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CHAPTER 4 METHODOLGY
Introduction
The study sought to capture and document teachers‟ views regarding contexts and mathematics. The research approach chosen for the study was informed by not only the purpose of the study but also the understanding that generalisability of the envisaged findings was not sought. The importance of eliciting teachers‟ views on the relationship between the intended curriculum and teachers‟ interpretation of the curriculum for implementation regarding mathematics and contexts was viewed as critical and as something that could not be over-emphasised. Noting that it was not only possible but also very likely to encounter teachers that would totally ignore the use of contexts in their teaching and learning of mathematics, other means of gathering this information were employed. Teachers were therefore exposed to situations that required them to identify and recognise the mathematics in the context(s) presented to them. The identification and recognition of the mathematics in the context would have been reflective of only what they regarded as mathematics or non-mathematics in the information they were provided with. Their views on what and how they identified and recognised as mathematics in the contexts they were provided with would have been giving an indication as to how they transited from context(s) to mathematics and vice versa. These matters necessitated and informed the choice of mixed methods as a research design for the study.
Research Design
Mixed methods approach is a method in which both the quantitative and qualitative methods are used to seek answers to the research questions (McMillan & Schumacher, 2010). The approach is necessitated when the sole use of either the quantitative or the qualitative method is inadequate to provide plausible data to fulfill the goal or the purpose of the study. In the current study, data were gathered from teachers using a survey on how much mathematics they considered as present in the context(s) they are provided with. In seeking the answers to the questions on how and why they identified and recognised whatever they considered mathematics/non-mathematics in the given context(s), semi-structured interviews were conducted.
Tashakkori and Creswell (2007) indicate that when a researcher collects and analyses data, integrates the findings and draws inferences using both quantitative and qualitative approaches or methods in a single study or a program of enquiry then the researcher is involved in mixed method approach. The combination of elements of the quantitative and qualitative research approaches are used for the broad purposes of breadth and depth of understanding of the phenomenon under investigation (Johnson, Onweugbuzie & Turner, 2007). For a study that sought to use data collected through a survey in order to gain a deeper understanding of the participants‟ responses by conducting interviews so that inferences can be drawn on the two sets of data, it is apparent that a mixed method approach was appropriate.
McMillan and Schumacher (2010) indicate that mixed methods are also appropriate when there are individuals or a small group whose thinking differs significantly from that of the majority. The methods are able to provide insights into the convergence of philosophies, viewpoints, traditions, methods and conclusions which otherwise would have been presented as separate had a either quantitative or qualitative approach been used independently. The insight gained from this approach enhances and clarifies the observed results. An important advantage of using a mixed methods approach is that it can elicit results (quantitative) which can then be explained in terms of how and why these results are obtained (qualitative).
The use of mixed methods is supported by Yin (2006) who indicates that the mixing of „group‟ designs and „single subject‟ research can produce a powerful, single study of human behaviour. Arguing for the freedom from viewing mixed methods only in terms of the qualitative-quantitative dichotomy, he points out that variants exist within specific research methods such as in-person and mail surveys, ethnographic and case study fieldwork, and laboratory and „natural‟ experiments. He further highlights that mixed method can exist whenever a single study includes method within or between different research designs.
Ivankova, Creswell and Stick (2006) indicate that there is approximately forty mixed-methods research designs reported in the literature. They further say that the six designs that are highly popular and most frequently used by researchers out of these are categorised into two. The first three are called concurrent and other three are called sequential. The design that was used in this study is called the mixed-methods sequential explanatory design. The design is characterised by the collection and analysis of data, first quantitatively and then qualitative in two consecutive phases within one study (Creswell, Plano Clark, Gutmann and Hanson, 2003).
McMillan and Schumacher (2010) point that when methods are used sequentially such that quantitative data are collected first and then using the results to gather qualitative data, elucidate, elaborate on or explain the quantitative findings, then one is involved in an explanatory design. In the current study, the main thrust of sequential explanatory design (see figure 4.1) was quantitative with the qualitative results being secondary. The first phase involved the collection and analysis of the views of the teachers regarding the mathematics that they encountered in the questionnaires and in the second phase two teachers who rated the sentences either highly or lowly were purposefully selected for further investigation. One of the teachers was selected incidentally in that she availed herself to be interviewed for the study. The teachers were interviewed using qualitative methods in order to determine how and why they elicited such views and ratings in the questionnaires.
