UNIVERSITY OF IBADAN
THIS. ™ I S  SUBMITTED BY
_ _ DR. ADEBAYO, _ OLURANTI^ _ ifflEM ELE
WAS ACCEPTED IN PARTIAL FULFILMENT 
OF THE REQUIREMENTS FOR THE 
DEGREE OF DOCTOR. OF PHILOSOPHY 
IN THE FACULTY OF EDUCATION
OF THIS UNIVERSITY
THE EFFECTIVE DATE OF THE AWARD IS
15DK SEPTEMBER, 1995
UNWFRSt rv Of IBADAN.
UNIVERSITY OF IB
DAN LIBRARY
COMPARATIVE EFFECTIVENESS OF LECTURE AND MATERIAL-BASED
INTERACTIVE METHODS ON SELF-CONCEPT, TEST ANXIETY AND 
ACHIEVEMENT IN MATHEMATICS IN LAGOS STATE
BY
OLURANTI ADENRELE ADEDAYO 
B.Sc Ed/Mathematics (Ife) 
M.Ed Evaluation (Ibadan)
A THESIS SUBMITTED TO THE INTERNATIONAL CENTRE FOR 
EDUCATIONAL EVALUATION (ICEE), INSTITUTE OF EDUCATION 
UNIVERSITY OF IBADAN.
IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE
OF
DOCTOR OF PHILOSOPHY 
OF THE
UNIVERSITY OF IBADAN.
AUGUST 1995
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ABSTRACT
This study was carried out with the aim of 
investigating the effect of three instructional methods 
and gender on achievement, test anxiety and self-concept 
in mathematics of NCE year one Business Education 
Students. The design used was a 3 x 2 pre-test/post-test 
quasi-e>iperimental design.
A sample of 165 first year NCE Business Education 
Students comprising 71 males and 94 females, with mean 
age of 22.96 years, was used in this study. The subjects 
were all full-time students from two colleges of 
Education in Lagos State who registered for the 1992/93 
session.
A total of nine hypotheses were tested with respect 
to the interactive effect of instructional method and 
gender on each of the three dependent variables. The 
instructional methods occurred at three levels as 
follows: Interactive learning with individual use of 
materials, interactive learning with group use of 
material and the lecture method. The materials were 
prepared from the second semester business mathematics 
course outline.Five instruments were used for data
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collection. They were:
1. A multiple choice test in mathematics to test pre­
requisite skills.
2. A multiple choice post achievement test in 
mathematics covering the course objectives of the 
second semester business mathematics course.
3. The Inventory of Test Anxiety in Mathematics (ITAM) 
by Osterhouse,.
4. A mathematics self-concept scale.
5. A learning package.
Analysis of covariance was done for each of the 
three dependent variables. Where significant interaction 
was found the hypothesis on main effect was not tested. 
Rather, Scheffe multiple Range Test was used to identify 
the source of variation. In case of no interaction, 
Multiple
Classification Analysis was used to determine the 
magnitude and direction of the effect as well as the 
amount of variation accounted for by each variable.
The result showed significant interactive effect of 
gender and treatment on mathematics achievement. Inter­
active learning with individual use of materials
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favoured females while interactive learning with group 
use of materials favoured males. There was no 
interactive effect of gender and instructional method on 
the other two dependent variables. However, with test 
anxiety as the dependent variable, it was found that 
instructional method had significant effect, with 
students exposed to the lecture method having the 
highest anxiety level when compared with others. The 
hypothesis on self-concept showed that gender had 
significant effect, favouring females. Instructional 
method and gender accounted for a total variance of 
24.3% in mathematics test anxiety and 22.8% in self 
concept in mathematics.
Recommendations are made as to how interactive 
learning could be used to meet students individual needs 
in the teaching of tertiary mathematics.
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DEDICATION
TO MY LATE FATHER PA SAMUEL ADEBIYI OLALEMI 
Who desired to witness the completion of this study
and
N M A M I ,
My darling mother, whose encouragement contributed to 
the success of this work.
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ACKNOWLEDGEMENT
I give glory to the Almighty God for giving me the 
opportunity of beginning and completing this work. At 
every stage of the work, I could feel His divine 
assistance.
My sincere and profound gratitude goes to my 
indefatigable and able supervisor, Dr. C. 0. Onocha, 
whose expert advice and patience were quite helpful in 
streamlining the report of this thesis as well as 
ensuring a successful defence of the thesis. I also 
appreciate the invaluable contributions of Dr.(Mrs) 
Chacko my first supervisor, who was always available and 
willing to offer assistance especially at the initial 
stage of this study. The kindness, patience and 
thoroughness of these twosome guided me through the 
genesis and completion of this work. My heart’s desire 
is for God’s abundant blessings on them. I have 
thoroughly enjoyed working under their guidance.
I am grateful to Professor Yoloye of Amoye 
Institute, who not only contributed to the development 
of this thesis, but made his well-stocked library 
available for my use. I quite appreciate the
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constructive criticisms made on the proposal of the 
study. The constant encouragement given to me by his 
wife, Dr (Mrs) Yoloye is highly appreciated.
I am also very grateful to the academic staff of 
the Institute of Education, who gave their expert advice 
on the original proposal of this study. Special thanks 
go to Dr Yoloye, our seminar coordinator, 
Professors Obemeata and Onibokun, Dr Ogunranti and Dr 
Okpala. Words cannot express my profound gratitude to 
Professor Falayajo, the director of the Institute, who 
gave expert advice on the analysis of the data.
I thank the Provosts of the Federal College of 
Education (Technical), Akoka Lagos, Mr Auta and Lagos 
State College of Education, Ijanikin, Dr Carew for their 
cooperation in the collection of data for this study.
The prayers, love and concern of friends and 
colleagues are greatly appreciated. I owe special 
thanks to the Amao-Kehindes, the Adewoles, the Adenibas, 
Mr Henry Owolabi, Mr Ikwuagu, Mrs Oni and Mrs Keshinro. 
Special thanks go to Yemisi Komaiya and Kehinde Bello.
Family members have also been of tremendous 
assistance. I thank Reverend and Mrs Lasebikan of All
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Souls Vicarage, Bodija, Ibadan who were kind enough to 
accommodate me throughout the period of study. My 
sisters, Mrs Babatunde and Mrs Durojaiye were always 
there to give their moral support. To Bola Foye, Kemi 
Foye, Victoria Kolade and Bisi Aina who held the home 
fort while I was away, I say a big thank you. I am 
grateful to my children, Tope, Yemi, Bayo, Temilade and 
Adeola for their moral and spiritual support. Finally, 
I thank my dear husband, Bisi, for providing the needed 
moral and financial support.
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CERTIFICATION
I certify that this study was carried out by 
Mrs. Oluranti Adenrele Adedayo at the International 
Centre for Educational Evaluation, Institute of 
Education, University of Ibadan.
DR C. 0. ONOCHA
B.Sc (Hons), M.Ed.,Ph.D.(Ibadan) 
Senior Research Fellow 
Institute of Education 
University of Ibadan.
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CONTENTS
Chapter one: Introduction ........................  1
1.1 Background to the problem...............  1
1.2 Statement of p r o b l e m ...................  12
1.3 Research questions ...................... 13
1.4 Statement of hypotheses.................  15
1.4.1 Achievement as the dependent variable . . 15
1.4.2 Mathematics test anxiety as the dependent
variable..............................  15
1.4.3 Mathematics self-concept as the dependent 
variable
.........................................  16
1.5 Significance of the s t u d y ...............  16
1.6 Scope of s t u d y ..........................  19
1.7 Definition of t e r m s ...................... 2 0
Chapter two: Review of literature ................  23
2.1 Introduction............................  2 3
2.2 The lecture m e t h o d ...................... 23
2.2.1 The lecture method versus other methods of
instruction..............................  28
2.3 Mathematics achievement, instructional
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techniques and gender ................. 4 4
2.4 Mathematics test a n x i e t y ............... 49
2.4.1 Mathematics test anxiety and achievement 54
2.4.4 Test anxiety and instructional strategy . 56
2.4.3 Gender and mathematics test anxiety . . .  58
2.5 Mathematics self c o n c e p t ............... 59
2.5.1 Academic self-concept and achievement . . 63
2.5.2 Academic self-concept and instructional method .. 66
2.5.3 Self-concept and gender................. 67
2.6 Summary ................................... 68
Chapter three: Research design and methodology . 72
3.1 Introduction...........................  72
3.2 Research design.........................  73
3.2.1 Variables in the s t u d y ................. 75
3.2.1.1 Independent variables .................... 75
3.2.1.2 Dependent variables ...................... 76
3.2.2 Sensitivity, internal and external validity 77
3.3 Population and sample...................  79
3.4 Instrumentation.........................  82
3.5 Procedures.............................  84
3.5.1 Construction and validation of the instruments. ... 84
3.5.2 Procedures for the main st u d y..........  96
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3.6 A n a l y s i s .................................  100
Chapter four: Results and discussion ...........  102
4.1 Descriptive statistics of the data . . . 103
4.2 Hypotheses testing ...................... 108
4.2.1 Achievement as the dependent variable . 109
4.2.2 Test anxiety as the dependent variable . 116
4.2.3 Self concept as the dependent variable . 122
Chapter f i v e .....................................  130
5.0 Summary of findings, educational implications
and recommendations.................... 13 0
5.1 Summary of findings...................... 13 0
5.2 Educational implications and recommendationdL36
5.3 Limitations of the s t u d y ...............  141
5.4 Suggestions for further studies ......... 141
R e f e r e n c e s .......................................  143
Appendix 1 .......................................  161
Pre-test achievement test in mathematics . . . .  161
Appendix 2 .......................................  165
Post-test in mathematics ........................  165
Appendix 3 .......................................  169
Correct options for pre-test in mathematics . . . 169
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Appendix 4 ............................... 170
Correct options for post test in mathematics . . 170
Appendix 5 ............................... 171
Item analysis of pre-test in mathematics . . . .  171
Functions of distractors ......................  171
Appendix 6 ............................... 174
Item analysis of post test in mathematics . . . .  174
Appendix 8 ............................... 178
Appendix 9 ............................... 179
Inventory of test anxiety in mathematics (itam) . 179
Appendix 10 Mathematics self-concept items .......  181
Appendix 1 1 ..................................... 183
Reliability analysis of i t a m ............  183
Appendix 1 2 ..............................  185
Reliability analysis of the self-concept questionnaire^
Appendix 1 3 ..............................  187
Factor analysis of the self concept scale . . . .  187
Appendix 1 4 ..............................  192
Letter to provost........................  192
Appendix 1 5 ..................................... 193
Learning materials ............................. 193
Unit 1. F r a c t i o n ........................  193
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Unit 2. D e c i m a l s ............................. 201
Unit 3. Percentages........................... 209
Unit 4. Discount - profit - loss.............. 215
Unit 5. Simple interest......................  221
Unit 6. Compound interest and depreciation . . .  226
Unit 7. Ratio and p r o p o r t i o n ................ 232
Unit 8. Partnerships........................  238
Unit 9. Stocks and shares....................  244
Unit 10 Income tax and pay roll computation . . . 252
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List of Tables
Table 3.1 Design of the s t u d y .......  74
Table 3.2 Number of business education in the achieved
sample...................  81
Table 3.3 Table of specification of the pre-test in
mathematics...............  85
Table 3.4 Table of specification of the post-test
achievement test in mathematics . . 87
Table 3.5 Psychometric properties of the achievement
t e s t s .....................  90
Table 3.6 Psychometric properties of the psychological
t e s t s ...................  95
Table 4.1 Descriptive statistics of the overall dathQ3
Table 4.2 Group means of post-test mathematics
achievement scores ...................  105
Table 4.3 Group means of post test anxiety scores 106
Table 4.4 Group means of post-test self-concept
scores..................................  107
Table 4.5 Ancova table for achievement in mathematiTCK) 
Table 4.5a Scheffe mulitiple range comparison test on 
achievement in mathematics of males in the 
three instructional groups............ 114
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Table 4.5b Scheffe multiple range comparison test on
achievement in mathematics of females in
the instructional groups........... 115
Table 4.6 Ancova summary table for mathematics test
anxiety ..........................  117
Table 4.7 Multiple classification analysis (mca) on
mathematics test anxiety........... 119
Table 4.8 Ancova table for mathematics self-conceptL24 
Table 4.9 Multiple classification analysis (mca) on 
mathematics self-concept........... 125
A - 5 (i) Function of distractors of pre-test in
Mathematics 171
A - 5 (ii) Discriminating power and difficulty level
of pre-test in Mathematics 173
A - 6 (i) Function of distractors of post-test in
Mathematics 174
A - 6 (ii) Discriminating power and difficulty level
of post-test in Mathematics 175 
A - 11 Correlation matrix of ITAM 183
A - 12 Correlation matrix of self concept 
questionaire 185
A - 13 Correlation matrix of self concept and ITAMg 187
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A - 13 Rotated factor matrix 189
A - 13 Factor Transformation matrix 190
A - 13 Factor score coefficient matrix 190
LIST OF FIGURES 
FIGURE DESCRIPTION
4.1 Interaction of treatment and gender on 
students’ achievement in Mathematics 112
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1
CHAPTER ONE 
INTRODUCTION
1.1 BACKGROUND TO THE PROBLEM
Knowledge of mathematics is basic to science and 
technology. It serves the unique purpose of being the 
foundation upon which all forms of scientific knowledge 
are built. History shows that most of the scientific and 
technological discoverers are found to be mathematically 
well-equipped.
However, the need for sound knowledge of
mathematics is no longer restricted to mathematicians, 
engineers and physicists. The biologists find statistics 
and biomathematics very useful in such topics as 
mendelian genetics. In the field of Geography, the use 
of mathematics includes map projections, statistics of 
settlement patterns of villages and towns, analysis of 
surfaces, shapes and network as well as prediction of 
population. Mathematics has also been recognised as 
being useful in the humanities. The study of law makes 
use of Logic which involves mathematical reasoning. In 
linguistics, mathematics cannot be avoided. For example 
the linguists use Chomsky’s mathematical theory of
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2
languages in order to enhance their knowledge of modern 
languages. Ethnographers are not left out. They need 
mathematical knowledge to create mathematical models of 
primitive society. Holt and Majoram (1973) asserted that 
even in Christian Religious study, biblical analysts 
invariably employed mathematical techniques to assign 
authorship to disputed passages. The use of mathematics 
in Business Studies is very vast. High aptitude in 
knowledge and use of the various aspects of Business 
mathematics will enhance effective business 
transactions. A nation having students with good 
mathematical knowledge will have improved technological 
development and higher standard of living. Ibiejugba 
(1990) sums it all in his theory of optimization. As a 
subject of intellectual reasoning the theory is 
mathematical in nature and helps in the art of selecting 
the best possible decision in any circumstances 
including decisions on business transactions. The theory 
involves identifying what quantity to maximise or 
minimise and mathematical knowledge is needed for this. 
Thus, the study of mathematics is emphasised in the 
Nigerian educational system.
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3
In Business Education, it is recognised that 
modern business needs sound background of mathematical 
knowledge to solve a variety of everyday problems. 
Although calculating devices are available there is need 
for the business students to be exposed to sound 
knowledge of basic mathematical procedures in order to 
use the calculating tools intelligently and efficiently. 
A sound background in mathematics is essential for 
effective solution of mathematical problems encountered 
in business and everyday life. It will also lead to the 
students being adequately equipped to pursue more 
advanced mathematics in statistics, econometrics etc. 
when the student is exposed to university education. 
Hamburger (1972) emphasized the importance of 
mathematics in Business when he commented that:
"It takes the conquest of mathematics to 
master each Business subject and the knowledge 
of Business subject to conquer mathematics. 
Successful operation in any line of Business 
calls for the ability to comprehend and solve 
its particular and specialised problems: 
problems that invariably involve mathematical
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4
concepts".
Thus, mathematics and business subjects are not 
mutually exclusive in terms of useability.
The NCE level is an important aspect of tertiary 
education in the Nigerian educational system. As Fafunwa 
(1990) said, "the NCE will ultimately become the minimum 
basic qualification for entrance into the teaching 
profession in Nigeria". Graduates of the various 
colleges of education are being trained to teach in the 
primary and secondary schools in Nigeria within the 
context of the National Policy on Education as ammended 
in 1981. Among the NCE program is a vocational course on 
Business Education whose philosophy and objectives are 
as follows:
1. To make the business educators understand the 
concept and philosophy of the National Policy on 
Education as regards Business Education and the 
importance of Business Education in National 
Development.
2. To produce NCE Business teachers who will be able 
to inculcate the vocational aspect of Business
UNIVERSITY OF IBADAN LIBRARY
5
Education into the society.
3. To produce NCE Business teachers who will start the
so much desired revolution of vocational
development right through Primary and secondary 
schools.
4. To prepare students in Business Education with 
necessary competencies to qualify them for a two 
year
post-NCE degree programme in Business Education in 
Nigeria.
The inclusion of Business Education in our 
educational system especially at the secondary level is 
very crucial because of the usefulness of Business 
Education in the society which includes making students 
became very adept in marketable skills. Harms and Stehr 
(1965) also opined that it "creates an awareness of the 
need of certain business learning to properly conduct 
personal business effect". For the teacher to 
effectively groom students in Business Education, he 
needs to be mathematical competent.
Coyle (1976) described mathematics in Business as
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6
a way of visualising problems and exploring structures 
in order to make effective decisions in business. Thus, 
for effective and rational decision in Business and 
everyday life, one needs, a sound knowledge in basic 
mathematics.
It is in recognition of the importance of 
mathematics in Business Education that the National 
Commission for Colleges of Education (NCCE) makes it 
compulsory for all first year NCE students in Business 
Education to obtain at least a pass in Business 
mathematics as one of the conditions for the award of 
the National Certificate of Education. It has been 
observed, though not empirically, that many of these 
students have problems with their Business mathematics 
courses. The high number of students with weak passes 
and carry-overs in the courses points to the urgent need 
for intervention programmes to improve the situation.
A number of factors interfere with the learning of 
any subject and as such, business mathematics is no 
exception. One of the factors that has been identified 
as affecting performance in mathematics is mathematics 
test anxiety (Clute, 1984). This stems from mathophobia,
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7
that is, the fear of mathematics. Majority of the 
students in the school of Business Education barely 
managed to scale through their 0 ’ Level mathematics. 
They are aversive to the subject which they view as a 
thorn in the flesh. They exhibit fear of mathematics 
test and examination. This may be due to lack of 
preparations on their part and lack of confidence on 
their ability to pass the mathematics course. Cronbach 
(1970) asserts that although test anxiety is common 
among all kinds of pupils, it is more common among the 
low achievers. The outcome of this that students go to 
all extent to be involved in various forms of 
examination malpractices in order to pass mathematics 
test or examination.
The Business Education students especially the low 
achievers in mathematics have low self-concept about 
their mathematical ability. This has been observed by 
the researcher through verbal discourse with these 
students. Results of various research studies have 
consistently shown significant negative relationship 
between academic self concept and achievement (Hansfort 
and Hattie, 1982; Bryne, 1984; Marsh, 1992). A student
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8
who wants to achieve well in mathematics must also have 
high self concept of his ability. Unfortunately, this is 
not the case with these students who tend to have low 
self concept about their mathematical competence.
The problem of negative effect of test anxiety and 
low self concept which various studies have shown as 
having relationship with mathematics achievement needs 
urgent attention. The students need to be encouraged to 
develop a positive attitude towards mathematics. There 
is an urgent need to address the issue of low 
achievement and related variables like mathematics self 
concept and test anxiety.
One way of looking at this problem is to examine 
the instructional strategies used in the Colleges of 
Education. The lecture method appears to be the 
conventional method of instruction used in the tertiary 
institutions worldwide, Nigeria inclusive. Costin (1972) 
confirms this in his statement that "of all methods in 
college teaching, lecturing is the most widespread". 
Lecturing involve verbal presentation of facts from the 
instructor to the students whose participation in the
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9
proceed involves intense concentration and taking down 
of notes. This method has been criticised by Akin-Aina
(1989) as for being teacher-centred, monotonous and 
failing to permit active learning".
Studies have been carried out pointing out the 
inadequacies of the lecture method when compared with 
some teaching methods such as discussion method (Bloom, 
1953); group study method (Ward,1956); use of audio tape 
(Harding et al,1981), guided discovery method (Clute, 
1984). All these obtained significant superiority of 
each of the methods over the lecture method. Although 
the lecture method is widely used, the constructive 
criticisms and disbelief about its usefulness and 
effectiveness point to the need for finding alternatives 
to this method.
Modern research advocates the need for an active 
mode of teaching whereby students are given opportunity 
for active engagement in learning. Collier (1980) in his 
study of higher education found that tertiary level 
students were successful in developing higher skills 
when understanding of course contents was combined with 
active involvement of students. While investigating the
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10
effect of individualized method as opposed to the 
traditional lecture method in tertiary mathematics, Wine 
and Olan (1983) found significant effect between the two 
methods for students with weak mathematics background in 
favour of the individualized method. File (1984) 
advocated that learning difficulties experienced by 
students could be overcome once these difficulties were 
recognised and once effort and time were devoted to 
overcoming them. The question that arises is: can the
learning difficulties experienced by students in
business mathematics be overcome if instructional
techniques which are quite different from the prevailing 
"teacher-centred" lecture method is adopted?. Although 
Nickson and Smith (1973) found no clear differences in 
the various teaching methods employed in teaching 
mathematics to students of economics at tertiary level, 
the effectiveness of the methods found by later research 
poses a challenge of investigating with Nigerian 
students especially as Robinson and Mansour (1991) found 
the interactive method to have more successful impact on 
cognitive variables.
Most of the studies on mathematics test anxiety and
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11
self concepts are correlational in nature. For example, 
Alpert and Haber (1960) found test anxiety to be 
negatively related to academic achievement while 
Hansford and Hattie (1982) found academic achievement to 
be more highly correlated with academic measures of self 
concept than with generalised measures of self concept. 
Few research aim at finding the impact of teaching 
methods on these psychological measures. The only 
research work identified by the researcher was that of 
Georgewill (1990) who
advocated the need for sequential teaching on order to 
reduce test anxiety in secondary school students.
On the issue of gender effect on achievement in 
mathematics, Marshall and Smith (1987) found that boys 
had better understanding of problem structure than 
girls. Earlier, Benbow and Stanley (1980) confirmed that 
gender was a significant predictor of high school 
mathematics. However, Feingold (1992) pointed out a 
decrease in gender related differences in cognitive 
variables. It will be interesting to find out if gender 
has significant effect on performance of NCE business
education students.
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12
Research on gender differences on the two 
psychological measures i.e. on mathematics test anxiety 
and mathematics self concept, showed that females scored 
higher in test anxiety (Benbow and Stanley, 1980) while 
boys had higher self concept in mathematics than girls 
(Marsh and Smith, 1990; Marsh 1989).
This study hopes to make a comparison between the 
effects of the interactive learning strategy and lecture 
methods on achievement in mathematics, test anxiety and 
self concept of Business Education students. Gender 
issue
is also included in order to study its effect and 
interaction with teaching methods.
1.2 STATEMENT OF PROBLEM
The issue of alternatives to lecture poses a 
problem of educational research. In the normal 
mathematics lecture room student’s role is inactive. The 
students are presented with so much factual material in 
an attempt by the lecturer to "cover the syllabus". 
According to Hubbard (1990) this usually results in 
students losing trend of the mathematical arguments
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13
especially if the pace is to^fast. They may omit some 
crucial points or write incorrect statements, thereby 
creating problem of revision.
The poor performance in mathematics by non 
mathematics majors at tertiary level poses a problem of 
finding appropriate instructional strategies to enhance 
students’ achievement and mathematics self concept and 
reduce their test anxiety.
This study will thus attempt to evaluate the effect 
of material-based instructional strategy (treatment) as 
well as gender on mathematics self-concept, mathematics 
test
anxiety and achievement in mathematics of NCE Business 
Education students.
1.3 RESEARCH QUESTIONS
The study aims at providing answers to the 
following questions:
1. Is there any difference in mathematics achievement 
between NCE business education students exposed to 
material-based instructional methods and those not
exposed?
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15
exposed.
9. Is there any difference in mathematics self-concept 
between male and female NCE business education
students?
1.4 STATEMENT OF HYPOTHESES
Based on the research questions, the hypotheses in
this study will be tested under the following
subheadings:
1.4.1 Achievement as the dependent variable
1. There is no significant main effect of
instructional methods on NCE business education
students mathematics achievement.
2. There is no significant interactive effect of
instructional methods and gender on NCE business 
education student’s achievement in mathematics.
3. There is no significant main effect of gender on 
NCE business education student’s achievement in 
mathematics.
1.4.2 Mathematics Test anxiety as the dependent
variable
4. There is no significant main effect of
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16
instructional methods on NCE business education 
student’s mathematics test anxiety.
5. There is no significant interactive effect of 
instructional methods and gender on NCE business 
education student’s mathematics test anxiety.
6. There is no significant main effect of gender on 
NCE business education student’s mathematics test 
anxiety.
1.4.3 Mathematics Self-Concept as the dependent
variable
7. There is no significant main effect of 
instructional method on NCE business education 
student’s mathematics self-concept.
8. There is no significant main effect of gender on 
NCE business education student’s mathematics self- 
concept.
9. There is no significant interactive effect of 
instructional methods and gender on NCE business 
education student’s mathematics self concept.
1.5 SIGNIFICANCE OF THE STUDY
Research findings have consistently emphasised
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17
correlational studies involving mathematics test 
anxiety, mathematics self-concept and achievement in 
mathematics. For example, Baya’a (1990) conducted a 
correlational study on mathematics anxiety, achievement, 
gender and socio-economic status among Arab students and 
found substantial relationship between the variables. No 
study known to this present researcher has investigated 
the combined effects of gender and instructional 
strategies on the afore-mentioned variables. This lapse 
emphasises the need for this study.
The issue of high mathematics test anxiety and low 
self-concept in mathematics is an educational problem in 
mathematics education. Research has shown that these two 
variables have effects on mathematics achievement. The 
application of experimental method to this educational 
problem is of high significant value and it is in line 
with the editorial comment of the 1991 edition of the 
Journal for Educational Psychology which stated that 
"there is need to stimulate interest in the progress of 
experimental pedagogy: For we believe that the time is 
ripe for the study of school room problem in the school 
room itself and by the use of experimental method". This
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18
experimental study, conducted in the classroom 
situation, is an answer to this clarion call for 
experimental research to study educational problems.
The inadequacies of the traditional lecture method, 
especially, in mathematics and science has been the 
focus of many educational researchers. Clement and 
Wrights (1983) rightly observed that:
"particularly in the scientific and technical areas, the 
use of lectures as a mechanism in the transferance of 
materials from the notes of the lecturer to the note 
pads of the students without any critical involvement or 
significant intellectual efforts on the parts of either 
has been seen to be a misuse of capabilities of staff 
and students".
There is need for alternative methods of teaching 
students at tertiary level in order to effect adequate 
transfer of mathematical knowledge. Instructional 
techniques need to be evolved that can optimize students 
involvement in the learning process. This study is of 
significant value since it attempts to take up this 
challenge.
