

females in ratio test.

PRE TEST POST TEST

SEX MEAN STANDARD

DEVIATION

MEAN STANDARD

DEVIATION

MEAN

GAIN

MALE =

( NO 60)

67.96 15.91 70.36 15.74 2.40

FEMALE =

(NO 60)

61.57 21.69 64.23 21.76 2.66

DIFFERENCE

IN MEAN

6.39 6.13

Table 5: Shows that for the pretest administered on RAT. The male pupils

taught using graphical symbol had a mean of 67.96 with the standard deviation of

15.91 where as the female pupils taught using graphical-symbol approach had a

mean 61.57 and standard deviation of 21.69.

59

For posttest, the male pupils had a mean of 70.36 with the standard devotion

of 15.74 while their female counterparts had a mean of 64.23 and standard

deviation of 21.76. The mean gain of the pre and post tests were 2.40 and 2.66

respectively.

Table 5 indicates that there is a little difference in the mean achievement

score of male and female pupils taught ratio with graphical-symbol approach as

their difference is recorded as 6.39 and 6.13 respectively

Research Question 4:

To what extent does gender influence the interest rating of pupils taught

ratio using graphical symbol approach?

TABLE 6: Mean and Standard Deviation of male and female pupils in ratio

interest inventory.

PRE TEST POST TEST

SEX MEAN STANDARD

DEVIATION

MEAN STANDARD

DEVIATION

MEAN

GAIN

MALES

(NO 60)

18.03 5.28 37.58 7.40 19.55

FEMALES

( NO 60)

16.97 3.36 35.53 8.83 18.56

DIFFERENCE

IN MEAN

1.06 2.05

60

Table 6: The result presented on pretest administered on R11. The male and

female pupils taught using graphical-symbol approach had the means 18.03 and

16.97 respectively with the standard deviation 5.28 and 3.36 respectively.

In posttest, male pupils had a mean of 37.58 with the standard deviation of

7.40 and female pupils had a mean of 35.53 and standard deviation of 18.56. Their

mean gains were 19.55 and 18.56 where as their mean difference of pre and post

test were 1.06 and 2.05.

Table 6: indicates that there is a little differences between the mean interest rating

of male and female pupils taught using graphical-symbol approach as their

difference is observed as 1.06 and 2.05 respectively.

Hypothesis 1

There is no significant difference between the mean achievement score of

pupils taught ratio with graphical – symbol approach and those taught with

conventional method.

TABLE 7: Analysis of Covariance for Hypothesis 1 and 3 on pupils achievement

score in ratio

61

SOURCE TYPE III

SUM OF

SQUARE

DF MEAN

SQUARE

F SIGNIFICANCE DECISION

AT 0.05

LEVEL

CORRECTED

MODEL

40220.025 4 10055.006 336.365 .000 S

INTERCEPT 575.577 1 575.577 19.255 .000 S

PRE TEST 22541.911 1 22541.911 754.084 .000 S

GROUP* 306.528 1 306.528 10.254 .002 S

SEX* 5.211 1 5.211 .174 .007 S

GROUP *

SEX

24.885 1 24.885 .832 .363 NS

ERROR 3437.707 115 29.893

TOTAL 587127.667 120

CORRECTED

TOTAL

43657.732 119

a R squared = .921 (adjusted R squared = .919)

Table 7: shows the ANCOVA results of pupils who were taught ratio with

graphical -symbol approach and conventional approach. From the table the group

62

has significant value of .oo2 while F-cal = 10.254. This shows that the null

hypothesis of no significant difference in the pupils achievement was rejected.

Hypothesis 3

There is no significant difference between the mean achievement score of

male and female pupils taught ratio with graphical – symbol approach.

With regards to table 7, the ANCOVA result also indicate on sex value that

F-cal =.174 and is more than the significant level which is .007 at α = 0.05.

Therefore the null hypothesis of no significant difference for gender was rejected

Hypothesis 2

There is no significant difference in the mean interest rating of pupils taught

ratio with graphical - symbol approach and those taught with conventional method.

Table 8: Analysis of Covariance for Hypothesis 2 and 4 on pupils and

gender interest rating.