The qualitative approach of the study was informed by its exploratory and interpretative nature. Leedy (1997) describes a qualitative study as an inquiry process of understanding a social or human problem that is conducted in a natural setting and is based on building a complex, holistic picture by forming words that report detailed views of informants. The study was not intended to look for the generalisation of the results but to highlight, through in-depth exploration, how teachers, when confronted with situations where they are expected to identify and/or recognise mathematics in the contexts they are provided with, actualise this process.
The quantitative element of the study took the form of inferential statistics using data collected through a descriptive survey design. According to Cohen, Manion and Morrison (2011) descriptive statistics visual techniques are used to analyse and interpret presented data. In the current study, the data were presented as frequencies, percentages, cross tabulations, bar charts and pie charts (Cohen et al., 2011). Additionally, the Pearson chi-square test was used to test for association (Cohen et al., 2011) between the different sections of the questionnaire. Teachers‟ biographical details and views about context in mathematics teaching and learning were associated with how they rated the sentences in section C of the questionnaire. The qualitative element of the study took the form of a case study. According to Opie (2004), a case study is an in-depth study of a single instance, in an enclosed system where certain features of social behaviour or activities in particular settings together with other factors influence the situation.
Creswell and Plano Clark (2007) provide the term embedded design to refer to a study in which one set of data is used as supportive or secondary in another set of data, in this case, the qualitative data gathered through interviews were used to inform and supplement the data gathered in the questionnaires of the survey.
Yin (2006) suggests the following five procedures to tighten the use of mixed methods so that it should occur as part of a single study: the research questions, units if analyses, samples for study, instrumentation and data collection and lastly, analytic strategies. The first to consider namely, the research questions, is to ensure that they address both the outcome questions (quantitative) and the process questions (quantitative) in an integrated manner. The research questions in this study covered the “what” (outcomes) and the “how” and “why” (process) of teachers‟ engagement with using context for teaching mathematics The next one namely, the units of analysis refer to the importance for researchers to consistently maintain the same point of reference when it comes to what is analysed. According to Yin (2006), persistent reference to the same unit of analysis creates the much needed force of integration that blends the different methods into a single study. This can also be done by deliberately covering the same question in the different methods. This was done in this study in that same questions were covered in both the survey and the case study. In this way, the responses were integrated into one form of analysis.
Another procedure that this study was to contend with was in ensuring that samples were deliberately nested within the different methods (Yin, 2006). The case studies of three teachers were coming from the sample of teachers that were part of the survey. The procedures of instrumentation and data collection methods (also listed as ones that can enhance the use of mixed methods) were also attended to (Yin, 2006). This entailed ensuring that the different instruments contained direct analogous variables if not the actual items. According to Yin (2006: 44), “the more that the items overlap or complement each other, the more that mixed methods can be part of a single study”.
In the interview phase, the participants were asked to respond to the question of how they rated certain items in the questionnaire and thus they were responding the same question more than once. The creation of direct comparable items ensured that the desired common scope of data collection and observed variables was attained. The cross-walking relationships between the different instruments were, through this process, also established. Cross-walking refers to the connection of one item or construct of an instrument to another item/construct of another instrument. Described by Yin (2006) as the trickiest of all the approaches, the analytic integrations or analytic strategies refer to the formulation of analyses in an analogous manner. This could be achieved by examining the relationship between the same dependent variable and the associated independent variable. This was achieved by ensuring that the typologies that were generated and studied were the same for both the quantitative and the qualitative phase. A visual model of the research design adapted from Ivankova et al. (2006) and used in this study is captured in Figure 4.1.
Sampling
Trochim (2006) indicates that in applied social research it is not feasible, practical or theoretically sensible to do random sampling. Nonprobability sampling which does not involve randomly selected respondents is considered in the current study. Nonprobability sampling methods are divided into two broad types namely, accidental and purposive. With most sampling methods being purposive in nature because the approach pursued is with a specific goal in mind, the study sought specific predefined groups, namely, practicing teachers, for investigation. Voluntary participation was sought from practicing teachers who are involved in a professional advancement developmental course at a particular university in South Africa. The requirements for registering for this course were that candidates should hold a senior certificate as well as a recognised three-year professional qualification. In addition, the candidates were to have registered for specialisation of the school subject of their first professional education qualification. Respondents were selected non-randomly in what is known as non-proportional quota sampling which requires just enough sample of the population (McMillan & Schumacher, 2010). In the current study, quantitative data in the form a survey were gathered and some of it was used to generate questions for the interviews that formed the qualitative data of the case study. McMillan and Schumacher (2010) call it the concurrent quantitative and qualitative sampling.