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The sample used in this study are students at the 
tertiary level. Literature points to a lot of research 
on higher studies but based on non-Nigerian samples. 
There is therefore the need to combat the issue of 
scanty studies on higher education in Nigeria and this 
study aims at meeting this need.
The prevailing economic situation in Nigeria is 
such that textbooks are extremely costly and beyond the 
reach of most students at tertiary level. If available, 
most are outdated and foreign-based. Nigerian authors 
are making efforts to combat this problem. However, 
publishers favour academic materials for primary and 
secondary level since this will sell faster. The new 
learning package developed in this study will be of 
significant use to help the students offering a first 
year Business mathematics course at NCE level.
1.6 SCOPE OF STUDY
The subject used in this study were students at 
tertiary level. This limits the scope of this study to 
higher education. Furthermore, the students used were 
non-mathematics majors. They were those who were
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compelled to undertake a course in mathematics as a 
condition for their graduation. The scope in not widened 
to include mathematics majors who are likely to have low 
mathematics test anxiety and high mathematics self- 
concept .
It should also be noted that the study does not aim 
at examining the inter-relationship between achievement 
in mathematics, mathematics test anxiety and mathematics 
self-concept. Rather, it compares the effectiveness of 
some attribute and active variables on the three 
dependent variables mentioned earlier.
1.7 DEFINITION OF TERMS
This section deals with definitions of some 
important terms used in this study. Examples are given, 
where necessary, in order to clarify the definitions.
(i) Independent variable: This is a variable which is
not affected by changes in the other variables. 
In this study gender and teaching methods are
the independent variables.
(ii) Dependent variable: This is the variable that is
affected by changes in the independent
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affected by changes in the independent variable. It is 
the "presumed effect" of the independent variable. 
Achievement test in mathematics, mathematics test 
anxiety and mathematics self concept are the three 
dependent variables studied.
(iii) Active variables: These are variables that can be 
manipulated. The methods of teaching are the active 
variables in this study.
(iv) Attribute variables: These are variable that can 
not be manipulated. The example used in the study is 
gender. Both the active and attribute variables are the 
independent variables used in this study.
(v) Target sample: This consists of all NCE students 
offering a first year course in Business mathematics.
(vi) Achieved sample: This consist of all the students 
used in this study.
(vii) Achievement in mathematics: This is a measure 
used to obtain firm responses of students on a carefully 
constructed and validated objective test in 
mathematics.
(viii) Mathematics Test Anxiety: This deals with scores 
obtained by students on a mathematics test anxiety
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scale which contains items describing test anxiety 
situations.
(ix) Mathematics Self-Concept: These are the scores of 
students’ responses on a questionnaire that describes 
students’ self-perceived ability in mathematics.
(x) Lecture method: This is a teacher-directed mode 
of teaching at tertiary level and it involves oral 
presentation of mathematical facts with little or no 
active participation by students.
(xi) Interactive method: This is a students-centred 
type of cooperative learning in which students work 
interactively with little assistance from the teacher.
(xii) Materials: These are structured instructional 
package based on the syllabus of the NCE Business 
mathematics course.
(xiii) Pre-tests: These are tests given before the 
commencement of the treatments.
(xiv) Post-tests: These are tests given at the 
conclusion of the experiment.
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CHAPTER TWO 
REVIEW OF LITERATURE
2.1 INTRODUCTION
The review of relevant literature to the study is 
presented in this chapter. The purpose of the study as 
well as the variables involved in the study have been 
taken into consideration in the review. Thus the 
literature review is presented under four major 
headings: the lecture method; mathematics achievement, 
instructional techniques and gender; mathematics test 
anxiety and mathematics self concept.
A summary of findings is given at the end of the 
review. This summary include appraisal of the reviewed 
literature.
2.2 THE LECTURE METHOD
The lecture method is the typical instructional 
strategy used at tertiary level in most subjects, 
mathematics inclusive. This was confirmed by Hoover 
(1981) that "the lecture is currently the most widely 
used instructional method in colleges and universities". 
Robinson (1991) gave a vivid illustration of this method 
when he described it as "more or less a continuous oral
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presentation of information and ideas by instructors 
with little or no participation by the students". This 
method has been regarded by students as being 
unfavourable (McLeish, 1968). According to File (1984) 
31% of students at undergraduate level sampled by him, 
claimed lecturers assumed more knowledge on part of 
students during lectures than they actually possessed. 
Unfortunately, lecturers on their part hardly realise 
just how little of what they say is understood by 
students (Abercombie, 1971). The lecture method, 
described by Marris (1962) as being "most universal and 
most impersonal method of post secondary instruction" 
often involves large group with limited scope for 
interaction.
The strengths and weaknesses of the lecture method 
have been identified by Gregory (1975). Among the 
strengths of the lecture method identified by him were: 
Coverage of more factual materials, uniform 
transformation of information, usefulness to large 
group, economic use of instructor’s time and a 
relatively easy instructional technique to adopt. The 
weaknesses included: not being able to cater for
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individual differences of learners, little or no active 
participation by students, and development of few high 
intellectual skills. To be effective, the instructor 
needs careful preparation and presentation on his part 
while the student needs increasing concentration in 
order to imbibe the concepts taught. As an impersonal 
method of teaching, it inhibits feedback and promotes 
covert rather than overt engagement in learning tasks.
This method is widely used in tertiary mathematics 
instruction. At this level, the lecture method is used 
as a mechanism in the transference of materials from the 
notes of the lecturer to the note pads of the students, 
without any critical involvement or significant effort 
on part of either the instructor or the learner 
(Clements and Wrights, 1983). Making wrong assumptions 
about the mathematical level, experience and competence 
of their students, the mathematics lecturer goes 
straight to confusing definitions, theorems and proofs 
while the student’s inactive role is to copy these 
concepts from the board. Lecturers need to be aware that 
little of what they teach during lectures is understood 
by the students. Morgan(1990), while expressing concern
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about the general poor level of mathematical competence 
of some selected engineering students in tertiary 
education, found out that the development of 
mathematical ability was not being achieved since 
students could not generate solutions to basic problems 
of the important concepts involved in mathematics.
Hubbard (1990) opined that must lecturers of 
tertiary mathematics adopted the lecture method because 
it was the method used in instructing them while they 
too were students at tertiary institutions. The feeling 
was that since they were taught that way and succeeded, 
their own students must succeed if taught in the same 
way. The fact that using different methods might imply 
more successful learning of the taught mathematics 
concept and reduce the drop-out rate in mathematics is 
not often considered. Furthermore, in an attempt to 
"cover the syllabus" students are presented with too 
many factual materials with lecturing done at a fast 
pace. This fast pace, coupled with the sequential nature 
of mathematics make many of the students get lost and 
the longer the mathematics lecture, the more serious the 
student’s incapability to absorb what is being taught.
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Students may omit crucial statements, copy down 
incorrect or incomplete statement thereby getting more 
confused about concepts.
Although lecturing in mathematics provides a fast 
way of reaching a large audience (Flexer, 1978), how 
effective is the individual in the audience being 
reached? This made November (1976) to comment that the 
lecture method, "while it is a good method for one 
person to transmit information to many, it is a very 
poor method for individual to receive learning". A good 
mathematics instructor at any level of the educational 
system, must not just be a transmitter of knowledge, but 
a manager of knowledge.
In most tertiary institutions in Nigeria, lecturing 
is supplemented with tutorials organised for the 
students. The problem is, how effective are these 
tutorials, if at all they exist in a given institution, 
in helping the low achievers in mathematics. In most of 
these tutorials post graduate students or high achievers 
among the students solve some problems on the board. 
Little or no pain is taken to ensure good understanding 
by the students (Adedayo,1994). Attendance at these
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tutorials are not encouraging. Some lecturers just give 
out tutorial questions to students to practise without 
making any attempt to assist them in their areas of 
difficulty.
Researchers involved in studies at high educational 
level have persistently shown concern about the 
inadequacies of the lecture method and hence have been 
involved in carrying out studies comparing the lecture 
method against other methods of instruction some of 
which are cited here.
2.2.1 The lecture method versus other methods of 
instruction
The studies comparing the lecture method with other 
strategies are very interesting. While many of the 
studies found significant differences when compared with 
the lecture methods, some did not find any difference. 
The other methods include lecture combined with 
tutorials, guided reading, learning packages, laboratory 
strategies and other methods.
(i) Lecture versus lecture with tutorials
Tutorials, if well organised, are good avenues of
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identifying and overcoming students learning 
difficulties in mathematics. These learning difficulties 
can be overcome once they are recognised rather than 
ignored and once time and efforts are devoted to 
overcoming them (File, 1984). The aims of tutorial 
classes as outlined by Searl (1979) are:
(i) making personal contacts with students.
(ii) Clearing up difficulties of lecture notes.
(iii) Solving problems.
(iv) meeting with fellow students.
(v) providing feedback to effectiveness of lectures.
(vi) Returning and correcting assignments
(vii) Ensuring that class is keeping up with tutorial
sheets.
(viii) Problem solving by students to give experience
and confidence.
Tutorial classes when held as supplements to lecture 
enhance learning. Hubbard (1990) investigated other 
teaching alternatives different from the normal lecture 
with first year undergraduates offering courses in 
mathematics and emerged with a format involving more 
active participation by students. The procedure involved
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provision of study guides for students on weekly basis 
containing extra explanations, in formal languages, for 
known areas of difficulties as well as exercises for 
testing students understanding. Tutorials were held in 
groups of 15-20 since a large class was involved. 
Frequent testing was also done with feedback and the 
outcome was an improvement in students attitude and 
achievement in mathematics.
Goldschmid and Goldschmid (1976) combined lecturing 
tutorials and students working in pairs on written 
materials in tertiary instruction and found this 
teaching method to be very effective. Wood (1991) 
examined the effect of computer laboratory tutorials 
when combined with cooperative learning on mathematics 
achievement, retention rate, mathematics anxiety, 
mathematics confidence and success in future mathematics 
among some tertiary students. This combined tutorial 
method was compared with the traditional lecture method 
which served as the control group. The control group 
showed greater increases in post-course confidence 
ratings than the experimental group. They also showed 
greater reductions in anxiety ratings than the
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experimental group. However, 69% of the experimental 
group received a course grade of A, B or C as compared 
with 52% of the control group while 87.5% of the control 
group students were successful in the subsequent 
mathematics course compared to 80% of the experimental 
group students.
One of the advantages of tutorials is providing 
feedback of students’ errors and correcting these 
errors. Boonruangratana (1980) observed that testing and 
subsequent discussion focusing on particular problem 
identified by the testing has significant effect in the 
maximum achievement of the class. Students showed 
positive reaction when provided with corrective testing, 
which comes up during tutorials especially students with 
high test anxiety (Arkin et al, 1984) . Also, Lyte and 
Kulhavy (1989) observed that undergraduates who received 
repeated feedback showed the greatest probability of 
correcting instructional errors. Johnson (1990) while 
investigating the effect of frequent testing on the 
learning of mathematics found that short tests improved 
students achievement in final examinations. So if short 
tests are given and correction of errors done during
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tutorials, students will be made to work hard on a 
continuous basis instead of cramming when examination 
draws near.
(ii) Lecture versus use of learning materials
Studies have also compared the lecture method with 
methods that involve use of learning materials. The 
results of these studies have been quite interesting.
November (1978) used a tutorial tape learning 
package as compared with the traditional lecture method. 
He was not able to prove quantitatively that the 
learning package produced better result than the lecture 
method although students using the learning package had 
a higher mean score. He attributed this high mean score 
to either light marking on part of the instructor or the 
package making the subject inherently more interesting. 
Harding et al (1981) also compared tape slide teaching 
with normal lecture and found no observable differences 
between the two groups taught.
Romberg (1969) used programmed instructional 
materials and investigated its effect on low-achievers. 
Although no significant difference in achievement was 
found, those exposed to the learning materials had
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better retention. With respect to its usage, Jones 
(1989) advocated the use of programmed instruction in 
cooperative groups in Science teaching as being very 
effective in promoting achievement. Wine and Olan (1983) 
compared the effect of the traditional method of 
instruction with individual type involving the use of 
programmed instruction for year one students in a 
tertiary mathematics course. His technique comprised 15- 
30 minute of lecture followed by individual use of 
supplementary text.
The use of the individual material was more 
effective in improving the performance of low achievers 
than the traditional method. Even for the handicapped, 
Slavin (1982) found that an instructional method making 
use of individualized learning in mathematics compared 
with the traditional method of instruction was more 
effective in increasing the sociometric status of main 
streamed academically handicapped student and the 
individualized learning involved use of materials.
Clements and Wright (1983) found guided reading very
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effective in engineering mathematics degree course. 
Students were guided through the reading and study of a 
particular textbook or several textbooks by a set of 
rules provided by the instructors and also supplemented 
with a series of discussion and tutorial classes run by 
the lecturers.
The use of new learning materials in improving 
students achievement was also investigated by Cohen and 
Ben-Zvi (1992). They found the new learning materials to 
be very effective in improving students’ achievements.
(iii) Lecture method versus guided discovery
The guided discovery method, another approach to 
teaching, has been defined as a unique individual 
experience by which concepts evolve in the mind of the 
learner than being transmitted ready-made. It is evident 
from the ideas of Gagne (1966) that the discovery 
approach does not imply students just sitting down and 
paying attention but rather involves the process of 
searching and selection. Bruner (1960) identified some 
of the advantages of the discovery method as:
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1. resulting in better retention since no memorisation 
of isolated information takes place. Rather, 
discovery of principles connecting the information 
is got.
2. It enhances motivation, interest, and satisfaction. 
If the student is satisfied there is tendency for 
competency.
3. It enhances the development of intellectual 
capacities, information and problem solving skills.
4. A general heuristics of discovery enables students 
to solve problems in new contexts and hence increase 
transfer of learning.
A lot of work has been done on its effect on 
learning of subject especially mathematics and science 
at various levels of the educational system. When 
carefully used, it has also been found to have positive 
effect on tertiary mathematics.
Flexer (1978) used this mode of teaching that enabled 
pre- service teachers to discover mathematical concept 
through exploration with manipulative materials such as 
dice, blocks and abacus. This laboratory strategy (as he 
chose to call it since it took place in the mathematics
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laboratory) was found to be very effective in achieving 
the desired goal. Clute (1984) compared a lecture method 
with a guided discovery method comprising of questioning 
sequences that guided tertiary mathematics students to 
discover mathematical principles in one of the 
mathematics course to which they were exposed. The 
result obtained showed that the discovery approach was 
of great benefit to students with low mathematics 
anxiety. The lecture or exposition method favoured 
students with high mathematics anxiety.
(iv) Lecture versus Interactive method
Recent findings lend support to a student-centred 
model of teaching (as opposed to the teacher-centred 
lecture method) which involves active engagement of 
students in learning task. Unlike the normal lecture 
which limits students to covert participation, this mode 
of teaching encourages pupil-pupil interactions as well 
as teacher-pupil interactions. Wanskowski (1982) asserts 
that "human interaction is most important in assisting 
students in learning how to learn." Peer teaching, group 
discussion and cooperative learning are some of the 
techniques used in the interactive learning processes.
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Cooperative learning has been advocated as promoting 
enhanced understanding and providing guidance for 
teachers in higher education (Todd, 1981). There is need 
to increase the involvements of and participation of 
students in the teaching - learning processes as well as 
give the control over the learning processes as 
advocated by Stephens (1981) and Castro (1991). Robinson 
(1991) emphasised the need for this model of learning by 
saying that "perhaps the most acceptable indirect method 
for the post-secondary instructor may be discussion or 
interactive instruction". Interactive instruction 
combines components of discussion, group interaction and 
a defined problem. The main aim is to maximize students 
involvement while the instructor minimizes his 
involvement by briefly presenting a topic, engaging 
students in class activity which generally involves 
group work or individual work with materials. Feedback 
must be given on the students ’ work so that students ’ 
involvement in the learning process is thus optimized. 
Bossert (1988) opined that cooperative learning method 
promotes students achievement that is at least as high 
and often higher as in traditional classroom. From the
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views of Conwell et al (1993) cooperative learning can 
be seen as implying shared leadership where there is not 
just one leader but each member of the group is a 
leader. A leader thus demonstrates academic as well as 
collaborative skills in helping the group achieve its 
goal.
Interactive learning involves small groups of 
students doing some peer teaching. Channon and Walker 
(1984) found that most lecturers at tertiary level 
agreed with small group teaching but made excuse of not 
doing so because of the large number of students they 
had to face. However, they advised that with proper 
planning it could be fully implemented with larger 
groups. The interactive technique is applicable to all 
age levels (Davidson,1989).
Small group interactive learning provides an 
alternative to both the traditional teacher-centred 
expository instruction and individual instruction 
systems (Davidson, 1990). Mavarech et al (1991) in an 
experimental study of learning computers in small group 
used ANCOVA to arrive at the conclusion that students 
who used computer assisted instruction for drill and
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practice in pairs performed better than students who 
used the program on individual basis.
The realistic, practical strategies for using small 
groups in mathematics teaching and learning for all age 
levels, curricula levels and mathematical topic areas 
include: problem solving and exploration with 
manipulative materials in groups of four, team learning 
approaches based on individual accountability and team 
recognition, procedure, for problem solving and inquiry 
in algebra, geometry and trigonometry, free explorations 
and guided discovery in cooperative groups. Factors 
affecting the implementation of these learning 
strategies among others have been extensively discussed 
by Davidson (1990).
Good et al (1990) sum up the advantages of this 
method of learning. According to them, the active 
learners are more motivated and enthusiastic about 
mathematics than those exposed to traditional 
techniques. However, well prepared materials are needed 
for its effective implementation. This interactive 
strategy has been greatly explored at various studies on 
the teaching of mathematics at all level of the
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educational system.
Nickson and Smith (1973) investigated alternative 
methods of teaching elementary mathematics to students 
of economics at first year level in Cambridge 
University. Three methods were compared, namely:
(i) Self instructional learning packages written
specially for the course.
(ii) a lecture course together with classes involving
use of computers and
(iii) a straight lecture course.
The results obtained showed no clear differences in 
effectiveness between teaching methods.
Collier (1980) combined understanding of course 
content with a student directed learning group and 
concluded that peer group teaching, which is an 
interactive technique, is a good technique even at 
tertiary level. Todd (1981) used a method he called 
collaborative learning. As an interactive strategy, it 
involved small groups of students working on learning 
assignment independent of tutor. The tutor put in his 
advice from time to time but left student 
to work alone. The result was very successful and he was
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able to conclude that this technique improved learning.
Goldberg (1981) investigated problem solving in 
small groups comprising four or five members among 
students in tertiary institution offering courses on 
algebra, logic, calculus and number theory. He found 
that this technique promoted more understanding and 
greater incentives for the students to do their 
assignment. They were to work individually but were 
allowed to seek help from their colleagues.
Sherman (1986) used secondary school students as her 
sample. One of her three studies involved comparison of 
the interactive cooperative technique with the 
individualistic method. The result favoured the 
cooperative group who demonstrated significantly higher 
post test achievement scores than the individualistic 
group. The second study involved cooperative learning in 
remedial mathematics for ninth grade students. One class 
used a modified team approach (interactive), while the 
other used the individualistic goal structure including 
class lecture, seat work and home work. The mean post­
test group scores was high for the cooperative group. 
The third study was done with a biology class and no
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difference was obtained.
Phelps and Damon (1989) assessed the effects of 
collaborative (an interactive method) on spatial 
reasoning. They found the technique to be effective for 
tasks requiring reasoning but not effective for those 
tasks that require rote learning or copying. The group 
used were also secondary school students.
Brickie and Woodrow (1990) investigated problem solving 
skills of "at risk" high School mathematics Students 
through cooperative work groups and computer assisted 
instruction. For a period of three months) computer 
manipulative, cooperative groups and Socratic 
questioning in the mathematics classroom were used to 
address a variety of learning preferences. No measurable 
differences could be noted from the pre- and post-test 
interest responses. However, improved performance and 
documented increases in class attendance made them 
conclude that the design of mathematics instruction to 
accommodate differences in students learning preferences 
must be an effective strategy for addressing the unique 
needs of the at-risk alternatives high School Student.
Pratt et al (1990) investigated the effect of
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cooperative learning as compared with the traditional 
approach among low ability fifth - graders. The 
achievement tests given included topics on mathematics. 
Students in the cooperative learning class scored higher 
than those in the traditional class with statistical 
significant differences occuring in favour of those in 
the cooperatives learning class on mathematics and total 
battery Scores.
Back to study on tertiary level mathematics, Dees
(1991) used a group of students in a college remedial 
mathematics course to find out if the cooperative 
technique would improve their problem solving skills in 
mathematics and found cooperative learning to be 
effective in solving worded problems in algebra and 
geometry. Cooperative learning at tertiary level has not 
only been effective in mathematics. Cohen and Ben-Zvi
(1992) tried it an a chemistry class and found that 
students involvements in the learning process were 
optimized.
Ur ion (1992) tried the small group interactive 
techniques on groups cutting across secondary and 
tertiary students in mathematics. The results showed
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that in all the levels compared there was no single case 
of the small group students performing poorly than those 
exposed to traditional technique. Thus we have seen that 
majority of the studies in interactive or cooperative or 
collaborative or group study or peer teaching reviewed, 
on the average, the interactive technique, when used in 
tertiary mathematics is quite rewarding.
2.3 MATHEMATICS ACHIEVEMENT. INSTRUCTIONAL TECHNIQUES
AND GENDER
Mathematics achievement deals primarily with the 
performance of students on either teacher made test or 
standardized achievement test administered by 
researchers and by examination bodies.
It is generally believed that students’ love, 
interest and achievement in mathematics at all levels of 
the educational system are not encouraging. Hatred for 
and poor achievement in mathematics increase with age as 
cited by Carpenter et al (1980).
Here in Nigeria, we are constantly confronted with 
statistics that show general low performance in 
mathematics, when compared with other subjects, at the
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National Common Entrance level, the Junior Secondary 
Mathematics level, as well as the Senior School 
Certificate mathematics examination level (Adedayo, 
1994) . The situation at the tertiary level is not any 
better. The high number of students with weak passes and 
carry overs in mathematics at the polytechnics, 
Universities and Colleges of Education show that 
students are not achieving high enough in mathematics.
Research have focused on both teachers and students 
in an attempt to find out if improvement in mathematics 
learning and achievement can be attained by manipulating 
some of the alterable variables such as teaching 
methods, time on task e.t.c. The studies on the 
teachers, among others, concentrate on instructional 
technigues being employed. As reviewed earlier in this 
chapter, various teaching strategies such as peer 
teaching, individualized method of instruction, guided 
discovery method, in which students play more active 
role, have been of great use in improving students 
achievement in mathematics. Frequent testing with 
corrective feedback measures has also been proved 
effective in improving students achievement in
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mathematics (Keats, 1972; Lyte and Kulhavy, 1987).
The use of humour in teaching has been advocated by 
Boughman (1979) as a means of promoting understanding 
and hence achievement in any academic subject, 
mathematics inclusive. Humour in teaching increases 
students attention and retention and interest and 
reinforce what is taught.
Concerning the issue of gender and achievement in 
mathematics, the general belief is that boys are 
superior than girls (Muscio, 1962) . Research results 
available show two sides of the coin - sometimes 
favouring males and sometimes females. Benbow and 
Stanley (1980) emphasised that gender differences in 
mathematics achievement is "huge and remarkable" with 
boys showing superior ability than girls. Later research 
conducted by these two researchers in 1982 and 1983 
using different samples confirmed this assertion. To 
support this finding, Marshall and Smith (1987) found 
that boys had better understanding than girls in 
mathematics. Mills et al (1993) also found that boys 
performed better than girls on tasks requiring
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mathematical concepts. For students at tertiary level, 
Boli et al (1988) found undergraduate males to be 
superior only in a calculus course. Bridgement et al 
(1991) however found that in a given mathematics course 
at the university level, the average grades of women 
were about equal or slightly higher than men’s average 
grades. Kimball (1989) found no gender differences in 
most university courses and that where differences 
occur, there was female superiority. On the Nigerian 
scene Uka (1966) found no sex difference while Obioma 
and Ohuche, (1980) obtained results which showed male 
superiority over females. However, Onibokun (1979) 
asserted that male superiority in numerical ability 
during the early year might be a myth after all. Adedayo 
(1982) found no significant difference between the 
scores of male and female students in standardised 
mathematics tests. Meta-analytic studies carried out on 
gender differences in mathematics achievement (Feingold, 
1988, Friedman, 1989) conclude that simple 
generalizations concerning superiority of either sex are 
impossible. Fennema and Sherman (1974) asserted that 
sex-related differences in mathematics were not as
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prevalent as had been believed. Rather, they found 
mathematics achievement to be age related. Fennema and 
Sherman (1977) also showed that universal sex related 
differences did not exist in mathematics. However, Hyde
(1990) emphasised that age, type of task and other 
factors determine gender differences and these factors 
should be taken into consideration in the issue of 
gender differences in achievement in mathematics.
Halton and Gosta (1974) investigated sex type 
interest as possible causes of difference in mathematics 
achievement between the sexes in different grades. They 
found no significant difference between the sexes up to 
grades 5, but after grade 5 the boys achieved higher 
than the girls. They cited the earlier results of Tyler, 
Anastasi, and Maccooby who surveyed sex differences in 
achievement of boys and girls and concluded that boys 
did better in numerical and spatial tests and tests of 
arithmetic reasoning. Wolleat et al (1980) also found 
that females have lower confidence and ability in 
mathematics than males. The concern about the gender 
issue is very essential and should be the focus of 
researchers (Jackline, 1989).
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At tertiary level, Betterly and Clarke (1974) 
carried out their study using undergraduate students in 
mathematics. The conclusion reached was that there was 
no evidence existing to show that members of either sex 
obtained better results. Deboer (1984) confirmed that 
fewer women took Science and mathematics courses than 
men but when they did they performed at a higher level 
in both high school and colleges.
Using Arab students in an Israeli secondary schools 
Baya’a (1990) found gender to be a significant predictor 
for course plans for high school students in mathematics 
and also that differences between males and females 
depended on their socio-economic status.
2.4 MATHEMATICS TEST ANXIETY
Anxiety describes the individuals level of 
emotionality. The behaviourist employ the terminology 
"anxiety" as a short hand label for complex pattern of 
responses which may be self-reports, physiological or 
somatic-motor in nature. Psychologists have pointed out 
that anxiety could affect the cognitive aspect of a 
human being. This implies it could affect performance in
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tests, especially mathematics tests.
Mathematics test anxiety is a measure of students 
responses to items invoking anxiety related conditions 
in testing situations. Scales have been designed to 
measure this test anxiety. A popular test by Richardson 
and Suin (1972) is the Mathematics Anxiety Rating Scale 
(MARS) which has been used extensively in America and 
other parts of the world. Fenneman (1977) modified this 
Scale to produce the Mathematics Anxiety Scale. Cronbach 
(1970) confirmed the prominence of testing anxiety in 
general among dull students.