S0URCE TYPE III

SUM OF

SQUARES

DF MEAN

SQUARE

F SIGNIFICANCE DECISION

AT 0.05

CORRECTED

MODEL

4418.584 4 1104.646 35.855 .000

INTERCEPT 8961.319 1 8961.319 290.869 .000

PRE

INTEREST

4.826 1 4.826 .157 .693

63

GROUP 4039.425 1 4039.425 131.113 .000

SEX 118.340 1 118.340 3.841 .002

GROUP*SEX 220.121 1 220.121 7.145 .009

ERROR 3543.007 115 30.809

TOTAL 168343.000 120

CORRECTED

TOTAL

7961.592 119

a R squared = .555 (adjusted R squared = .540)

From table 8, the result of ANCOVA indicate that the interest rating of

group at F-cal = 131.11 which is more than F-critical which has the value of 0.000.

at α level of 0.05. Following the decision rule, since F-cal > F-critical therefore the

null hypothesis of no significance was rejected. We therefore conclude that the

mean interest rating of pupils taught ratio with graphical - symbol approach is

statistically significant

Hypothesis 4

There is no significant difference in the mean interest rating of male and

female pupils.

Table 8: shows that F-cal of sex which is 3.841is more than F-critical which

is 0.002 at α = 0.05. This shows that the interest rating of male and female pupils

64

taught ratio with graphical – symbol approach is statistically significant. Therefore,

the null hypothesis of no significant difference of gender was rejected.

SUMMARY

In summary, the following observations were made from the findings:

1) There is a significant difference between the achievement score of pupils

taught ratio with graphical – symbol approach and those taught with

conventional method.

2) The pupils interest was highly aroused in ratio learning with the use of

graphical – symbol approach.

3) Achievement was greatly improved by the use of graphical – symbol

approach on both male and female pupils in ratio learning.

4) Both male and female pupils developed more interest in ratio learning with

the use of graphical – symbol approach, than the conventional approach.

65

CHAPTER FIVE

DISCUSSION, CONCLUSIONS, RECOMMENDATION AND SUMMARY

OF THE STUDY

In this chapter the discussion, conclusion, recommendation and summary of

the study from the study.

Discussion:

The findings of this study showed that there was a significance difference

between the mean achievement score of pupils taught ratio with graphical-symbol

approach and those taught with conventional method.

The pupils taught ratio with graphical-symbol approach performed better

than those taught with conventional method. It has also proved that manipulative

materials enhance mathematics learning. This study is similar to the early research

of Ohuche (1990), who investigated on the use of graph in teaching of some

concept in mathematics. The concepts included area, fraction and proportion. The

study showed that the use of graph enhanced the learning of some mathematics

concepts.

The study is similar to the study of Ogwuche (2002) who investigated age

and sex as correlates of logical reasoning and mathematics achievement in ratio

and proportion tasks. The finding indicated that male students perform better than

females.

66

Also in this study, the Piagetian theory on learning and instructional

materials was also proved. Based on the theory, primary school pupils who are at

concrete operational age learn more by doing and seeing than by listening only.

This study showed and supported the initial ideas which emphasized that

mathematics teaching and learning mostly at primary school needs to be taught

with relevant instructional materials in order to help the pupils develop interest in

mathematics learning.

Conclusion:

From the result of this study, the following conclusions were made:

1. Pupils learn mathematics more by the use of manipulative materials.

Because the performance of the pupils who were taught ratio with graphical-

symbol approach was significantly better than their counterparts who were

taught with conventional method, when the ratio achievement tests were

administered.

2. There was also a significant difference in the mean interest score and rating

between those pupils taught ratio with graphical-symbol approach and those

taught with conventional method.

3. It was observed that males performed better than females when graphical-

symbol approach was used in teaching of ratio despite that males do better

than females, it was observed that females taught ratio with graphical

67

symbol approach performed better than their counterparts who where taught

with conventional method.

Therefore, graphical symbol approach has a positive affects on the

interest of females in ratio learning.

4. It was also observed that graphical-symbol approach motivated the interest

of pupils in ratio and mathematics learning as a whole. Females also were

motivated more through graphical-symbol approach in mathematics

learning.

Implications:

From the findings of this study, it was very clear that those pupils who were

taught ratio with graphical-symbol approach performed better than those who were

taught with conventional method. That showed that manipulative materials are

very important in teaching and learning of mathematics mostly at primary level.

