This project is a research work carried out
to investigate into the difficulties encountered by students in solving
problems when differential calculus in senior secondary schools. The population
used was randomly selected from ten secondary schools in five Local Government
areas of Lagos state.
A total of two hundred (200) students and ten
(10) further mathematics teachers served as the respondents of this study from
ten selected schools. Diagnostic test and questionnaires were designed and
administered to the students. Also, questionnaires were designed and administered
to the teachers.
The results of the diagnostic test were
analyzed using frequency distribution and t-test. The teacher’s and students’
questionnaires were analyzed using the opinions based on each items. Findings
from the study revealed that students have problem of understanding questions,
they have difficulty in interpreting questions correctly. The findings equally
showed that students have problem recalling formulas. It also revealed that
there are insufficient teaching materials for the teaching and learning of
The study also revealed that there is no
significant difference between male and female students’ performance when
solving problems in differential calculus. Relevant recommendations were made
in the light of the research findings.
TABLE OF CONTENT
of content v
of tables vi
to the study 1
of problem 4
of study 5
of the study 7
of the study 8
and Sample 10
CHAPTER TWO: LITERATURE REVIEW
of Teaching further 15
Mathematics in the senior secondary school
further mathematics curriculum 15
and learning of further mathematics 18
further mathematics Students 19
Education and conditions of service 21
Teachers‘ attitude towards mathematics and 24
2.4 Students’ attitude towards mathematics
further mathematics and it’s effect
encountered by students when solving 30
problems in some further mathematics topics
aside differential calculus
Differential calculus 33
Calculus: An indispensable mathematics tool 35
CHAPTER THREE: RESEARCH METHODOLOGY
3.1 Research design 36
3.2 Population and sample 36
3.3 Research instruments 37
3.4 Method of
Data Administration (Procedure) 39
3.5 Method of Data Analysis 40
CHAPTER FOUR: ANALYSIS AND PRESENTATION
4.0 Introduction 41
4.1 Results from Diagnostic test and test of
4.2 Findings from the difficulties encountered by
the students in the diagnostic test 51
4.3 Analysis of students’ and teachers’
4.3.1 Introduction 63
of students questionnaire 64
4.3.3 Analysis of teachers’ questionnaire 66
CHAPTER FIVE: DISCUSSION, RECOMMENDATIONS,
SUMMARY AND CONCLUSION
5.0 Introduction 69
of findings 69
for further research 77
LIST OF TABLES
Tale 4.1 The general performance of the students in the
4.2 The scores of male and female students in the
4.3 The nature of difficulties encountered by the students in the diagnostic
1.0 BACKGROUND TO THE STUDY
In the words of Fafunwa
(1974) “Education is seen as the best means for developing the potentialities
of young and adult learners so that they can in-turn make meaningful
contribution to the development of the society”. As an instrument of change, it
is generally recognized that no society can make any significant progress
without relevant education.
Mathematics is one of the
essential subjects taught in the school curriculum at all levels of education.
It is needed in all aspect of our daily lives e.g engineering, science,
agriculture, philosophy medicine etc. It is the body of knowledge centred on
such concepts as quantity, structure, space and change and also the academic
discipline that studies them. Benjamin Pierce (1809-1880) called it “the
science that draws necessary conclusions”.
Mathematicians seek out
patterns whether found in numbers, space, science, computers, imaginary
abstractions or elsewhere. They explore such concepts, aiming to formulate new
conjectures and establish their truth by rigorous deductions from appropriately
chosen axioms and definitions.
Through the use of
abstraction and logical reasoning, mathematics evolved from counting,
calculation, measurement and systematic study of the shapes and motions of physical
objects. Knowledge and use of basic mathematics have always been an inherent
and integral part of individual and group life.
In accordance with the
national policy on education 1977, mathematics is regarded as a compulsory
subject for every student at the secondary school level. This is attributed to
the fact that mathematics is most applicable to real life situations and all
The general objectives of
mathematics at the senior secondary school level are:
To enable students develop further
To develop the ability to think deductively.
To provide the mathematical background for
the application of mathematics in other subjects.
To develop the ability to solve mathematical
To stimulate and encourage creativity
To generate students’ interest in mathematics
and provide a solid foundation for those who may want to continue with
It has been argued by
numerous mathematics teachers that the content of the curriculum is inadequate
to cater for the
more- able students in
mathematics. Hence, the need for the further mathematics curriculum.
The aims of the further
mathematics syllabus as quoted from the National curriculum for senior
secondary school volume 5, of the Federal Ministry of Education are:
To help the students develop further
conceptual and manipulative skills, and its application.
To provide an additional intermediate course
of study that bridges the gap between elementary mathematics and higher
To meet the needs of potential mathematicians,
engineers, scientists and other professionals such as business administrators,
curriculum has three major components, which include pure mathematics,
mechanics and statistics. Differential calculus belongs to the pure mathematics
family. The content of differential calculus includes differentiation of
explicit, algebraic, circular, logarithm, implicit functions and its
Hence, providing the panacea
to the difficulties encountered by students when solving problems in
differential calculus will go a long way to improve and enhance the scientific
and technological advancement of the society.