Anxiety can be grouped into two main divisions: the 
high-anxious or those are mainly tense and worried and 
the low-anxious or those who are cool. According to 
Dececco et al (1974), low anxious students perform 
better when challenged by a given task in which their 
performance will be assessed while high anxious students 
perform worse under the same condition and perform 
better when they are not threatened with evaluation and 
faced with a challenging or difficult task. Hallworth 
(1964) emphasized that the anxiety experienced by a 
person when taking test or examination could play an
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important factor in his success or failure. Over­
anxiety could lead to being too confused to put in his 
best, while not being anxious at all could lead to poor 
performance due to insufficient effort. He defined 
general anxiety in terms of typical symptoms recognised 
by clinicians and concluded that general anxiety of 
which test anxiety is a subset, correlated highly with 
anxiety specifically related to tests and examinations.
It is generally believed that anxiety could perform 
dual role which is either positive or negative in 
nature. Test anxiety could be an aid to achievement. A 
student who is anxious about an impending test could be 
so eager to perform well that anxiety will stimulate him 
into putting more efforts into his preparation. The 
positive outcome is that he would do better than if he 
had not been anxious. This anxiety has a limited level 
after which it will have negative side effect: being too 
anxious will make the student so nervous as to forget 
some facts which he already knows. Alpert and Haber 
(1960) worked in line with this theory. They believed 
that anxiety would not always lead to poor performance. 
It could have a positive effect on performance up to a
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certain level after which it could impede performance. 
This led them to constructing and validating the 
facilitating and debilitating anxiety scales. The 
facilitating scale makes the student report how anxiety 
improves his performance while the debilitating anxiety 
scale measures how anxiety interferes with test 
performance.
A recent theory based on the interference model
asserts that anxiety produce test irrelevant responses 
that divert attention from test relevant thoughts and 
hinders retrieval of information needed to properly
answer questions (Tyne, 1980). The high test anxious 
student occupies himself with task-irrelevant responses 
such as worry and do not devote attention to test-taking 
tasks.
Culler and Holahan (1980) gave alternative meaning to 
the interference model as follows:
(i) Test anxiety represents a problem of broader
behaviourial scope and the test anxiety
relation-ship is at least partially a function 
of differential study related behaviours between 
high and low test anxious individuals.
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(ii) Their assertion attests to the fact that study 
habit has a part to play in test anxiety since 
it could occur as a result of inadequate test 
preparation on part of the students. In fact 
their views are that high test anxiety is caused 
by inadequate test preparation.
Since poor study habits has been traced to high test 
anxiety, researchers such as Allen (1991), Mitchell 
(1972) have recommended study counselling and 
desensitization therapy in the treatment of test anxiety 
among students. Quite recently Moshe (1991) made a 
comparison of training programs for students who have 
been identified as being test anxious. Desensitization 
or study skills training were given to different groups 
of this set of high test-anxious students. He found 
desensitization to be more useful for those with 
retrieval problem while the study skills training was 
more beneficial to those with problem in all stages of 
information processing. Eddy (1985) has a lot of 
suggestion to offer on effective treatment of 
mathematics test anxiety.
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2.4.1 Mathematics test anxiety and achievement
Sarason et al (1959) investigated the relationship 
between anxiety and performance on the 11* examination 
in Britain. They predicted that the 1l*examination would 
increase test anxiety scores of incumbent testees and 
that children with low test anxiety would perform more 
effectively on the examination than those with high 
test-anxiety. They reasoned that the freer a child is 
from anxiety the less his intellectual functioning would 
be impaired by the disabling effects of anxiety. The 
result they obtained was contrary to their predictions 
and they were able to give reasons for the 
discrepancies. Alpert and Haber (1960) found that test 
anxiety scales had greater negative correlations with 
college entrance examination than general anxiety scale 
and that they made a good contribution to academic 
knowledge. Turnbull (1965) also used primary school 
student in his research on test anxiety and achievement. 
He found that test anxiety could be associated with 
certain impairment in performance and that such anxiety 
lessened after the test. The conclusion is that there is
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tendency for anxiety about test to have adverse effects 
on the performance of able students.
Szeteta (1973) while studying the effects of test 
anxiety and success-failure in mathematics performance 
and mathematics anxiety among eighth grade students 
hypothesized that the performance of the high anxiety 
students would be inferior to those in the low-anxiety 
group. However his results showed that "the guestion of 
the effects of test anxiety on learning of mathematics 
still remains cloudy".
Ohlson and Mea (1977) used the State Trait Anxiety 
Inventory (STAI) to measure anxiety levels of 
mathematics and non mathematics majors who were 
undergraduates. Using multiple linear regression model 
they confirmed that mathematics majors were no more or 
less anxious than non mathematics majors. They also 
found that being in a mathematics class did not lead to 
more anxiety than being in a non mathematics class.
Here in Nigeria, Abadom (1980) used students from 
Ibadan Grammar school and found that performance in 
mathematics was not independent of anxiety towards 
mathematics. The low anxious students performed
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significantly better than other students. There was no 
significant difference between the mathematics
performance of those rated high or moderate in anxiety 
towards mathematics: rather both group had tendency to 
f ail.
Clute (1984) using a sample of American students 
found that students with high level of mathematics test 
anxiety had significantly lower achievement than 
students with low level of anxiety. Druva (1984) while 
investigating problem solving by mathematics anxious and 
non-mathematics anxious students enrolled in college 
mathematics course in the U.S found that high 
mathematics anxious students did better in computational 
visual type problems than did non-mathematics anxious 
students. Wigfield and Eccles (1989) showed test anxiety 
to be negatively related to achievement at the Primary 
and Secondary levels.
Baya’a (1990) found significant correlation between 
mathematics test anxiety and achievement.
2.4.4 Test Anxiety and Instructional Strategy
Different instructional strategies have been 
employed in studies on mathematics test anxiety. Clute
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(1984) compared the effect of the lecture method and a 
method based on questioning sequences that guided 
students to discover mathematics principles on test 
anxiety and achievement. The results showed that 
students with mathematics anxiety had significantly 
lower achievement. Students with high test anxiety 
benefited more from the lecture method while those with 
low anxiety benefited more from the discovery approach. 
Lipsett et al (1988) investigated the effects of two 
instructional interventions on achievement of students 
at tertiary level. The two levels of anxiety used were 
high and low and the instructional techniques were the 
experiencing mathematics instructional method and 
expository method. The conclusion was that either of the 
two instructional interventions could be used to improve 
achievement. Georgewill (1990) advocated sequential 
teaching of mathematics in order to reduce mathematics 
test anxiety and fear of mathematics.
The cooperative method of instruction has been found 
to be favourable in reducing anxiety. Blum-Anderson 
(1992) described 10 teaching strategies that gave 
attention to affective variables in mathematics and
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among the affective variables was mathematics anxiety. 
Also Wood (1992) used the traditional lecture method and 
cooperative learning to investigate their effects on 
test anxiety. The result showed greater reductions in 
anxiety ratings of those exposed to the lecture method. 
2.4.3 Gender and mathematics test anxiety
Two main group of results are evident from past 
students on gender differences in mathematics test 
anxiety. One of this is that boys tend to have low 
mathematics test anxiety than girls. Sarason et al 
(1952) obtained results which showed that boys got lower 
anxiety scores than girls in mathematics test anxiety. 
The result was explained as being probably due to the 
fact that it was harder for the boys to admit anxiety 
than for girls because a greater defensiveness could be 
aroused in boys in the process of answering questions 
than for the girls. Szetela (1973) treating mathematics 
anxiety as a quadratic function of test anxiety found 
significant interaction between test anxiety and gender 
due to the tendency of girls being more test anxious 
than boys at the highest test anxiety level.
Baya’a (1990) found that females scored higher in the
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mathematics anxiety rating scale and can thus be said to 
be more test-anxious than males. However, Turnbull 
(1965) did not obtain any significant difference in the 
mathematics anxiety levels of the university students he 
used. Ohlson (1977) also confirmed this. His data 
analysis led to his concluding that males and females 
"do not differ in anxiety levels".
2.5 MATHEMATICS SELF CONCEPT
In this section attempt will be made to explain the 
concept of academic self concept and especially 
mathematics self concept. This self concept will also be 
discussed with respect to academic achievement and also 
with respect to teaching methods. Finally, gender 
differences in mathematics self concept will be 
examined.
Self concept, in general has been defined by Helmke 
(1994) as "the relatively stable picture people have of 
themselves and their attributes". It is a desirable 
educational goal that acts as a mediating variable 
facilitating the attainment of desired outcomes such as 
academic achievement. Cooley (1902) as cited by Helmke
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(1994) said that a person’s self concept is in large 
part the result of interaction with significant others". 
Thus a person’s self concept is influenced by the 
opinion others have of him.
Self concept deals with a person’s belief or idea 
about his ability or feeling towards the items on the 
self concept scale and researchers have claimed some 
elements of biasness in terms of this self perception. 
This has raised some questions on the stability of self- 
concept measures.
Some researchers advocate that self concept is 
totally directed by situational influences and so is 
only stable as long as the situations themselves are 
stable. Other school of thought view it as a highly 
stable personality trait and that right from its 
inception in infancy, self concept becomes increasingly 
stable.
Rosenberg (1986) asserts that a state of destabili­
zation is found during adolescence. Young children tend 
to over estimate their own ability when measured against 
test of academic achievement.
An unrealistically high measure of self concept or
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perceived ability has been found in many areas of 
competence especially those relating to academic 
achievement. Marsh (1990) identifies two theoretical 
approaches to academic self concept. The approaches are 
the skill development approach and the self enhancement 
model. The skill development approach regards self 
concept as primarily the result of past achievement 
rather than a cause of subsequent achievement while the 
self-enhancement model views it as a cause rather than 
an effect of academic achievement. Thus academic self 
concept more or less reflects past achievement related 
successes and failures. The conclusion is thus that self 
concept of ability must be viewed as both a cause and a 
consequent of achievement. High self concept ability can 
be obtained if there is increase in intrinsic motivation 
and decrease in test anxiety.
Of importance in educational research is the 
specific self-concept known as academic self-concept 
rather than the global self-concept. Such academic self- 
concept, of which mathematics self-concept is an 
example, are known as evaluative self concepts and they 
have uni-dimensional constructs according to Coopersmith
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(1967). Academic self-concept, according to Marsh (1990) 
is subject specific and so when investigation is to be 
done an academic self-concept one must chose scales 
which are specific to those subject areas.
How accurate are the responses given by students on 
specific self-concept scales? Accuracy of academic self- 
concept deals with the agreement between self- 
assessment of academic ability and an independent 
criteria such as teachers rating or performance in a 
test. Two views are held by research results: the view 
regarding self assessment of child and youth as 
realistic description of academic ability and school 
performance, and the second view that says self concept 
can only be better understood in terms of systematic 
bias of students involved. Shraughter and Osberg (1981) 
observed that "self-ratings of academic ability consist 
of quite accurate description of actual academic 
performance".
Academic self concept scales have been developed in 
most subject areas and items on these domain-specific 
self-evaluation could be negative or positive. In 
summing up a child’s self concept the nature of the
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items are considered in the scoring procedures.
2.5.1 Academic Self-concept and Achievement
Review of literature predominantly point to the 
existence of significant correlations between carefully 
constructed measures of academic self-concept and 
achievement in the relevant subject area although a few 
reported no significant correlations. The general view 
is that high achievers are more accurate in their self- 
perceived ability and that students tend to rate 
themselves above class average than below it.
Marsh (1988) used High School students and found 
that mathematics and English self-concepts were 
uncorrelated despite a substantial correlation between 
mathematics and English test scores and that these self 
concepts were negatively affected by school-average 
achievement. Furthermore, a student’s self-concept in a 
particular academic subject area is formed in relation 
to performances by other students in the same subject 
(external frame of reference) and in relation to the 
performance by the same student in other academic 
subject area (internal frame of reference).
Chambers et al (1991) investigated the relationship
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between team learning outcomes, achievement and other 
measures of which academic self concept was one, and 
found that team outcome was related to achievement as 
well as academic self concept.
Yohanna and Kurman (1991) compared academic self 
concept of 287 elementary school pupils with teachers 
ratings of their academic ability and school 
performance. Using regression and path analyses they 
were able to establish that accuracy of perceived 
ability had an independent unique effect on academic 
attainments beyond the effects of academic self-concept.
Kruger et al (1992) investigated self-concepts of 95 
special needs students in grades 3 through 12 who 
received mathematics instruction in special education 
classes. The relationship between the Piers-Harris self- 
concept scale scores and mathematics achievement was 
significantly stronger for the elementary students than 
for Secondary students. The percentage of school day 
spent in special education however had negative 
relationship to general self concept scores although the 
relationship between the period had positive 
relationship to mathematics self-concept scores.
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Quite recently Marsh (1992) found academic self 
concept scale to be related to school achievement in 8 
school subjects. The correlation between matching areas 
of achievement obtained in this study was higher than 
those in his past studies. High correlation was obtained 
for self concept scales reflecting academic subject 
areas rather than non academic subject areas.
Some research findings have found results contrary 
to those discussed thus far. For example, Hansford and 
Hatlie (1982) found no substantial relationship between 
self-concept and achievement although academic 
achievement correlated more with measures of academic 
self-concept than generalised measures of self concept. 
Bryne (1984) supported this view in his finding that 
general or global self-concept was uncorrelated with 
achievement. Using a sample of students at tertiary 
level, Perney and Ravid (1990) found that academic 
performance in a statistic course was not related to 
mathematics self-concept. Low mathematics self concept 
has also been identified as one of the correlates of 
test anxiety (Eddy, 1985).
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2.5.2 Academic self-concept and Instructional method
Studies reviewed earlier show evidence that increase 
in achievement will also bring about an increase in 
academic self concept. Thus, the studies on appropriate 
strategy for enhancing achievement in mathematics will 
also bring about enhanced academic (mathematics) self 
concept. Very scanty literature specifically focused on 
instructional strategy and self concept per se. In the 
few studies identified, achievement is somehow brought 
in. Skilful teaching in a diagnostic sense will bring
9
about better achievement and improved and healthier 
self-concept. Schurer and Kraut (1979) attempted to 
increase educational achievement via self concept 
change. The desired objective was not attained probably 
due to the weak design, smallness of the sample and lack 
of multi-dimensional instruments with adequate 
construct, validity. Marsh and Richards (1988) however 
found that instructions that made frequent use of praise 
and performance feedback greatly enhanced students 
self-concept.
Lawal (1987) using a sample of Nigerian students 
investigated the effect of participation in small groups
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on academic self concept of students. He found that this 
interactive instructional strategy increased the 
students self-concept as well as his achievement in the 
subject area used.
2.5.3 Self concept and gender
The studies reviewed on gender differences point out 
that boys tend to have higher self concept in 
mathematics than girls. Marsh and Smith (1987) confirmed 
that apart from boys having better understanding of 
problem structure than girls they also tended to have 
higher mathematics self-concept than girls. Maqsud et al
(1991) investigated the relationship of some factors 
which included mathematics self concept on mathematics 
achievement of secondary and university students in 
Bophuthatswana. The findings indicated that boys scored 
significantly higher than girls on the self concept as 
well as achievement measures.
Dickens et al (1990) examined parental influence on 
mathematics self-concept of 165 high achieving 
adolescent girls. The research assessed, among other 
things, the degree of parent-child identification on the 
girls’ own mathematics concepts. A series of analysis
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showed that parent mathematics self concept had little 
direct effect on daughter’s mathematics self concept. 
Yee (1986) also found that parents strongly influenced 
their children’s mathematics attitude and mathematics 
self-concept. The available scanty literature points to 
the need for more investigation by researchers in 
education on issue of gender and mathematics self- 
concept.
2.6 SUMMARY
This chapter has reviewed some studies that are 
relevant to the variables in the study.
The lecture method was examined in terms of its 
advantages and disadvantages. Studies comparing its 
effect with respect to other techniques of teaching at 
tertiary level were also reviewed. The trend is that the 
lecture method is too teacher centred and does not 
promote active involvement of students. The need for the 
use of interactive learning which is a type of 
cooperative learning was discussed.
Mathematics achievement has been discussed with 
respect to different instructional techniques and
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gender. The teaching method employed either at primary 
or secondary or even tertiary level plays a part on 
student’s achievement. When gender is considered some 
studies found significant difference between males and 
females while some did not find any difference.
Test anxiety and the theory behind it has also been 
discussed in this chapter. The trend is that test 
anxiety is negatively related to achievement and females 
tend to have higher test anxiety especially at the 
primary level than males, although some studies at 
tertiary level do not indicate any significant 
difference.
From the general self-concept theory, academic self- 
concept is identified as student’s self-perceived 
ability in the particular subject area. As a personality 
trait some researchers have complained about self 
concept measures not being accurate because of it being 
subject to bias on the part of the respondent to the 
given self-concept scale. Studies show that results are 
stable while some do not agree on its stability. Some 
studies found a strong relationship between academic 
self concept and achievement in the particular subject
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area while some did not find substantial significant 
correlation. Marsh, who has researched extensively on 
self concept, advised that in investigating self-concept 
and achievement one must not use the global self concept 
scale but the specific scale. Not many studies have 
focused directly on teaching methods and gender but the 
few that did pointed out that any teaching method that 
improved academic achievement would also improve 
academic self-concept. Sequential teaching of subjects 
and arrangement of problems from easy to difficult would 
boost up the self concept of a students since success at 
tackling easier problems could increase his confidence. 
High academic self concept has been associated more with 
male students than with females. Furthermore, some 
studies have linked high academic self concept with low 
test anxiety.
The review of literature has shown evidence of 
research efforts being made to improve students 
achievement in mathematics not only at the primary and 
secondary level, but also at the tertiary level. However 
most of the studies revealed that the efforts on 
exploring alternatives to the lecture method have been
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done outside the Nigeria setting. No study in Nigeria, 
to the knowledge of this researcher, has been undertaken 
with students at tertiary level. This points to the need 
to investigate other appropriate methods to be used in 
helping students undertaking one mathematics course or 
the other at tertiary level.
When the two psychological measures in this study 
(i.e. mathematics test anxiety and mathematics self- 
concept) are considered, the review points to the dearth 
of literature on how appropriate instructional 
strategies could be employed to promote the 
psychological effects of these variables. Most of the 
studies reviewed are correlational in nature. This calls 
for the need for an experimental research involving 
these variables. Also, the review has pointed out the 
need to investigative how gender affects both the 
psychological and cognitive variables as well as to 
study the interactive effect of gender and instructional 
technique on these variables since literature in this 
aspect is very scanty. This inadequacy has led to the 
purpose of carrying out this study.
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Chapter three
RESEARCH DESIGN AND METHODOLOGY
3.l Introduction
This chapter gives a detailed description of the 
design and procedures used in this study.
The chapter is divided into five main sections as 
follows:
(i) Research Design
This section describes the design adopted in 
this study. Included are the types of variables 
used in the study as well as the issues of 
sensitivity and internal and external validity.
(ii) Population and Sample
The population and sample used in the study are 
described in this section.
(iii) Instrumentation
The instruments used in the study are listed here. 
The procedures employed in constructing and
validating the instruments are also described.
(iv) Procedures for data Collection
This describes how the data for the study was
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collected.
(v) Procedures for data analysis
This section gives a description of the statistical 
techniques used in the analysis of the data 
collected in the study.
3.2 RESEARCH DESIGN
A 3 x 2 non randomised control pre-test/post-test 
quasi-experimental factorial design was used for this 
study. In experimental research, the subjects used 
should ideally be randomly assigned to different groups. 
However, in most experimental research in education in 
which the classroom is the formal setting, it is not 
easy to disturb the existing classroom setting and 
intact classes are used as suggested by Lovell and 
Lawson (1970) and Van Dalen (1973). Thus, the quasi 
experimental design was employed in this study. The 
design was such that instructional strategy (Treatment) 
was crossed with students’ gender (male, female). The 
choice of this design is based in its distinctive 
advantages which are:
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1. It provides opportunity for the researcher to study 
the interactive effect of gender and method of 
instruction on each of the dependent variables.
2. It permits the disentangling of the independent 
variables that are intertwined naturally and to 
establish a causal link between each of the 
independent variables and the dependent variable.
3. It is much more economical (in terms of energy time 
and number of subjects) than corresponding multiple 
single factor experiments.
The design is illustrated on table 3.1.
Table 3.1
Design of the study
GENDER
MALE FEMALE
INSTRUCTIONAL Expt. 1
STRATEGY Expt. 2
Control
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This design can also be represented in symbolic form as 
follows:
Group Pre-test Treatment Post-test
Experiment 1 mu Ti Mu
Experiment 2 *̂21 t2 m 21
Control ®31 t 3 m 31
where m11# and ir̂  represent pre-tests^, T2, T3 
represent the treatment and Mxl/ M21, M31 represent post­
tests .
3.2.1 Variables in the Study
As is the case with experimental research, there are 
two types of variables involved: namely the independent 
variables and the dependent variables.
3.2.1.1 INDEPENDENT VARIABLES
There are two independent variables used in this 
study. They consist of an active variable and attribute 
variable. They are as follows:
1. TREATMENT
This is the variable that was manipulated and it is
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thus the active variable. Treatment occurred at 
three levels: Experimental group 1 [Tj experimental
group 2 [T2] and control [T3] .
In the experimental group 1, the subjects were 
exposed to interactive learning plus individual 
"hand-on-experience" with learning materials. In the 
experimental group 2, the subjects were exposed to 
interactive learning plus group "hand-on-experience" 
with learning materials. In the control group the 
subjects were exposed to the conventional lecture 
method. The investigator conducted the lessons in 
the three treatment groups. The treatment lasted for 
10 weeks at the rate of two hours per week.
2. GENDER
Naturally, this attribute variable occurred at two 
levels: male and female.
3.2.1.2 Dependent Variables
The three dependent variables used in this study 
were achievement in mathematics, mathematics test 
anxiety and mathematics self-concept.
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3.2.2 Sensitivity, internal and external validity
Sensitivity, according to Lipsey (1990) refers to 
the likelihood that an effect, if present will be 
detected. Sensitivity is based upon what is "true" but 
unknown state of affairs in a population and that state 
as observed by the researcher. A design is thus said to 
be sensitive to the extent that the researcher’s 
conclusion mirrors the "true" state. If the difference 
between the "true" and experimental state (known as 
error) is reduced, the sensitivity will thus be 
increased (Lawson, 1994).
The researcher adopted two strategies to ensure 
systematic differences in the treatment conditions. 
First, a large sample size (N = 165) was used with the 
hope of increasing the size of the obtained F (Keppel 
and Sanfley, 1980). The use of a large sample increases 
the number of degree of freedom for the denominator 
which leads to a smaller critical value of F (Cox, 
1958) .
The sensitivity of the experiment was also increased 
by ensuring the existence of within groups homogeneity
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among the experimental lectures to be received by each 
treatment group. The investigator undertook the 
provision of treatment across the groups and ensured 
that the mode of the treatment that characterized each 
group was maintained within the group. It was thus 
possible for the study to ensure a good measure of 
intra-lecturer reliability which, if not maintained, 
could be a potential source of experimental error 
(Lovell and Lawson, 1970; Fricks. 1978).
Internal validity deals with the extent to which the 
observed changes or findings are attributed to the 
manipulation or treatment. It is concerned with the 
confidence that can be placed on the assertion that the 
treatment was solely responsible for the effect that was 
observed. The factorial design and the analysis of 
covariance technique adopted in this study were expected 
to have depreciating effects on internal validity 
threats.
Compte and Goetz (1982) emphasized the fact that the 
concept of external validity "focuses on 
generalizability, comparability and translability" of
research results.
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This present study attempted to satisfy all these 
conditions through the use of all Business Education 
students as well as the use of highly reliable and valid 
instruments as suggested by Cook and Campbell (1979). It 
is thus expected that within the limitations of this 
present study its findings could be extrapolated to 
other similar educational settings.
3.3 POPULATION AND SAMPLE
The target population consists of all students 
offering Business Education in Colleges of Education in 
Lagos state.
The sample used were all first year full time 
students of Business Education in Colleges of Education 
in Lagos State. At the time the study took place, there 
were only two Colleges of Education in Lagos State 
offering full time courses leading to the National 
Certificate of Education (NCE) in Business Education. 
The institutions were: The Federal College of Education 
(Tech), Akoka and the Lagos State College of Education 
Ijanikin. The Business Education students in these 
Colleges offered the same courses in Business
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mathematics as stipulated in the NCCE syllabus. The 
first year students in the two institutions in Business 
Education were admitted through the Joint Admission and 
Matriculation Board examination for Polytechnics and 
Colleges of Education and the requirements for admission 
to the two Colleges of Education were similar. Both 
Colleges had links with University of Lagos which was 
responsible for moderation of their examination papers.
At Akoka the Business Education students are 
admitted into the departments of accounting and 
secretarial education and each set of students receives 
lecture as intact classes on departmental basis. These 
two groups of students offer the same courses throughout 
their first and second year. At Ijanikin the division 
into the departments is done in their third year. This 
was the condition that
existed during the experimental period and still exists 
today.
The students in accounting department formed 
Experimental 1 while the students in the Secretarial 
group comprised the Experimental 2 by random assignment 
of groups. Thus experimental one and two were from
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Akoka. All the year 1 Business Education students at 
Ijanikin comprised the control group.
Table 3.2 shows the distribution of the sample.
Table 3.2
Number of Business Education in the achieved sample
Experimental Experimental Control Total
Group 1 Group 2 Group
Male 39 10 22 71
Female 24 44 26 94
Total 63 54 48 165
These 165 students consisted of 71 Males and 94 
Females and had mean age of 22.96 years.
Excluded in the sample were all students in year 2
and year 3 with carry-over in year one Business
Mathematics since this group of students were not
regular at lectures.
Ijanikin and Akoka are far apart. This arrangement 
ensured that there was no contamination between the 
experimental and control groups.
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3.4 INSTRUMENTATION
The following are the instruments used in this 
study.
1. Pre-test: A test of pre-requisite skills in 
mathematics. This is a 30 item objective test 
based on the syllabus of the course done by all 
the students in the semester preceding the 
commencement of the study. As the pre-test of 
the study, it measured the initial ability of 
the students and thus served as the covariate.
2. Post-test: A 28 item post achievement test 
covering the course content of the Business 
Mathematics course to which all the students 
were exposed during this study.
This test was constructed and validated by the 
researcher.
3. Structured learning materials on the topics in 
the syllabus carefully developed by the 
researcher and
whose content validity was ensured with the help 
of experienced mathematics lecturers and
evaluators.
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4. A mathematics test anxiety questionnaire known 
as the Inventory of Test Anxiety in mathematics 
(ITAM) by Osterhouse, Obe (1981) validated this 
on a group of Nigerian secondary school 
students. The instrument consists of 16 items 
describing anxiety situations to which the 
respondents react as to the degree of their 
agreement. All the items describe how anxiety 
hinders their performance. It is thus a 
debilitating scale. The maximum Score on this 
scale is 80 while the minimum is 16.
