Study population and sample size

The study population refers to the larger representative group from which the study participants are to be selected (sampled) on the basis of their homogeneous characteristics, qualities, or attributes (De Vos et al., 2011). The core of the study population in this research consisted mainly of grade 11 learners at four high schools of the Mthatha district. In each school, 30 volunteering participants were invited to take part in the research, but only 112 learners from four secondary schools took part in the test as eight learners decided to withdraw in the middle of data collection. They felt the questions asked were too difficult for them. The test was conducted in the very schools the learners attended. The researcher chose their schools because they were familiar to learners. Such an environment was conducive for the learners to express themselves freely. The test was written under examination conditions, and was of 50 minutes’ duration. The researcher invigilated the test in all the four (4) schools. The answer sheets were marked soon after the test, and 10 learners were selected for the task-based interview from each school. The researcher then gave general feedback to the Head of the Mathematics department at the school on learners’ areas of strength and weakness.

4.3.1.1 Sample size and sampling method (strategy/ technique)

In total, 112 grade 11 learners were selected from the four Mthatha high schools for participation in the empirical (interview- and test-based) aspects of the study.

The researcher used the purposive sampling technique in this study on the basis of the selected research participants’ possession of similar (homogeneous) representative qualities with the study population (Saunders et al., 2012). Purposive sampling is a sampling technique according to which the researcher relies on his/ her judgement and knowledge or maximum familiarity with the research milieu (environment or setting) to select the relevant research participants (Black, 2010). The other advantage of purposive sampling is that it is cost and time effective as judgement on selection is usually based on previously available data (Black, 2010).

Most of the learners do not perform well in functions and English is not their first language. They learn English at school as a second language, therefore selection focused on learners who were willing to participate in the study voluntarily. Learners who are not perfoming well in Mathematics tend not to answer questions accordingly, and the researcher did not wish to risk having many blank spaces in the test-based tasks.

57 4.3.2 Sampling criteria

As a concept, ‘sampling criteria’ refers to the researcher’s own predetermined range of considerations or norm (standard) according to which the selected research participants were regarded as either relevant or unsuitable for inclusion in the empirical aspect of the research (De Vos et al., 2011; Kumar, 2012).

In the selection of learners for the study’s task-based interviews, the researcher presented the names of learners who would take part in the interviews. Learners then discussed among themselves and recommended names of six volunteers for the next stage of the data collection. The other four (4) learners were placed as possible substitutes in the event that one of the selected participants did not come or decided to withdraw from participation. In each of the four schools, no selected participant withdrew from the task-based interviews, and three pairs of learners participated in the task-based interviews at their school after the designated teaching hours. Prior to the interviews, the researcher explained the expected level and nature of participation to the learners (Creswell, 2014; Saunders et al., 2014). Learners were also allowed to write some additional notes and bring them with if they desired to do so, in addition to being allowed to talk during the recorded interview sesssions. Only a pair was interviewed in the presence of the researcher.

The best performing 30 learners in grade 11 were selected from each of the four schools. The researcher and teachers in each school used the average of the marks obtained at the time of the study to select the best 40 learners to participate in the test. The researcher earmarked 30 learners and the other 10 (ten) were placed as substitutes in the case of learners who were not willing to participate in the study. In total, 112 learners participated in the test.

The researcher further selected 6 (six) best performing learners from each of the participating 4 (four) schools. These 24 best performers took part in a paired task-based interview. Only one selected learner dropped out because he changed schools, but was replaced by the learner ranked number seven at his school. The characteristics of these learners were such that they were the best in each school. In school A, the selected learners’ marks ranged from 40% to 60%. In school B, the top six learners’ marks ranged from 75% to 80%. In school C, the marks ranged from 85% to 90% for the best six; and at school D, the best six learners’ marks ranged from 65% to 80%. Although 24 research participants were the best 6 from each school, their marks in the test ranged from 40% to 90%. Since this was a very wide range, the researcher did not expect their mathematical discourse to be the same. At face value, their mathematical discourse looks the same, but a closer analysis using the DPHEF exposes some differences that are discussed in the data analysis section below (Section 4.4).

58 4.4 Data Management and Analysis

Data management focuses on the treatment or preparation of the collected raw data and its preservation from contamination, thus ensuring its practical usability and translation for the resolution of the research problem; as well as for the maximum achievement of the stated research objectives (De Vos et al., 2011). The management and analysis of data are two sequentially and symbiotically interconnected processes (Marshall & Rossman, 2011). The latter authors allude to this inter-connectedness thus: “Data collection and analysis thus typically go hand in hand in order to build a coherent interpretation of the data. The researcher is guided by initial concepts and developing understanding, but shifts or modifies them as he collects and analyses the data” (p. 18). Therefore, despite their concurrence, data management precedes data analysis (Holloway & Wheeler, 2010; Marshall & Rossman, 2011). Following the treatment/ preparation and analysis of the qualitative and quantitative raw data, it was backed digitally with a master copy and kept in a secured place. Only the researcher has password access to the digitally secured information files. Hard copies are also kept safely at a secure location, where they will be shredded after a period of five years. In this regard, data management reflects and ensures a standardised monitoring and evaluation quality assurance mechanism.

In this study, the researcher personally collected the answer sheets that learners used during the interviews as evidence of their participation. Teachers in the selected four schools were only involved in ensuring the availability of fully furnished and conducive venues, and assisted with coordinating learners for tests and interviews. The data collection times differed from school to school. For instance, in some schools data was collected after official school hours when other learners had left for home. In other schools, data was collected during the study time after the daily lessons had ended. For effective quality assurance purposes of the interview, the teacher ensured that he was not drawn to into superfluous probing. The latter could inadvertently turn the interviews into teaching sessions and defeat the goal of these interview sessions.

Twenty (20) of the participants were video recorded with their informed consent, while four (4) participants did not agree to video recording; instead, they were only audio-recorded. The advantages of video recording was that it captured the original/ authenticated voice of the learners, their actions, their demeanour; as well as their verbal (e.g. talking) and non-verbal communication (e.g. writing, gestures and facial expressions). The researcher has realised that the video recording may have affected the manner in which some learners responded to the questions. Some of the learners were too aware of the presence of the recordings that happened. The four learners who opted for audio recording only did so as they were not comfortable with

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their faces being recorded. For ethical reasons, both the video and audio recordings were strictly for the purpose of this study, and not for viewing by unauthorised persons.