To answer the first critical question of “What mathematical problems or challenges does the teacher encounter and how the teacher engages with these problems and challenges as he implements a section in geometry”. What is of great concern is how the teacher teaches the section on polygons in particular, quadrilaterals in grade 10 to meet the requirements of the National Curriculum Statement? This was ascertained via pre and post semi-structured interviews, classroom observations and field notes. A general discussion follows about the
16 Mr. Ken is a pseudonym that refers to the teacher and all names used for the learners are also pseudonyms.
17 Third term: according to the public school calendar the school year is divided into four terms.
data collection techniques employed and the reasons for the range of techniques that were employed.
3.3.1. Interviews:
An initial and post semi-structured interviews (see appendix A, B) were conducted once to establish, from the perspective of the teacher, why certain tasks were chosen, the purposes they intended to serve and how the teacher intended to use the tasks in class to teach geometry. According to Maykut and Morehouse (1994), the interview is a form of discourse that is shaped and organized by asking and answering questions.
A semi-structured interview was most appropriate as there are clear areas of focus and concern for me, which allowed for a rich discussion of thoughts and feelings. At the same time, I needed to be open to interpretations and comments from the teacher that might not have been anticipated. The teacher was an active participant in the interviews, that is, the teacher was given a voice, since the teacher has his own ideas, feeling, insights, expectations and attitudes.
The interviews consisted of broad, open questions that allowed me the opportunity to investigate, explore, probe and develop a conversation with the teacher. That is, I was able to probe the teacher’s verbal responses to the approaches he used in choosing mathematical tasks and the mathematical work he did when teaching polygons in particular, quadrilateral related tasks. The interview was also aimed at investigating how the teacher worked with learners’ errors (common errors) and misconceptions that surface during teaching; how the teacher was able to explain the concepts he intends learners to understand and what he hoped learners would be able to do in order to complete and learn from the set tasks. The interviews also explored what the teacher knew about his learners in relation to the tasks prepared for the lessons, which illuminated the second research question “what knowledge resources (mathematical and other) does the teacher call on as he goes about this pedagogical work to enhance geometric development?” The teacher could say what he was thinking and why he chose these particular tasks that contributes to the richness and spontaneity (Oppenheim, 1966) of the data. Each response generated more information, particularly as the teacher was encouraged to elaborate on his ability to cope with the demands from the learners, his positive and negative feelings towards the new curriculum and his expectations and fears.
Thus before the implementation of the first lesson and after the implementation of the last lesson, reflective semi–structured interviews were conducted with the teacher, to ascertain his experience of teaching, the challenges he faced and so the problems he might have confronted. The reflective pre-interview schedule (Appendix A) was designed before the observation of the lessons took place and the types of questions that are included in this interview schedule are informed by the first two critical questions. The post–interview schedule was designed (Appendix B) after the data was collected from the lessons that contributed to the answers of the third critical question “ how does this work and the resources called on in this class, relate to new curriculum goals for mathematical proficiency and how can this relationship be explained?”
3.3.2. Lesson Observation:
In addition to interviews with the teacher, a period of one week was spent with the teacher to observe the lessons in which mathematical work took place. All classroom observations were video taped to capture uninterrupted raw data and the videos were transcribed into full lesson excerpts. The analytical and theoretical framework then informed the choice of which excerpt to use in the analysis and it also had an impact on me wanting to see what happens in the classroom over time. In conjunction field notes and tape recordings was used as back up during the observation phase.
Since the purpose of this study is to find answers to the burning question, the literature that possibly would provide insight into what might be observed for example, Ball et al. (2004), Ball & Bass (2000), Shulman (1986, 1987) and Adler (2005) as well as from observing the teacher when he attempts to fix meaning for his learners. The lesson observation was thus used to illuminate the intended and implemented task so as to be able to relate this to the problem–solving the teacher was doing, and the resources called on for this.
In conclusion, the focus of the interviews was on how the teacher reflected on his teaching, both prior to the first lessons and after observing the last lessons. From the observation I identified, teaching work in action and (at a theoretical level) the recontextualisation of mathematical knowledge into pedagogic communication (Bernstein, 1996). The analysed data was compared and contrasted to previous research evidence. Thus triangulation took place by cross – sectioning (cross checking) the interview data, field notes and the observation data to qualify the results. Three established researches verified the analysis of the data tables by
listening to the lesson transcripts and checking the coding of the data tables. This preliminary analysis informed me how to redesign the critical questions to focus on the central topic. After the post interview a more refined analysis of all the observed data of the lessons took place.