5. A mathematics self-concept scale constructed and
validated by the researcher on NCE Business 
Education students. It consists of 17 items 
expressing students feeling about their
perceived ability in mathematics. The responses 
are scored on likert five points scale with 
options ranging from strongly agree to strongly 
Disagree. The first part of the questionnaire 
contains background information of the
respondents while the second part contains the 
statements of self-concept. The validation
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procedure for all the instruments is described 
later in this chapter.
The instruments can be found in the appendices 
at the end of this thesis.
3.5 PROCEDURES
This study took place in two parts namely
1. The construction and validation of the instruments
2. The main study itself.
3.5.1 Construction and Validation of the Instruments
This section deals with construction of the 
instruments, where necessary, and the validation 
procedure used.
(i) THE ACHIEVEMENT TESTS
The two achievement tests used in the study are both 
multiple choice objective tests and were constructed in 
line with the principles of test construction.
A good test instrument must possess three basic 
attributes which are validity, reliability and 
useability according to Lyman (1963). Validity of a test 
deals with its ability to measure what it is expected to 
measure while reliability deals with obtaining
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dependable or consistent scores and useability is 
concerned with practical factors such as ease of 
scoring, practicality of time and all these three 
attributes were borne in mind by the researcher while 
constructing the achievement tests.
The first step taken was to determine the objectives 
of instruction and items included in the test were drawn 
from the objectives of instruction of the respective 
courses. Content validity is related to how adequately 
the test sample the domain about which inferences are 
made (Mehrens and Lehman, 197 5) . A good table of 
specification of course content by objectives ensures 
adequate content validity. Thus a table of specification 
was constructed for each of the two tests in other to 
ensure that the tests measured the learning outcomes and 
content in a balanced manner as suggested by Ali (1989). 
The table of specification or test, blue print was then 
given to experienced mathematics teachers and evaluators 
and modifications done based on their advice. Tables 3.3 
and 3.4 show the tables of specification for the two 
tests.About total of 80 items were written for each of 
the tests and face validity was ensured with the help of
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the experts mentioned earlier. The items were then 
pilot-tested on seventy-five NCE Business Education 
Students who have already covered the course and were 
typical of those for whom the test is built. This pilot 
testing is very essential in the validation of 
achievement tests as it helps in item analysis and item
Table 3.3
Table of specification of the pre-test in 
mathematics
LEVEL OF KNOWLEDGE COMPREHENSION APPLICATION
OBJECTIVE
CONTENT AREA item no item no item no
1. Indices - 3 28
2. Logarithm 5 4
3.Descriptive 2, 19, 29 1, 7 16, 21
statistics
4. Fraction 22, 27 20, 10, 17, 25
26
5. Decimals 8, 9, 14, — 24
23
6. Percentage 11, 15 18 12, 13
7. Algebra 6 30
Total 13 (43.3%) 10 (33.3%) 7 (23.3%)
(percentage)
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Table 3.4
Table of specification of the post-test achievement test
in mathematics
LEVEL OF KNOWLEDGE COMPREHENSION APPLICATION
OBJECTIVE
CONTENT AREA (item no) (item no) (item no)
1. Fraction and 8 If 2 7
decimals
2. Percentages 13, 14 20, 11 3, 12,19
Profit and loss
3. Simple 5, 21 6 16
interest
4. Compound 22 23 24
interest 
Depreciation
5. Ratio and 17, 18
Proportion 
Partnership '
6. Rates and 25 4 10
Tax
7. Stocks and 27 28 -
shares
8. Payroll
computation _ 9, 15 26
Total 8 (28.6%) 12 (42.8%) 8 (28.6%)
(percentage of 
total)
selection procedure (Ohuche and Akeju, 1977). A scoring 
stencil was used in scoring the items. The tests have 
four options with numerical options written in ascending
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order. The tests were scored and item analysis was done 
on the best —1 3  and the worst 
1
3—  of the testees as 
suggested by Ali (1989). The item analysis was done with 
respect to
(a) the difficulty level of each item
(b) the discriminating power of each item
(c) and the functioning of the distractors. The 
difficulty level dealt with the percentage of students 
who correctly answered the given test item.
Although Educational Testing services (ETS) recommends 
that item difficulty level of 0.30 to 0.80 is good, the 
researcher decided to select items with difficulty level 
of between 0.3 to 0.7 as suggested by Ali (1989).
The discriminating power was also investigated. 
According to Ahmann (1962), a test item possesses 
adeguate discriminating power when it is capable of 
differentiating between superior and inferior students. 
He recommended D values greater than +0.4 as being good 
while those between +0.4 and +0.2 are satisfactory and 
anything below 0.2 as being poor. The researcher decided
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to select only items whose discriminating power was 
greater than 0.2.
The analysis of the distractor was also done to 
determine its functionality and also its different 
attractiveness. The functioning distractors showed 
greater attractiveness to the lower than to the upper 
group and those who did not were either modified or 
discarded.
The result of the item analysis led to the emergence 
of 30 items in the pre-test and twenty eight in the post 
test. A full detail of the item analysis can be found in 
the appendix at the end of this report.
The two achievement tests were not speeded tests and 
so estimation of reliability through the method of 
internal consistency was appropriate as advocated by 
Mehrens and Lehman (1975).
Table 3.5 gives the psychometric properties of the
two tests.
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Table 3.5
Psychometric Properties of the 
Achievement Tests
Psychometric measure Pre-test Post-test
Range of 0.40 - 0.72 0.40 - 0.64
discriminating 
power (D)
Item Difficulty Range 0.32 - 0.70 0.30 - 0.70
(P)
Coefficient of 0.87 0.88
internal consistency
(ii) THE MATHEMATICS TEST ANXIETY SCALE
The Inventory of Test Anxiety in mathematics as 
modified and adapted by Obe (1981) was administered 
to the seventy-five NCE Business Education Students. 
The scores for each item ranged from 1 to 5 with a 
maximum score of 80. Students with scores of 16 - 37 
are classified as low test anxious while those with
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scores between 38 - 59 have moderate test anxiety 
and those with 60 and above are high anxiety 
students. Four weeks later, the same instrument was 
administered to the same set of students and the 
estimate of coefficient of stability was obtained. 
The coefficient of internal consistency as measured 
by Cronbach a was 0.87.
The coefficient of stability was 0.85 while the 
criterion validity with mathematics test scores as 
the criterion yielded 0.63.
(iii) THE SELF-CONCEPT SCALE
This scale was constructed by the researcher. Thirty 
items were written describing self concepts in 
mathematics. The items were content validated by 
specialists in mathematics education, evaluators and 
two educational psychologists for review. This 
review was done in order to determined the item to 
be retained, reframed or rejected. The review led to 
the emergence of seventeen items. The aim was to 
ensure face and content validity.
The final form of the questionnaire was then 
administered to the sample of Business Education
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Students used in the pilot study.
Likert five point scale with options ranging from 
strongly agree to strongly disagree was used to 
obtain the responses. The items were scored by 
weighting positive items from 5 for strongly agree 
to 1 for strongly disagree. The weights were 
reversed for negative items. The respondents score 
was the sum of the weighted options selected by him. 
A maximum of 85 and a minimum of 17 is expected from 
the instrument. Scores under 40 depict low 
mathematics self concept while those between 4 1 - 6 3  
depict moderate self concept and 64 and above shows 
high self concept.
Reliability of an instrument deals with the 
consistency of the results obtained by using the 
instrument while validity deals with the extent to 
which the measuring device measures what it is 
expected to measure. Validity and reliability 
estimates were computed for this instrument. The 
coefficient of stability was computed by the test- 
retest method. A coefficient of 0.75 was obtained. 
Correlation between a particular score and a
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criterion score gives rise to validity index (Lyman, 
1963) . The scores on this instrument were correlated 
with their scores in Business mathematics. A 
validity coefficient of 0.68 was obtained. This is 
quite in order since Cronbach (1970) has confirmed 
that it is unusual for validity coefficient to rise 
above 0.6 and that any positive correlation is an 
indication of the accuracy of the instrument. Thus 
the fairly high validity coefficient of this 
instrument is highly significant in pointing out its 
criterion related validity.
Construct validity is the extent to which test 
performance can be interpreted in terms of certain 
psychological constructs. One of the methods 
identified by Gronlund (1976) of establishing 
construct validity of a test is to relate the test 
with other known test of that ability. To this end, 
the researcher computed the correlation between this 
mathematics self-concept scale and one of the 
mathematics self-concept scale used by Eshel and 
Kurman (1991). A correlation of 0.82 was obtained. 
The factorial validity (Lyman, 1963) was also
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established through the use of factor analysis. The 
items in this test were factor analyzed with those 
of the Inventory of Test Anxiety in Mathematics 
(ITAM). Literature has shown that this has negative 
relationship with mathematics self-concept. This 
test anxiety scale was thus expected to measure 
different constructs. The purpose of doing this was 
to give evidence of divergence of indicators as 
advocated by Cronbach (1970). This method, employed 
in the factor analysis was the Principal Component 
Analysis with iteration. An initial number of eight 
factors were extracted. Using the Kaiser 
normalisation criteria of eigenvalues greater than 
1, only two factors were obtained when the eight 
factors were subjected to orthogonal varimax 
rotation. The final statistics showed that the self- 
concept items were generally positively heavily 
loaded on factor 1 while the test anxiety was 
heavily positively loaded on factor 2. The self- 
concept items all have positive loadings on factor 
1 and negative loadings on factor 2 while ITAM had 
negative loadings on factor 1 and positive loadings
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on factor 2. Thus factor 1 deals with self-concept 
while factor 2 deals with test anxiety. This, 
coupled with the high coefficients of internal 
consistency obtained in the two tests ensures that 
the factorial validity is established. The results 
are in appendices 13 and 14 at the end of this 
report.
Cronbach alpha was also computed for the self 
concept scale.
The SPSS programmed was used in the validation 
process. Table 3.6 gives a summary of the 
psychometric properties of the inventory for Test 
Anxiety in mathematics and the self-concept scale.
Table 3.6
Psychometric properties of the psychological tests
Magnitude
Psychometric measure Test Anxiety Self concept
Standardized item a 0.87 0.90
Cronbach a 0.87 0.90
Validity Coefficient 0.63 0.68
Coefficient of 0.85 0.75
stability
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(iv) The learning materials
The materials used were structured learning units 
covering topics in the syllabus of the second 
semester Business mathematics for year 1 students as 
stipulated by the NCCE syllabus. Each unit starts 
with the objectives for that topic and contains some 
explanation of the lectures. There are questions at 
the end of each unit which students were expected to 
attempt. Face and content validation of the 10 unit 
materials were done with the helpful contributions 
of two experienced mathematics lecturers in the 
course and two educational evaluators.
The learning material can be found in appendix 15.
3.5.2 PROCEDURES FOR THE MAIN STUDY
Before the commencement of the main study official 
permission was sought from the authorities of the two 
institutions.
At the beginning of the second semester, the 
mathematics test anxiety scale and self concept 
questionnaires were administered to the groups of 
students before the achievement pre-test to prevent the 
inflation of the scores by the pre-test results and then
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the treatment commenced. The time table was so arranged 
that the Akoka students were taught on mondays while the 
Ijanikin students had their lectures on Wednesdays. 
Efforts were made to prevent contamination by ensuring 
that the two groups at Akoka had the lectures almost one 
after the other. At the time of research the distance 
between Akoka and Ijanikin was not easy to cover in less 
than two hours. The students in the three groups were 
tested after each lecture. Thus chances of contamination 
was controlled. Attendance was kept on each students. At 
the end of the treatment a post test was administered on 
each of the following.
(i) the Inventory of Test Anxiety in mathematics
(ii) the mathematics self concept scale
(iii) the mathematics test on topics taught.
DESCRIPTION OF THE INSTRUCTIONAL PROCEDURES
1. The lecture method: This usually began with a 
summary of the previous lesson and introduction of 
the topic for the day (5 minutes) . This was then 
followed by gradual and systematic presentation of 
the topic for the lesson, with illustrative examples
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by the lecturer (1 hours, 30 minutes). During this 
period students were allowed to ask questions. The 
responses to these questions were supplied by the 
instructor alone. The lecture method thus involved 
more of oral presentation by the lecturer. The 
lecture was followed by a short fifteen minutes 
test. The summary of the lesson and the corrections 
of the test usually ended the lecture for the day 
(10 minutes).
2. Interactive learning: The students involvement was 
maximised by this type of strategy. The instructor 
started by reviewing previous lessons through asking 
of questions (5 minutes).Efforts were made to 
involve non volunteer students. Correct answers were 
acknowledged while partially correct and incorrect 
responses were tackled with cues and probes and help 
from other students. The instruction for the day was 
then delivered with active contribution from the 
students until the concept was well understood (1 
hour 15 minutes).
During this period students were encouraged to solve 
problems on the board. This was followed by
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distribution of the materials which the students 
were expected to work upon either on individually or 
an group basis (30 minutes). The materials contained 
some exercises which the students attempted and 
submitted. The lecture concluded with a summary and 
correction of the assignments (10 minutes).
Experimental 1
Students in this group worked individually on the 
learning materials. The seating arrangement while 
working on this material was such as to minimise 
contacts with other students. They were not allowed to 
seek help from their mates. The assignments were scored 
on individual basis.
Experimental 2
Students sat in groups of five as suggested by Goldberg 
(1981) and Channon and Walker (1984) who advocated for 
groups of four or five for problem solving if 
cooperative
learning was to be initiated at tertiary level. Each 
group contained at least one of each gender. A leader
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who was noted for having high achievement in mathematics 
as evident from the pre-test was included in each group. 
Conwell et al (1993) advocated that in small group 
learning each group should have a leader with strong 
academic as well as collaborative skills in helping the 
group achieve a goal. This was taken into consideration 
in allocating leaders into each group. Each group thus 
contained high, low and moderate achievers. The 
materials were used by students in their various groups. 
Every member of each group was given opportunity to 
contribute his idea to the problem as well as to seek 
clarification from his colleagues within the group. The 
test at the end of the material for each lecture was 
done on group basis and feedback was also done on group 
basis.
The whole treatment lasted for a period of ten weeks 
after which post-tests on each of the dependent variable 
were administered.
3.6 ANALYSIS
The experimental and control groups constituted the 
units of observation. The analysis thus focused on the
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group mean scores in mathematics test anxiety, 
mathematics self concept and achievement in mathematics. 
The three groups were compared on each of the three 
dependent variables. The scores of individual student 
formed the unit of analysis. Data analysis was done 
using SPSS sub-programme (Nie et al, 1975) on analysis 
of covariance with the pre-tests as covariates. The 
ANCOVA was expected to correct for any initial 
differences in the dependent variables and other 
extraneous factors that could compound treatment effect 
(Lovell and Lawson, 1970).
Each of the hypotheses was examined. In the case of 
no significant interactive effect, Multiple 
Classification Analysis (MCA) was used to determine the 
magnitude and direction of the effect as well as the 
amount of variation due to each independent variable. 
Where an interaction was significant, Scheffe Multiple 
Range test was used to identify the source of the
interaction.
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Chapter four 
RESULTS AND DISCUSSION
The results obtained from the analysis of the 
collected data are discussed in this chapter.
The descriptive statistics of the data are first 
discussed followed by the testing of the various 
hypotheses. Once the presence of interaction was 
established in the various results of analysis of 
covariance the main effects were not tested since the 
two independent variables jointly affect the dependent 
variables under focus as suggested by Kim and Kohout 
(1988) . This is to prevent the risk of erroneous or 
misleading interpretation of results (Wright, 1976). The 
post test mean scores for measures with significant 
interaction were examined carefully by studying the 
simple effects of the independent variable with 
significant interaction. The cell means of the variables 
were also plotted in order to give more visual impact.
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4.1 DESCRIPTIVE STATISTICS OF THE DATA
Table 4.1 gives some descriptive statistics of the 
data in terms of the mean, standard deviation, Kurtosis 
and Skewness.
Table 4.1
Descriptive Statistics of the Overall data.
PRE-TESTS POST-TESTS
DESCRIPTIVE ACHIEVE TEST SELF ACHIEVE TEST SELF
STATISTICS -MENT ANXIETY CONCEPT -MENT ANXIETY CONCEPT
MEAN 48.339 34.648 43.921 55.370 29.091 49.728
STANDARD
DEVIATION 14.189 10.903 7.924 12.573 11.051 6.581
STANDARD ERROR
OF MEAN 1.105 0.849 0.617 0.979 0.860 0.517
KURTOSIS -.338 -0.133 0.408 0.877 0.310 -0.081
SKEWNESS .427 0.615 0.115 -0.277 0.977 -0.389
S.E. KURT .376 0.378 0.376 0.376 0.376 0.379
S. E. SKEW .189 0.189 0.189 0.189 0.189 0.191
It is evident that the experiment has resulted in an
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increase in general level of mathematics achievement 
from 48.339 to 55.370. Both tests were converted to 
percentages prior to analysis for ease of comparison. 
There has also been a reduction in the debilitating 
anxiety level of the subjects. The self concept has also 
improved as anticipated. However the result of the 
hypothesis testing would prove to be of statistical 
significance than the general descriptive statistics. 
The negative values of Skewness obtained showed that the 
means of these values were lower than their respective 
medians.
Kurtosis deals with the degree of peakness of a 
distribution. The variables with positive values of 
Kurtosis have relatively high peaks and are known as 
leptokurtic distribution while those with negative 
values have flat-topped peaks and are called 
platykurtic.
Tables 4.2, 4.3 and 4.4 also show the average composite 
scores for each of the three variables namely 
achievement in mathematics, mathematics test anxiety and 
mathematics self-concept. The scores on each table has 
been divided by the three categories of treatment and
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two categories of gender.
Table 4.2
Group means of post-test mathematics achievement scores
GENDER Experimental Experimental Control Grand
1 2 total
MALE 54.85 63.50 52.23 55.25
FEMALE 56.38 56.18 53.38 55.46
TOTAL 55.43 57.54 52.85 55.37
The students exposed to interactive learning with use of 
materials in group had the highest scores in the 
achievement test (57.54) while those exposed to only 
lectures had the lowest (52.85). The females had a 
slightly higher mean score of 55.46 than the males 
(55.25). Males in experimental 2 had the highest mean 
score (63.50) when compared with other males while 
females in experimental one had the highest mean score 
(56.38) when compared with females in the other groups.
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Table 4.3
Group means of post test anxiety scores
Experimental Experimental Grand
Gender 1 2 Control mean
Male 29.36 32.76 31.55 30.51
Female 25.46 29.18 28.42 28.02
Total 27.87 29.83 29.85 29.09
Those exposed to lecture method alone had the highest 
test anxiety scores (29.85). Males had higher test 
anxiety mean score (30.51) although the significance of 
this will be discussed under hypothesis testing. Also, 
both males and females in experimental 2 obtained the 
highest test anxiety score when compared with those in 
other groups.
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Table 4.4
Group means of post-test self-concept scores.
EXPERIMENTAL EXPERIMENTA CONTROL GRAND
GENDER 1 L 2 TOTAL
MALE 50.56 47.40 46.43 48.87
FEMALE 50.25 49.41 52.29 50.38
TOTAL 50.44 49.04 49.56 49.73
Students exposed to interactive learning with individual 
use of material had the highest self concept score 
(50.44). Contrary to expectation those exposed to the 
group use of material had the lowest mean (49.04). 
Females had higher self concept (50.38) than males 
(48.87) . Males in group one had the highest self concept 
score (50.56) when compared with other males in the 
three groups. The female students in the lecture group 
(the control group) had the highest self concept score 
when comparison is made among the females in each of the 
three groups.
The section on hypothesis testing which deals with 
inferential rather than descriptive statistics will
enable
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appropriate decisions to be taken with respect to the 
significance of the observed differences.
4.2 HYPOTHESES TESTING
Altogether nine hypotheses were formulated and 
tested using ANCOVA for each of the three dependent 
variables. The first column of each of the ANCOVA table 
lists the sources of variation. The second column deals 
with the sum of sguares attributable to each of the 
components. The "explained" sum of squares is the total 
sum of squares for the covariates, the main effects and 
interaction terms in the model. The degrees of freedom, 
listed in the third column of each table, are one fewer 
than the number of categories. Thus gender has one 
degree of freedom while the three levels of treatment 
lead to two degrees of freedom. The degree of freedom 
for the interaction is the product of these two. The 
degrees of freedom for the residual are 
N - 1 - K where K is the degree of freedom for the 
explained sum of squares. The mean squares in column 4 
are obtained by dividing each sum of squares by its 
degrees of freedom. The ratios of the mean squares of
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each source of variation to the mean square for the 
residual gives the F ratios upon which hypothesis tests 
are based. The level of significance used in testing 
each of the hypothesis is 0.05.
4.2.1 ACHIEVEMENT AS THE DEPENDENT VARIABLE
Hypotheses 1-3 deal with the tests of hypotheses 
with achievement as the dependent variable. These three 
hypotheses are based on the main effect of each of 
gender and treatment as well as the interactive effect 
of the two.
Table 4.5 shows the output of the ANCOVA test on 
these three hypothesis. The table shows a significant 
interactive effect of treatment and gender on students 
achievement in mathematics (F(l, 158) = 3.166, P < 
0.05).
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Table 4.5
ANCOVA Table for Achievement in mathematics
SOURCE OF SUM OF MEAN
VARIATION SQUARES DF SQUARE F
COVARIATES 2587.714 1 2587.714 19.525
(PRE-TEST)
MAIN EFFECTS 1559.522 3 519.841 3.922
INSTRUCTIONAL
METHODS (V2) 1422.705 2 711.353 5.367
GENDER (V3) .284 1 .284 .002
2- WAY
INTERACTIONS 839.263 2 419.632 3.166
V 2 x V 3 839.263 2 419.632 3.166
EXPLAINED 4986.500 6 831.083 6.271
RESIDUAL 20939.949 158 132.531
TOTAL 25926.448 164 158.088
Thus the hypothesis that "there is no significant interactive 
effect of instructional methods and gender on NCE Business Education 
Students achievement in mathematics" was rejected since a 
significant effect was found. Although the main effect on treatment 
was significant [F = 5.367, P < 0.05] while that of gender was not 
significant
[F = 0.002, P > 0.05] the fact that interactive effect was 
significant implies that the other two hypotheses need not be tested 
since the two variables jointly affect achievement in mathematics. 
The small F value associated with gender does not necessarily imply
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that achievement was unaffected by gender, since gender was included 
in the significant interaction terms. Rather, it showed that if the 
scores were averaged over treatment levels, the two gender category 
means were not significantly different.
According to Kerlinger (1973) it is also possible, and often 
very profitable to graph interactions. This will give a visual 
impact on the result obtained. By plotting on the horizontal axis 
treatment positions (T1# T2, T3) and the mean values in the cell
means table (Table 4.2) at the levels of the other independent 
variable (i.e. male, female) as suggested by Kerlinger (1973), a 
graph of the interaction was obtained. The horizontal axis denotes 
the treatment level while the vertical axis shows the mean scores. 
The mean score for each of the gender was plotted with respect to 
the treatment level. Figure 4.1 shows the graph of the interaction.
*
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INTERACTION OF TREATMENT AND GENDER ON STUDENTS ACHIEVEMENT IN 
MATHEMATICS.
T 1 = Experimental Group 1 
T 2 = Experimental Group 2 
T 3 = Control
_____  = male
- - - = female
Figure 4.1
From the diagram it is obvious that the mean scores did not only 
relate to the treatment group and the gender of the student, but 
also to the combination of the values of the variables. The males 
in Experimental group 2 obtained the highest scores (63.50) while 
the females in experimental 1 had the highest scores (56.38). Thus, 
the scores for each level of treatment depend on the gender 
variable. If there was no interaction the plot for the males and
females would never cross.
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In order to explore the significance of interaction of 
instructional method and gender, a post hoc test, the Scheffe 
Multiple Range Comparison test, was performed separately for the 
males and females on the three instructional methods. The results 
are displayed on tables 4.5a and 4.5b.
For the males, the interactive method with group use of 
material was significantly better than the other two methods and the 
pairwise difference between the group use of materials and the 
individual use of materials as well as the difference between the 
group use of materials and the lecture method were significant. For 
the females the source of variation was due to the superiority of 
the individual use of material over the lecture method since the 
test gave evidence of significant pairwise difference between these 
two methods.
♦
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Table 4.5a
Scheffe mulitiple Range Comparison Test on Achievement in 
mathematics of MALES in the three instructional groups.
G G G
r r r
P P P
Mean Group 1 2 3
54.85 Grp 1
63.50 Grp 2 *
52.23 Grp 3
Group 1 = Interactive with individual use of materials.
Group 2 = Interactive with group use of materials.
Group 3 = Lecture method.
* Pairs of groups significantly different at 0.05 level.
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Table 4.5b
Scheffe Multiple Range Comparison Test om Achievement in Mathematics 
of FEMALES in the instructional groups.
Group 1 = Interactive with individual use of materials.
Group 2 = Interactive with group use of materials.
Group 3 = Lecture method.
* Pairs of groups significantly different at 0.05 level.
The fact that individual use of material was more favourable 
to the female students while the group use of material was more 
favourable to the male students in terms of achievement could be 
explained. The females in experimental group 1 were fewer in number 
than the males. Similarly the males in group 2 were fewer than the 
females. This could result in their putting in their best in order 
boost up their sex ego. During group work females tend to chip in 
some irrelevant discussion which might be a hinderance to their
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performance. In the experimental group 2 only one male was attached 
to each group because of the fact that they were few in number. The 
male ego of wanting to put in one's best in the midst of the 
opposite sex may account for the higher performance of the males in 
this group. On the other hand, individual use of materials would 
favour the females more since there will not be time for non 
academic activities as they would be compelled to face their 
studies.
Furthermore the result confirmed the finding of Li and Georgina 
(1992) that boys preferred both the individualistic and competitive 
styles in mathematics while females preferred the individualistic 
styles. Although the sample used by these two was based on gifted 
secondary school pupils this study has shown that it could hold for 
NCE students.
4.2.2 TEST ANXIETY AS THE DEPENDENT VARIABLE
(HYPOTHESIS 4 - 6 )
Hypothesis 4: There is no significant interactive effect of 
instructional methods and gender on the test anxiety of NCE Business 
Education Students.
■t
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Table 4.6
ANCOVA summary table for mathematics test anxiety
SOURCE OF SUM OF DF MEAN F
VARIATION SQUARES SQUARE
COVARIATES 3951.685 1 3951.685 41.277
(PRE-TEST)
MAIN EFFECTS 915.884 3 305.295 3.189
INSTRUCTIONAL 691.959 2 345.980 3.614
METHODS (V 2)
GENDER (V3) 273.456 1 273.456 2.856
2 WAY 35.662 2 17.831 .186
INTERACTIONS
V 2 x V3 35.662 2 17.831 .186
EXPLAINED 4903.231 6 817.205 8.536
RESIDUAL 15126.405 158 95.737
TOTAL 20029.636 164 122.132
Table 4.6 shows the results of all the tests of 
hypothesis based on test anxiety as the dependent 
variable. In order to test for significant
interactive effect between gender and treatment the F 
value of the 2 way interaction was examined. The
obtained F of 0.186 showed that there was no ti significant 
interaction. Thus the above hypothesis was not rejected.
