The Drivers for this Research

Understanding how pedagogical approaches are used in the teaching of division of fractions and the influences of lesson designs are the central themes of this research. As stated in chapter 1, the two main research questions are

What are the influences of lesson design on pupil learning? What are the implications of a change of approach to teaching fractions for teacher training?

with an additional two subsidiary research aims:

What types of learning episodes could support better pupil understanding of mathematics?

and

What apparent mathematical misconceptions and barriers prevent pupil progress?

My conjectures at the outset of the research were that the types of work given to pupils and the teaching approaches taken by teachers have an influence on learning. I also conjectured that good subject knowledge and receptiveness to a change of mathematical approach by teachers might also influence pupil

learning. I wondered, therefore, what might be the effect of a change of mathematical teaching approach on

(a) a teacher’s design of a lesson (b) pupil learning.

116 It is inevitable that some of my beliefs about the impact of lesson design and

terminology on learning have been formed and reformed during thirty years of secondary school mathematics teaching. Recently eight years of working as an initial teacher educator have also further impacted on my beliefs. Trainee teachers, in my experience, often do not equate lesson design with learning as they tend to concentrate on the factual content or mathematical processes. The design for this study is based on a descriptive, interpretive, participatory case study in a single setting that then analyses the data through qualitative methods. Questionnaires were used to evaluate trainee teachers’ beliefs and views before and during the research. Descriptive statistical methods were used to analyse the questionnaires. Teacher participants were involved in the classroom fieldwork, face-to-face semi-structured interviews, and video recordings of lessons were used as data collection tools. Pupils were obviously involved, but not as active researchers, however their views were sought to enrich the narrative (section 3.3.2).

My ontological and epistemological assumptions, and my views concerning learning, teaching and mathematics, will impact directly on my selection of an appropriate research methodology. Cohen et al. (2013, p. 8) claim that how you view the world will impact on “the choice of problem; the formulation of questions to be answered; the characterisation of pupils and teachers; methodological concerns; and the kinds of data sought and their mode of treatment”. A researcher’s epistemological assumptions arise from their beliefs where a positivist view is questioned by Cohen et al. (2013, p. 11) because of the “complexity of human nature and the elusive and intangible quality of social phenomena”. Hammersley (2002) suggests that adopting a positivist epistemological stance is “inappropriate in educational investigations” as there is a “values dimension” in education research that a scientific model fails to address (Burton and Bartlett, 2005, p. 5). Given my positivist mathematical background my ontological beliefs of the social world and education have led me to an interpretive, qualitative approach to this educational research. Using an interpretive paradigm will still aim to apply the same rigour as the natural sciences whilst being concerned with explaining human behaviour to emphasise “how people differ from inanimate natural phenomena and indeed, from each other” (Cohen et al., 2007, p. 7).

117 My two research questions led me to consider both quantitative and qualitative

methodologies. Bryman (2008) asserts that the selection of a specific methodology should be firmly linked to the research questions. Denzin and Lincoln (2011) also remind us that qualitative research emphasises discovery of social meaning and stresses the relationship between the researcher and the topic studied. Aliaga and Gunderson (2002, in Muijs, 2004, p. 1) define

quantitative research as the exploration of a “phenomenon by collecting numerical data that are analysed using mathematically based methods”. Quantitative research, by contrast, tends to investigate causal relationships between variables by taking an empirical stance through measurements. Reflecting on the two research questions it was considered to be extremely unlikely that numerical quantities would form a huge part of the research. This would be because of the very nature of the research questions and the need for a more descriptive, in-depth approach rather than a measuring of variables. A number of quantitative studies into teachers’ beliefs (Ball, 1992; Nisbet and Warren, 2000; Yates, 2006), using numerical data as a means of extracting teacher beliefs have been conducted. These studies have been found to be of limited use because they tend to describe what is happening rather why it is happening. A fuller, richer dataset can be achieved through mixed methods approach to gain a deeper understanding of what is happening and why. As a former practising mathematics teacher with some working knowledge of research methodologies, I naturally gravitated towards a quantitative approach. So I searched for a methodology that would be both flexible and supportive whilst accommodating both qualitative and quantitative approaches. This, I felt, would allow for the gathering of rich data, with an increasing refinement of both insights and detail as the research progressed. As Major and Savin-Baden (2010, p. 14) reminds us “the two approaches taken together could provide powerful

information”.

For Wright (1995) qualitative research means less reliance on number counting and statistical techniques in favour of getting closer to the data collected from natural settings. The emphasis of qualitative research is often to understand comprehensive, interdependent, holistic structures that are dynamic and

118 variable or combinations of variables are important in the interaction, a view

shared by Kleiner and Okeke (1991). For Van Maanen et al. (1982) a

characteristic feature of qualitative research is the reliance on multiple sources of data, rather than just one source, so as to be able to explain events.