The implication of this is that primary school teachers who are more of

female teachers should use adequate approaches in teaching any mathematics

concept to enable the pupils learn and develop interest in mathematics learning and

also to help females who are not lovers of mathematics to develop interest in

mathematics and mathematics related subjects.

Educational planners should develop mathematics curriculum that will

emphasized more in the use of different approaches such as graph-symbol

approach in teaching of any mathematics concept. They should also visit and

68

supervise teachers when they are teaching mathematics in classrooms. They should

provide suitable instructional materials for schools mostly the primary schools.

It was observed that pupils dislike mathematics because of poor method of

teaching and lack of adequate approaches in teaching of some abstract concept

such as ratio. For the teachers to make their mathematics teaching enjoyable,

models should be used in teaching of mathematics to make the pupils active

learners not passive learners. Some abstract concepts like ratio and others should

be taught with manipulative materials.

Limitation of the study:

This study experienced some limitation

(1) Because of the introduction of private schools in Nigeria today, public

schools are mainly for poor people and maids who are not mainly

financially up-to-date. Such pupils find it difficult to provide graph-sheets

and other school material for their study.

(2) The study faced some problems based on poor environmental factors.

Government neglect public school, so, because of that there were no good

blackboard, tables, chairs, desks for the children to have a comfortable

seating arrangement. That also affected the study.

69

Recommendation:

The following recommendations have been made in this study,

(1) From the findings of this study, the mathematics educators (N.C.E.

teachers) should teach mathematics in the primary school, as specialist

teachers. This is because the primary school teachers who are mainly

female teachers do not have good mathematics background, because of

that, some of them cannot use some adequate approaches correctly and

some do not know how to use some material such as graph, in teaching of

mathematics.

(2) The government should provide instructional materials, mathematics

textbooks, graph sheets, workbooks for the pupils, and teachers manual for

primary schools for effective teaching and learning of mathematics.

(3) Mathematics work-shops, seminars and conferences should be organized

frequently for primary school teachers by the professional bodies like

Mathematics Association of Nigeria (MAN), Science Teacher‟s

Association of Nigeria (STAN). To enable them learn some new

approaches in teaching of mathematics.

(4) To enable them learn some new approaches in the mathematics teaching

system. In-service training should be encouraged for mathematics teachers

from N.C.E. level to Ph.D. level to enable mathematics teachers to up-date

70

themselves. So that the teachers will be exposed to the new innovational

approaches in teaching of mathematics.

(5) Government should provide Audio-visual aids such as power point,

computer, video to schools for mathematics teaching and learning.

Suggestions for further research:

The following suggestions have been made for further studies.

(1) The effect of graphical-symbol approach in base-system.

(2) With the use of computer in the society. It was discovered that children are

now more exposed than before. Some topics, in the junior secondary

schools mathematics curriculum such as Quadratic Equation, factorization

can be introduced in the primary school mathematics curriculum to find out

how it will work out.

Summary of the study:

(1) The study was on effect of graphical-symbol approach in achievement of

ratio in upper primary school, some schools were randomly selected in

Nsukka Central Local Government Area of Enugu State.

Basically, it was carried out to find out the effect of models and

manipulative materials with special reference to graph in teaching and

learning of mathematics in primary schools, specifically on primary six

pupils. It was undertaken to find out in particular the effect of graphical-

71

symbol approach in ratio concepts in upper school. Gender was also taken

into consideration.

The study was also used to find out the extent on pupils performance

in the test of ratio achievement test and interest rating on pupils attitude

toward‟s mathematics learning. Four research questions and four research

hypothesis were used to carry out the study.

(2) The literature review for the study was carried out on relevant areas under

two headings – Theoretical and Emperical studies on model, modeling and

graphical representation in primary school mathematics learning. From the

review of the related literatures it was discovered that manipulative

materials enhance the teaching and learning of mathematics in primary

schools.

(3) The population of the study consists of 1067 primary six pupils in the 46

primary schools in Nsukka Central Local Government Area of Enugu State.

The total number of schools randomly selected were 2. Intact class of 60

primary school pupils from each school were used. In the schools selected,

one intact class was used where one school serves as experimental group

and the other as control group.

The design of the study was Quasi – experimental research design.

