The design of this study

By using the term ‘design’, I mean the ‘framework for the collection and analysis of data’ (Walliman, 2006, p.42), which links the study questions with the data (Roberts-Holmes, 2011). Scholars have proposed several kinds of research design: cross-sectional or survey, experimental, longitudinal and case study (Walliman, 2006; Vaus, 2001). In addition, Bryman (2004) suggests a fifth design, namely, comparative design.

There are various types of design that are suited to the qualitative approach. Robson (2002) identified the ‘flexible design’, which can be changed during the research process and is associated with the qualitative approach; this includes grounded theory, ethnographic and case study designs (Punch, 2000).

For this study I have selected a case study approach. In order to justify my choice, I focused heavily on Yin’s book (2009) Case Study Research in which he elaborates on the case study research design. He states that the case study has a notable strength when, ‘a ‘how’ or ‘why’ question is asked about a contemporary set of events, over which the investigator has little or no control’ (Yin, 2009, p:9). In this research, both how and why questions are used, although other factors affecting the way that teachers deal with their students’ contributions are addressed. Furthermore, it is necessary to find specific details regarding the social situations within the context of mathematics classes in Saudi Arabia. I am therefore addressing the quality of the findings, not seeking generalizations. In the case of this study, the case study is the most appropriate design, according to Walliman (2006).

One of the most frequently used qualitative research methods, which this study used, is the case study (Yazan, 2015). Case is defined by Yin (2014, p:2) as ‘a contemporary phenomenon (the “case”) in its real-world context, especially when the boundaries between phenomenon and context may not be clearly evident’. Stake (1995, p:2) conceptualises case as ‘a specific, a complex, functioning thing,’ more specifically ‘an integrated system’ which ‘has a boundary and working parts’ and ‘in social sciences and human services, [it] is likely to be purposive’. These definitions emphasise the importance of cases being specific and in a real context and they require researchers to pay attention to their boundaries to control the focus of the study.

To design a case study, researchers should consider the five components of a case study: ‘a case study’s questions; its propositions, if any; its unit(s) of analysis; the logic linking the data to the propositions; and the criteria for interpreting the findings’ (Yin, 2014, p:29). After writing the research questions researchers should think about the study propositions and unit of analysis to specify their cases. The research questions help scholars to define their cases (unit of analysis) which could be individuals, groups, organisations, programmes and so on. These units of analysis cannot be studied fully so propositions direct the attention to the things that should be studied in detail.

In this research, to decide what this case study is a case of I thought about the unit of analysis and propositions as, according to Yin (2009), identifying the propositions and the unit of analysis are important components for case studies. As the research focuses on teacher knowledge (proposition) I took this into account and I thought about how the variety of teacher knowledge could affect their responses to the students’ contributions. I believed that the individual teacher was the unit of analysis so I decided to form case studies of trainees with differences in terms of their SMK and PCK.

Once the case and its boundaries have been determined, the type of case study must be considered depending on the overall study purpose. There are several types of case study proposed by scholars. For example, Yin (2014) identifies three types in terms of their outcomes: descriptive, explanatory and exploratory case study. He defines each of them respectively as ‘a case study whose purpose is to described a phenomenon (the “case”) in its real-world context’, ‘a case study whose purpose is to explain how or why some condition came to be’ and ‘a case study whose purpose is to identify the research questions or procedures to be used in a subsequent research study, which might or might not be a case study’ respectively (ibid, p:238). Also, he suggests four types of design for case studies. They include a single holistic design, multiple holistic design (holistic designs require one unit of analysis), single embedded design and multiple embedded design (embedded designs require multiple units of analysis) (ibid, p:50). Stake (1995) uses three terms to describe case studies: intrinsic, instrumental, and collective. He suggests that if the researcher is interested in a particular case and needs to learn about it, not because he/she wants to know about other cases but because he/she is interested in it, he/she can use the intrinsic case study. Also, if the researcher wants to know about a problem in general an instrumental case study can be used by choosing a case then studying it in order to understand the phenomenon, not the particular case. The collective case study is similar to

the instrumental case study but it contains more than one case where, in order to achieve full understanding, each is an instrumental case study.

This research can be considered as a descriptive and explanatory case study. It has some element of description of how trainee teachers deal with their students’ contributions in the real life context and it tries to explain the reasons behind the ways of response. Yin(2014) argues that it is a misconception that case studies are appropriate for the exploratory phase only as they can serve two other purposes: descriptive and explanatory. In addition, this research is of a multiple holistic design (Yin, 2014) as it contains more than one trainee teacher with one unit of analysis (a person). It is also in the form of a collective case study design (Stake, 1995) as it includes many instrumental cases to draw a big picture of the phenomenon under investigation. In this multiple case study design, I examine several cases to understand the similarities and differences between the cases then draw a cross-case analysis to compare them. Each case must be selected so that it either, ‘(a) predicts similar results (a literal replication) or (b) predicts contrasting results but for anticipatable reasons (a theoretical replication)’ (Yin, 2014, p:57). In this study, contrasting results had been predicted between teachers with differences in terms of knowledge which may be considered as a theoretical replication between cases.

This project focuses on case studies of three male trainee mathematics teachers (chosen from five participants the project started with) who are practising in three primary schools in Arrass city during the second term of 2013/2014, and each student teacher forms a case study as outlined in the next section.