STATEMENT OF PROBLEM
Differential calculus being
one of the branches of mathematics with varied application, in physical
sciences, engineering, medicine, computer sciences, business administration etc
ought to be properly taught to the students. However, students are faced with
difficulties when solving problems in differential calculus over the years, and
this leaves much to be desire. This could be as a result of:
Lack of problem solving technique
Neglect of application of pre-requisite topics
such as indices, logarithm, exponentials, simplifications, factorization
polynomials, trigonometry functions etc.
1.2 PURPOSE OF STUDY
As explained above, the difficulties
encountered by students when solving problems in differential calculus needs to
be studied vividly. Therefore, the study is aimed at:
To ascertain students’ level of performance
in differential calculus.
To investigate the difficulties students
encounter when solving problems in differential calculus.
To find out the factors that contributes to
the difficulties encountered by students when solving problems in differential
To proffer and recommend workable solutions
to the difficulties identified during the course of this study.
To achieve the objectives of
this study, an attempt was made to provide answers to the following questions:
What is the students’ level of performance in
What are the difficulties encountered by students
when solving problems in differential calculus?
What factors contribute to the difficulties
encountered by students when solving problems in differential calculus?
Do students show positive attitude towards
the learning of differential calculus?
Is there any justification for the inclusion
of differential calculus as a topic in further mathematics?
Do teachers make use of enough instructional
materials in teaching differential calculus?
Is there difference between the mathematical
achievement of senior secondary school two (SSS 2) male and female students in
How can the difficulties encountered by
students when solving problems in differential calculus be minimized?
The hypothesis below will be
tested during the course of this study.
H0: There is no significant difference
between male students’ performance and female students’ performance when
solving problems in differential calculus.
H1: There is significant difference between
male students’ performance and female students’ performance when solving
problems in differential calculus.
will be tested against H1
at 5% level of significance.
1.5 SIGNIFICANCE OF THE STUDY
study will focus on:
Providing dependable solutions to problems
associated with the teaching and learning of differential calculus.
Guiding the teachers in the effective
teaching of differential calculus.
Encouraging proper learning approach in the
study of differential calculus.
SCOPE OF THE STUDY
The study will be carried
out in five local Government Area in Lagos State (Somolu LGA, Mainland LGA,
Agege LGA, Kosofe LGA, Surulere LGA). It will cover ten randomly selected
senior secondary schools and they will be the representation sample for the
whole Local Government Area.
The data will be collected from
senior secondary school two (S.S 2) students, a total of 200 students, that is,
20 students randomly selected from all existing arms of S.S. 2 classes in each
A more comprehensive
research work will entail visit to all schools in every Local Government Area
and administration of test to all the SS 2 students in each school. But due to
time factor and financial constraint, a random sample as stated above will be
The following instruments
will be used in the course of this study:
Students’ Diagnostic test on differential
Questionnaire for students
Questionnaire for teachers.
The student’s diagnostic
test on differential calculus shall consist of five theory questions selected
to cover all aspects of differential calculus and its applications.
In accordance with S.S.2
further mathematics syllabus, the questions will cover areas on:
Differentiation from first principle
Function of a function
Differentiation of algebraic functions.
Differentiation of exponential functions.
Differentiation of inverse functions.
Differentiation of logarithm functions.
Differentiation of trigonometry functions.
Application of differential calculus
The questions will be set to
identify areas of students’ understanding or areas of teachers ineffective
teaching and to find out specifically how much of the concepts the students
understand. Time duration of one hour thirty minutes will be given. Objectives
will be avoided since the focus is on ascertaining the errors students make
when solving problems in differential calculus.
The questionnaire for
students will be open-ended, and will consist of 5 well-structured items will
be administered. For teachers’ questionnaire, 4 well structured items will be
administered, and will also be open-ended. These questionnaires will be
designed to investigate some of the difficulties that occur in the teaching and
learning of differential calculus in senior secondary schools (SS2 Students).
POPULATION AND SAMPLE
The population under study
will be students and teachers selected from schools in five Local Government
Areas of Lagos State (Somolu LGA, Mainland LGA, Agege LGA, Kosofe LGA, Surulere
A random sample of ten
schools will be selected from the State Local Government Areas as the
representation sample. A total of two hundred students will be used as the
respondents and ten further mathematics teachers will be used (one from each
school). The students will be randomly selected with quota sampling of twenty
students from each of the schools. The students will be from Senior Secondary
School two (SSS2) classes.
Two hundred students will be
randomly selected from ten schools with twenty students from each school. The
twenty students will be selected from each of the schools and the students’
diagnostic test will be administered. With adequate invigilation, the students
will be made to solve the problems by showing the workings to each question in
their answer scripts.
Questionnaire will also be
administered to the students and the teachers so as to gather more information
on the difficulties encountered by students when solving problems in
differential calculus. After administration, all data will be gathered for
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