Twenty-four tests were set from three types of ratio concepts and

administered.

72

(1) Division of ratio

(2) Equivalent ratio

(3) Missing addends in ratio relationship.

Twelve checklists were used to find out the attitude of the pupils towards

mathematics and learning in primary schools.

The data were collected and analyzed using mean score and standard

deviation in answering the research questions. Research hypothesis were tested

using analysis of covariate (ANCOVA) at p <.05. The pretest used as covariate to

the posttest scores. And posttest mean and standard deviation were used to draw

conclusion of the findings. And finally it was discovered that graphical-symbol

approach has a positive effect on pupils performance in ratio achievement tests.

73

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79

APPENDIX A

The distribution of the schools

Towns No of schools

Nsukka town

Lejja

Obukpa

Obimo

27

6

7

6

Total 46

80

APPENDIX B

INSTRUMENT I

DIVISION OF RATIO

1. Ngozi and Ada are to share some oranges in the r atio 4:5. Ngozi gets 12

oranges. How many oranges does Ada get?

2. Femi and Ayo are to share some apples in the ratio 3:1. Femi gets 12

sweets. How many apples does Ayo get?

3. The side or a rectangle are in the ratio 5:2. Find the longer side, when the

shorter side is 10cm.

4. Okon and Etteh are to share a sum of money in the ratio 1:3. If Etteh gets

N24.00. How much does Okon gets.

5. The sides of a rectangle are in the ratio 2:5. The longer side is 20cm. Find

the shorter side.

6. In a class the ratio of the number of boys to the number of girls is 3:4. There

are 18 boys in the class. How many girls are in the class?

7. The ratio of the length to the width of a rectangle is 2:3. The width is 12cm.

Find the length.

8. If a man shares his cows to his two sons in the ratio of 3:5. If the first son

gets 12 cows. How many cows has the second son?.

81

APPENDIX C

Instrument II

Equivalent Ratio

1. N5 to N10 =

2. 20k to N2.00 =

3. 8c to 12c =

4. 10mins to 1hr. =

5. 25cm to 100cm =

6. 4m to 16m =

7. 10k to N1 =

8. 6g to 10g=

: = : : : : = = =

: = : : : : = = =

: = : : : : = = =

: = : : : : = = =

: = : : : : = = =

: = : : : : = = =

: = : : : : = = =

: = : : : : = = =

82

APPENDIX D

INSTRUMENT III

Missing Addends

Find the Letters

(1) 8

6

4

x

(2) 305

2 y

(3) 405

4 a

(4) u

18

3

1

(5) 5

315

x (6) w

12

3

2

(7) 153

2 x

(8) y

24

14

3

83

APPENDIX E1

QUESTIONNAIRE

Sex

Male Female

1. I enjoy to solve ratio problems

2. I find ratio problem difficult

3. I prefer solutions only division ratio.

4. I have solving ratio missing addends

5. I do not enjoy solving problems on equivalent ratio

6. I enjoy solving all mathematics problems

7. Ratio problems demoralized my interest in mathematics

learning.

8. Ratio problems motivate me in mathematics learning

9. I enjoy learning of mathematics when it is been taught

with materials.

10. I dislike mathematics learning when the teacher uses

materials in teaching it.

11. I ignore mathematics learning whenever it involves ratio

problems

12. I enjoy mathematics learning more whenever the teacher

uses materials in teaching it.

SA A D SD

84

APPENDIX E2

TABLE OF SPECIFICATION

Ratio

Problem

Ability Process Dimension

Topics % Low cognitive

process (60%)

Higher

process (40%)

Division of

ratio

30 4 3 7

Equivalent

ratio

30 4 3 7

Missing

addends

40 6 4 10

Total 100 24 16 24

85

APPENDIX FI

A LESSON NOTE ON MATHEMATICS

Conventional Method

Class: Primary 6

Age: 11 years

Duration: 30 minutes

Date:

Topic: Division of ratio

Instructional Objectives: At the end of the lesson, the pupils are expected to:

solve at least five problems out of ten problems involving division of ratio

correctly.

Entry Behaviour: It is assumed that the pupils have learnt addition, multiplication

and division of whole numbers.

Test of Entry Behaviour: Children solve the following sums.

