4.9 Data collection procedures
4.9.2 Sampling
In mixed methods research studies, Bronstein and Kovacs (2013) identify three types of samples:
single sample where the same sample is utilised for both quantitative and qualitative segments of the research; single sample with subset in which data from the quantitative component of the study is used to qualitatively investigate another phenomenon; and, more than one sample which describes a mixed methods study that uses one sample for a quantitative component and seeks additional information from a subset of a different sample. In this sequential explanatory study mixed methods, I adopted a single sample with a subset approach in which the single participant in the subsequent qualitative component was drawn from the same larger sample after the completion of the quantitative phase of the study. Specifically, in line with the logic of sequential explanatory designs in which the quantitative component is dominant, after administering surveys to one hundred and thirty five (135) learners at selected Dinaledi schools, I invited an extreme case (one learner) to participate in a semistructured interview based on the survey results.
4.9.2.1 Schools
I selected a sample of Dinaledi schools to administer two survey questionnaires in order to answer the first two quantitative research questions. As Wagner, Kawulich, and Garner (2014) suggest, I randomly surveyed three schools from a population of ten Dinaledi schools (Motshekga, 2015) in the Pinetown school district in KZN, South Africa to accommodate the limited resources available for this study. In cluster sampling, convenient and naturally occurring groups are randomly selected which is followed by a selection of individuals in the groups (McMillan & Schumacher, 2010). This sampling method ensured that the fundamental premises of probability sampling,
Research methodology Data collection procedures
namely, that every of the Dinaledi schools must have an equal chance of being included in the sample, was not violated.
In the pursuit of increasing the participation and performance in Mathematics and Physical Sciences of historically disadvantaged learners, the Department of Basic Education (DBE) established the Dinaledi School Project, in 2001 (Department of Basic Education [DBE], 2009).
The initiative involved selecting certain secondary schools for Dinaledi status that demonstrated their potential for increasing learner participation and performance in mathematics and science (Department of Basic Education [DBE], 2009). These schools were provided with resources (for example, textbooks and laboratories) and other related resources to improve the teaching and learning of mathematics and science. The ultimate intention was to improve mathematics and science results and thus increase the availability of key skills required in the South African economy (Department of Basic Education [DBE], 2009). The rationale for selecting Dinaledi schools for the investigation was that these schools were monitored by a team that included senior education department officials and individuals with an interest in educational research.
However, only three Dinaledi schools were sampled for the main study and the another one accordingly served as a prelude to the main study (Cohen, Manion, & Morrison, 2011). These were public schools with two of them located in a township and the other two in a suburban area. On the one hand, a township is a residential area previously designated for blacks16 and characterised by poor socioeconomic conditions whose schools lack resources (for example, qualified mathematics and science teachers, science and computer laboratories, and sports fields). On the other hand, a suburban school has adequate facilities, teachers and educational opportunities for learners.
16 The use of race as a form of classification and nomenclature in South Africa is still widespread in the academic literature with the four largest race groups being Black (African), Indian, Coloured (mixed-race) and White. This serves a functional (rather than normative) purpose and any other attempt to refer to these population groups would be cumbersome, impractical or inaccurate (Spaull, 2013, p. 437).
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4.9.2.2 Learners
Mathematics learners attending in three schools were subsequently sampled. The sample comprised a total of 135 culturally and linguistically diverse and inclusive Grade 11 mathematics learners (seventy eight female and fifty seven male with an average age of 17.4 years and 17.8 years, respectively). The ages of this group ranged from 15 to 18 years. Although learners were informed of the right of their parents to refuse them participation, all of them participated in the study. In each of the three schools, all the learners were studying mathematics, physical sciences, life orientation and at least four other subjects, including two compulsory official South African languages at first- and second-language level.
Survey data was collected, presented, analysed, and discussed to inform both sampling and the development of the Interview Schedule for the subsequent qualitative phase. The size of the sample needed to be large due to the extent of the heterogeneous nature of the population of Grade 11 learners in Dinaledi schools. That is, a bigger sample was required to draw reasonably accurate inferences in light of variation in characteristics of Grade 11 learners in every respect; namely, language, resources, and gender. A summary of the participants across the Dinaledi schools is shown in Table 4—3. By the time of the research, the sampled learners had finished the prescribed Euclidean geometry.
The choice of a single case was guided by two key considerations: appropriateness and adequacy (Morse & Field, 1995). According to Morse and Field (1995), the former implies the identification of participants who can best inform the study, and the latter relates to adequate sampling of participants so as to address the research questions and develop a full description of the phenomenon being studied. Because I was interested in examining a “successful” participant where successful meant holding informed functional understanding of proof as judged by their high LFUP score, the extreme case sampling strategy was used. The term “case” refers to the single participant who took part in the semistructured interview.
Research methodology Data collection procedures
Table 4—3. Summary of demographic characteristics of the three schools and participants.
School code
Gender Home Language School
Location Total Female
(54.1%)
Male (45.9%)
IsiZulu English
A 22 16 10 28 Suburban 38
B 29 21 46 4 Township 50
C 27 20 36 11 Township 47
In this mixed method design, the qualitative component was subsumed within a primarily quantitative project. The qualitative phase of this study relied exclusively on purposive sampling because I needed that participant whose information was likely to give deeper insight into the factors affecting functional understanding of proof in mathematics. Purposive sampling refers to a sampling technique for the identification and selection information-rich individuals for the most effective use of resources (Patton, 2002). This sampling design involved the selection of a deviant participant for the purpose of learning from an unusual manifestation of functional understanding of proof. Following Guest, Greg, Arwen, Johnson, and Laura’s (2006) evidence-based recommendations regarding nonprobabilistic sample sizes for interviews, a semistructured task- based interview was conducted with a single participant, Presh N, judged to be holding an informed belief about the functions of proof. The single case study was adopted on the rationale that the depth of data collected is more important than recruiting large samples (McMillan &
Schumacher, 2010).