Drawing division: emerging and developing multiplicative structure in low-attaining students' representational strategies

Student thesis: Doctoral ThesisDoctor of Philosophy


This thesis examines the particular difficulties with multiplicative thinking experienced by students with very low attainment in school mathematics, and the representational strategies they use for multiplication and division-based tasks.
Selected students in two mainstream secondary schools, all performing significantly below age-related expectations in mathematics, placed in ‘bottom sets’, and described by their teachers as having particularly weak numeracy, received a series of tuition sessions (individual or paired). These involved ongoing qualitative diagnosis of their arithmetical strengths and weaknesses, and personalised, flexible learning support, delivered by the author. Students engaged mainly in division-based scenario tasks designed to encourage their engagement in multiplicative thinking, and explored various visuospatial representational strategies tailored to their specific areas of conceptual and procedural difficulty.
Multimodal audiovisual data collected from tuition sessions was analysed qualitatively across multiple analytic dimensions using a microgenetic approach. This led to the development of an adaptable framework for the analysis of nonstandard visuospatial representations of arithmetical structures and relationships. Analysis of changes in individual students’ strategies provided insight into some possible learning trajectories for multiplicative thinking. Parallel comparison of students’ varied representational strategies resulted in evidence for the psychological power of certain fundamental representation types, such as unit arrays and containers.
The main findings of this thesis concern: the fundamentally componential nature of the concept and practice of division, the potential difficulties this causes in understanding, and the importance of modelling and manipulating unitary multiplicative structures; and the relationship between representational strategies, economy and efficiency in carrying out multiplication and division-based tasks.
Conclusions are drawn on the relationship between the development of representational strategies and multiplicative thinking. Recommendations are given regarding learning and teaching practice for students with severe and milder difficulties in mathematics, and particularly the nature of 1:1 support provision for those considered to have Special Educational Needs.
Date of Award2014
Original languageEnglish
Awarding Institution
  • IoE Institute of Education
SponsorsESRC Economic and Social Research Council
SupervisorMelissa Rodd (Supervisor)

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