(1) 24 + 6 = , (2) 7 x 9 = (3) 49 ÷ 7 =

Instructional Techniques: Explanation and illustration

Instructional Materials: Chalkboard

Step 1: The teacher will explain to the pupils, that ratio is a way of comparing two

or more things depending on the number of times one is bigger or smaller than the

other(s).

86

Example 1:

(1) Bola and Olu are to share some oranges in the ratio of 3:1. Bola got 12 oranges,

How many oranges did Olu gets.

Solution:

3:1, 6:2, 9:3

or

:- Olu = 4414

1

12

3

1xx

:- Olu will get 4 oranges.

Step III: The teacher will set some sums on the board and ask the pupils to come

to the board one after the other and solve them.

Evaluation: Some written exercise will be given to the pupils from the lesson

taught in order to assess the lesson.

Exercise:

(1). A woman shared some oranges to Ada and Uju in the ratio of 3:4, Uju got 16

oranges. How many oranges did Ada get?

(2). In our class our teacher shared pencils to the boys and girls in the ratio of 3:5

and the boys received 21 pencils. How many pencils did the girls received?

(3). A man shared some coconut to his two sons in the ratio of 2:3. The first son

got 12 coconuts. What will be the share of the second son?

4

87

APPENDIX F2

A LESSON NOTE ON MATHEMATICS

Graphical-symbol approach

Class: Primary 6

Age: 11 years

Duration: 30 minutes

Date:

Topic: Division of ratio

Instructional Objectives: At the end of the lesson, the pupils are expected to

solve problems on division of ratio by use of graphical-symbol approach.

Entry Behaviour: It is assumed that the pupils have learnt division of ratio by

conventional method.

Test of Entry Behaviour: Divide 12 oranges to Ada and Uche in the ratio of 1:3.

Solution: Total ratio = 4. Ada = ;3

1

12

3

1x

Uche = ;9

1

12

4

3x

Instructional Materials: Graph-sheet, ruler, pencil and eraser.

Instructional Techniques: Demonstration, explanation and illustration

Instructional Procedure:

Step I: The teacher will introduce a new method in solving ratio. The use of

graphical-symbol approach.

Example 1: Bola and Olu are to share some oranges in the ratio of 3:1. Bola gets

12 oranges. How many oranges will Olu get?

3 3

88

Solution:

Let y – axis = Bola, and x – axis = Olu.

See the Graph-sheet I.

Step II: Some exercises will be given to the students to solve on the board. They

will be solving the problems one after the other on the board while the teacher will

be guiding them where necessary.

Example II: John and Ann share some oranges in the ratio of 3:1, John gets 12

oranges. Find the number of oranges Ann will get.

Example III: A farmer shares some cows between his two sons, Aliyu and Madaki

in the ratio of 6.5. Aliyu gets 30 cows. How many cows Madaki gets.

Step III: The teacher will guide the children on how to plot the graph.

Evaluation: Some written exercises would be given to the pupils in order to assess

the lesson taught. Graphical-symbol approach is to be used to solve the problem.

Exercise:

(1). A woman shared some oranges to Ada and Uju in the ratio of 3:4, Uju got 16

oranges. How many oranges did Ada get?

(2). In our class our teacher shared pencils to the boys and girls in the ratio of 3:5

and the boys received 21 pencils. How many pencils did the girls received?

(3). A man shared some coconut to his two sons in the ratio of 2:3. The first son

got 12 coconuts. What will be the share of the second son?

89

Bola

x

x

x

x

x

APPENDIX F3

Graph I: Division of ratio

Let y axis = Bola

Let x axis = Olu

Summary

Olu got 4 oranges when Bola has 12 oranges.

15

12

9

6

3

0 1 2 3 4 5

Olu

90

APPENDIX GI

Conventional Method

A LESSON NOTE ON MATHEMATICS

Class: Primary 6

Age: 10years

Duration: 30 minutes

Date:

Topic: Equivalent ratio

Instructional Objectives: At the end of the lesson, the pupils are expected to find

out equivalent ratio of the given ratios correctly.

Entry Behaviour: It is assumed that the pupils have learnt fraction and decimal.

Test of Entry Behaviour: Change the following to decimal 1000

72,

100

1,

10

7

Instructional Techniques: Explanation and Illustration

Instructional Materials: Chalk-board

Instructional Procedure

Step I: The teacher will explain the meaning equivalent to the pupils-Equivalent

means the same.