The population in the study was teachers who were all enrolled for Advanced Certificate in Education6 (ACE) programme in mathematics education. The programme consisted of five modules. Teachers registered for the module, Algebra for Intermediate and Senior Phase Teachers (Module 2) were identified for the study. One of the purposes listed for this module was to enable teachers to use mathematical models to represent and understand quantitative relationships as well as to analyse change in various contexts. The expectation of the teachers to analyse contexts and use mathematical modelling rendered this group appropriate for investigation in the current study. Teachers enrolled for the module were approximately seven hundred and of these 220 (31%) were randomly sent the questionnaires to complete. The study was of such a nature that such a sample was adequate to provide data needed to address the research questions since generalisation from the data gathered was not sought.
Research Instruments
The questionnaire
The questionnaire consisted of three sections, namely, section A where biographical details of the respondents was sought, section B in which teachers expressed the extent to which they were capable of accessing mathematics in a given context, and section C where the teachers had to identify or recognise mathematics in the texts they were provided with (see Appendix A). This provided for opportunities of correlation that Frankel and Wallen (1990) indicated were not uncommon in survey research where the relationship of one set of questions could be compared to another set.
Of the four basic ways of data collection in survey research – with „live‟ administration of a survey instrument to a group, using a telephone and face-to-face interaction with individuals being amongst them – the mailing one was used. The advantage of using the mail was that respondents were more easily accessible through this mode of exchange. One of the strengths of survey research is that the researcher will have access to samples that are hard to reach, as was the case with the teachers in the current study. Another advantage of this approach was that it permitted the respondents to gain sufficient time that enabled them to reflect freely on the questions. The respondents were as a result, in a position to provide acceptable answers to the questions, under the assumption that they completed the instrument without being assisted. A weakness to this approach was that there were less opportunities to encourage the cooperation of the respondent as the prospects of building rapport with them were severely minimised. There was also the prospect of eliciting low response rates, as it is normally the case with this kind of approach (Frankel & Wallen, 1990). There was also no opportunity of elaborating on the questions in case of misunderstanding on the part of respondents.
CHAPTER 1:
1.1 Introduction
1.2 Historical background
1.3 The mathematics curriculum in South Africa
1.5 Definition of terms
1.6 The objectives of the research
1.7 Motivation for the study
1.8 Structure of the Thesis
1.9 Summary
CHAPTER 2
2.1 Introduction
2.2 Realistic Mathematics Education (RME)
2.3 Mathematical knowledge for teaching
2.4 The use of context in the teaching and learning of mathematics
2.5 Implementation of the curriculum
2.6 Problem Solving
2.7 Mathematical Proficiency
2.8 Teachers‘ views and beliefs
2.9 Summary of Chapter
CHAPTER 3
3.1 Introduction
3.2 Research on learning
3.3 The modelling approach
3.4 Bernstein‘s constructs
3.5 Mathematical processes
3.6 The mathematical participation model (MP-model) .
3.7 Summary of chapter
CHAPTER 4.
4.1 Introduction
4.2 Research Design
4.3 Sampling
4.4 Research Instruments
4.5 Pilot Study
4.6 Procedure/ Data analysis
4.7 Ethical considerations
4.8 Summary
CHAPTER 5
5.1 Introduction
5.2 Survey results of section A of the questionnaire
5.3 Results for Section B of the questionnaire
5.4 Results for section C of the questionnaire
5.5 Statistically significant association between the sections of the questionnaire
5.6 The interviewees
5.7 Comparing Bongani‘s and Kelebogile‘s ratings of sections B and C of the questionnaire with the rest of participants in the survey
5.8 Analysis of the interviews
5.9 Summary of chapter
CHAPTER 6
6.1 Introduction
6.2 Using the sentences to transit from contexts to mathematics
6.6 Summary of chapter
CHAPTER 7
7.1 Introduction
7.2 Summary and synthesis of the study
7.3 Conclusion
7.4 Recommendations
7.5 Limitations of the study
7.6 Areas for future research
7.7 Epilogue
REFERENCES
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TEACHERS’ VIEWS ON THE USE OF CONTEXTS IN TRANSITION TO MATHEMATICS
