In the present chapter, details of the different stages involved in carrying out the case study will be given, including the design of the case study, the sample used and the data-collection methods applied in order to illuminate the above research questions. Also, ethical considerations were also taken care of in the current study.
3.2 The Case Study
A case study, according to Robert Yin (2003), can be described as a "story about something unique, special, or interesting". This something can either be an individual, an organisation, a process, a program, a neighbourhood, an institution or even, an event.
3.2.1 Elements of the Case Study
The elements of a case study are not fixed as each case study is different from each other. Hence, depending on the chosen story, the elements will vary as the data to be collected and the purpose of the case study are not the same. However, case studies normally describe a program or intervention put in place to address a particular problem. Neale et al. (2006, p.7) provided the following elements for a case study:
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
3.2.2 Presenting a Case Study
Case studies are said to be flexible as they can be presented in a number of ways. In fact, there is no definite format to follow. Still, 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 just like for all evaluation results. A case study can be a stand-alone report or can be included in a larger evaluation report. Neale et al. (2006, pp.10-11) suggested the following outline report whenever a case study is presented as a stand-alone report:
How was the process carried out? (Describe the process of selecting the case and data collection sources, as well as how data was collected.)
What assumptions are there (if any)?
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
3.3 Sample Description
The researcher conducted the study in his place work which is a private fee-paying secondary school. Since the chapter 'Logarithm' is taught at Form IV level, the researcher conducted his study with the Form IV students, including boys as well as girls, in order to suit his purpose of the study. The sample consisted of 16 boys and 8 girls of mixed abilities, that is, they are low-, regular- or high-achieving students, coming from different social background. The age group of the sample is between 15 and 16 years old. In fact, this particular group of students was chosen as it would help the researcher to cater for the diverse needs in learning and sharpen the focus on individual strengths and any short-comings in the cognitive domain. It is worth pointing out that, previously in Form III, the mathematics classes were streamed, that is, the best students were in Group I and the low-achievers were in Group II, and not all students in the sample were formerly in Group I.
3.4 Assumptions of the Study
The researcher assumed the following while carrying out this case study:
The sample was representative and adequate, that is, the sample had more or less the same characteristics as that of the whole population.
The case study did not have any negative impact on the students.
Instrumentation refers to the process of collecting data at an appropriate time for the purpose of undertaking a study. Data consists of facts and findings obtained by the researcher from the individuals under study. Data is normally collected by the help of different tools known as instruments. Some commonly used instruments are questionnaires, interviews and standardized tests among others. In this study, the researcher has made use of the following instruments:
The researcher has opted for a triangulation method, that is, the use of more than one data collection method. Triangulation is when "the strengths of one method offset the weaknesses of the other, so that together, they provide a more comprehensive set of data. To the extent that the results from each method converge and indicate the same result, there is triangulation and thus greater credibility in the findings" (McMillan and Schumacher, 2006, p.6).
Observation is often considered to be the best technique in determining the usability requirement of those being studied. The researcher has opted to be a participant observer since he is carrying out the study with his students. Ethical considerations were taken in to account before starting the observation. The researcher explained his students about the study being undertaken, its aim and objectives, and lastly asked for their approval to carry out the observation process. The researcher afterwards noted down all the problems faced by the students and necessary remedial was applied.
3.5.2 Standardised Test
The researcher chose to give the standardised test in the form of an aptitude test whereby the goals were, primo, to measure the ability of the participants to solve problems directly related to the laws of logarithm and secondo, to measure the ability of the participants to apply their knowledge of laws of logarithm in different situations. This data collection tool was chosen for the following reasons:
The fact that it is standardised renders it to provide the same stimulus, that is, the same problems are given to all the participants in the sample.
It possesses a strong psychometric property. In other words, standardised tests provide high measurement validity.
Analysis of data is easy due to the quantitative nature of the data obtained.
The test can be administered to a group. Hence, it helps save time.
The cost of producing standardised test is low compared to other data collection tools such as questionnaires.
18.104.22.168 Method of Administering the Standardised Test
Since the researcher was in fact the Additional Mathematics teacher of the boys target sample, the standardised test was administered during the class period. In fact, after being granted permission by the Rector of the school to carry out the research, this standardised was given as an assessment for the third term of the school calendar. Hence, for the good conduct of the research, this ensures that the students would give their best when taking the test. At the same time, this guarantees the researcher to have reliable and valid data. For the girls target sample, the standardised test was administered with the help of one of the researcher's colleague in the girls' department.
22.214.171.124 Ethical Issues
Throughout the course of the study, great care was taken not to lack respect to any of those involved, be it the school, the management of the school, the target sample or anyone who were directly or indirectly related to the research work. First, permission was sought to the Performance Monitoring Office (PMO) - who are responsible for monitoring the performance of students at the school - and to the Deputy Rector by the researcher to carry out the standardised test as an assessment for the third term. Hence, the researcher was able to keep the scripts for the purpose of his study.
Furthermore, before carrying out the test, students were made aware of the aims, objectives and purpose of this research. It was made clear to them that the data received would be kept strictly confidential and would be used solely for the purpose of the study. The students were also informed about the fact that participating in the study is only voluntary.
As for administering the standardised test to the girls target sample, the researcher respected the school's philosophy of segregation of sexes and so, arrangement was done with a female colleague from the Mathematics department to undertake the test with the girls.