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This shows that the two independent variables, treatment 
and gender do not interact significantly with 
mathematics test anxiety with mathematics test anxiety. 
So the treatment does not affect mathematics test 
anxiety differentially at different facets of gender. 
Hypothesis 5; There is no significant main effect of 
instructional methods on NCE Business Education 
student’s mathematics test anxiety. As the table shows, 
the F value for the effect of instructional method is 
3.614 which has significant effect. Thus the hypothesis 
of no significant main effect of instructional methods 
was rejected. Instructional method used had effect on 
mathematics test anxiety. This supports Clute’s (1984) 
findings that the instructional method does have an 
effect on mathematics test anxiety.
In order to understand this result fully, there was 
need to carry out a multiple classification analysis. 
This is because there was no interactive effect of 
treatment and gender. The multiple classification 
analysis table,
according to Kim and Cohort (1988) is of interest only 
when no interactive terms are significant.Table 4.7
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shows the
output of the multiple classification Analysis on the 
Test Anxiety Scores.
Table 4.7
Multiple Classification Analysis (MCA) on Mathematics
test anxiety.
GRAND MEAN = 29.091
VARIABLE + N UNADJUSTED ETA ADJUSTED FOR BETA
CATEGORY DEVIATION INDEPENDENTS 
+ COVARIATES 
DEVIATION
INSTRUCTIONAL
METHODS
EXPERIMENTAL 1 63 -1.22 -2.15
EXPERIMENTAL 2 54 0.74 -0.20
CONTROL 48 0.76 3.04
.09 0.19
GENDER
MALE 71 1.42 1.60
FEMALE 94 -1.07 -1.21
.11 0.13
MULTIPLE R 2 0.243
MULTIPLE R 0.493
The table consists of the grand mean of the test anxiety
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scale, a table of deviation from the grand mean for each 
factor level and some measures of association.
The fact that the treatment (instructional methods) 
was significant only showed that the mean of at least 
one category of the instructional methods was different 
from the grand mean, after appropriate adjustments have 
been made. Hence it is very important to examine the 
pattern of the relationship of the instructional methods 
to the test anxiety variable. The first column of the 
table shows a description of the independent variables 
and their categories. The next column shows the means of 
each category expressed as deviation from the grand 
mean. No
adjustment of covariates or other independents was made 
for in calculating these values.
From the table it can be seen that treatment 
accounted for 3.6% (0.19)2 of the variation in test
anxiety. The adjusted post test mean score for any of 
the category was obtained by addition of the grand mean 
and the adjusted mean. The result showed that the 
control group had the highest adjusted post test mean
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score of 32.131 followed by Experimental 2 with 28.89 
and Experimental 1 with 26.94. Thus, students exposed to 
the lecture method had the highest anxiety and so 
obtained significantly higher anxiety than those exposed 
to interactive learning. The multiple R of 0.493 at the 
bottom of table indicates the overall relationship 
between mathematics test anxiety and the two independent 
variables. A value of R2 = 0.243 was obtained which 
showed the proportion of variation in mathematics test 
anxiety students as explained by the additive effect of 
all factors and covariates.
That students exposed to the interactive methods did 
not score highest in the mathematics test anxiety is 
quite expected. Artzl and Newman (1990) suggested that 
the interactive methods make students enthusiastic about 
mathematics since they enjoy discussing mathematical 
problems with other students and benefit from their 
interaction with colleagues and instructors. This 
enthusiasm could reduce their fear of mathematics and 
hence bring about lower anxiety than those exposed to 
the lecture method. Lipsett (1988) found that different
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instructional strategies, favouring group work could be 
used to reduce anxiety and this result has confirmed 
this although the result obtained by Wood (1992) 
contradicts this and implies that lecture method aids 
greater reduction in anxiety.
Hypothesis 6: There is no significant main effect of 
gender on NCE Business Education Students mathematics 
test anxiety. The result on table 4.6 showed that the F 
value of 2.856 was not statistically significant at the 
level of 0.05. Hence the above hypothesis was not 
rejected the conclusion is that gender has no 
significant effect on students mathematics test anxiety.
The multiple classification Analysis showed an 
initial difference of 2.49 between males and females in 
anxiety scores. When the confounding effects of 
covariates and treatments were controlled for the 
difference between males and females increased slightly 
to 2.81. Gender accounted for 1.69% (0.132) of the 
variation in test anxiety scores.
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4.2.3 SELF CONCEPT AS THE DEPENDENT VARIABLE
Three hypothesis 7, 8 and 9 were tested here. Table 
4.8 shows the output of the ANCOVA.
Hypothesis 7. There is no significant interactive effect 
of instructional methods and gender on NCE Business 
Education Students mathematics self concept.
A study of table 4.8 showed that there was no 
significant interactive effect of treatment and gender 
on the students mathematics self concept since an F 
value of 1.569 at P > 0.05 was obtained. Thus the above 
hypothesis was not rejected. This shows that gender and 
instructional methods do not have joint effects on the 
mathematics self concept of students. The type of 
instructional method to which a boy or girl is exposed 
does not have any effect on his or her self concept.
Since no interaction was found the multiple 
classification Analysis was done and proved useful in 
studying the pattern of the changes of the effects of 
each of the two independent variables as more variables 
are introduced as control. This analysis can be found on
Table 4.9.
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Table 4.8
Ancova table for mathematics self-concept
SOURCE OF SUM OF MAIN
VARIATION SQUARES DF SQUARE F
COVARIATES 1275.867 1 1275.867 37.478
(PRE-TEST)
MAIN EFFECTS 312.749 3 104.250 3.062
INSTRUCTIONAL
METHODS (V 2) 33.664 2 16.832 .494
GENDER (V 3) 311.465 1 311.465 9.146
2-WAY
INTERACTIONS 106.795 2 53.398 1.569
V 2 x V 3 106.795 2 53.398 1.569
EXPLAINED 1695.412 6 282.569 8.300
RESIDUAL 5276.638 155 34.043
TOTAL 6972.049 161 43.305
P < 0.05
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Table 4.9
Multiple Classification Analysis (MCA) on mathematics
self-concept.
GRAND MEAN = 49.728
ADJUSTED
EQREPENDENT
+
CDEARZZTBN
A VARIABLE + UNADJUSTED
CATEGORY N DEVIATION ETA BETA
INSTRUCTIONAL
METHODS
EXPERIMENTAL
1 63 0.72 0.55
EXPERIMENTAL
2 54 -0.69 -0.62
CONTROL 45 -0.17 -0.03
0.09 0.08
GENDER
MALE 70 -0.86 -1.73
FEMALE 92 0.65 1.31
0.11 0.23
MULTIPLE R 2 0.228
MULTIPLE R 0.477
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Hypothesis 8. There is no significant main effect of 
instructional methods on NCE Business Education Students 
mathematics self concept.
The F value of 0.494 showed that the above 
hypothesis was not rejected. This implies that the type 
of instructional method used had no significant effect 
on the students mathematics self-concept. The MCA table 
(See Table 4.9) shows that the percentage of variation 
accounted for by the instructional methods was too low 
(0.08)2 i.e. 0.64% . This backs up the result that there 
was no significant effect of treatments group on the 
student’s self-concept. The type of instructional 
strategy adopted thus had no effect on the students self 
concept.
That instructional method had no effect on student’s 
self concept is contrary to the findings of Wood (1992) 
who found that traditional lecture method made students 
have post confidence ratings in mathematics than the 
group method. We know that self concept deals with a 
student’s self perceived ability in the particular 
subjects area. The higher the level of confidence a
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student has about his mathematics ability the higher his 
level of self-concept. One expects that interactive 
learning would boost up students activity in mathematics 
and hence result in increased mathematics self concept. 
This study, however did not show this expected 
prediction. The reason may be that as matured students 
they had already made up their minds as to their 
perceived ability in mathematics and no amount of 
variation in instructional method could affect their 
self concept. This may not be case with younger 
students.
Hypothesis 9. There is no significant main effect of 
gender on NCE Business Education Students mathematics 
self-concept.
The F value obtained on table 4.8 was 9.149 and this 
was significant at p < 0.05. Thus the above hypothesis 
was rejected and it could be concluded that gender does 
have effect on the student’s self concept.
The multiple classification Analysis on Table 4.9 
throws more light to this result. As initial difference 
of 1.51 between males and females increased to 3.04 as 
more variables (treatment and covariates) were
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introduced as controls. The adjusted post mean score of 
50.38 for the females as against 48.87 for the males 
showed that the females had significantly higher mean 
scores than the males as far as self- concept is 
concerned. Furthermore gender accounted for 5.3% of the 
variation in mathematics self concept. The multiple R of 
0.477 at the bottom of the output showed the overall 
relationship between self concept and the two 
independent variables. The value of R2 = 0.228 showed 
the proportion of variance in self concept "accounted 
for" by all factors, covariates and interaction terms.
Unlike the treatment factor, the partial beta for 
the gender factor increased from 0.11 to 0.23. (That of 
treatment decreased from 0.09 to 0.08) as other 
controls
were introduced. Thus the variance proportion explained 
by gender increased as more controls were being 
introduced.
The findings that females had higher self concept, 
which was even significant, than males contradicts the 
findings of Marsh and Smith (1990) who found that boys
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had higher self concept in mathematics than girls. 
However the sample was obtained with students in 
secondary school. Self concept, has been defined as the 
relatively stable picture people have of themselves and 
of their own academic ability when the focus of 
attention is mathematics ability. Self concept theorists 
have advocated that a phase of destabilisation of self 
concept is found during adolescents. The sample used by 
Marsh and Smith were adolescents. The sample used in the 
study were matured students with mean age of 
approximately 2 3 years. They are more likely to be 
stable in their self perceived ability or self concept 
in mathematics. The skill developmental approach to the 
theory of self-concept advocates that self concept is 
primarily the result of past achievement rather than a 
cause of subsequent achievements. The females in this 
sample have higher achievement in mathematics as evident 
from the means of their achievement test scores than the 
males. Thus since achievement has been found to be 
related to self concept it can be argued the finding is 
not contrary to expectations as far as this sample is
concerned.
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Chapter five
5.0 SUMMARY OF FINDINGS. EDUCATIONAL IMPLICATIONS AND
RECOMMENDATIONS
This chapter contains summary of the findings of the 
study, the educational implications, the limitations and 
recommendations, as well as suggestions for further 
research.
5.1 SUMMARY OF FINDINGS
Over the years there has been an increase in 
emphasis on the study of mathematics at all levels of 
the educational system. At primary and secondary levels 
in Nigeria and other parts of the world, mathematics is 
compulsory for all the students and it is one of the 
core subjects offered at the end of their final 
examinations. Its impact is also felt in almost all 
courses taken at tertiary level. At the tertiary level, 
it is no longer in the sciences that mathematics is 
intensely studied as a pre-requisite for graduation. For 
courses in social sciences and in the humanities, 
mathematics is being made compulsory for the students.
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In Business studies in particular, its impact in areas 
of business decisions is indisputable. Thus, 
mathematics, is one of the courses offered in tertiary 
institutions offering courses in Business studies. At 
NCE level in Nigeria, a look at the stipulated minimum 
standards of NCE education shows that mathematics is 
compulsory for all students in Business Education.
Despite this emphasis on the teaching of mathematics 
at all levels, the cognitive and affective outcomes of 
the study of mathematics by students have not been 
encouraging. In terms of popularity, mathematics is 
generally not liked by students. The poor performance in 
mathematics at primary and secondary level is well known 
and well documented in research reports by the various 
testing organisations, researchers in mathematics 
education and even the mass media. At the tertiary 
level, the low passes in mathematics courses and the 
high number of students with carry over in the 
mathematics courses is a pointer to poor performance in 
mathematics. Reports on the affective aspect vary from 
lack of interest towards mathematics to hatred, fear and 
anxiety about mathematics. Most students will readily
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admit having poor self perception about their ability in 
mathematics. This poor outcome of cognitive and 
affective variables, as far as mathematics is concerned, 
has engaged the attention of most researchers in 
mathematics education over the years. Attempts have been 
made at how some identified alterable variables, 
especially teaching methods, can be manipulated in order 
to bring about a favourable increase in both cognitive 
and affective outcomes. The effect of some innate 
attribute variables such as gender have also been 
examined. However, most of these studies have focused on 
the primary and secondary levels. The study at tertiary 
level has received least attention than that at primary 
and secondary levels. The studies on tertiary level need 
to be increased since this area has been neglected. This 
is because some of the products of the tertiary level 
end up as teachers of students at the lower level of the 
educational system and their cognitive and affective 
outcomes in mathematics will in one way or the other 
affect the students they have to teach and also affect 
their own progress in everyday challenges requiring 
mathematical competence.
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Thus the researcher was motivated to carry out this 
study in order to see the impact of the popular 
instructional strategy, the lecture method, and an 
attribute variable, gender, on students’ cognitive 
outcome
(achievement in mathematics), and selected effective 
outcomes (mathematics test anxiety and self-concept).
The two institutions in Lagos State offering full­
time courses in NCE Business Education were used in this 
study.The first year Business Education Students on full 
time NCE studies in these two institutions were assigned 
to three groups comprising two experimental and one 
control groups. The final sample was 165 students 
consisting of 71 males and 94 females.
The independent variables were instructional 
strategies and gender. The instructional strategies 
occurred at three levels namely: interactive learning 
with individual use of materials (Experimental 1) , 
interactive learning with group use of materials 
(Experimental 2) and the lecture method (control). 
Gender occurred naturally at two levels, male and 
female. Thus the study was a 3 x 2 non-randomised
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control group pre-test/post-test quasi-experimental 
design.
Two achievement tests, two psychological tests and 
instructional materials were the instruments used. The 
achievement tests comprised an objective pre test of 
pre-requisite skills and an objective post test based on 
the syllabus of the second semester Business mathematics 
course for year 1 NCE Business Education Students. The 
two psychological tests were the Inventory of Test 
Anxiety in Mathematics (ITAM) by Osterhouse and a 17- 
item mathematics self concept scale constructed by the 
researcher. A total of nine (9) hypotheses were tested 
consisting of three hypothesis on each of the three 
dependent variables namely achievement in mathematics, 
mathematics self-concept and mathematics test anxiety. 
The whole study took place for a period of ten weeks.
The 3 x 2  Analysis of Covariance (ANCOVA), with pre­
test scores on each of the dependent variables as 
Covariates, was employed in the analysis of the data. 
Where there was no significant interactive effect, 
multiple classification Analysis (MCA) was done to find 
the magnitude of the effect of each factor level of the
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independent variable. Where there was a significant 
interactive effect the main effects were not tested. 
Rather the post test mean scores were carefully examined 
by studying the effects of the variables having
significant interaction.
The findings in this study were as follows:
1. There was significant interactive effect of 
treatment and gender on mathematics achievement of 
these NCE students.
2. There was no significant interactive effect of
instructional method and gender on students
mathematics test anxiety.
3. There was no significant interactive effect of
instructional method and gender on students
mathematics self concept.
4. Post test mean score of students exposed to
interactive learning with group use of materials was 
higher than those in the other two groups. The
students exposed to the lecture method alone had the 
lowest post test mean achievement score.
5. Students exposed to the lecture method had
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significantly higher test anxiety scores than those 
exposed to interactive learning.
6. Post test self concept mean score of females was 
significantly higher than that of males.
7. The interactive learning with individual use of 
materials favoured females while interactive 
learning with group use of materials favoured the 
males.
8. Treatment and gender accounted for a total variance 
of 24.3% in mathematics test anxiety and 22.8% in 
mathematics self concept.
9. Instructional method (Treatment) accounted for 3.61% 
of the variation in test anxiety and 0.6% in 
mathematics self concept.
10. Gender accounted for 1.69% of the variation in 
mathematics test anxiety and 5.29% in mathematics 
self concept.
5.2 EDUCATIONAL IMPLICATIONS AND RECOMMENDATIONS
The results obtained in this study have some
implications for the educational system especially at
tertiary level. The result obtained from the testing of
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hypothesis with achievement in mathematics as the 
dependent variable pointed out the superiority of the 
interactive method of teaching over the lecture method. 
Students exposed to the lecture method had the lowest 
post test mean score in mathematics achievement while 
those exposed to the group use of materials had the 
highest. This calls for the urgent need for teachers of 
tertiary mathematics to look into the inadequacy of the 
lecture method as the sole means of instruction at 
tertiary level. Experts on Nigerian educational 
practices in mathematics have constantly pointed out the 
inefficiency of this method at secondary level. Ali 
(1980) advised that:
"It will help if we vary our teaching methodology or 
better still use activity teaching methodology to 
determine whether mathematics achievement in schools 
can improve. The use of lecture approach is a 
mundane teaching technique; this effort, in the 
Nigerian experience has continued to translate into 
very poor achievement in secondary school 
mathematics education programmes".
This study has shown that this advice is not only for
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secondary school mathematics but also affects 
mathematics at tertiary level. The interactive method of 
learning, which is not a new method (Ajose, 1990), could 
be explored further and its implementation in tertiary 
mathematics imbibed. To this end, it is recommended that 
workshops and seminars be held for teachers of tertiary 
mathematics as to how interactive and cooperative 
learning could be inculcated in the teaching of 
mathematics in order to promote students achievement in 
mathematics at tertiary level. Research grants should be 
made available to those who wish to explore the 
practicability of this in the tertiary institutions.
The obtained significant interaction of 
instructional method and gender on mathematics 
achievement of the students points out the possibility 
of differential method of instruction to cater for the 
male and female students whenever interactive method 
with use of materials is being adopted. The fact that 
individual use of materials favoured the females while 
the group use favoured the males could be examined 
further. While not advocating that males and females 
should be separated for instructional purposes at
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tertiary mathematics, the researcher would recommend the 
technique of individual use of materials in institutions 
that are strictly for females. Here in Nigeria we have 
a Federal College of Education for girls at Gusau in the 
Northern part of the country. In institutions such as 
this, the use of both individual and group of materials 
should be tried out to find out the extent to which 
achievement in mathematics of females could be improved.
This study also showed those exposed to the lecture 
method as having higher mathematics test anxiety than 
those exposed to interactive learning when the 
significant of treatment on mathematics test anxiety 
obtained was more closely examined. This also implies 
that instructional method not only affects the cognitive 
aspect of the student at level tertiary but also the 
affective aspect especially that dealing with the 
debilitating aspect of mathematics test anxiety. This 
further strengthens the need for adopting the 
interactive method of learning at tertiary level.
There is evidence that this technique has been 
practised in some tertiary institution outside Nigeria 
(Hong and Neil, 1992; Cohen and Ben-Zvi, 1992) with some
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measure of success. We need to incorporate this in the 
Nigerian educational system at all levels.
The interactive learning used in this study involved 
the use of materials. The implication of this for the 
educational system is that teachers at tertiary level 
need to introduce the use of instructional materials at 
this level. Text books are costly and the few available 
for Business mathematics and other courses are foreign 
inclined. There is need for workshops and avenues for 
teachers of tertiary mathematics to develop 
instructional materials that could be used by students.
That gender had an effect on self concept with 
females having higher self concept than males showed 
that females could be encouraged to participate more in 
mathematics related courses at tertiary level. It is 
well known that fewer females than males opt for 
mathematics courses. With proper counselling, their 
self-perceived ability about mathematics could be 
influenced positively and thus they will be encouraged 
to have a more favourable attitude towards mathematics. 
This will remove the myth of mathematics being a "male" 
subject.
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5.3 LIMITATIONS OF THE STUDY
In this study, intact classes were used in order not
to disrupt the academic practices of the two
institutions used. This poses a limitation of complete
randomisation of subjects as expected in pure
experimental research.
Although all the full time students in the
institutions formed the initial sample in this study, 
the final sample was less because some of the students 
were dropped due to a number of reasons including 
sickness, missing of final examination and inadequate 
information of the questionnaires and withdrawal from 
the institutions. This reduction in sample poses a 
limitation to the study.
The scope was limited to students at NCE 
institutions in Lagos State. The fact that other states 
in Nigeria were not involved could pose a limitation to 
the generalizability of the result.
5.4 SUGGESTIONS FOR FURTHER STUDIES
The sample used in this study comprise Business 
Education Students who are not majoring in mathematics.
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It would be interesting if the study is replicated using 
sample of mathematics majors at tertiary level. The 
sample could also be varied to comprise non mathematics 
majors at Polytechnics, Universities and Colleges of 
Education.
The dependent variables could be varied. In this 
study, achievement in mathematics was the only cognitive 
variable. This could be varied to include other 
variables such as cognitive styles and retention. 
Similarly, the affective measures could be varied to see 
the effect of the independent variable on these 
affective measures.
The design of the experiment could be altered to 
include more dependent measures such as locus of 
control.
This will give the opportunity to examine more 
interactive effects among the variables.
The study could also be carried out in many 
institutions of learning and the results pooled together 
in order to obtain a more global effect. However those 
to be involved in the treatment must be trained as to
the three methods of instruction.
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APPENDIX 1
PRE-TEST ACHIEVEMENT TEST IN MATHEMATICS
1. The median of the numbers 8,5,11,6,8,13,6, is
A. 13 B. 8 C. 7 D. 6
2. The mean of the number 2,4,6,8, is
A. 2 B. 4 C. 5 D. 6
3. Simplify 2° + 2'1
A. 0 B. 1 C. 1—2 D. 2
4. What is log636?
A. 36 B. 6 C. 2 D. 1
5. Simplify log 4 + log 3
A. log B. log C. log 12 D. log l
6. At what points does (x - 1)(x + 2) cut the x-axis.
A. x = 1 and 2 B. x = 1 and -2
C. x = -1 and 2 D. x = 1 and -2
7. Which of the following is a measure of central 
tendency?
A. variance B. mode C. Range D. Mean deviation
8. Subtract 2.02 from 20.2
A. 1.782 B. 16.82 C. 17.82 D. 18.18
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A. 40O,'O B. 45% C. 50% D. 80%.
10. What is a third of one quarter of 48?
A. 16 B. 12 C. 4
11. Express 72 —4 % in decimal
A. 0.724 B. 0.7225 C. 0.0724 D. 0.07225
12. A workman had 5k deducted from every naira he earned 
in 1977. By the end of the year he had paid a total 
tax of 886.25. How much did he earn during the 
period?
A. 81725.00 B. 8862.50 C. 8431.25 D. 8345.00.
13. A second hand car dealer sold a Volkswagen car for 
816,800 at a loss of 20%. What did he pay for the 
car?
A. 83360 B. 812,000 C. 813,440 D. 821,000
14. What is the difference between 75 and 0.75?
A. 75.75 B. 74.25 C. 74.15 D. 0.7425
15. Express —9  as a percentage
20
A. 9% B. 18% C. 45% D. 90%.
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16. The mean age of the three children in a family is7 
years. If Bola is 10 and Kike is 8, how old is Kemi.''
A. 2 —1 2  years B. 4—
1
2  years 
1 C. 8—2  years 
D. 22-|- years.
17. If seven - tenths of a number is 2.10. What is the
number
A. 1.47 B. 3 C. 7 D. 147.
18. An amount of money earned N24 in four years at a 
rate of 5% per annum simple interest. What was the 
amount of money?
A. HI.20 B. H4.80 C. M120.00 D. H240.
19. Find the mean of 30,25,20,70,96,75 and 55.
A. 55.8 B. 53 C. 50 D. 45.
20. Express 0.6 as a fraction in its lowest terms.
A. 1-5|  B. 5 C. —5  D. —10
21. A man pays a total of H5,430 as wages to his workmen 
each month. If the number of workmen is 12, what is 
the mean monthly wage of the workmen?
A.H142.50 B. H447.50 C. N450 D. N452.50
22. 90.3 is equal to
A. 19 C. 3 D. 30.
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164
23. Express 0.0067368 to three significant figures
A.0.00673 B. 0.00674 C. 0.007 D.0.0674
24. If 6 oranges cost 30 kobo, then the cost of x 
oranges is
A. x kobo B. 5x kobo C. 5 kobo D. —5x kobo.
25. By how much would you multiply 3 6 to make the 
product equal the sum of 36 and 72?
A. —4 9  B. 2
1
2—  C
3. 2— 4 D. 3.
26. If my pocket money were cut by half I should have 24 
naira a week. How much do I get now?
A. H12 B. N24 C. N48 D. H96.
27 . Simplify 3i  * 10 —2 + 124 
A. — 120 B. 18 C. 20 D. 30.
Find x if 42X = 2
A. -2 B. _ 1 2 C. —2 D. 2 .
29 . If the scores of 3 students are 5,6 and Find the
variance.
A. 13 B. c. 3 D. 2
30. x + y = 5 x - y = 1
A. x — 2, y = 3 B. x = -2, y = -3
C. x = 3, y = 2 D. x = 4, y = 1.
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APPENDIX 2
POST-TEST IN MATHEMATICS
1. Express 0.09 as a percentage
A. 90% B. 9% C. 0.09% D. 0.0009%.
2. Express 37—i% as a fraction
A. 175 D.
3
8
3. A salesman, working on 15% commission sold M240,000 
worth of services for his company in May. Find his 
commission for the month of May.
A. M3,000 B. M3,600 C. N30,000 D.N36,000
the amount of his rent. How much will he pay half 
yearly for rates.
A. M3 2 5 B. M650 C. M3,900 D. M7,800.
5. Find the simple interest on M780,000 for 6 years at
10%.
A. M39.000 B. M78,000 C. M108,000 D. M468,000
6. How much will M15,000 amount to is 2 years at 5% 
simple interest?
A. 15,150 B. M16,5000 C. M156,000 D.M165,000
7 . A worker spends 15 of hi
. s i. ncome on housi. ng, —1  of
it on food and 1.4 on transport. How much does he
have left?
C. A D. 195 20 '
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8 . Add —3  to the difference between 3-2i  and 1 6
A. 3—  B. 3 C. 2—  D. 2-i
A man’s monthly salary is Ml,200; he does not pay tax on 
the first 8600 of his salary. For the remaining, he pays 
10k on every naira. Use this information to answer 
question 9 and 10.
9. How much tax does he pay every month?
A. 860 B. 8120 C. H600 D. 1,080.
10. What percentage of his salary goes on tax?
A. 2.5% B. 5% C. 10% D. 50%.
11. An article costing H90 was increased by 10%. If the 
increased price was later reduced by 10%, find the 
new price.
A. 889 B. 890 C. 899 D. 8100.
12. After deducting 10% from a certain amount of money 
81,620 is left. How much is the money?
A. 81,458 B. 81,610 C. 81,782 D. 81,800.
13. A man bought a car for 830,000 and later sold it at
872,000, what was his percentage profit?
A. 41.7% B. 58.3% C. 71.4% D. 140%.
14. Potatoes are bought at 815 per kilo and sold at 821 
per kilo. What is the profit per cent?
A. 28.6% B. 40% C. 60% D. 71.4%.
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15. Find the net pay of a man who pays a tax of 2-i% on 
his salary of M4,000.
A. Ml,000 B. M3,000 C. M3,890 D. M3,900.
16. A man borrowed M20,000 and agreed to pay an interest 
of 10 k per naira for month. How much did he pay 
back if he paid the loan and interest two months 
after taking it?