Step II: Problem solving-Find three equivalent rations

(1) 1:2

(2) 5:10

We solve it by multiplying or dividing each part by the same non-zero number.

91

Example

Solution

50:25

5

10:5

40:20

4

10:5

30:15

3

10:5

20:10

2

10:5

Equivalent ratios of 5:10 = 10:20 = 15:30, 20:40 = 25:50.

Step III: Pupils activity:- The teacher will give the pupils some problems to solve

on the board. Example. Find the first three equivalent ratios of (a) 2:3 (b) 1:3

(c) 3:4

Evaluation: The teacher will assess the lesson taught by giving the pupils some

exercise based on the lesson.

Find the first five equivalent ratios of the following.

(1). 3:5 = : = : = : = : = :

(2). 4:7 = : = : = : = : = :

92

APPENDIX G2

A LESSON NOTE ON MATHEMATICS

GRAPHICAL-SYMBOL APPROACH

Class: Primary 6

Age: 10years

Duration: 30 minutes

Date:

Topic: Equivalent ratio

Instructional Objectives: At the end of the lesson, the pupils are expected to:

solve at least five questions correctly from the eight given problems on equivalent

ratio with the use of graphical-symbol approach.

Entry Behaviour: It is assumed that, the pupils have learnt conventional method

in solving equivalent ratio.

Test of Entry Behaviour: Find three equivalent ratios of the following

1:2 = 2:4 = 3:6 = 4:8.

Instructional Techniques: Graphical – illustration

Instructional Materials: Graph-sheet, pencil, ruler.

Instructional Procedure:

Step I: The teacher will explain the meaning of equivalent ratio to the pupils. The

teacher will also explain to the pupils that the multiples of the given numbers to be

used to find the equivalent ratio.

93

Example 1:

1:2 = 2:3 = 3:6 = 4:8.

Step II: The teacher will illustrate equivalent ratio graphically.

Example:

Find the first five equivalent ratio of 5:10

See Appendix G3.

Step III: The teacher will appoint the pupils one after the other to come and solve

the problems on the board. They will be given the graph-sheets and also be asked

to represent the information graphically.

Problems: Find the three equivalent ratio of

(a) 2:3 (b) 1:3

Evaluation: The teacher will give the pupils some problems on equivalent ratio to

solve graphically.

Exercise: Find the 1st five equivalent ratios of the following

(1). 3:5 = : = : = : = : = :

(2). 4:7 = : = : = : = : = :

94

Multiples of 5

x

x

x

x

x

x

APPENDIX G3

Graph 2: Equivalent ratio

Let y axis = multiples of 5

x axis = multiple of 10

Summary

The first fifth equivalent ratios are illustrated graphical 5:10, = 10:20, 15:30

= 20:40 = 25:50.

30

25

20

15

10

5

0

10 20 30 40 50 60

Multiples of 10

95

APPENDIX HI

A LESSON NOTE ON MATHEMATICS

Conventional Method

Class: Primary 6

Age: 11 years

Duration: 30 Minutes

Date:

Topic: Missing Addends in ratio relationship

Instructional Objectives: At the end of the lesson, the pupils are expected to be

able to find at least two out of the three numbers (addends) in ratio form.

Entry Behaviour: It is assumed that the pupils have learnt to find missing

numbers in simple addition.

Test of Entry Behaviour: Find the missing numbers

(1) + 3 =

(2) 5 + =

Instructional Techniques: Illustration and explanation

Instructional Materials: Chalkboard

Instructional Procedures:

7

12

96

Step 1: The teacher will explain to the pupils the meaning of missing number.

There are numbers we do not know. It is being represented with alphabets such as

x, y, m, r, etc.

Step II: Problems. Find the ratio of (1) ,

8

6

4

x

(2) 305

2 x

the teacher will

explain to them that, to find the missing number, one has to check the given

number.

Example II:

What was that number used to multiply 4 that gives us 8? The number is 2. Then

what is that number that x represents so that if you use 2 to multiply it you get six.