126.96.36.199 Justification of Questions in the Standardised Test
Each question in the standardised test has been carefully and purposefully been chosen based on the research being undertaken, that is, they will help the researcher to answer the research questions. The table below gives the source of each question and the reason for choosing the particular question:
Source of Question
Reason for Choosing Question
Cambridge O-level Additional Mathematics (June 2009 Paper 2 No. 7)
To test for the cognitive of students ability in recalling and applying the laws of logarithm for simple manipulation. Furthermore, the Cambridge O-level report (2009) states that "Answers to this question ranged from those with complete accuracy to those with muddled ideas on combining logarithms."
To test for the cognitive of students ability in recalling and applying the laws of logarithm for evaluating logarithm without the use of calculators.
Cambridge O-level Additional Mathematics (November 2008 Paper 2 No. 5(ii))
To test for the problem-solving ability of students to solve equations involving logarithm to base . As for the Cambridge O-level report (2008), it states that "Many candidates made no attempt to answer part (ii), but some who did made little progress, writing , and others did not use ."
Cambridge O-level Additional Mathematics (November 2010 Paper 13 No. 5)
To test for the computational skill of students in formulating two linear equations from two logarithmic equations with two unknowns. Furthermore, the logical reasoning ability of students was also required in applying the conditions for a logarithm to exist. The Cambridge O-level report (2010) states that: "Some candidates were quite comfortable with logarithms but many would benefit from more practice on this topicâ€¦.very few realised that a = 1, b = -1 was not a possible solution."
Cambridge O-level Additional Mathematics (November 2011 Paper 12 No. 3)
To test for the translating ability of students between words and mathematics so as formulate an appropriate quadratic equation in three terms. Also, the Cambridge O-level report (2011) states that "Most candidates were able to obtain the correct result which, as stated in the second part of the question, was meant to help candidates. In spite of obtaining a correct result for part (i), few candidates were able to use the given substitution correctly and were thus unable to obtain the intended three term quadratic equation."
Cambridge O-level Additional Mathematics (November 2010 Paper 23 No. 12 (i),(ii) OR)
To test the ability of students to represent data spatially and thereof formulate an appropriate equation involving two variables. With regards to the Cambridge O-level reports, it states that "(i) Points were usually plotted with some degree of accuracy. Some graphs were spoilt by one poor point leading to a line of best fit which caused accuracy problems later. (ii) The accuracy depended greatly on the quality of the graph with candidates often avoiding reading errors by using tabulated values. The better candidate often jumped in one step from an equation linking to to a correct equation involving and . Some candidates stopped after the first step and need to reflect on the wording which implies an answer of the form was required. A significant number incorrectly wrote an answer in this form which should have been the form."
Frey and Oishi (1995, p.1) define an interview as "a purposeful conversation in which one person asks prepared questions (interviewer) and another answers them (respondent)". The researcher opted to use a one-on-one interview style which was semi-structured. In other words, a set of five questions was prepared beforehand but at the same time further spontaneous questions were asked based on the respondent's answer in order to obtain clearer answer or to ask for further elaboration of the answer. The set of questions were:
What was your first impression about logarithm?
What teaching styles were used by the teacher in explaining the different parts of logarithm?
What learning techniques did you use?
What difficulties did you encounter and what difficulties do you still face?
What proposals would you suggest to make the teaching and learning of logarithm to be easier?
188.8.131.52 Reasons for Choosing Interview
The researcher opted for interview for the following reasons:
Interviews are said to be flexible, that is, "if a respondent has misunderstood a question, the interviewer can clarify, thereby obtaining relevant responses" (Babbie, 1992).
The researcher while conducting his interviews in private ensures that the respondents will give unbiased answers as there is no consultation with their peers.
The response rate for interviews is high relative to other data collection tools such as questionnaire.
After conducting the first interview, necessary changes can be made to the interview schedule.
184.108.40.206 Justification of Questions in the Interview
Just like for the standardised test, questions were also meticulously chosen for the needs of the study. The following reasons were behind the choice of the questions:
As it is a well-known fact, the first impression is often long-lasting and can only be changed by appropriate behaviours. So, the researcher was hoping to relate this first impression with the learning experience of the respondent regarding the topic logarithm.
Research has shown that different teaching styles deployed by the educator affect different students in different ways (Wetzel et al., 1982, p.34). Hence, the researcher is hoping to find a relationship between the teacher style used and the problems encountered by the students in the learning of logarithm.
Each student has a different ability in relation to leaning. However, a student may, knowingly or unknowingly, make use of different learning strategies even though some would argue that one particular learning style would be dominating. Generally speaking, Wetzel et al. (1982, p.33) argues that students can be classified as a "dependent learner, an independent learner, or a collaborative learner".
This question will partly help the researcher to achieve his objective of investigating the difficulties students have. The first part of the question tries to find out the initial problems students were facing and how did they remedy to the problems and the second part ought to find out the problems that students are still facing after the completion of the chapter.
The researcher is trying to look for the creative and imaginative ideas of the students in enhancing the teaching and learning experience in relation to logarithm.
220.127.116.11 Reliability and Validity in the Interview
For the quest for validity and reliability in his collected data, the researcher has preferred to carry out his interview one-on-one with the respondents and the same questions was asked to each as under the influence of their peers, the respondents may give biased information.
3.6 Statistical Analysis
For the purpose of analysing the collected data, various statistical tools, such as descriptive statistical tools, were used to display the results and thereof, conclusions were made.