GAINING ACCESS AND LOCATION OF RESEARCH SITES

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CHAPTER 3 RESEARCH DESIGN AND METHODOLOGY

Introduction

This chapter provides an outline of the research design, methods used for data collection and the methodical frame that guided data analysis. The chapter further discusses issues pertaining to rigour in research to ensure that factors affecting validity and reliability were minimized. The chapter concludes by discussing ethical issues that were considered before data collection. The context of the practical arrangements made with research participants and the practical decisions made regarding interpreting and synthesizing the data are also described. The classroom arrangements and interpretive decisions made created an opportunity to see mathematics classrooms in a way not seen before. A strong feature of the approach used is the decision for me as a researcher to work closely and listen to the mathematics teachers and immigrant learners.
The study was guided by the following research questions:

1. How do teachers create opportunities for the participation of immigrant learners when learning linear programming?
2. How are languages within the immediate environment of immigrant learners used during teaching and learning linear programming?
3. How do immigrant learners take up these learning opportunities when solving a linear programming task?

Research approach

This was an empirical case study which used qualitative methods of data collection and analysis. Merriam (1998: 19) defines a case study as “an intensive description and analysis …of a bounded system such as an individual, a program…” She offers examples of such bounded phenomena in education thus, “a program, an event, a person, a process, an institution or a social group”. Hitchcock and Hughes (1995) argue that in a case study an individual’s or social group’s constructions are elicited by interactive dialogue between the researcher and the participants. Opie (2004) elaborates this point further by arguing that a case study can be viewed as an in-depth study of interactions of a single instance in an enclosed system.
The main focus of a case study is to organize social data for the purpose of viewing social reality in order to gain a wealth of detailed information on a small sample size (Patton, 2002). Therefore, the purpose of the in-depth study of interactions reported in this thesis was to gain insight into the opportunities created by teachers for the participation of immigrant learners when learning linear programming in a Grade eleven mathematics classroom. As will be reported, the concern was the meaning in context of how teachers created opportunities as well as how immigrant learners take up these opportunities in a country that has been reported to be hostile to foreigners.

Different types of case studies

Yin  (1994)   identifies  three  types  of  case studies,  all  of  which  seem  to  form  important components of my study. He identifies them in terms of their end product as exploratory, descriptive and explanatory. According to these identifications, exploratory case studies have been used as pilots, mainly to generate further research questions or try out data collection methods (Yin, 2009). Descriptive case studies are aimed at giving a narrative account of social situation (Yin, 2009). The approach is able to generate a lot of detail; however it is low in theory building opportunities making it less suitable for some research studies. The explanatory case study tends to be used either to generate a new theory or test an existing one.
Another variation in case studies also comes from Stake (1994) who classifies case studies into intrinsic and instrumental. Intrinsic case studies are considered suitable when intentions are to understand the particular case in question whilst the latter is used when the intentions are to examine a particular case to gain insights into a certain issue. Stake (1994) also gave a third category which involves a study of multiple cases. But all these classifications highlight some overlap and the implication being that, drawing from more than one model would be an advantage.

Difference in case studies

A case study can follow a determined study of a single case or be a multiple case study project, which is a collection of single case studies (Yin, 2009; Stake, 1994). In the study described in this thesis, multiple case studies have been considered a more relevant methodology than a single case study. This was mainly because I considered the diversity of school situations due to their locations. Multiple case study designs have distinct advantages suitable for such situations. This is so because it allows for investigations of what is particular to individual persons, to individual classrooms or individual schools. Differing views are permitted as they lead to multiple realities that become visible in each of the case studies. In this study, I identified, explored and described issues and existing practices with the intent of providing a link between current literatures on teaching mathematics to immigrant learners. The evidence from multiple cases is often considered more persuasive and considerable (Yin, 2009) than from single cases.

Why use a case study?

