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The main objective of this research is to study Students' Understanding of the chapter "Normal Approximation to the Binomial Distribution". The research questions for this study are:
What type of difficulties do students encounter while attempting questions on 'Normal Approximation to Binomial Distribution'?
How can teachers help to improve students understanding of Normal Approximation to Binomial Distribution'?
Compare the performances of boys and girls from the two different schools?
What are the proposed solutions to improve students' performance of 'Normal Approximation to Binomial Distribution'?
These questions would be addressed by using different instruments such as student's and teacher's questionnaires. This chapter gives details of the stages involved in the case study procedure, showing how the research was carried out, its design, the sample and the data collected methods used. Ethical considerations were also taken care of in the present study.
4.2 What is a case study?
A case study gives the strong behind the result by capturing what happened to bring it about, and can be good opportunity to highlight a project's success, or to bring attention to a particular challenge or difficulty in a project (Yin, Robert K. (2003)). Case studies are appropriate when there is a unique or interesting story to be told. Case studies are usually used to provide context to other data (such as outcome data), offering a more complete picture of what happened in the program and why.
4.2.1 Advantages and limitations of a case study
The primary advantages of a case study is that it provides much more detailed information then what is available through other methods, such as surveys. Case studies also allow one to present data collected from multiple methods, which is surveys, interviews, document review and observation to provide the complete story.
There are a few limitations
Can be lengthy: Because they provide detailed information about the case in narrative form, it may be difficult to hold a reader's interest if too lengthy. In writing the case study, care should be taken to provide the rich information in a digestible manner.
Concern that case studies lack rigor: Case studies have been viewed in the evaluation and reach fields as less vigorous than surveys or other methods. Reasons for this include the fact that qualitative data research in general is still considered unscientific by some and in much case, case study researchers have not been systematic in their data or have allowed bias in their findings. In conducting and writing case studies, all involved should use care in being systematic in their data collection to ensure validity and reliability in the study.
Not generalisable: A common complaint about case studies is that it is difficult to generalize from one case to another. But case studies have been prone to overgeneralization, which comes from selecting a few examples and assuming without evidence that they are typical or representative of the population.
4.2.2 What are the elements of a Case study?
Case studies do not have set elements that need to be included; the elements of each will vary depending on the case or story chosen, the data collected, and the purpose (for example, to illustrate a best case versus a typical case). However, case studies typically describe a program or intervention put in place to address a particular problem. A case study can consist of the following elements:
Identify the problem
Explain why the problem is important
How was the problem identified?
Was the process for identifying the problem effective?
Steps taken to address the problem
Challenges and how they were met
4.2.3 How are case studies presented?
Case studies are flexible in that they can be presented in a number of ways. There is no specific format to follow. However, like all evaluation results, justification and methodology of the study should be provided, as well as any supporting information (that is copies of instruments and guides used in the study). Case studies may stand-alone report, the following outline is suggested:
How was the process carried out?
What assumptions are there?
Are there any limitations with this method?
What instruments were used to collect data?
What sample is being used?
The steps taken to address the problem
The challenges and how they were met
4.3 Sample description
This study was conducted with a group of students studying at form VI level in two state secondary schools. The group consists of 57 girls and 25 boys of mixed abilities that are, a group with low, regular, and high achieving students, in the age group between 17 and 18 years. This group was chosen as it would help the teacher to cater for diverse needs in learning and sharpen the focus on individual strengths and short comings in the cognitive domain. These students studied Mathematics and Additional Mathematics at Form V level whereby all of them obtained a credit in Mathematics but some did not obtain a credit in Additional Mathematics in the 'O' level Cambridge Examinations.
4.4 Assumptions of the study
The following points have been assumed during the case study:
The sample was representative and adequate, that is, the sample had more or less the same characteristics as that of the population.
The case study did not have any negative impact on the students.
The entire process of data is defined by instrumentation. Data includes facts and findings obtained by a researcher from the individuals under study. Data can be collected by different tools known as instruments. Some commonly used devices are questionnaires, interviews, tests and so on. In this study, observation and questionnaire was used.
4.6 Classroom Observation
A group of students were observed in their normal class time in the presence of their teacher. In this process of observation, I addressed the understanding of Normal Approximation to the Binomial Distribution concepts through the six stages of Bloom's taxonomy, namely knowledge, comprehension, application, analysis, synthesis, and evaluation.
4.7 Primary data
The Cambridge reports were of great help in identifying the difficulties students encountered. I have decided to perform a survey in order to gather information. Since lots of information is obtained from questionnaire. Two types of questionnaires have been prepared:
A first one for students
A second was designed for teachers
The questionnaire has been chosen for data collection in this research because
The respondents are left in anonymity and privacy. Students feel free to express their views since they know their names cannot be identified.
The responses obtained from the questionnaire are easy to process and analyze.