A. M20,000 B. M22,000 C. M24,000 D. M44,000
17. M60,000 is to be shared between two people in the 
ratio 2:3. Find the smaller share.
A. M20,000 B. M24,000 C. M36,000 D. M40,000.
18. If A :B = 3:4 and B:C = 5:6. Find A:B:C.
A. 3:20:6 B. 3:9:6 C. 15:20:24 D. 12:20:30.
19. The market price of a refrigerator is M8,400. If the 
buyer is given 15% cash discount. Find the cash 
price.
A. Ml,260 B. M6,900 C. M7,140 D. M8,385
20. A man paid Ml,400 for a suit after 5 k in the naira 
had been deduced from the market price. Find the 
market price.
A. Ml,330 B. Ml,333 C. Ml,405 D. Ml,474
21. The sum on which interest is payable is known as.
A. Amount B. Balance C. Principal D. Interest.
22. A loss in value is known as:
A. Discount B. Depreciation 
C.Demotion D. Appreciation.
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23. Find the compound interest on 8500 for 2 years at 
10% per annum.
A. H50 B. 855 C. 8100 D. M105.
24. A manufacturer bought a machine for 850,000 and 
wrote off 10% depreciation at the end of each year. 
What was its book value after two years?
A. 830,000 B. 840,000 C. 840,500 D. 845,000
25. Income tax is an example of
A. Indirect tax B. Tax free pay
C. direct tax D. Taxable pay.
26. A man’s basic monthly salary is 82,500. He pays 5% 
tax on this and his non taxable payment is 82,000, 
find his take home pay.
A. 84,275 B. 84,375 C. 84,400 D. 84,500.
27. A payment of apportioned profits to stock holders is 
known as :-
A. Dividend B. Stocks
C. Shares D. Brokerage.
28. How much Stock at 62-i-k can be bought for 85,000?
A. 8 2 , 6 2 5  B. 8 3 ,1 2 5 C. 8 8 ,0 0 0 D. 8 8 ,1 2 5 .
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APPENDIX 3
CORRECT OPTIONS FOR PRE-TEST IN MATHEMATICS
1. B 2 . C
3 . C 4 . C
5. C 6. B
7. B 8. D
9. D 10. C
11. B 12 . A
13 . D 14. B
15. C 16. B
17. B 18. C
19. B 20. B
21. D 22 . D
23 . B 24. B
25. D 26. C
27 . C 28. B
29. A 30. C
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APPENDIX 4
CORRECT OPTIONS FOR POST TEST IN MATHEMATICS
1. B 2. D
3 . D 4 . A
5. D 6. B
7. A 8. B
9 . A 10. B
11. A 12. D
13 . D 14 . B
15. D 16. C
17. B 18. C
19. C 20. D
21. C 22 . B
23 . D 24. C
25. C 26. B
27 . A 28. C
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APPENDIX 5
ITEM ANALYSIS OF PRE-TEST IN MATHEMATICS 
FUNCTIONS OF DISTRACTORS
HIGH GROUP LOW GROUP
Item A B* C D Omits A B C D OmitsNo.
1 - 22* - 3 - 2 12* 3 7 1
2 1 1 22* 1 - 2 7 12* 4 -
3 - 1 22* 1 1 3 6 11* 5 -
4 1 4 19* 1 - 3 11 7* 2 2
5 - 1 23* 1 - 5 1 12* 3 4
6 1 20* 2 1 1 4 3* 12 3 3
7 - 30* - 12 - 2 3* 2 17 1
8 - 1 1 23* - 5 2 6 11* 1
9 1 1 1 21* 1 4 4 5 11* 1
10 7 3 15* - - 10 6 4* 1 4
11 2 17* 2 2 2 6 5* 8 4 2
12 15* 3 4 1 2 5* 6 6 4 4
13 - - 1 23* 1 3 2 4 12* 4
14 1 21* 1 1 1 3 11* 5 2 4
15 - 1 14* 10 - 1 6 3 15 -
16 1 18* 5 1 - 6 5* 10 2 2
17 2 18* 3 1 1 6 3* 9 5 2
18 2 5 16* 2 - 6 10 5* 4 -
19 - 24* - 1 - 3 11* 5 4 2
20 1 24* - - - 5 7* 7 5 1
21 1 1 2 20* 1 4 6 5 9* 1
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I t e m A B* c D O m i t s A B C D O m i t s
No.
22 - - 3 22* - 2 3 2 4* 4
23 1 19* 2 3 - 2 9* 5 7 2
24 1 15* 7 2 - 4 5* 12 4 -
25 1 1 1 22* - 5 6 8 5* 1
26 2 2 19* 1 1 7 7 6* 3 2
27 i - 21* 1 2 5 6 7* 4 3
28 2 16* 2 4 1 3 6* 8 8 -
29 14* 1 9 1 - 5* 4 14 2 -
30 2 2 20* 1 4 3 10* 8 -
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(ii) DISCRIMINATING POWER AND DIFFICULTY LEVEL OF
PRE-TEST
MATHEMATICS.
Item Difficulty Discriminating Item Difficulty Discriminating
No Level (P) Power (D ) No level (p) Power (D)
1 .68 .40 16 .46 .52
2 .68 .40 17 .42 .60
3 .66 .44 18 .42 .44
4 .52 .48 19 .70 .52
5 .70 .44 20 .62 .68
6 .46 .68 21 .58 .44
7 .32 .40 22 .52 .72
8 .68 .48 23 .56 .40
9 .64 .40 24 .40 .40
10 .38 .44 25 .54 .68
11 .44 .48 26 .50 .52
12 .40 .40 27 . 56 .56
13 0.34 .44 28 .44 .40
14 0.70 0.44 29 .36 .40
15 0.64 0.40 30 .60 .40
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APPENDIX 6
ITEM ANALYSIS OF POST TEST IN MATHEMATICS: 
(i) FUNCTIONS OF THE DISTRACTORS
HIGH GROUP LOW GROUP
Item A B C D O Item A B C D O
No No
1 2 20* - 3 - 1 4 10* 2 7 2
2 1 - 3 21* - 2 6 5 8 5* 1
3 - 1 2 22* - 3 3 5 3 11* 3
4 15* 4 5 1 - 4 5* 7 9 3 1
5 4 1 1 19* - 5 10 3 3 9* -
6 1 20* 3 1 - 6 3 9* 6 7 -
7 15* 3 3 4 - 7 5* 4 5 11 -
8 1 20* 2 2 - 8 5 6* 7 5 2
9 18* 7 - - - 9 7* 10 4 4 -
10 1 16* - 8 - 10 3 5* 3 14 -
11 16* 8 - 1 - 11 4* 18 2 1 -
12 1 2 3 18* 1 12 7 6 7 2* 1
13 4 - 2 19* - 13 7 5 5 7* 1
14 1 19* 2 2 1 14 6 7* 6 4 2
15 4 3 - 18* - 15 9 7 4 4* 1
16 4 2 17* 2 - 16 8 5 6* 4 2
17 2 22* - 1 - 17 5 12* 5 3 -
18 - 7 16* 1 1 18 3 10 6* 4 2
19 4 - 20* 1 - 19 8 4 7* 5 1
20 - - 12 13* - 20 3 8 10 2* 2
21 1 1 22* 1 - 21 5 5 11* 3 1
22 - 25* - - - 22 3 10* 4 4 4
23 - 1 9 15* - 23 3 6 11 5 -
24 1 9 14* 1 - 24 4 14 4* 2 1
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HIGH GROUP LOW GROUP
25 1 - 21* 3 - 25 5 3 11* 5 1
26 - 22* 1 - 2 26 4 7* 12 1 1
27 24* - 1 - - 27 11* 3 5 4 2
28 2 3 16* 3 1 28 3 8 6* 8 -
(iiDISCRIMINATING POWER AND DIFFICULTY LEVEL OF POST
MATHEMATICS
ITEM
Item No Difficulty- Discrimina Item No Difficul Discrimina 
level (p) ting Power ty level ting B̂ wer 
(D) (P) (D)
1 . 60 .40 16 .46 .44
2 .52 . 64 17 . 68 .40
3 . 66 .44 18 .44 .40
4 .40 .40 19 . 54 52
5 .56 .40 20 .30 .44
6 .58 .44 21 . 65 . 44
7 .40 .40 22 .70 . 60
8 . 52 .56 23 .40 .40
9 . 50 .44 24 .36 .40
10 .42 .44 25 . 69 .40
11 .40 .48 26 .58 . 60
12 .40 . 64 27 .70 . 52
13 . 52 .48 28 .44 0.44
14 .52 . 48
15 .44 . 56
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APPENDIX 7
Formulae for Item Analysis and Reliability.
The difficulty level can be calculated for each item 
with the formula
p = R 100%
T 1
where R = no of students who got the item right and 
T = total number of one third of the candidates.
The popular formula used to obtain the 
discriminating power was
where U = number of students in upper one third 
group who answered items correctly
L = number in lower one third group who answered the 
item correctly and
N = number of students in each group.
The coefficient of internal consistency can be 
computed with the formula
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KR X(n -  X) 21 n S*
where n = number of items in the test. 
X = mean of test scores and
Sx = variance of test scores.
Cronbach alpha can be obtained using the formula.
Sum S2Xi 
a = nn - 1 1 - ---S--2X-.-- i
where n = number of items
S2X1 = variance of part scores and
S2Xt = variance of the sum of n part scores.
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APPENDIX 8
NCCE SYLLABUS. BED 122.
BUSINESS MATHEMATICS
Percentages and their application to discount,
depreciation
etc.
Ratio and proportion.
Simple and compound interest up to partnership, hire 
purchase, and annuities. Payroll Computation and 
application to PAYE 
Shares and Stocks.
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APPENDIX 9
INVENTORY OF TEST ANXIETY IN MATHEMATICS (ITAM)
Direction: Read each of the following statements
carefully.
In the space in front of each, write the letter 
corresponding to the way you actually felt during your 
mathematics examination. Use the following scale for 
your answers.
a. The statement did not describe my feeling or 
condition.
b. The feeling or condition was barely noticeable.
c. The feeling or condition was moderately intense.
d. The feeling or condition was strong.
e. The feeling or condition was very strong.
1. I felt terrified while taking this examination.___
2. I felt during this examination that I wouldn’t be
able to finish the examination on time. _______
3. My mouth got dry during this examination. ______
4. Prior to taking this examination, I felt that other
students were better prepared for this examination 
than I was. ________
5. My mind went blank at the beginning of this
examination. It took me a few minutes to recover.
6. I feel that I disappointed myself and other persons 
by my poor performance on third examination. ______
7 . I felt my heart beating fast during this 
examination.
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8. I found myself worrying about a low mark before this 
examination. ________
9. During this examination, I found myself thinking
about the consequences of failure. ________
10. I got so tense during this examination that my
stomach became upset. ________
11. After finishing this examination, I feel that I
could have done better than I actually did. _______
12. I got a headache during this examination. _______
13. While taking this examination, I found myself
thinking of how much brighter other students are 
than I am _________
14. My hands were sweating during this examination.
15. I did not feel very confident of my performance
before I took this examination that I forgot facts 
which I really knew. ____________
16. I got so nervous during this examination that I 
forgot facts which I really knew.
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APPENDIX 10
MATHEMATICS SELF-CONCEPT ITEMS
Instructions: The following are statements. Concerning
your feelings about mathematics.
Please write strongly Agree, Agree, "Disagree or
Strongly disagree according to your feeling.
1. I always scores high marks in mathematics ______
2. I can never pass mathematics no matter how hard
I try __________
3. I answer correctly most questions asked by my 
mathematics lecturer ___________
4. Most of the time I cannot work a single problem in
mathematics without seeking for help _____________
5. I am confident enough to volunteer to solve
problems on the black board ____________
6. My colleagues recognise me as a good mathematics 
student
I enjoy practising mathematics problems 
I like taking mathematics test ________
During mathematics examination I believe I would
fail even before the results are out _____________
10 I believe I am competent to understand any
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mathematics topic _____________________
11. I can never be good at solving mathematics problems
12. I believe I am specially gifted in mathematics____
13. Everybody regards me as a poor student in
mathematics ___________________
14. Mathematics is only good for the gifted students and
not for people like me _____________________
15. I always struggle to pass mathematics examination
16. Which of the following would describe your ability 
in maths.
A. Outstanding B. Good C. Average
D. Highly below average E. Not sure.
17. How would you rate your general performance in 
mathematics.
A. 70 - above B. 60-69 C. 50-60 D. 40-49
E. Below 40.
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APPENDIX 11
Reliability Analysis of ITAM 
CORRELATION MATRIX
ITEM 1 ITEM 2 ITEM 3 ITEM 4 ITEM 5
ITEM 1 1.0000 .5718 .3165 .1345 . 3475
ITEM 2 .5718 1.0000 .3554 .13078 .2871
ITEM 3 .3165 .3554 1.0000 .2173 .4987
ITEM 4 . 1345 .3078 .2173 1.0000 .3220
ITEM 5 .3475 .2871 .4987 .3220 1.0000
ITEM 6 .4266 .4477 .2030 .3208 .4243
ITEM 7 .3021 .2636 .3320 .2816 .3966
ITEM 8 .2651 .2955 . 1726 .5663 .2108
ITEM 9 .2771 . 1485 .2400 .2509 . 3981
ITEM 10 .2570 .2682 . 1149 . 1575 .2938
ITEM 11 .0170 .3140 .3363 .5314 . 1716
ITEM 12 . 1568 .0702 . 1344 . 1202 .5618
ITEM 13 .3089 .4619 .4467 . 3470 . 6537
ITEM 14 .0015 -.0092 .2235 .2804 .3036
ITEM 15 .0848 .2108 .3103 .2092 .4609
ITEM 16 .3532 . 3912 .4986 .2311 .4922
ITEM 6 ITEM 7 ITEM 8 ITEM 9 ITEM 10
ITEM 6 1.0000 . 1888 .4691 .2219 . 6178
ITEM 7 . 1888 1.0000 .2864 .4856 .3087
ITEM 8 .4691 .2864 1.0000 .4732 .4235
ITEM 9 .2219 .4856 .4732 1.0000 .4248
ITEM 10 . 6178 .3087 .4235 .4248 1.0000
ITEM 11 . 0809 . 2318 . 1014 . 0955 -.0318
ITEM 12 .4549 .2793 .1187 .2650 .4312
ITEM 13 .2959 . 2524 .2499 .3448 .3135
ITEM 14 .0898 . 1173 . 0409 .2614 . 1774
ITEM 15 .3389 .3300 . 1788 . 3240 . 5744
ITEM 16 .3609 .3610 -.0388 .2970 .3277
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I T E M I T E M I T E M I T E M I T E M I T E M
1 1 12 1 3 1 4 1 5 1 6
I T E M  1 1 1 . 0 0 0 0 - . 0 2 7 7 . 3 8 3 5 . 2 1 2 3 . 2 0 8 3 . 3 2 1 9
I T E M  12 - . 0 2 7 7 1 . 0 0 0 0 . 3 8 3 7 . 3 5 4 4 . 4 6 5 3 . 2 7 4 0
I T E M  13 . 3 8 3 5 . 3 5 3 7 1 . 0 0 0 0 . 2 6 3 0 . 4 4 4 6 . 4 2 1 5
I T E M  1 4 . 2 1 2 3 . 3 5 4 4 . 2 6 3 0 1 . 0 0 0 0 . 2 6 8 5 . 3 3 9 5
I T E M  1 5 . 2 0 8 3 . 4 6 5 3 . 4 4 4 6 . 2 6 8 5 1 . 0 0 0 0 . 3 4 3 5
I T E M  1 6 . 3 2 1 9 . 2 7 4 0 . 4 2 1 5 . 3 3 9 5 . 3 4 3 5 1 . 0 0 0
ITEM TOTAL STATISTICS
SCALE MEAN SCALE CORRECTION SQUARED ALPHA
IF ITEM VARIANCE IF ITEM TOTAL MULTIPLE IF ITEM
DELETED ITEM DELETED CORRELATION CORRELATION DELETED
ITEM 1 31.4464 132.3971 .4334 .4791 .8628
ITEM 2 31.5357 131.0896 .5135 .5538 .8589
ITEM 3 31.7857 132.0623 .5129 .4756 .8590
ITEM 4 31.3393 130.4828 .4961 .6393 .8598
ITEM 5 31.8214 128.0039 .6741 .6993 .8519
ITEM 6 31.8393 130.6828 .5616 .6739 .8568
ITEM 7 31.6071 131.9156 .5067 .4324 .8592
ITEM 8 31.6786 134.1130 .4323 .7109 .8625
ITEM 9 31.6071 130.8610 .5110 .5191 .8590
ITEM 10 31.8214 132.2221 .5228 . 6735 .8586
ITEM 11 30.9821 134.1633 .3413 .5260 .8680
ITEM 12 32.0536 136.0516 .4457 .5431 .8620
ITEM 13 31.5893 126.1010 .6448 .6183 .8523
ITEM 14 31.9286 137.2312 .3276 .3345 .8670
ITEM 15 31.5536 131.6698 .5371 . 5274 .8579
ITEM 16 31.3929 127.0792 .5747 .5957 .8558
A L P H A 8 6 7 2 S T A N D A R D I Z E D  I T E M  A L P H A 8 6 8 7
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APPENDIX 12
RELIABILITY ANALYSIS OF THE SELF-CONCEPT QUESTIONNAIRE
CORRELATION MATRIX
ITEM 1 ITEM 2 ITEM 3 ITEM 4 ITEM 5
ITEM 1 1.0000 .5904 .5368 .5125 .7086
ITEM 2 .5904 1.0000 .4868 . 6006 . 6289
ITEM 3 .5368 .4868 1.0000 .3760 .4324
ITEM 4 .5125 .6006 .3760 1.0000 .5935
ITEM 5 .7806 .6289 .4324 .5935 1.0000
ITEM 6 .5733 . 5336 . 4236 .4302 .5523
ITEM 7 .5506 .4400 . 3331 .3038 . 6387
ITEM 8 . 6229 .5329 .4201 .4712 .7028
ITEM 9 .3099 . 6018 . 1954 .3608 .4403
ITEM 10 .2435 .4392 .2027 .2465 .3199
ITEM 11 . 1880 .4775 .21107 .2178 .3205
ITEM 12 .4041 . 0363 .2042 . 1715 .3596
ITEM 13 .4467 .5494 .3665 .3007 . 5398
ITEM 14 . 3322 . 6652 .3304 .4902 .4671
ITEM 15 .2900 . 0926 . 0710 .1476 .3022
ITEM 16 . 3964 .4086 . 1861 .4012 . 5422
ITEM 17 . 5019 .4344 .2861 .3750 . 5969
ITEM 6 ITEM 7 ITEM 8 ITEM 9 ITEM 10
ITEM 6 1.0000 . 6039 .4250 . 3271 .2758
ITEM 7 .6039 1.0000 .4964 .2728 .3390
ITEM 8 .4250 .4964 1.0000 .4331 .3204
ITEM 9 .3271 .2728 .4331 1.0000 .3215
ITEM 10 . 2758 .3390 .3204 . 3215 1.0000
ITEM 11 .2428 . 1449 . 3712 . 6027 .2926
ITEM 12 .3150 . 1848 .3923 .2071 . 1628
ITEM 13 .4327 .3855 .5054 . 6094 . 1782
ITEM 14 .3518 .2114 .5682 . 6329 .2046
ITEM 15 .2277 .3442 . 2793 . 1287 .2427
ITEM 16 .3063 .3212 .4196 .4188 .2367
ITEM 17 . 3959 . 5091 .4320 .2974 . 1348
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I T E M  1 1 I T E M  12 I T E M  13 I T E M  14 I T E M  1 5
I T E M 1 1 1 . 0 0 0 0 . 1 3 5 7 . 4 5 2 3 . 5 7 1 8 . 0 3 4 0
I T E M 12 . 1 3 5 7 1 . 0 0 0 0 . 4 7 9 3 . 0 1 7 1 . 3 5 9 4
I T E M 1 3 . 4 5 2 3 . 4 7 9 3 1 . 0 0 0 0 . 5 0 0 8 . 1 3 5 6
I T E M 1 4 . 5 7 1 8 . 0 1 7 1 . 5 0 0 8 1 . 0 0 0 0 . 0 0 1 3
I T E M 1 5 . 0 3 4 0 . 3 5 9 4 . 1 3 5 6 . 0 0 1 3 1 . 0 0 0 0
I T E M 1 6 . 2 4 9 7 . 3 9 5 7 . 4 3 5 2 . 2 7 6 5 . 2 7 3 5
I T E M 1 7 . 1 3 4 4 . 3 7 5 8 . 3 3 9 7 . 2 1 2 5 . 2 4 2 7
I T E M  1 6 I T E M  1 7
I T E M  1 6 1 . 0 0 0 0 . 4 2 9 9
\ I T E M  17 . 4 2 9 9 1 . 0 0 0 0
ITEM - TOTAL STATISTICS
SCALE MEAN IF SCALE CORRECTION SQUARED ALPHA IF
ITEM DELETED VARIANCE IF ITEM-TOTAL MULTIPLE ITEM
ITEM DELETED CORRELATION CORRELATION DELETED
ITEM 1 46.5357 72.3987 .7218 .7378 .8879
ITEM 2 45.8393 71.2282 .7136 .7793 .8872
ITEM 3 46.4107 75.5919 .4782 .3869 .8948
ITEM 4 46.2857 72.8623 .5992 .5322 .8910
ITEM 5 46.4464 70.1062 .8247 .7903 .8838
ITEM 6 46.7857 72.4987 .6268 .5488 .8901
ITEM 7 46.2500 72.0455 .6019 .6544 .8908
ITEM 8 46.2321 72.4360 .7298 .6513 .8877
ITEM 9 45.6786 74.6584 .5977 .6311 .8919
ITEM 10 46.2143 75.9896 .4034 .3861 .8669
ITEM 11 45.9286 74.5766 .4261 .4766 .8967
ITEM 12 46.6964 75.0153 .4183 .6537 .8968
ITEM 13 45.9286 73.9948 .6667 .6749 .8902
ITEM 14 45.9286 73.7766 .5429 .6902 .8928
ITEM 15 46.9821 73.6906 .3110 .2700 .9058
ITEM 16 46.2500 69.5727 5615 .4231 .8934
ITEM 17 45.8929 70.8247 . 5543 .4681 .8930
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APPENDIX 13
FACTOR ANALYSIS OF THE SELF CONCEPT SCALE
(i) CORRELATION MATRIX OF SELF CONCEPT AND ITAM
SELF ITEM 1 ITEM 2 ITEM 3 ITEM 4 ITEM 5 ITEM 6 ITEM 7
CONCEPT
ITEM 1 1.00000 .59043 .53671 .51247 .78044 .57327 .55058
ITEM 2 .48677 1.00000 .148677 .60058 .62893 .53363 .43999
ITEM 3 .53677 .48677 1.00000 .37599 .43240 .42356 .33309
ITEM 4 .51247 .60058 .37599 1.00000 .59352 .43023 .30576
ITEM 5 .78064 .62893 .43240 .59352 1.00000 .55225 .63874
ITEM 6 .57327 .53363 .42356 .43023 .55225 1.00000 60395
ITEM 7 .55058 .43999 .33309 .30376 .63874 .60395 1.00000
ITEM 8 .62294 .53294 .42010 .47119 .70279 .42495 .49635
ITEM 9 .30992 .60176 .19537 .36078 .44026 .32711 .27276
ITEM 10 .24346 .43919 .20267 .24653 .31992 .27578 .33902
ITEM 11 .18804 .47751 .21075 .21781 .32046 .24281 .14486
ITEM 12 .40415 .03633 .20424 .17149 .35963 .31502 .18475
ITEM 13 .44667 .54937 .36648 .50073 .35984 .43267 .38548
ITEM 14 .33222 .66524 .33036 .49021 .46706 .35185 .21141
ITEM 15 .29004 .09259 .07099 .14755 .30217 .22769 .34422
ITEM 16 .39636 .40865 .18613 .40118 .54222 .30631 .32115
ITEM 17 .50193 .34443 .28613 .37503 .59690 .39587 .50907
ITAM ITEM 1 ITEM 2 ITEM 3 ITEM 4 ITEM 5 ITEM 6 ITEM 7
ITEM 1 -.41840 -.41737 -.21558 -.57811 -.59310 -.37341 .-.35258
ITEM 2 -.50036 -.43512 -.23580 -.45911 -.52045 -.36705 -.30590
ITEM 3 -.22140 -.47257 -.32648 -.46715 -.23143 -.35226 -.16821
ITEM 4 -.37693 -.35255 -.38857 -.24977 -.38850 -.21827 -.29979
ITEM 5 -.30035 -.57426 -.27927 -.40689 -.51805 -.30836 -.43826
ITEM 6 -.48007 -.45533 -.15669 -.42044 -.46148 -.26183 -.22168
ITEM 7 -.296-8 -.29325 -.42839 -.52052 -.45528 -.33834 -.22796
ITEM 8 -.407-1 -.34341 -.20826 -.30707 -.39147 -.26339 -.22040
ITEM 9 -.391.5 -.26197 -.31023 -.42800 -.49872 -.25515 -.26181
ITEM 10 -.283-6 -.24122 -.04957 -.28488 -.39560 -.20330 -.07581
ITEM 11 -.188.1 -.31437 -.45787 -.24897 -.25619 -.14645 -.00675
ITEM 12 -.139-4 -.23402 -.16388 -.13508 -.30151 -.00881 -.20712
ITEM 13 -.298-9 -.47411 -.23723 -.42400 -.44142 -.34507 -.33687
ITEM 14 .05305 -.09875 -.06087 -.11616 -.13156 -.05429 -.05180
ITEM 15 -.26289 -.23092 -.27946 -.29744 -.32699 -.18975 -.19741
ITEM 16 -.31157 -.33710 -.29651 -.42434 -.41430 -.23682 -.24848
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ITEM 8 ITEM 9 ITEM 10 ITEM 11 ITEM 12 ITEM 13 ITEM 14
ITEM 8 1.00000 .43312 .32044 .37120 .39232 .50538 .56818
ITEM 9 .43312 1.00000 .32152 .60269 .20705 .60939 .63293
ITEM 10 .32044 .32152 1.00000 .29255 .16277 .17819 .20457
ITEM 11 .37120 .60269 .29255 1.00000 13566 .45234 .57179
ITEM 12 .39232 .20705 .16277 .13566 1.00000 .47929 .01711
ITEM 13 .50538 .60939 .17819 .45234 .47929 1.00000 .50081
ITEM 14 .56818 .63293 .20457 .57179 .01711 .50081 1.00000
ITEM 15 .27928 .12866 .24272 .03393 .35940 .13561 .00130
ITEM 16 41961 .41884 .23670 .24972 .39565 43519 .27648
(ii)_____________________ FINAL STATISTICS
VARIABLE 
SELF CONCEPT COMMUNALITY I TAM COMUNALITY
ITEM 1 .69154 ITEM 1 .46319
ITEM 2 .57529 ITEM 2 .45392
ITEM 3 .29194 ITEM 3 .36809
ITEM 4 .49331 ITEM 4 .27225
ITEM 5 .76804 ITEM 5 .57390
ITEM 6 .49342 ITEM 6 .41784
ITEM 7 .50182 ITEM 7 .30369
ITEM 8 .58766 ITEM 8 .24444
ITEM 9 .48758 ITEM 9 .31026
ITEM 10 .19877 ITEM 10 .38808
ITEM 11 .39418 ITEM 11 .18210
ITEM 12 .39778 ITEM 12 .30904
ITEM 13 .58745 ITEM 13 .49471
ITEM 14 .52205 ITEM 14 .32744
ITEM 15 .19659 ITEM 15 .41953
ITEM 16 .40436 ITEM 16 .40994
ITEM 17 .46744
FACTOR EIGENVALUE PCT OF VAR CUM PCT
1 11.45304 34.7 34.7
2 2.44459 7.4 42.1
Varimax Rotation 1, Extraction 1, Analysis 2- Kaiser Normalization. 