The case study approach was used because it is flexible and adaptable. It involves the researcher participating openly in people’s daily lives for an extended period of time, gathering evidence that may be available to explain the issues under study. The data may be gathered by watching what happens, listening to what is said and / or asking questions through informal and formal interviews (Hammersley & Atkinson, 2007). The basic premise for the method is that people (e.g. teachers and immigrant learners) do not act in isolation. Their behavior and actions are shaped by social groups, cultures and institutions (Gee, 2005a) and therefore may be studied in their natural settings.
Therefore, the case study offered opportunities to take account of life experiences of teachers and immigrant learners while focusing on their schools and classroom processes. The method allows people to tell stories about their situations to others in order to make sense of the worlds they inhabit (Yin, 2009). Furthermore, it is more focused and intensive as it is conducted in clearly bounded environments (Denzin & Lincoln, 2000, 2006).
According to Denzin & Lincoln (2006), a case may be simple and straight forward or may be complex and extended, but its centrality lies in the focus and emphasis upon the clearly bounded and unique nature of a setting having some kind of internal coherence. The core for its identity rests in its notion of unity or overall that bounds the system like a school and it is a part of this unity that is studied. According to Opie (2004: 74):
…its aim then is to provide a picture of a certain feature of social behavior or activity in a particular setting and the factors influencing this situation. In this way the interactions of events, human relationships and other factors are studied in a unique location.
It is clear here that the interest in case studies is on the activities of the case and not on generalising the results across teachers and immigrant learners in South Africa. Hitchcock & Hughes (1995) claim that a case study is handy in that it not only gives an opportunity to look at the common, widespread and general, but it also allows for attempts to identify anything unique or particular to an individual teacher or immigrant learner. It is not a problem accomplishing these in case studies as it allows for employing a variety of data from a variety of sources.
These features allowed me to ask important questions during clinical interview with immigrant learners as well as when interviewing the teachers. They also allowed intensive analysis (Cohen Manion, 1994) of data in that they provided an understanding of the opportunities created for the participation of immigrant learners when learning linear programming.

 Identifying boundaries of the study

Identification of the case meant locating the boundaries, schools, teachers and immigrant learners learning linear programming in a Grade eleven mathematics classroom. This involved a purposeful sampling strategy which I discuss in the next section.

Sampling

Patton, (2002: 70) defines purposeful sampling as “a strategy in which particular settings, persons, or events are selected deliberately in order to provide important information that can’t be gotten anywhere from other choices”. In support, Merriam (1998) emphasizes that this purposive sampling strategy is used when the researcher wants to discover, understand, and gain insight into a phenomenon (for example immigrant learners learning linear programming). The advantage of purposeful sampling is that a researcher handpicks the cases to be included on the basis of his or her judgment of their typicality (Cohen & Manion 1994). Patton (1990: 70) develops the argument as follows, “the logic and power of purposeful sampling lies in selecting the information rich cases for study in depth. Information rich cases are those from which one can learn a great deal about the issues of central importance”.
In this study, purposeful sampling strategy was employed because of the kind of data that was to be collected, the data collection instruments used, and also due to the data sources which did not lend themselves to random sampling. So in line with my original interest in how teachers create opportunities for the participation of immigrant learners, it was necessary to observe mathematics classrooms where this occurred. Thus, the main consideration centered on choosing schools which enroll immigrants.

 The selected cases

Merriam (1998) argues that in order to use purposive sampling one must decide the selection criteria. Cohen, Manion & Morrison (2000) broaden this argument further by pointing out that the quality of any research writing may be affected by the precision of sampling strategy employed. I designed a selection criterion that was to assist me in hand picking the ‘information rich case’.
The criteria entailed were: schools enrolling immigrant learners in an urban, township and rural environments; teachers teaching linear programming to Grade eleven learners and teachers willing to participate in the study. Grade eleven was included because; according to the Department of Education document (2003) linear programming is introduced at this level.
Three schools were involved in the study. Each was carefully selected so that it could predict similar results often called a literal replication or produces contrasting results but for expected reasons. I purposely selected the three schools drawn from three different multilingual contexts, three mathematics teachers from these schools and their Grade eleven classrooms with immigrant learners learning linear programming. The number of teachers and schools participating in this study is small. I wanted to select a manageable group of cases such that each case would illustrate the diversity of the different context in which the schools are located and yet show the need to examine the complex nature of creating learning opportunities for the participation of immigrant learners in multilingual classrooms in South Africa. These schools were identified through a process which is discussed in the next chapter (Chapter 4).
The sample is opportunistic and purposive in that I was looking for immigrant learners who might have first enrolled at their country of origin and entered school in any of the grades in South Africa. My interest was to gain insight into the opportunities created by teachers for the participation of immigrant learners in a country whose citizens have been reported to be hostile to foreigners. Reading from literatures (Landau & Kabwe-Segatti, 2009; Crush & Williams, 2003; Landau, 2007), I have become aware that some of these immigrant learners or teachers are from French, Portuguese and Spanish speaking countries. Furthermore, they cannot speak nor understand some of the official languages of South Africa. So I wanted to explore the resources the teachers recruit when engaged in the teaching of linear programming to this particular group of immigrant learners. However, the main purpose of the study was not to evaluate the teaching but to learn from it by making analytic judgments of teachers creating learning opportunities for the participation of immigrant learners when teaching linear programming.
The three selected schools were such that: one is located in an urban environment in central Johannesburg neighborhoods, which I will refer to as School A, the other school in a township environment located west of Johannesburg, School B and the third school located in a rural environment in the Limpopo province, School C in this thesis. The mathematics teachers from these schools are experienced and were willing to participate in the study. I was fortunate to work with teachers at these schools who are highly qualified and competent. They all have more than eight years teaching mathematics at secondary school level. Two are immigrants, one from the DRC teaching at School A and has been teaching in South Africa for more than ten years. The other teacher teaching at School C is from Zimbabwe, she started teaching in South Africa in 2010 while the third teacher is a South African citizen teaching at School B.