The number is 3 x = 3

Examples:

(b) ,

18

9

6

y (c) ,

25

15

5

x

Find the missing addends. The teacher will also solve

them on the board

Step III: Pupils activity:- The teacher will allow the pupils to ask questions where

necessary and also give them some problem on the board to solve.

Evaluation: The teacher will assess the lesson taught with some exercise.

15·5 3

255 5 (b)

6x3 18

9x3 27 (a)

97

Exercise: Find the missing numbers which are represented with letters.

1. a = 21

7 49

2. 4 = 24

12 y

3. 6 = 18

W 27

98

APPENDIX H2

A LESSON NOTE ON MATHEMATICS

GRAPHICAL-SYMBOL APPROACH

Class: Primary 6

Age: 11 years

Duration: 30 minutes

Date:

Topic: Missing Addends in ratio relationship

Instructional Objectives: The end of the lesson, the pupils are expected to:

(1) Find the missing numbers in ratio relationship with the use of graphical-symbol

approach.

Entry Behaviour: It is assumed that the pupils have learnt missing addends in

ratio relationship by conventional method. And will also have the idea of factors

and multiples.

Test of Entry Behaviour: Find the missing numbers

(a) ,

8

6

4

x

(b) y

18

9

6

Solution: (a) ,

8

6

4

3

(b) 27

18

9

6

Instructional Techniques: Graphical illustration

Instructional Materials: Graph-sheet, ruler, pencil, eraser.

Instructional Procedures:

99

Step 1: The teacher will explain to the pupils that missing addends can also be

found through graphical representation. Example:

Step II: Problems solving (a) ,

8

6

4

y

(b) 305

2 x

she will explain to them that the

factors of the given numbers are to be used to find the missing number if the

missing number is to be less than the given numbers while the multiples of the

given numbers are to be used to find the missing number if the missing number is

to be greater than given numbers.

Example II:

(a) ,

18

9

6

y (b) 25

15

5

x

, Find the missing addends. With the use of graphical-

symbol approach, see Appendix H3

Step III: The teacher will set some problems on ratio and appoint the pupils one

after the other to come out and solve them on the board while other pupils will be

listening and the teacher will be guiding the pupils on how to do it.

Evaluation: The teacher will give the pupils some written work based on the

lesson taught in order to assess the lesson taught. (See instrument III Appendix I).

The teacher will mark and score the children.

Exercise: Find the missing numbers which are represented with letters.

1. a = 21 2. 4 = 24

12 y 7 49

3. 6 = 18

W 27

100

x

x

x

x

APPENDIX H3

Graph 3: Missing addends in ratio relationship

Let y axis = Factors of 6

Summary

From the graphical illustration, when x = 4, y = 3.

6

3

2

1

0 1 2 4 8

8 Let x axis = Factors of 8

6

101

APPENDIX H4

TABLE OF SPECIFICATION

Ratio problem Ability process dimension

Topics % Low cognitive process

(60%)

Higher process

(40%)

Division of ratio 30 4 3 7

Equivalent ratio 30 4 3 7

Missing

addends

40 6 4 10

Total 100 24 16 24

102

APPENDIX I.1

MARKING SCHEME

Division of ratio

1.

2.

3.

12

8

4

0

5 10 15

x

x

x

Ngozi

Ada

12

9

6

3

1 2 3 4

x

x

x

Femi

Ayo

x

0

25

20

15

10

5

2 4 6 8 10

x

x

x

Longer

side

Shorter

side

x

0

x

Ayo will get 4 apples

Ada will get 15 oranges

The longer side will be

25cm

Ada will get 15 oranges

103

Okon will get N8

4.

5.

20

15

10

5

2 4 6 8

x

x

x

Longer side

Shorter side

x

0

The shorter side will 8cm

I. 2

Etteh

Okon

3 6 9 12 15 18 21 24

x

x

x

x

0

x

x

x

x

104

6.

7.

8

6

4

2

3 6 9 12

x

x

x

Length

Width

x

0

18

15

12

9

6

3

0

4 8 12 16 20 24

Boys

Girls

x

x

x

x

x

x

There are 24 girls

The length will be 8cm

I.3

105

8.

12

9

6

3

5 10 15 20

x

x

x

1st son

2nd

son

x

0

I.4

The Second Son

will get 20 cows

106

APPENDIX J.1

Equivalent Ratios

1.

2.