Varimax converged in 3 iterations.
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Rotated Factor Matrix:
SELF FACTOR 1 FACTOR 2
CONCEPT
ITEM 1 .81153 -.18152
ITEM 2 .55028 -.52199
ITEM 3 .44842 -.30143
ITEM 4 .54503 -.44300
ITEM 5 .78821 -.38310
ITEM 6 .68118 -.17152
Varimax converged in 3 iterations. Rotated Factor Matrix:
SELF CONCEPT FACTOR 1 FACTOR 2
ITEM 1 .81153 -.18152
ITEM 2 .55028 -.52199
ITEM 3 .44842 -.30143
ITEM 4 .54503 -.44300
ITEM 5 .78821 -.38310
ITEM 6 .68118 -.17152
ITEM 7 .70120 -.10068
ITEM 8 .64075 -.42084
ITEM 9 .34830 -.60520
ITEM 10 .41929 -.15155
ITEM 11 .15825 -.60757
ITEM 12 .54485 -.03045
ITEM 13 .49076 -.58873
ITEM 14 .31437 -.65055
ITEM 15 .44133 .04268
ITEM 16 .55580 -,30894
ITEM 17 .67475 -.11025
ITAM
TEST ANXIETY 
ITEM 1 -.61911 .28267
ITEM 2 -.60042 .30564
ITEM 3 -.25908 .54861
ITEM 4 -.28282 .43848
ITEM 5 -.31691 .68809
ITEM 6 -.41814 .49295
ITEM 7 -.29415 .46601
ITEM 8 -.34425 .35486
ITEM 9 -.32390 .45315
ITEM 10 -.10809 .61351
ITEM 11 -.09083 .41696
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SELF CONCEPT FACTOR 1 FACTOR 2
ITEM 12 -.04910 .55374
ITEM 13 -.25753 .65451
ITEM 14 -.18918 .54004
ITEM 15 -.03936 .64651
ITEM 16 -.25021 .58935
Factor Transformation Matrix:
FACTOR 1 FACTOR 2
FACTOR 1 .71341 -.70075
FACTOR 2 .70075 .71341
Factor Score Coefficient Matrix:
SELF CONCEPT FACTOR 1 FACTOR 2
ITEM 1 .16988 .08496
ITEM 2 .05103 -.04254
ITEM 3 .06152 -.00355
ITEM 4 .06245 -.02355
ITEM 5 .13173 .03060
ITEM 6 .13951 .06651
ITEM 7 .15582 .08751
ITEM 8 .08949 -.00260
ITEM 9 -.01191 -.09592
ITEM 10 .07848 .02939
ITEM 11 -.05891 .12709
ITEM 12 .12876 .07999
ITEM 13 .02569 -.06887
ITEM 14 -.02752 -.11277
ITEM 15 .11513 .08170
ITEM 16 .08665 .01184
ITEM 17 .14779 .08085
I TAM -.10641 -.02861
ITEM 1
ITEM 2 -.09813 -.01984
ITEM 3 .02469 .09606
ITEM 4 .00115 .06460
ITEM 5 .03294 .12178
ITEM 6 -.02328 .05651
ITEM 7 .00280 .06996
ITEM 8 -.02737 .03372
ITEM 9 -.00657 .06167
ITEM 10 .07217 .13665
ITEM 11 .04479 .09008
UN
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SELF CONCEPT FACTOR 1 FACTOR 2
ITEM 12 .07703 .13113
ITEM 13 .04210 .12290
ITEM 14 .13328 .16602
ITEM 15 .09434 .15599
ITEM 16 .03342 .10772
Covariance Matrix for Estimated Regression Factor Scores;
FACTOR 1 FACTOR 2
FACTOR 1 1.00000
FACTOR 2 .00000 1.00000
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APPENDIX 14 
LETTER TO PROVOST
International Centre For Educational
Evaluation
Institute of Education,
University of Ibadan,
25th April 1993.
The Provost,
Lagos State College of Education,
Ijanikin,
Lagos.
Application For Permission To Use NCE Year One Business 
For Research Purpose.
I hereby apply for permission to use all the NCE 
year one Business Education Students for research study 
of my PHD thesis.
The study involves the use of all NCE students at 
the two colleges of Education in Lagos State to collect 
data on my research design. It consists of lecturing in 
business mathematics (BED 122) as well as administering 
some tests and guestionnaires.
Enclosed is a summary of my research proposal and a 
letter of introduction from my university.
Thank you for your anticipated co-operation.
Yours Faithfully,
0. A. Adedayo.
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APPENDIX 15 
LEARNING MATERIALS 
UNIT 1 FRACTIONS
A. OBJECTIVES.
Students will be to recall how to
1. Add, subtract, multiply and divide given 
fractions.
2. Use the concept of BODMAS.
3. Apply fractions in solving practical problems.
B. INTRODUCTION
A fraction is part of a whole number. Examples of 
fracti. ons are —5 9 , —
7 
10 , -
1
1-
3
7 etc. Thus, a fraction is
of the form -E where p and q do not have any factor
in common. Pq is the NUMERATOR while q is the
DENOMINATOR. In the fraction —19 , 12 is the
numerator while 19 is the denominator.
Exercise 1.1 
Gi. ven fracti. on -54-1, (a) the numerator is
(b) the denominator is __________
C. ADDITION AND SUBTRACTION OF FRACTION.
The important point note is the need to find the LCM of 
the denominator.
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For example, to fi. nd —2 3  + —
4
5 , one needs to find the
L.C.M of the denominators i.e. the L.C.M of 3 and 5. 
Their LCM is 15. You then obtain the addition by bring 
the two fractions to a common denominator
i.e. 1  + 1 10 ♦ 12 22 _ 1 _ 1_  
3 5 15 15 15
Now, try this problem:
2 + 3 2 7—  —  + —  + ——  .
3 4 5 12
1. What is your L.C.M?
2. Did you obtain 60?
3. The sum of the numerators should be 
40 + 45 + 24 + 35 = 144.
4. The addition is thus 60 = 2%5.
For subtraction, the principle is the same with 
addition.
e.g^. 7—  9- —  
Your LCM is
Your numerator should be 45 - 14.
Thus the final answer is 3163
For mixed numbers, the procedure is the same. You only 
need to deal with the whole numbers first.
e.g. 2—5  + 1—6  - 1—3 . = (2 + 1 - 1) 
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18 + 20 - 20
30
D. MULTIPLICATION AND DIVISION OF FRACTIONS.
For multiplication, if the fraction is a proper 
fraction, you only need to cancel out the common factors 
in the numerator and denominator.
In case of mixed fraction, first convert to improper 
fraction and then cancel out common factors.
e.g. x _54
Exercise 1.2
If you cancel out common factors, you will obtain
For division, the rule is:"invert and multiply"
e.g. 1 x 1  = li = 3 1 .3 5 5 5
Exercise 1.3
Now try these:
(a) 4 —4  x 17 The answer is
(b) 15-2 + 2—
1
4 The answer is 
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E. BODMAS.
For problems involving mixed operations, the concept 
of BODMAS comes in very useful. It tells you which 
operation to perform first.
Recall that it means Bracket comes before Of, Division 
comes before Multiplication which comes before Addition 
and Addition comes before Subtraction.
2 4 —3  +  1 15 + (\ l—12  + 2 —3/\  - a6 4.
Exercise 1.4
f\ l—12  + 2 A3)/ = 7
. 4—  + 2—3 + 1—4 = 36
Exercise 1.5
4
15 o f I = 1l8 5 x 1  =8
the expression becomes
2 —3  +  —  - ? +6
If you simplify correctly you will obtain final solution 
of
6 13 '
/
e.g. - r
\
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Exercise 1.6
Exercise 1.7(i)
Exercise 1.7(ii)
Also 3—2  - 2—3 = 
Exercise 1.8
1—  + 2—  = ___?
The expression thus becomes 2 + —*—7 24  = 6
c
7—  Ans.
E. PRACTICAL PROBLEMS INVOLVING FRACTIONS.
e.g. A bath can be filled by three cold water tap in 
15 minutes and by the hot water tap in 21 minutes. How 
long would it take to fill the bath if both taps were 
turned on together.
Solution
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Fracti. on of bath fi. lled in 1 min by 1st tap = —1 15
Fracti. on of bath fi. lled i. n 1 m .m  by7  2nd tap 
1 
* = —21
Total 1 f 1 1215 21 105
Total time taken by the two taps if both were turned
on together = 1 + -12105
Now try this: 
Exercise 1.9
A bath can be filled by the hot-water pipe in 12 minutes 
and by the cold water pipes in 15 minutes.
1. How long will it take to fill the bath when both
taps are turned on? Ans = 6—23 minutes.
If the waste pi, pe can empty the bath in 7-1|  minutes,
how long will it take to fill the bath when both 
taps are turned on and the waste plug is drawn? Ans 
= 1 hr.
e.g. A man shared N240,000 such that —1 4  went to his wife.' 3-
9=•
to his sons and the remainder was divided equally 
between his two daughters. How much did each daughter 
receive?
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Solution
Total fraction for wife and son = —4  + —3  = l1i2-
Daughter received 1 - 11
Exercise 1.10
Each daughter received —12 of the fraction both
daughter
1
24
Amount received by each daughter 240,00024
M10,000.
F. REVISION EXERCISES
1. Simplify the following:
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2. If takes a man 6 hours to finish a job and another 
man 4 hours to finish the job, how low will it take 
the two man to finish the job if they work together?
3. A man spent —  of his money and then gave away —  of
£ -J
the rest. If he had M3,600 left, how much had he at 
first?
4. Three men agreed to pay a bill. The first man paid—25
of i• t, the second pai. d —14 of it plus an extra M2
whi. le the thi. rd pai. d —1 of it plus an extra M3. How 
much did the first man pay?
5. In an election, there are two candidates and —25 of
the people who could have voted did not do so. Of 
the remai• nder, —4 voted for the winning candidate
who had a majority of 3,696. How many voters were 
there?
Answers:
A. 17 B. 1 —20 C. 8 D. . 2 —7 E. 7 —3
2-5| hrs
3. M10,800.
4. M12 0.
5. 7,700 voters.
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UNIT 2. DECIMALS
A. OBJECTIVES
Students will be able to:
1. Find the place value of any digit in a given 
number containing decimals.
2. Express a number correct to a specified number 
of decimal places.
3. Use the balancing rule to evaluate problems 
involving combination of division and 
multiplication of decimals.
4. Covert fractions to decimals and vice verse.
5. Solve worded problems involving decimals.
A. PLACE VALUE
To find the place value of a given number containing 
decimals, recall that each place value increases ten 
fold as one goes from right to left and decreases by 
one-tenth from left to right.
Hundreds Tens Units Tenth hundredth thousandth
e.g. 378.429 = 3  7 8 4 2  9
Thus digit 3 actually represent 300.
7 represents 7 tens i.e 70
8 represents 8 units.
4 is four tenth 1/—"
2 is 2 hundredth j
9 is 9 Thousandth
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e.g. 9451.63.
9 represents 9 thousands 
5 represents 5 tens 
3 represents 3 hundredths etc.
Exercise 2.1
Find the place value of the number in bracket for each 
of the
following:
a. 14.058 (5) .
b. 127.563 (3) .
c. 968.43 (6)
d. 7925.814 (7,1)
B. Addition and Subtraction of decimals
This can be easily done with your calculator. If no
calculator is available, the key point to note is 
correct placement of decimals of the numbers concerned. 
All decimals must tally!
e.g. 0.92 + 450.007 + 0.0056.
To do this, arrange the numbers in such a way that the 
position of the decimals with tally.
So above becomes
0.92
+ 450.007 
. 0056
450.9326
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e.g. Subtract 3.092 from 51.78. 
Arrange as follows:
51.780 
3.092 
48.688
Now, do the following: 
Exercise 2.2
0.1305 + 121.0027 + 42.1002 + 0.005.
Exercise 2.3
9781.0403 - 103.00049.
C. Multiplication and Division
For multiplication, treat the number as whole then, do 
long multiplication and then round off to the total 
number of decimal places, e.g. 53.47 x 0.042.
Use long multiplication to obtain product of 5347 and 
42 .
Counting from left to right, the total number of decimal 
places for rounding up is 5.
So,
5 3 4 7 
x 4 2
1 0 6  9 4  i.e. 5347 x 2
2 1 3 8  8 i.e. 5347 x 4
2 2 4 5 7 4
Ans = 2.24574.
Now try this: Check result with the calculator.
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Exercise 2.4
0.00413 x 461.2.
For division, the divisor is changed to a whole number 
by moving the decimal point to the right, with the 
divisor being used at the reference point.
e.g. 0.46 0.00023.
The divisor, which is 0.00023 needs to be moved 5 
points.
Thus 0.46 will have its decimal moved five places. The 
problem thus transforms to 6000 -5- 23 .
2000
23|46000 
46000
e.g. 0.252 -5- 3.15 = 25.2 + 315
The long division yields 0.08. 
Now do this:
Exercise 2.5
78.678 0.00348.
For combination of product and division the "balancing 
rule" is used. This rule deals will balancing up the 
decimal places of the given quantities.
0.168 x 5.04 
*g 6.3 x 1.344
The total number of decimal places in the numerator = 5 
while that of the denominator = 4. One needs to balance 
up as follows:
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0.168 X 5.04 
6.30 x 1.344 '
Addition of 0 to either 6.3 or 1.344 balances up the 
decimal places.
Thus the expression becomes
168 x 504 
630 x 1344
which can be easily cancelled to obtain —10 or 0.1
e a 1.21 x 1.44 _ 1.2100 x 1.44
*9’ 0.022x0.528 0.022x0.528'
This has balanced up the decimal places in the numerator 
and denominator. We now remove the decimals to get
12100 x 144 
22 x 528
which can be easily cancelled to obtain 150.
Now do this:
Exercise 2.6
0.156 x 3.36 
8.4 x 0.0468 Ans 1-3i or 1.333 
Exercise 2.7
0.169 x 3.43 
0.091 x 0.637 Ans 10.
D. Expressing a number to a given number of decimal 
places:
When expressing a quantity to x number of decimal 
places, examine the (x + l)ch digit. If it is more than, 
5 add 1 to the xth digit.
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e.g. Express 2.7893 to two decimal places. The third
digit after decimal is 9 and is more than 5. So add 1 to 
8 and obtain 2.79 as answer.
e.g. Express 0.000612 to 5 decimal places. The 6th 
digit after decimal is 2 and this is less than 5. So the 
solution is 0.00061.
Now do this:
Exercise 2.8
(a) 561.087 to 2 decimal places.
(b) 9.00305 to 3 decimal places.
E. Fractions to decimals and vice-versa.
A fraction can easily be changed to decimal by dividing 
the numerator by the denominator using long division.
e.g. —4 = 0.75
4 | 30 
28 
20 
20
—  = 0.75
To convert decimals to fraction count the number of 
decimal places and then covert to fractions by 
multiplying up and down by the powers of 10 of the 
number of decimal places.
e.g. Convert 0.025 to fractions. There are 3 decimal
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places. So multiply up and down by 103
' 0 025 = Q-02150 0x0 1000
25
1000
1
40 '
e.g. 0.00125 = 0■00112050 0x0 1000QQQ
125
100000
1
800
F. Worded problems involving decimals.
e.g. A pole is 6.8 metres long. What length will be 
left after cutting off 8 pieces each 0.58 metres long.
Solution
Total number of length cut off = 8 x 0.58
= 4.64.
.1 Length left = 6.8 - 4.64 = 2.16m .
Exercise 2.9
What is the greatest number of books 3.5 cm thick can be 
placed on a shelf 5 metres long. (Hint: Convert metres 
to centimetres and divide. The whole number obtained is 
the solution.
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Exercise 2.10
(a) Convert 0.00562 to fraction
(b) Convert —1315 to decimals.
Revision Exercises
la. Express 62 3 to 3 decimal places
b. Express 0.000842 in standard form.
2a. 1.1816 + 0.00978 + 120.12
b. 81.125 - 79.0064
3. Simplify 2-3l x 1.28 x Q.275 ^ express your answer
0.77 x 0.55 x 1.6 
as a vulgar fraction.
4. A tank full of spirit lost 0.025 of its content by 
evaporation. 9 gallons were then drawn off leaving
the tank —34 full. How many gallons does the tank
hold?
5. Poles at the side of the road are 3.75 m apart and
extend for —3 of a kilometre. How many poles are 
there?
Answers: la. 20.667 lb. 8.42 x 10“4
2a. 121.31138 2b. 2.1186
3 . 1 —5 . 4. 40 gallons
5. 201 poles.
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UNIT 3 PERCENTAGES
A. Objectives:
Students will be able to
1. Know the various ways percentages are used in 
business.
2. Convert fractions to percentages and vice versa.
3. Convert decimals to percentages and vice versa.
4. Find the value of a percentage of a quantity.
B. Introduction
As a student in Business Education you cannot avoid 
the use of percentages. The next few weeks will express 
you to some of the various ways in which it is used in 
business, among which are:
1. Commission paid to a salesman, (which is usually a 
percentage of good sold).
2. Interest paid by borrower to lender. This is 
usually a specified percentage for a specified 
period.
3. Discount, a decrease in price, is also expressed as 
percentage of price.
4. Depreciation. This deals with reduction in value of 
an asset and is usually expressed as a percentage of 
its original value.
5. Population changes are expressed as percentage 
increase or decrease in population.
6. In insurance, the premium paid to insure against a 
loss is usually expressed as a percentage of amount 
insured.
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7. Dividend paid to share holder are also expressed as 
percentage of capital. Thus, percentage is very 
useful to you as a business students.
C. Changing fractions to percentages and vice-versa.
To change a fraction to percentage, one needs to 
just multiply by 1 0 0 .
e.g. 14  = 24 x 100% = 75%.
Exercise 3.1
Convert 35 to percentage.
To change percentage to fraction, you need to realise 
that percentage involves the use of a common denominator 
of 1 0 0 .
So just divide by 100 and cancel out common factors.
e.g. 55% 55 _ _11
Exercise 3.2 100 20 ‘
Change the following percentages to fractions:
(a) 37 2 %  (b) 72% (c) 64%.
D. Changing decimals to percentages and vice versa.
To convert a decimal to percentage, one needs to 
multiply by 1 0 0 .
e.g. 0.35 = (0.35 x 100)%
= 35%.
e.g. 0.0465 = 0.0465 x 100
4.65%.
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Exercise 3.3
Convert 0.00025 and 0.19 to percentages.
To convert a percentage to decimal you need to divide 
the given percentage by 100 and obtain your answer in 
decimals.
e.g. 97— % 9752 100 100 0.975.
Exercise 3.4
Convert the following percentages to decimals, 
(a) 2-|% (b) 45% (c) 48-|%.
E. To find the percentage of a given quantity
In order to do this, you will first express the 
percentage as a fraction and then multiply by the given 
quantity.
e.g. find 40% of M72,000.
This becomes 100 x 1 = M28,800.
e.g. The population of a town was formerly 576, 307 but 
it showed an increase of 47.65 per 1,000 in the recent 
census. Find the population at the recent census.
Solution
Increase = 47.65 per 1,000 
= 4.765 per 100.
.'. percentage of new population = 104.765%.
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New rate = -104,765 x 57630
= 603768 to the nearest whole number. 
NEoxwer cdio set he3 .5following:
In the last census the population of a town was 386,475. 
This was an increase of 17,453 over the figure for the 
previous census. Find the increase per 1,000.
If you do it carefully you will obtain 47 per 1,000.
Exercise 3.6
In a school of three departments there were47—1% 
infants,
42 girls and 147 boys.
(a) what percentage of the students was non infants?
(b) How many infants were there.
Did you get 171 infants?
e.g. An oil exporter exported oil this year of the 
same value as last year but the amount exported was 
17—1% less. By how much per cent was the price per 
barrel increased?
Solution
Increase
x 100%
21.2%
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Exercise 3.7
If the amount of oil exported in the last example was 
6—3 %, by how much was the price per barrel increased? 
Ans 7.2%.
The following figures are given by a company:
1992 1993
Gross receipts 6,758,500 9,860,000
working 5,479,000 6,254,500
expenses
Find the percentage increase in the net receipts in 
1993 .
Solution
Net receipts 
Exercise 3.8
1992 = N6758500 - M5479000 =
Exercise 3.9
1993 = N9,860,000 - 6,254,500 = ___________
Increase = net receipt in 1993 - net receipt in 1992 
= N2326000
Exercise 3.10
Percentage increase = .2326 000 x ^00%
181.79%.
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F. Revision Exercises
1. Convert to percentages
(a) -|| (b) 0.0725.
2. Express the following as fractions in its simplest 
forms:
(a) 6 7 (b) 33-|%.
3. In a consignment of tomatoes, 14% are spoilt in 
transit. If a total of 2322 is left, how many are 
there in the original consignment?
4. A man bought a crate of 30 eggs at M4.50 per egg 
from a trader. He then paid the trader gave him a 
change of M25. Find the percentage error in the 
correct change to be given by the trader.
5. The following figures are given by a company.
1991 1992
Gross 5400000 4800000
receipts
working 3828600 3201600
expenses
Find (a) the net receipts for 1991 and 1992.
(b) The percentage increase in the net receipts in 
1992.
Answers
la. 24.2% lb. 7.25
2a. 2740 2b.
27
80
3 . 2700 4 . 66.7
5a. N1571400 and 1598400 5b. 1.7%
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UNIT 4 DISCOUNT - PROFIT - LOSS.
A. OBJECTIVES
Students are expected to
1. Know how to find the cash price of an article for 
which discount is allowed.
2. Calculate profit or loss or cost price or selling 
price under given conditions.
DISCOUNT
A deduction from a given price is known as discount 
given for an article. If the customer wishes to pay 
cash, a cash discount may be given by a trading 
company. It is the normal practice to express 
discount as a percentage.
e.g. 1. An article priced at N7,000 is subject to a
discount of 10% for cash. Find the cash price.
Solution
Discount = 10% of W7,000 
= N700.
Cash price = N7000 - 700 
= N6,300.
e.g. 2. A man paid N600,000 for a generator after 10% 
discount has been deducted from the list price. 
What was the list price?
Solution
Let the list price be 100%.
Price paid by the man in percentage = (100 - 10)%
= 90%
.*. 90% -+ 600,000
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Exercise 4.1
100% 600,00090 x 100 N
If a company allows both trade and cash discount, the 
trade discount is first deducted before deducting the 
cash discount.
e.g. A wholesaler allows a retailer 25% trade discount 
and 5% discount for cash. Find the cash price of an 
article which is priced in the wholesaler’s catalogue at 
N480,000.
Solution
The layout is as follows:
H
Catalogue 480,000.
Exercise 4.2
Less 25% trade discount -3-6-0,:-0-0-0
Less 5% cash discount PQQ- Ans.
Exercise 4.3
Cash Price ?____ Ans.
Now do this on your own:
Exercise 4.4
An article priced at N47,750 is subject to a cash of 5%.
Find the cash price.
Did you get N45,362.50?
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e.g. A fridge is listed in a wholesaler’s catalogue as
21,000 subject to a trade discount of 33-|-% and a cash 
di, scount of 2—1%. Find the cash price.
N
Catalogue 21,000
Example 4.4 Less 33-j% trade discount ?_______
Example 4.5 Balance ?
Example 4.6 Less 2—12%  Cash -----?--------
.’. Cash price = M13650 .
C. PROFIT AND LOSS.
In business enterprise one can make a profit or a loss. 
If the cost price is less than the selling price a 
profit is made, otherwise a loss will be incurred. 
Profit or loss per cent is usually calculated with 
respect to the cost price except if otherwise stated.
Profit (loss) percent = PrC°ofsitt .P(rliocses ) _iQ1Q %
e.g. The cost price of an article is N7,000 and the 
selling price is N8400. Find the profit percent.
The profit = 8400 - 7000
= 1400
Profit 14007000 x 100 = 20%.
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Exercise 4.7
Find the loss per cent if an article bought for 87,500 
was sold at 86,000.
To find the cost price or selling price it is essential 
that the cost price is denoted by 100% while the 
percentage of the selling price is found from the given 
information.
e.g. An article costing 8160,000 was sold at a loss of 
5%.
Find the selling price.
Solution
Let the price be 100%. 
SP = (100 - 5)%
= 95%
So, S.P 160,000100 x 95
= 8152,000.
e.g. By selling an article for 875,000, a trade made a 
gain of 15%. Find the cost price.
Solution
Let C.P = 100%
S = 100 + 15 
= 115%.
115% 75,000
100 75,000 x 100115 65,217.39.1
Now do the following:
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Exercise 4.8
Find the cost price of an item sold for 835,000 at a 
profit of 2-|%. (Ans. = 834,146.34)
Exercise 4.9
If the cost price of an article is 815,000 and the loss 
made on it is 8% find the selling price (813,800). We 
now go on to more interesting examples.
e.g. In selling an article to a wholesaler the
manufacturer makes a profit of 7—2 % on the actual cost
price. The wholesaler sells it to a retailer at a profit 
of 15% on the wholesale selling price. The retailer 
sells it to the customer at 81250 and makes a profit of 
10% on the retail selling price. What was the cost of 
the article to the manufacturer?
Solution
8
Customer pays 1250
Retailer’s profit (10%) _ 125
Retailer pays 1125
Wholesaler’ profit (15%) 168.75
Wholesaler pays 956.25.
Manufacturers cost price = 956.25 100107.5 x 1
8889.53
Exercise 4.10
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A man bought a bicycle for M2,100. The dealer made a 
profit of 20% on the marked price and the manufacturer
makes a profit of 33 on gross receipts, What was the
manufacturing cost of the bicycle?
(Ans: note that dealer pays M1680 
while manufacturer’s gross profit = M560 
and so manufacturing cost is Ml,120). 
Revision Exercises
1. A man bought a shirt at M150 and sold it for M120. 
Find the loss per cent.
2. By selling an article at M720 a trader made a profit 
of 8%. Find the cost price.
3. An article listed in a wholesaler’s catalogue at 
M4,800 is subject to a trade discount of 25% and a 
cash discount of 5%. Find the cash price.
4. A makes an article for M200 and sells it to B at a 
profit of 20%. B sells it to C at a profit of 25%
and C sells it to D at a profit of 33-j%. How much
did D pay for the article?