The respondents have adequate time to think about the questions before answering.
The cost of using a questionnaire is relatively low compared to other instruments.
4.8 Student's Questionnaire
4.8.1 Pilot test
Pilot test was done before designing the final questionnaire. This exercise was planned as a sort of pre-test (See Appendix 3) which was then given to a small group of students. As a result of the pre-test some questions were rephrased, the length, language, time to fill the questionnaire were modified in such a way that it was properly constructed. Hence the data collected are more valid and reliable.
The final format of the student's questionnaire (See Appendix 4) thus consists
A covering letter informing the respondents about the objectives of the survey and assuring him/her that all information obtained would be held in strict confidence.
In section A in order to maintain this confidentiality the respondents were requested to give only their class and gender.
Section B consist a list of criteria which was used to investigate the learning difficulties of the chapter.
Section c comprises of short multiples choice questions on the topic.
4.9 Teacher's Questionnaire
The second type of questionnaire was designed for teachers (See Appendix 5) comprise the covering letter as the one for the student questionnaire. The questionnaire itself is anonymous and includes 12 closed questions. The teachers were not required to work out problems. While preparing the questionnaire, the following criteria were considered:
The questions were kept simple to be understood with the help of only printed information.
The nature of the questions required a minimum amount of writing by the respondents.
4.10 Method of administering the questionnaire
As I was sent at one of the secondary school for this study and was assigned a mentor to help me to carry out this research. So with the help of my mentor the questionnaire was administered to the target sample and the teachers working with the respective classes were given a questionnaire each. I believe that the face to face method of contacting my mentor and head of department would give reliable and valid data. Similarly the survey was carried out in the second state school.
4.11 Ethical Issues
While conducting the research work, great care had been taken not to cause any sort of prejudice to anybody, be it students or teachers or simply anyone directly or indirectly related to the research work. I was also careful not to penalize a group of students or teachers at the expense of others. I met all the teachers in the Mathematics department and the students to inform them about the purpose, aim and objective of this research, including the rector of the school and the parents whose children are involved in this study.
Participants were free to withdraw from the study if, at any time they felt that any sort of prejudice was caused to them. The participants in the research work were assured of the fact that their views, opinions and written tasks would be strictly confidential.
4.12 Justification of Questions
Each questions in the questionnaire has been designed purposely based on the research questions and on the Cambridge A level reports. These questions were considered so as to highlight the weaknesses of students on the topic "Normal Approximation to the Binomial Distribution". The table below shows of the questions is classified:
Area of testing
Cambridge A level reports
Concepts of continuity Corrections
A significant number of students tend to forget to apply continuity correction when attempting questions on Normal Approximation to the Binomial Distribution.
2, 3, 4, 5, 6, 7
Applications of continuity corrections
8, 9, 10, 11, 12, 13, 14
Conditions when to apply Normal Approximation to the Binomial Distribution
Most all students know the conditions for Normal Approximation to the Binomial Distribution
15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25
Application of Normal Approximation to the Binomial Distribution
Some students do not know when and how to apply the Normal Approximation to the Binomial distribution and some students failed to identify that it is a Normal Approximation case.
4.13 Multiple Choice Questions
Since many years the use of multiple choice questions (MCQs) has been a very helpful for students as a means of assessment. MCQs are seen as a way of enhancing opportunities for quick feedback to students and also less for teachers to mark. Marking is mainly done by computer programs, thus speeding marking and test results. The use of MCQs promotes memorization and factual recall and do not encourage high level cognitive processes. Feedback provided by MCQs is limited as it is predetermined during construction of test and the use of MCQs is directed by the need for teacher efficiencies rather than pedagogical principles. The seven points listed below have helped me in preparing the MCQs for this survey and these points eliminate the limitations listed above. The seven points are
Clarifying goals, criteria and standards
Self assessment and reflection
Delivers high-quality feedback
Encourage dialogue around learning
Feedback and motivation
Closing the gap
Feedback shaping teaching
4.14 Reliability and Validity
Validity is commonly understood to mean that a certain measurement is able to measure what it is supposed or intended to measure whereas reliability denotes the extent to which a particular measurement instrument is capable of rendering similar results on a continual basis (Thorndike, 1991).
4.15 Statistical Analysis
The tools to be used are
Multivariate Central Limit Theorem
Using Excel to enter all the data collected during the survey. For instance to determine the number of correct answer, the following code was used
=COUNTIF(O2:083=11), where 11: option A and a correct answer
=(MOD(O2,10)+(MOD(P2,10))+...+(MOD(S2,10)) means total score for continuity correct.
For Multivariate Central Limit Theorem, equation 2.6, the following codes were used
The description of the sample, the instruments to be used and several points have been discussed in this chapter. The next chapter will deal with the chapter analysis of data collected and findings.