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# DIFFICULTIES ENCOUNTERED BY STUDENTS WHEN SOLVING PROBLEMS IN DIFFERENTIAL CALCULUS IN SENIOR SECONDARY SCHOOLS

## Content

**ABSTRACT**

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 differential calculus.

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.

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**TABLE OF CONTENT**

** **Pages

Title page i

Certification ii

Dedication iii

Acknowledgement iv

Table of content v

List of tables vi

Abstract vii

**CHAPTER ONE: INTRODUCTION**

1.0 Background to the study 1

1.1 Statement of problem 4

1.2 Purpose of study 5

1.3 Research Questions 6

1.4 Research Hypothesis 7

1.5 Significance of the study 7

1.6 Scope of the study 8

1.7 Research Instruments 9

1.8 Population and Sample 10

1.9 Procedure 11

**CHAPTER TWO: LITERATURE REVIEW**

2.0 Introduction 13

2.1 Further mathematics 13

2.1.1 Objectives of Teaching further 15

Mathematics in the senior secondary school

2.1.2 The further mathematics curriculum 15

2.1.3 Teaching and learning of further mathematics 18

2.1.4 The further mathematics Students 19

2.2 Teacher Education and conditions of service 21

2.3 Teachers‘ attitude towards mathematics and 24

further mathematics.

2.4 Students’ attitude towards mathematics and 28

further mathematics and it’s effect on their

performance

2.5 Difficulties encountered by students when solving 30

problems in some further mathematics topics aside differential calculus

2.6 Calculus 32

2.6.1 Differential calculus 33

2.6.2 Derivative 34

2.6.3 Calculus: An indispensable mathematics tool 35

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**CHAPTER THREE: RESEARCH METHODOLOGY**

3.0 Introduction 36

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

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**CHAPTER FOUR: ANALYSIS AND PRESENTATION **

** OF DATA**

4.0 Introduction 41

4.1 Results from Diagnostic test and test of hypothesis 42

4.2 Findings from the difficulties encountered by

the students in the diagnostic test 51

4.3 Analysis of students’ and teachers’ questionnaires 63

4.3.1 Introduction 63

4.3.2 Analysis of students questionnaire 64

4.3.3 Analysis of teachers’ questionnaire 66

**CHAPTER FIVE: DISCUSSION, RECOMMENDATIONS, **

** SUMMARY AND CONCLUSION**

5.0 Introduction 69

5.1 Discussion of findings 69

5.2 Recommendations 72

5.3 Recommendations for further research 77

5.4 Summary 77

5.5 Conclusion 78

**REFERENCES **79

**APPENDIX **84

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**LIST OF TABLES**

Tale 4.1 The general performance of the students in the

diagnostic test.

Table 4.2 The scores of male and female students in the

diagnostic test.

Table 4.3 The nature of difficulties encountered by the students in the diagnostic test.

**CHAPTER ONE**

**INTRODUCTION**

**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 academic disciplines.

The general objectives of mathematics at the senior secondary school level are:

- To enable students develop further computational skills.

- 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 problems.

- To stimulate and encourage creativity

- To generate students’ interest in mathematics and provide a solid foundation for those who may want to continue with mathematics.

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 mathematics.

- To meet the needs of potential mathematicians, engineers, scientists and other professionals such as business administrators, and architects.

Further mathematics 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 application.

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.

**1.1 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.

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**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 calculus.

- To proffer and recommend workable solutions to the difficulties identified during the course of this study.

**1.3 RESEARCH QUESTIONS **

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 differential calculus?

- 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 differential calculus?

- How can the difficulties encountered by students when solving problems in differential calculus be minimized?

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**1.4 RESEARCH HYPOTHESIS **

The hypothesis below will be tested during the course of this study.

**H _{0}**: There is no significant difference between male students’ performance and female students’ performance when solving problems in differential calculus.

**H _{1}**: There is significant difference between male students’ performance and female students’ performance when solving problems in differential calculus.

**H _{0}** will be tested against

**H**at 5% level of significance.

_{1}** **

**1.5 SIGNIFICANCE OF THE STUDY **

The 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.

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**1.6 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 secondary school.

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 used.

**1.7 RESEARCH INSTRUMENTS **

The following instruments will be used in the course of this study:

- Students’ Diagnostic test on differential calculus.

- 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.

- Implicit differentiation.

- Higher derivative.

- 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).

**1.8 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 LGA).

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.

**1.9 PROCEDURE **

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 analysis.

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