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Drawing division

Carla Finesilver

At a national and international level, research on very low attaining pupils of secondary age is rare compared to younger age groups, and while there has been increased interest in the past decade at national level, research on support for numeracy still lags significantly behind that of literacy. Various numeracy interventions have been deployed in primary schools for some years, and more recently at secondary level, but while there are now certain generally-accepted principles, there is little consensus on the finer details of exactly ‘what works’ for these students.  It is increasingly recognised that understanding of multiplicative structures is a major component of numeracy, and forms a significant milestone in – or in some cases, a barrier to – achieving basic arithmetical functionality. The role of visuospatial representations in learning (such as the use of visual imagery in mathematical thinking) has long been a personal interest; however, way these function in relation to number work are as yet under-theorised and under-evidenced, particularly their potential for supporting arithmetical understanding for those learners with atypical development. For my doctoral thesis I chose to combine these three aspects – low attainment at secondary level, multiplicative relationships, and visuospatial representation.

I also made the choice that my research would use only qualitative analysis. While quantitative data can show how many learners are achieving – or failing to achieve – specified criteria in school and internationally-standardised mathematics tests, qualitative research of equal rigour is a vital complement. This is particularly the case when trying to understand the nature of struggling learners’ difficulties, and the cognitive tools which can allow them to progress; it is especially difficult to observe progress in those pupils whose Special Educational Needs and disabilities make their progress particularly slow and nonlinear, or who do not have the linguistic or metacognitive skills to explain their thoughts. My thirteen case study pupils would be ‘outliers’ in a large quantitative study: this is what makes them interesting to research! Drawing on grounded theory, microgenetic, traditional and design-based practitioner-research methods, I designed a methodological and analytical approach to function as a theoretical microscope, trained on the arithmetical-representational behaviours of individuals and offering a rare window into their thinking.

The awarding of the BERA doctoral thesis award for my research means a great deal – not only to me personally, but in terms of bringing increased attention to the issue of (very) low attainment in maths, and for the validation of the importance of qualitative research in education. The feeling of achievement is heightened for me by personal circumstances: there were times when I had doubts I would even complete my PhD. I never once lost motivation or interest in my research, but at times my work was significantly impacted by ongoing medical issues. I would like to take a moment to publicly express my appreciation for all who provided support and understanding when it became clear that life and doctorate were not going as planned: my supervisor, Dr Melissa Rodd, for her patience and flexibility of supervisory approach; the IOE Disability Support team for their invaluable practical advice and accessibility provisions; and the IOE and ESRC (my funding body), for granting me time out when necessary, a switch from full-time to part-time scholarship, and an extension.

The teaching/learning of young people who struggle to become numerate will continue to be a pressing and concerning current issue for schools, and a fertile field for further postgraduate research, perhaps even following similar methodologies. For my fieldwork I worked with selected pupils from ‘bottom sets’ in mainstream Inner London secondary schools on (mostly) division-based tasks, taking the time and close observation necessary to qualitatively diagnose their arithmetical strengths and weakness. By exploring the representational strategies they found effective, and tailoring these based on the individuals’ specific areas of conceptual and procedural difficulty, I was able to see small changes and developments of a kind not possible when teaching a whole class. I hope my work will be useful for educational practitioners as well as researchers, in terms of using arithmetical imagery, interpreting low-attaining students’ microprogressions, and practical ways of tailoring numeracy support practices to the needs of individuals. 


Finesilver, C. (2014). Drawing division: emerging and developing multiplicative structure in low-attaining students’ representational strategies (PhD thesis). Institute of Education, London.