| Abstract: |
Research has shown, with respect to the learning of mathematics, that whole class interactive teaching, its form and function, is a complex phenomenon. Teachers develop and exploit pedagogical strategies, which they believe are effective either in engaging their children in mathematical learning or in presenting mathematics to learners. Such strategies, whether later shown to be effective or not, are typically assumed to develop during periods of teacher education or through practice after qualification. Alongside these assumptions is the belief that teachers who are enthusiastic about and have a secure subject knowledge with respect to mathematics will evoke similar enthusiasm, confidence and competence in their learners. However, observations during my years as a teacher educator have led me to conclude that trainee teachers, even those with similar qualifications, frequently behave very differently when put in front of children. Such differences confound the naïve assumption, for example, that similar enthusiasm and confidence will yield similar patterns of teaching practice. Thus, what primary teachers do and why they do it has vexed me for a number of years. I have wanted to know, in particular, what makes teachers teach differently during whole class episodes, not least because my experiences as both teacher and teacher trainer have led me to believe that it is during these periods that teachers induct their children into those mathematics-related beliefs and behaviours that will determine the extent to which they enjoy and engage meaningfully with the subject. Addressing such questions demands an appropriate methodological stance. Consequently an exploratory case study of six teachers, two during a first, essentially pilot phase, and four during a second, was undertaken. All teachers, to facilitate understanding of how exemplary practice differs from one person to another, were considered, against various criteria, as effective. The pilot enabled me to evaluate not only the effectiveness of extant frameworks for analysing classroom behaviour but also my skills as an interviewer and observer of classrooms. The second phase, drawing on what had been learnt during the first, was more open in that existing frameworks were abandoned in favour of allowing the data to speak for themselves rather than being constrained by others’ conceptualisations of effective teaching. Both phases, to examine teachers’ underlying beliefs about mathematics and its teaching, their classroom practice, particularly during whole class episodes, and their rationales for their actions, were addressed by means of a battery of data collection tools. Teachers’ backgrounds and underlying beliefs about mathematics and its teaching were examined through preliminary, life history, interviews framed by a loose set of questions derived from the literature. Interviews were video-recorded. Teachers’ classroom actions were captured by means of a tripod-mounted video camera placed discretely in their classrooms, augmented by a wireless microphone worn by the teacher and a separate, static microphone to capture as much of the children’s talk as possible. Finally, teachers’ rationales and explanations for their actions were examined through the use of video-recorded video stimulated recall interviews. All recordings, whether of classrooms or interviews, were transcribed for later analysis. Analysis during the first phase drew extensively on pre-existing frameworks. While they were helpful in identifying both similarities and differences in teachers’ beliefs, actions and rationales, it became clear that they failed to capture the subtleties and nuances of meaning embedded in the high quality data yielded by the approaches adopted. In so doing it became clear that while data collection approaches were appropriate, analyses needed to be more open in order to allow the data to give up the depth and complexity of their stories. During the second phase, while it was acknowledged that this was not a grounded theory study, analysis drew extensively on the coding strategies of the constant comparison procedures of grounded theory. This approach to analysis yielded results previously unknown in the literature. Quite unexpectedly two groups emerged from the data. Significantly, each was underpinned by teachers’ experiences as learners of mathematics and whether the enjoyment they had gleaned from those experiences was instrumentally located or relationally located. The first group, identified as the mediators, having been engaged, in various ways, with mathematics and derived pleasure from relational experiences expected their children to experience mathematics similarly. Their teaching was based on a desire to develop, in collaborative ways, a deep conceptual knowledge that would form the basis for later procedural skills and, significantly, problem solving. Teachers in the second group, identified as the mediated group, having derived pleasure from their procedural successes as children, saw mathematics and its teaching as skills-based. Their classroom actions and commensurate rationales were focused on surface learning and the replication of the pleasure they had experienced when young. Interestingly, the beliefs of both groups and, to an extent, their classroom actions were independent of any training they had received. The Mediators showed different signs of professional independence and autonomy. They had a clear articulation of their warranted principles and were able to exploit these in the ways that mediated the constraints within which they worked. Moreover, and this presents substantial implications for teacher education, teachers in the Mediated group, exhibited few signs of professional independence; their actions being constantly mediated by the constraints, whether institutional or governmental, within which they worked. They had few articulated principles around which they based their teaching. These differences permeated all aspects of their work.
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