25

20

15

10

5

0

10 20 30 40 50

Naira

Naira

x

x

x

x

x

100

80

60

40

20

0

200 400 600 800 1000

Kobo

Naira

x

x

x

x

x

5:10 = 10:20 = 15:30 = 20:40, 25:50

20k:N200, 40k: N400, 60k: N600, 80k:

N800, 100k: N1000

J.1

107

3.

4.

40

32

24

16

8

0

12 24 36 48 60

Liter

Liter

x

x

x

x

x

50

40

30

20

10

0

1 2 3 4 5

Minutes

Hours

x

x

x

x

x

J.2

8L:12L = 16L:24L =24L:36L = 32L:48L =

40L:60L

10mins:1hr, 20mins:2hrs, 30mins:3hrs,

40mins:4hrs, 50mins:5hrs

108

6.

125

100

75

50

25

0

100 200 300 400 500

cm

cm

x

x

x

x

x

20

16

12

8

4

0

16 32 48 64 80

Meter

Meter

x

x

x

x

x

J.3

25cm:100cm = 50cm:200cm =

75cm:300cm = 100cm:400cm =

125cm:500cm

4m:16m = 8m:32m = 12m:48m =

16m:64m = 20m:80m

109

7.

8.

50

40

30

20

10

0

1 2 3 4 5

Kobo

Naira

x

x

x

x

x

30

24

18

12

6

0

10 20 30 40 50

Grammes

Grammes

x

x

x

x

x

J.4

20k:N1, 20k: N2, 30k: N3, 40k: N4, 50k: N5

6g:10g = 12g:20g = 18g:30g =

24g:40g = 30g:50g

110

3

1

APPENDIX K

K.1

Missing Adding

1.

x = 6

2.

y = 12

6

2 4 8

Unknown axis

Known axis

x

x

x

12

10

8

6

4

2

0

5 10 15 20 25 30

Unknown axis

Unknown axis

x

x

x

x

x

111

K.2

3.

a = 32

32

24

20

16

12

8

4

0

5 10 15 20 25 30 35 40

Unknown axis

Unknown axis

x

x

x

x

x

28

x

x

x

112

Unknown

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

3

6

9

12

15

18

21

24

27

30

33

36

39

41

45

48

51

54

K.3

4..

U = 54

Known axis

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

113

K.4

5.

= 15

6.

15

12

9

6

3

0

5 10 15 20 25

x

x

x

x

x

Unknown axis

18

15

12

9

6

3

0

2 4 6 8 10 12

Unknown

x

x

x

x

x

x

Known axis w = 18

114

Known

Known

98

112

K.5

7.

x = 10

8.

Unknown

84

70

56

42

28

14

0

y = 112

10

8

6

4

2

0

3 6 9 12 15

x

x

x

x

x

14

Unknown

x

x

x

3 6 9 12 15 18 21 24

x

x

x

x

115

APPENDIX L

Department of Science Education

University of Nigeria,

Nsukka

The Head Master/Head Mistress

---------------------------------------

---------------------------------------

Sir/Madam

APPLICATION FOR PERMISSION TO CARRY OUT EXPERIMENT

I hereby apply for permission to carry out a mathematics experiment on ratio

with the grade six pupils in your school. I am a post-graduate student from the

above address and studying Mathematics Education. The research will last for two

weeks.

The purpose of the experiment is purely for research and not for the public use.

I will co-operate with rules and regulation of your school.

I will be glad if this request is granted.

Yours Faithfully

Nwoye M.N. (Mrs)

116

APPENDIX M

Department of Science Education

University of Nigeria,

Nsukka

24/4/2009.

Dear Prof/Dr./Mr/Mrs.

VETTING AND FACE VALIDATION OF ACHIEVEMENT TEST IN

MATHEMATICS

Kindly assist in the Validation of the following instruments for my research.

The instruments are:

(1) Ratio Achievement Test (RAT)

(2) Ratio Interest Inventory (RII)

(3) Making Guide

Content validation of RAT using test blue point. Criteria for face validation is

based on

(1) The appropriateness of the test items.

(2) Sequencing and coverage of ratio concepts and to certify that the instruments

are valid for the purpose of the present investigation.

I will be glad if you grant the request.

Yours Faithfully

Nwoye M.N. (Mrs)