5. On 50% of the turn over of a firm, there is a profit 
of 25%, on 35% there is a profit of 10% and on the 
remaining 15% there is a loss of 5%. Find the rate 
of profit on the whole turnover.
Answers:
1 . 20% 2 . M666.67 3. M3420 4. M400
5. 15-4%.
UNIVERSITY OF IBADAN LIBRARY
UNIT 5 SIMPLE INTEREST
A. OBJECTIVES
Students will be able to know the concept of Simple 
Interest and also various calculation involving the 
parameters on Simple Interest.
B. INTRODUCTION
People obtaining loans from individuals, banks or 
finance houses are made to pay interest of one form or 
the other.
Similarly, the banks pay a specified interest to
depositors. Interest has been defined as a charge for 
the use of money. There are 2 popular types of interest 
in business transactions. They are:
1. Simple interest in which the principal is fixed.
2. Compound interest in which the interest is added to 
the principal at the end of given periods and simple 
interest is calculated on the sum of the principal 
and interest of the previous year.
This unit deals with simple interest while Unit 6 
will focus on Compound Interest.
C. SIMPLE INTEREST
You may recall that Simple Interest is calculated
using the formula S.I = ~-X1 00 - where P is the
principal, R is the rate and is usually expressed in 
percentage and T is the time in years.
e.g. 1 Find the Simple Interest on M500,000 for3-j 
years at 1—1% per cent per annum.
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Solution
Principal = 500,000.
Time = 3 —1 2  years Rate = 1 
1
4—  %.
So, using the formula we obtain
500, 000 x 3-i x 1-i __J_ 2 7
100
= 821,875.0
Note that if the time is in days or months you will need 
to convert to years.
e.g. Find the simple interest on 820,000 at 1—1% p.a for 
8 months.
Note that time = —12 yr
20,000 x 1-i x 8 
S.I. =  2100 x 12 
= 8200.
Exercise 5.1
What is the simple interest on 825,000 at 6—1% per 
annum for 1 year 6 months?
(Ans. 82437.5).
When the principal is added to the simple interest we 
have the amount.
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A = P + I.
Exercise 5.2
What is the amount of the problem in e.g. 1? Ans. 
521,875.
Can you find the expressions for
(i) Principal (ii) Rate (iii) Time?
You should obtain: P = -■ R■x  xl TQQ .
R _ I x 100 . T _ I x 1QQ 
P x T ' P x R
Note that T is in years and can be converted to months 
by multiplying by 12 or to days by multiplying by 366.
e.g. In how many days will N43,800 amount to W44,556 at
4 —i% per annum simple interest?
Exercise 5.3
What is the principal? _______________
Exercise 5.4
What is the amount? ___________________
Interest = Amount - Principal 
= N756 
T 750 x 100 x 365 days
43800 x 4 —2 
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Exercise 5.5
= __________________  days.
e.g. Find the principal which earns 8750,750 in 11 years 
at 7% simple interest.
Exercise 5.6
What is the interest? ___________________
.*. P = 1 PX  x -T-- = 8975,000
e.g. Find the rate per cent annum at which 836,000 will 
earn 81,640 in 131 days.
Rate I x 100 P x T
Exercise 5.7
Substitute in the values and get Rate = 12 —1 %
e.g. A man borrows 8146,000 on the March and on 21st May 
he repays it, with interest at 11%. How much does the 
pay altogether to clear the debts?
Exercise 5.8
Days in March = ______?___________
April = 30 
May = 2 1
Exercise 5.9
Total •?
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I P x R x T100 M3,300.
Exercise 5.10
So altogether he pays A =
C. Revision Exercises
1. A sum of money was invested at 8% per annum simple 
interest. If after 4 years the money amounts to 
M26,400, find the amount originally invested.
2. A customer saved N375,000 with a bank and at the end 
of the year his money amounted to M390,000. Find the 
rate of interest.
3. Find the simple interest on M900 for 4 months at 8% 
per annum.
4. For how long must 1 leave M540 in a bank to earn an 
interest of M108, the rate being 12% per annum 
simple interest?
5. Find the total amount to be paid if M52,080 is 
borrowed on 19 April and repaid on 12 September, of 
the same year, with interest at 12% per annum.
Answers:
1. M20,000 2. 4%
3. M24 4. 1 yr 8 months
5. M54579.84
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UNIT 6 COMPOUND INTEREST AND DEPRECIATION
A. OBJECTIVES
By the end of this unit student will know
1. The concept of compound interest and two method of 
computing it.
2. How to find the depreciating value of a given asset.
B. Compound Interest
The principal in compound interest is not fixed.
Instead, the interest is added to the principal at end 
of the stated period and future interest is then 
calculated on the aggregate of the principal and 
interest of the previous period. The banks use compound 
interest in computing interests for customers.
Exercise 6.1
In a given data, which is bigger, simple interest or 
compound interest?
Answer: __________________
e.g. Find the compound interest on N400 for 3 years at 
10% p .a .
There are two methods of doing this.
Method 1
1st year : Interest in M400 at 10% = -■ 100
= N40.
Now add this to the principal and obtain a new 
principal of H440.
U
IVERSITY OF IBADAN LIBRARY
2nd year: Interest on 8440 at 10% p.a 
440 x IQ x 1 
100
844
New principal at end of 2nd year = 440 + 44
8484
Interest at end of 3rd year 484 x 1 x 10 100
848.40
Total interest = 8(40 + 44 + 48.40)
Compound interest = 8132.40 
Method 2
Formula method. This is a faster method and it involves 
finding the new amount as follows:
Amount = Pfl + 1
where P = principal, R = Rate and T is time in years.
Thus
compound interest = Amount - Principal 
Thus in the last example,
Amount = 400
Exercise 6.2
Amount 9
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Exercise 6.3
Compound Interest = Amount - 84 00 = ________________ ?
e.g. Find the compound interest on 824,000 for 3 years 
at 5% per annum, if interest is added half yearly.
Solution
There are 6 - half-yearly periods in 3 years
1st half-year Interest = 24Q1Q0Q0  x 52 *--- = 8600.
Amount at end of 1st period = 24,600.
Exercise 6.4
2nd half year, interest = _______________
Amount at end = 825215.
Exercise 6.5
3rd half year, interest = _________________
Amount = 825845.4.
Exercise 6.6
4th half year interest = 8646.1
Amount = 826491.50
5th half year, interest = 8662.3
Amount = 827153.8
3rd year interest = 867.88
Amount 827832.60
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Compound Interest = 827832.60 - 824,000
= 83832.60
To use formula method, note that there are six--|- years 
in 3 years, your T = 6.
Also since rate is half yearly, your rate = = 2-|-%.
/
A 24000 1 +
\ 100 827832.6
Compound Interest = 27,832.6 - 24,000 = 83832.6.
C. Depreciation
Assets or possession decrease in values for various 
reasons and the decrease in value of called 
depreciation.
It is usually calculated as a percentage decrease on the 
cost of asset.
The book value is the value of the depreciated asset at 
the end of the specified period.
e.g. Assuming that at the end of a year a machine is 
worth 10% less than at the beginning of that year, find 
the value after three years use of a machinery bought 
for 820,000.
Solution
8
Value 20,000
1 s t  y e a r  d e p r e c i a t i o n 2 , 0 0 0
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18,000
2nd year 1,800
16,200
3rd depreciation = 1620
.’.value at end of third year = 8(16200 - 1620)
= 14.580
e.g. A machine costs 872,500. It is decided to write off 
8% for depreciation at the end of each year. What will 
be its value at the end of 4 years?
Solution
Depreciation first year = 8% x 72500 = 85800 
Value at end of 1st year = 72500 - 85800 = 66700. 
Exercise 6.7
2nd year depreciation = __________ ?____________
.'. Value at end of 2nd year = 61,3 64 
Exercise 6.8
Depreciation at 3rd year = _______ ?________________
.'. Value at end of 3rd year = 856454.88 
Exercise 6.9
Depreciation at end of 4th year = ______ ?_______
.’. Value at end of 4th
year = ______________ ?_____________
If you do this correctly you will obtain 851,938.49 as
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the book value.
D. Revision Questions
1. What sum will amount to W3993 in 3 years at 10% per 
annum compound interest?
2. What is the difference between the simple and 
compound interest on W24 0 for three years at 5% 
annum?
3. A man borrows W2,000 at 4% compound interest per 
annum at yearly instalmental repayment of W500.00. 
How much is he still owing at the end of the second 
year?
4. A capital of W15,000 is deposited at a finance 
institution with an agreement to pay 5% interest 
rate on the money per annum, calculate the total 
amount payable to the owner at the end of the 5th 
year.
5. Calculate the book value of a machine after two 
years whose after two years price is N85,000 and 
which depreciates by 5% by the end of the year.
Answers:
1. W3,000 2 . Ml.83 3. N1143.20
4. W19,144.22 5. W76712.5
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UNIT 7 RATIO AND PROPORTION
A. OBJECTIVES
Students will be able to
1. Know the concept of ratio and its application in 
business.
2. Know the concept of proportion and its application 
in business.
3. Know how to calculate problems on ratio and 
proportion.
B. Ratio
When two quantities of the same magnitude are 
compared, then the relationship which one quantity bears 
to the other is known as ratio. It can be expressed as 
fractions.
e.g. If Mr A earns M3,600 monthly and Mr B earns N5,400 
monthly. Then the ratio of A ’s salary to B's salary is 
3600:5400 which when simplifies gives 2:3.
e.g. If 1 man in 15 and 1 woman in 9 failed to pass 
their driving test, how many of each were tested if 168 
men and 104 women passed?
For the men, 1 in 15 failed
14 in 15 passed
So —  - 168
Total men = 168 x —14 = 180 men.
Similarly for the women, 1 in 9 failed
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Exercise 7.1
_ in 9 passed
No of women 104 x —g
8
Exercise 7.2
= _________________________  women.
e.g. If A:B = 3:5, and B:C = 4:7, find A:B:C.
Solution
Look at the values allocated to B i.e. 5 and 4. Find the
LCM which is 20.
A :B = 12:20 and B:C = 20:35 
.*. A: B: C = 12:20:35.
Now do the following:
Exercise 7.3
If A: B = 5:9 and B:C = 12:13, find A:B:C. (Ans.
20:36:39)
Exercise 7.4
Find the smallest share if 45,000 is to be divided 
in the ratio 3:5:7. (Ans. N9,000)
e.g. A trader allows on the list price of his goods a 
trade discount of 20% and a cash discount of 5%. What is 
the ratio of the cash price to the list price?
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Solution
Let list price be 100
Trade discount 2080
Cash discount = 476
Cash price
Ratio is 76:100.
C. Proportion
Proportion deals with the equality of ratio
e.g. 2 4.3 6
and also 2:3 = 4:6
Problems in this section are usually worked by the 
fractional methods, bearing in mind that ratio can only 
exist between quantities of the same magnitude.
There are two types of proportion:
1. direct proportion in which the price will be 
directly proportional to the quantity.
e.g. If 2 oranges cost 10 k one expects 5 to cost 
25k.
2. The second is the inverse proportion in which the 
change in the ratio of the first quantity is 
followed by a change in the inverse ratio of the 
second quantity.
e.g. If 4 men can do a piece of work in 10 minutes,
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20 men will take less time i.e. 4 x 1020 = 2 minutes.
(Note Ratio of men is ) .
e.g. If 3 men take 15 hours to do a piece of work 
how long will 5 men take?
Exercise 7.5
Ratio of men in fraction is ______________
.. ti• me take = 15 -r rati. o = 15 x —35 = 9  hours.
Sometimes a problem may be a mixture of both direct and 
in direct proportion.
e.g. If x is directly proportional to the square of y 
and inversely as z and x = 36, when y = 3 and z = 4, 
find x when y = 5 and z = 2.
Solution
The equation is x = z
7.6 Substituting the values k = _________
x = bzd .
7.7 Substitute for k, y x z and obtain x = 
e.g. If 3:5 = 12:P find P
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Solution
3 _ 12 
5 P
3p = 12 x 5
Exercise 7.8
P = _______________ ?
e.g. Divide 8 50 into two parts so that one is two- 
thirds of the other
Solution
The proportion is 23 i. e. 2:3
Exercise 7.9
One part will be 25 x 50 =
Exercise 7.10
The other will be —5 x 50 =
Revision Exercises
1. Divide 814,000 among A, B and C so that A has twice 
as much as B and B has twice as much as C.
2. The ratio of men to women in a committee of 3 3 
people is 7:4. How many women must be added so that 
the ratio will be 7:7?
3. If a:b = 5:8 and x:y = 25:16 find —x : —y .
4. If x:y = 5:8 and y:z = 3:2 find x:y:z.
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5. Find the ratio between the selling price which will 
give a profit of 20% on the cost price and the 
selling prices which will give a profit of the 20% 
on the selling price. The cost price is the same in 
both cases.
Answers:
1.88000, 84000, 82000 2 . 9 3. 2:5
4. 15:24:16 5. 24:25.
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UNIT 8 PARTNERSHIPS
A. OBJECTIVES
Students will
1. Know the meaning of partnership in business.
2. identity the various ways of sharing profits in 
partnership.
3. acquire the ability of solving mathematical problems 
on partnership.
B. Introduction.
Whenever two or more people own a business jointly with 
the purpose of making profits, then they are said to 
from a partnership. Thus, partnership has been described 
as "the relation which subsists between persons carrying 
on a business in common with a view to profit". The 
profits made in the business are shared periodically 
according to the agreement made at the time the 
partnership is formed and a document (deed of 
partnership) usually gives the conditions governing the 
sharing of profits or losses. Partners usually specify 
the agreed proportion for profit sharing and the capital 
and human input is usually considered in doing this. If 
partners do not make agreement legally, then 
the profits must be shared equally. We are more 
concerned with the mathematics involved in profit 
sharing in this course.
C. Solved problems on partnership.
e.g. A net profit of N35,000 is to be shared equally 
between A and D. What is each person’s share?
Solution
All you need to do is to divide 35,000 by 2.
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Exercise 8.1
This gives _________________________
e.g. 2 A profit of 832,000 is to be shared between A, B 
and C in the ratio of their investments. The investments 
are 8100,000; 8200,000 and 8500,000 respectively. How 
much will each receive.
Solution
The r at i o  of t h e i r  i n v e s t m e n t  is 
100,000:200,000:500,000.
When reduced to lowest term the ratio is
Exercise 8.2 _________________________
Exercise 8.3
A ’s share = —18 x 32000 = 
Exercise 8.4
B ’s share = —8 x 32000 = 
C ’s share = _______________________________
Please check the addition of the three shares. They 
should add up to 832,000.
e.g. A, B and C are in partnership. It was agreed that
A should recei. ve a quarter of the profi. t, B, —1  and C
the remainder. The profit amounted to 837,458. What was 
the share of each?
Solution
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Total shared by A and B = —4  + —3  = —12
Exercise 8.5
Fraction shared by C =
A ’s shared = -4i x 37458 
= 89364.5
B ’s share = -3 x 37458 
812486
Exercise 8.6
C ’s share = ____________________________________
e.g. Davidson, Lee and Daniel’s partnership agreement 
calls for division of half the annual profit in the 
basis of their original investments and half on the 
basis of 40% for Davidson, 35% for Lee and 25% for 
Daniels. The original investments were:
Davidson - 824,000; Lee - 820,000 and Daniel - 812,000. 
The net profit of the business is 846,200.
What is each person’s share in the profit?
Solution
The arithmetic involved in this problem is quite simple!
It boils down to sharing 46^,000 23,100)
in two ways:
(a) Ratio 40:35:25 i.e. 8:7:5 and
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(b) 24000:20000:12,000 i.e. 6:5:3
M23,100 share in ratio:
Davidson Lee Daniels
M M M
a. 8:7:5 9240 8085 5775
b. 6:5:3 9900 8250 4950
19140
16335 10725
Now try the following:
Exercise 8.7
Two partners A and B are in business together
A ’s capital is 40,000 and B ’s N30,000. The profits of 
the firm are 21,000. How much should each partner 
receive?
Exercise 8.8
A, B and C are in partnership.lt was agreed that A
should recei. ve —1  of the profit, B, —2  and C the
remainder. If the profits were 14,550. how much should 
each partner receive?
Exercise 8.9
A and B started business with a capital of M17,820. A 
contributed M9,720. and B the rest. If profit of N19,492 
are shared in the ratio of their capital, how much will 
each receive?
Exercise 8.10
Share profits of N365,172 between two people in the 
ratio 5:9.
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D. Revision Exercises
1. Three partners Ade, Abu and Obi are in business 
will capitals of 8500,000 8400,000 and 8200,000. 
The profits amounted to 8279,576. How much will 
each person receive?
2. A construction company is owned by two partners 
P and Q and it is agreed that their profit will 
be divided in the ratio 3:4. At the end of the 
year, P received 84,000 less that Q. What is the 
total profit of the company for the year?
3. A, B and C are in partnership. It was agreed
that A should receive —1 6 , B 
1
5—  and C the
remiander. If the profits were 8290,100 how much 
should each partner receive?
4. The profits of a firm were 8127,500. A provided
3 times as much as B and —23 the amount of C ’s
capital. How much should they each receive?
5 . Two men Ayo and Akin form a business partnership 
Ayo provided 854,000 capital and Bayo 863,000
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W-i® cappital. It was decided that profit made each 
year will be divided according to their capital 
at the begining of that year. In the first year, 
the total profit was 87,800. At the end of the 
year, Ayo withdraw 81600 of his share of the 
profits and then invested the remainder in the 
business while Akin invested the whole of his 
share. If the profit for the second year was 89 
160.80, how much did each person get?
Answers
1. Ade - 8127080; 8101664 for Abu; Obi - 850832.
2. 82,800.
3. A receives 848,350. B receives 858,020 C receives 
8183730.
4. A receives 845,000 B recieves 815,000 C receives 
867,5000.
5. Ayo 84164 Akin 84996.8.
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UNIT 9 STOCKS AND SHARES
A. OBJECTIVES
1. Students will be able to know about stocks and 
shares and the difference bvetween them.
2. Know about ordinary and preference shares.
3. Solve simple problems on stocks and shares.
B. Introduction
A public company issues shares to which the public 
is invited to subscribe. This helps the company to 
acquire more capital to be incested in the business. The 
shares are issued in fixed units of say 50 kobo, N1 or 
M2 etc. At the end of the financial year the company 
then declares a dividend or profit to share holders 
which is taxable and is in the form of a percentage of 
the nominal value of the shares. A shareholder can only 
buy an exact number of shares e.g. 70 - Ml share whereas 
any amount of stock (e.g. N98.60 stock) can be 
purchased. The buying and selling of stock and shares is 
controlled by the Stock Exchange.
By owning a share in an incorporated company, it 
means the shareholder is prepared to prosper with the 
company in good times and also to suffer with it
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financially in bad times.
There are many types of shares, but the most common 
are the ordinary and preference shares. A preference 
share is issued at a fixed percentage and the payment of 
the dividend of this type of share is the first to be 
paid. The dividend is cumulative, unless the shares are 
expressly issued as non cumulative. If the dividend is 
not paid when due it must be added to the next dividend. 
Thus any time profits in a company are shared, the 
preference share holders are fist satisfied before the 
ordinary holders are considered.
When stocks are bought or shared at the stock 
market, the broker receives a commission known as 
brokerage. When stocks are at sale for a higher than the 
nominal or face value they are said to be at a premium; 
when at a lower price they are said to be at a discount 
and when the market price is the same as the face value 
they are said to be at par. Speculators buy shares in 
anticipation of a rise in price when they then sell out 
the shares at a profit while investors buy stock or 
shares with the aim of receiving a steady income.
The following are some examples involving stocks and
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shares.
C. Exercises
e.g. Bata Shoes Company 3—i % stock is quoted at
47 —2 . What will be the cost of N800 stock and what 
dividend is due on that stock?
Solution
Cost of stock = —1 00 x N47.50 
Exercise 9.1
= N _
Exercise 9.2
Dividend = 100 x 3—2 =
Now do this:
Exercise 9.3
War loan stock i, s quoted at 37 i . What will be the cost
of M400 stock and what dividend is due on that stock?
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e. g. How much stock at 89 can be bought for 8555?
Solution
Amount = 555 xlOO
= N600 nominal value.
Exercise 9.4
How much stock at 87 can be bought for 8350?
e.g. How much stock at 110 can be bought with proceeds 
of the sale of 81045 stocks at 95?
Solution
Proceeds of 81045 stocks at 95
= 1045 x -12
9
0-
 2-50 = 8992.75.
Amount of stock at 110 = 11.075 iqq
Exercise 9.5
=  8
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e.g. Standard motor 50 kobo shares are quoted at 75 kobo 
and a dividend of 15% is declared. How much does it cost 
me to buy 8000 shares and what dividend do I expect?
Solution
The cost = 8(8000 x -11050-) = 86000.
The dividend = 15% of the nominal value
= 100 x 84000
= 8600.
Exercise 9.6
John Holt 50 kobo shares stand at 75 k and declare a 
dividend of 20%. What does it cost me to buy 8000 shares 
and what dividend do I expect?
e.g. Find the profit made by buying 81000 stock at82
and selli. ng at 87—i2 .
Solution
Profit per 8100 = 87—2  - 82—2 = 8 5 .
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Profit made = M5 x 1000100 N50.
Exercise 9.7
Find the profit made by buying M8 000 of stock at87
and selli. ng at 88—3 .
e.g. How much stock at 90 can be bought for N7290. If 
the brokerage i, s 1—14 per cent.
Solution
Consideration = 7290 x ■ -
101 —4
Exercise 9.8
->
Exercise 9.9
Stock = ____ 10090
88000 nominal value.
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Exercise 9.10
How much stock at can I buy for M609. If the brokerage
is 1—2 per cent?
(Ans. 800 nominal value).
D. Revision Exercises
1. PZ 15—1 % preference shares of Ml norminal value are
quoted at W0.72-i. What will be the cost of 400
shares and what dividend is payable on them?
2. Find the profit made by buying M6000 stock at97-zi
and selli. ng at 98—34 .
3. A man sells M600 of stock at 82—l 4 . The brokerage i
. sl —14
per cent. How much will be receive?
4. A man has M8500 invested in 5—2 % Cumulative
Preference Shares. Last year he received a dividend
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of 8320. How much should be receive this year?
5. I buy 6000 shares of nominal value 50 k at 65 k. If 
the dividend declared is 4% what is the income 
derived?
Answers
1. 8290; 862. 2. 8750 3. 8487.33 4. 8615
5. 8120.
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UNIT 10 INCOME TAX AND PAY ROLL COMPUTATION
A. OBJECTIVES
Students will be able to
1. Distinguish between direct and indirect tax.
2. Know the procedures for payroll computation.
B. Introduction
Every worker pays tax which is deducted from his 
salary in pay roll computation. The direct tax paid on 
a person’s income is known as income tax. Other type of 
direct taxes are licences and death duties. Indirect 
taxes, on the other hand, are paid at source and later 
passed on to consumers e.g. excise duties on wines, 
spirits and beer and custom duties on imported goods.
Every worker is entitled to some allowances set 
against his income before whatever remains is subject to 
tax. Exercises of such are children allowance, 
contribution to pension, life assurance etc. After 
deducting these allowances, whatever is left is known as 
TAXABLE INCOME and it is on this income that tax is 
paid.
In payroll computation, you need to be familiar with
some terms:
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1. Gross earnings: This comprises basic salary,
overtime acting allowances and other taxable income.
2. Taxable Income is the income that is subject to tax 
after the allowances have been deducted.
3. Non taxable income: This is not subject to tax. 
Example are housing allowance, transport allowance 
and learned society allowance.
4. Pay roll deductions comprises income tax and loans 
or salary advances.
5. Take home pay is what is left after pay roll 
deductions and addition of allowances are done from 
the gross earnings.
C. Worked Problems
e.g. A man with an annual salary of N3 0,000 has
allowances of N12,000. If income tax is 35%,how much tax
does he pay each year?
Solution
Annual Salary = N30,000
Allowances = 12,000
Exercise 10.1
Taxable Income = ___________________
Annual Income tax = 35% of Taxable income = N6,300
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e.g. A tax payer is allowed 18 of his income tax - free,
and pays 20% on the remainder. If he pays 84 9 0 tax, find 
his income.
Exercise 10.2
What is his taxable income, given that 84 9 0 represents 
20% of his taxable income.
Exercise 10.3
If he is allowed —I8 of his income tax free, what
fraction of his income is subject to tax.
Exercise 10.4
Equate the fraction in exercise 10.3 to the taxable 
income in exercise 10.1 and use simple proportion to 
obtained an income of 828,000.
Now try this on your own:
Exercise 10.5
A man with an annual salary of 845,000 has allowances of
815,000. If income tax is 15% how much does he pay? 
e.g. A man having a wife and two children is on a 
monthly salary of 82000. He is allowed 8200 tax free in
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respect of himself, 8200 tax free in respect of his wife 
and 880 tax free in respect of each child. He pays tax 
at the rate of 10 k in the 8 on the first 8400 of his 
taxable income and 20 k in the naira on the rest. A loan 
of 8208 is deducted from his salary while his monthly 
non taxable allowances total up to 81802. Calculate
1. his taxable income
2. the total deductions in his pay slip.
3. his take home pay.
Exercise 10.6
What is the tax free amount in respect of the two 
children?
Exercise 10.7
What is the total tax free allowance (i.e. add his own, 
his wife and the two children’s allowances)
Monthly salary = 82,000 What then is his taxable income? 
His taxable income is 81440.
Exercise 10.8
Tax on 1st 8400 at 10 kobo per naira = __________
Tax on nex 1,040 at 20 kobo per naria = 8208 
Total tax paid = 8248.
Exercise 10.9
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Total deductions = tax plus loan = ______________
So take home pay = (salary and allowances - deduction)
= M3.346
e.g. A man with an annual salary of N37,000 has 
allowances of M9,000. What percentage of his salary does 
he pay annually as income tax, given that tax is 10% on 
1st M20,000 and 15% on the next W20,000.
Exercise 10.10
What is his income tax? __________________
Percentage = .Incoine tax x 100 = 8.6%.
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Revision Exercises
1. A man whose salary is 8200,000 has allowances of
850,000. Find the annual tax he pays if the rates 
are: 33% on the first 840,000
43% on the next 840,000 
53% on the next 840,000 
63% on any remaining.
2. A science teacher gets a basic salary of 82,000 
every month and taxable science teacher’s allowance 
of 8200. Tax is paid at a flat rate of 10%. His 
housing and transport allowances total up to 82,500. 
What is his take home pay?
3. If the take home pay of a civil servant is 82,825 
and deductions from his salary are 8250, while his 
housing and transport allowances are 81,500, what is 
his basic salary?
4. A man with an annual salary of 845,000 has 
allowances of 812,000. If income tax is 25%, how 
much tax does he pay each year.
5. A man’s tax is 8240 and this is 5% of his gross 
earnings. What is his gross earning?
Ans: 1. 870,500 2. 84,480 3. 81575 4. 8 8250 5. 84800.
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