Sampling techniques, sample and part

Sampling involves the selection of a number of study units from a defined study population. Fraenkel & Wallen (2006) define a sample as a small group drawn from a population on which information is obtained. Qualitative researchers usually work with small numbers.

Sampling is a process of selecting a group of people with whom to conduct a study (Burns and Grove 2003). The most common sampling methods used in qualitative research are purposive sampling, quota sampling and snowballing. IPA research usually works with purposive sampling because it seeks to obtain insights into particular experiences that occur within a specific location, context and time (Smith et al 2012 & Gray 2009). According to Parahoo (1997), in selecting a sample, researchers use their own judgement to select participants to be included in the study based on their knowledge of the phenomenon. Burns (2000), Miles & Huberman (1994), cited in Gray (2009:182), and Creswell (2008) advise that the best strategy is to initially target those cases that are most likely to yield the richest data or in-depth picture. Smith et al (2012) argues that participants should be selected on the basis that they represent a perspective rather than a population.

3.2.2.1 Sample:

This study was carried out in a Sixth Form College located in a City in England. The college serves approximately 1000 students, 150 of who study mathematics at AS, A2 and Further Mathematics Level. Sampling was theoretically consistent with the qualitative paradigm in general, and with IPA’s orientation in particular. The research sample was obtained from a group of ‘A’ level mathematics students, who volunteered to take part in the research. The topic under investigation was peculiar to a particular group of A Level students, therefore participant selection process considered two criteria. (1) All participants were drawn from volunteers studying GCE Core 3 mathematics module. (2) Participants were also supposed to be available during scheduled workshop sessions. This was to allow continuity and progression in activities. The workshop sessions are slots reserved on the college timetable, outside the normal mathematics timetable. The sessions are intended to offer students an opportunity to get extra help outside classroom time but during the college day.

Due to detailed case-by-case analysis of individual transcripts, IPA studies usually benefit from a concentrated focus on a small number of cases. Smith et al (2012) do not set a limit on the number of cases to be studied at doctoral level research, but however proposes a small number of between four and eight cases. I settled for a sample of six ‘A’ Level participants, selected from classes on my timetable, to develop an insight into: a) how they engaged GeoGebra in their learning of mathematics and (b) how GeoGebra, through its multiple representational facilities, related to and contributed to their whole learning experience?

With permission from the college head and the curriculum leader, I explained my research project and its objectives to all students in my A2 classes from whom I requested for volunteers. Initially eight students volunteered but two withdrew due to clashes on their timetable. Six students took part in the project. I made it clear that the project would be contacted outside their normal timetable. The six participants met as a small group during scheduled workshop sessions under my supervision and worked mostly in pairs using laptops. A sample size of six was also consistent with Holloway and Wheeler (2002:128) who assert that “sample size does not influence the importance

or quality of the study” and note that there are no guidelines in determining sample size in qualitative research. Onwuegbuzie & Leech (2007) suggest that sample sizes in qualitative research should not be so large that it becomes difficult to extract thick, rich data. On the other hand, Flick (2006) suggests that the sample should not be too small so that it becomes difficult to achieve data saturation.

The selected participants had not encountered any work on modulus functions in Core 1 or Core 2 modules. At the time of the study, they had been introduced to the concept of functions and had covered work involving definition of functions, types of functions (even, odd or periodic), domains, ranges, composition of functions and inverse functions and their graphs, including graphs of functions and their inverses. From the Core 1 and Core 2 modules, participants had encountered curve sketching of simple polynomials by identifying x and y intercepts and finding coordinates of the maximum or minimum point (Core 1) and by identifying turning points using calculus (Core 2).

3.2.2.2 Participant profiles:

The names used in these profiles are pseudonyms, for confidentiality, but the profiles are real.

John: John was 17 years old at the time of this study. He had passed AS mathematics with an overall grade C. In the initial interview, John had indicated that he had struggled with curve sketching in Core 1. However, he pointed out that he enjoyed curve sketching when calculus (Core 2) was used to identify stationary points. College records indicated that John had done well in his Core 1 and 2 modules averaging a score of seventy five per cent in both modules but had scored a grade E in the Statistics 1 module. On the functions topic, John indicated that he was still struggling to distinguish between even, odd and periodic functions.

Peter: Peter was18 years old. He had attained an overall grade B in his AS mathematics and was predicted to get a grade B in ‘A’ level mathematics. Peter was also studying Further Pure mathematics (FP 1) module and he was confident with his curve sketching including functions involving rational algebraic fractions, since this work is also covered in more detail in FP 1. Unlike John, classroom discussions and topic

assessment results indicated that Peter had mastered the aspects of functions covered by the time of the study (i.e. types of functions, inverse functions and curve sketching). From my own observations, Peter had no hesitations when it came to participating during lesson discussions.

Susan: Susan was 17 years old. She had attained an overall grade C in her AS mathematics and indicated that she needed some extra help in Core 3 mathematics. She indicated that she was fine with her mathematics during lessons but struggled to apply basic concepts when solving examination related questions. She came across as someone who lacked confidence and most of the time sought for second opinion from friends before answering a question.

James: James was 19 years old. He was on an intensive one year course covering AS and A2 mathematics. James had just joined the college and was doing all six ‘A’ level modules in one year. By the time of this study, James was still trying to come to grips with algebra at Core 1 and Core 2 level. He had been out of formal education for three years. James had already been identified by the department as a student who needed extra help. He lacked confidence and preferred to work on his own most of the time. Emma: Emma was 18 years old. This was Emma’s third year in college having repeated AS mathematics. She had attained an overall grade C on the second attempt of her AS mathematics. Emma claimed that she was comfortable with sketching graphs of algebraic functions from her Core 1 & 2 modules. She was still trying to figure out composition of functions and the types of functions (i.e. odd, even and periodic). Mid- topic assessment also indicated that she could find the inverse of a function but struggled to sketch the graph of the inverse function. Emma did not come across as a shy person. She enjoyed discussions.

Sophie: Sophie was 17 years old. Very confident and predicted a grade A in her A level mathematics. Sophie had attained an overall grade A in her AS mathematics. She enjoyed working on her own most of the time. Sophie had no problems with sketching graphs of algebraic functions. She could recognise links between topics previously learnt and current work. She came across as someone who could think outside the box.

None of the six participants had encountered modulus functions beyond its use in stating the validity of a binomial expansion as |x| < 1. The whole concept of graphs, equations and inequalities involving modulus functions was new to them. All had encountered the software GeoGebra while doing course work on numerical analysis in the Core 3 module a few weeks prior to the start of the research sessions. However, there were some aspects of GeoGebra pertinent to the learning of modulus functions that had to be introduced to them at the start of this study.