ABSTRACT
ACKNOWLEDGEMENTS 
CHAPTER 1 
1.1 Introduction
1.2 Establishing a focus
1.3 Selecting a suitable theoretical framework
1.4 Migration to South Africa .
1.5 My experience
1.6 Language policy in South Africa
1.7 Language in education policy (LiEP) in South Africa
1.8 Multilingualism in the global era
1.8.1 Multilingual classroom settings in South Africa
1.9 Conclusion
1.10 Structure of the report
CHAPTER 2 LITERATURE REVIEW AND THE IMPORTANT THEORETICAL ASSUMPTION
2.1 Introduction
2.2.0 Literature review
2.3 Strategies to support multilingual learners when learning mathematics
2.4 Challenges in multilingual classrooms with immigrant learners
2.5 The role of policy
2.6 Underlying principle for the study
2.7 Theoretical framework.
2.8 Linear programming
2.9 Conclusion
CHAPTER 3 RESEARCH DESIGN AND METHODOLOGY
3.1 Introduction
3.2 Research approach
3.3 Qualitative methods of data collection
3.4 Advantages and disadvantages of the methods of data collection
3.5 Piloting instruments
3.6 Data analysis
3.7 Rigor in qualitative/case study research
3.8 Ethical considerations
3.9 Conclusion
CHAPTER 4 GAINING ACCESS AND LOCATION OF RESEARCH SITES
4.1 Introduction
4.2 Gaining access
4.3 Immigrants in the rainbow nation
4.4 Locating the schools
4.5 The schools
4.6 Languages represented in Classrooms A, B and C
4.7 Conclusion
CHAPTER 5 TEACHING IN TWO ADDITIONAL LANGUAGES: FOCUS ON CLASSROOM A
5.1 Introduction
5.2 Teaching in Classroom A – urban environment
5.3 Overview of the linear programming lessons
5.4 Teacher’s language practices in Classroom A
5.5 Analysis of the teaching in Classroom A
5.6 Teacher’s Discourses in Classroom A
5.7 Conclusion
CHAPTER 6 TEACHING IN MANY LANGUAGES: FOCUS ON CLASSROOM B
6.1 Introduction
6.2 Overview of the linear programming lessons
6.3 Teachers’ language practices in Classroom B
6.4 Analysis of the teaching in Classroom B
6.5 Teacher’s Discourses in Classroom B
CHAPTER 7 TEACHING IN ENGLISH ONLY: FOCUS ON CLASSROOM C 
7.1 Introduction
7.3 Teacher’s language practices in classroom C
7.4 Analysis of the teaching in Classroom C
7.5 Teacher’s Discourses in Classroom C
7.6 Conclusion
CHAPTER 8 IMMIGRANT LEARNERS SOLVING A LINEAR PROGRAMMING TASK
8.1 Introduction
8.2 Framework of the discourse analysis .
8.3 Identifying linear programming content to be analysed
8.4 Framework of the analysis of the immigrant learners’ activities
8.5 John’s background
8.6 Analysis of the learners’ tasks
8.7 Conclusion
CHAPTER 9 SO WHAT AND WHERE TO FROM HERE? 
9.1 Introduction
9.2 An overview of the journey .
9.3 Summary of findings -teachers creating learning opportunities
9.4 The contribution of this study
9.5 Recommendations to mathematics education
9.5 Methodological reflections
9.6 Limitations of the study
9. 7 Conclusion
REFERENCES 
APPENDICES
GET THE COMPLETE PROJECT
IMMIGRANT LEARNERS LEARNING LINEAR PROGRAMMING IN MULTILINGUAL CLASSROOMS IN SOUTH AFRICA