Learning to ‘deal’: A microgenetic case study of a struggling student’s representational strategies for partitive division

International audienceThis paper focuses on the arithmetical understandings and behaviours of one fifteen-year old student with very low attainment in mathematics, as she worked on a sequence of scenario-based partitive division (sharing) tasks with individually-tailored verbal and visuospatial support. The student’s independent and co-created visuospatial representations of arithmetical structures, along with her verbal comments, were analysed qualitatively using a multimodal microgenetic approach. This paper focuses on three particular excerpts which illustrate the fundamentally componential nature of the concept and practice of division, some difficulties that may be experienced when modelling ‘sharing’ tasks, and the pedagogical importance of spatial structuring when a learner is moving between different kinds of representation

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

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.\ud 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.\ud 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.\ud 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.\ud 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

The “problem” of teacher quality: exploring challenges and opportunities in developing teacher quality during the Covid-19 global pandemic in England

Teachers and teacher education are often presented as “problems” to be solved, with policy solutions that focus on ways to make teachers “better” and improve teacher “quality” by introducing prescriptive strategies. We investigate the ways Covid-19-related changes to university and school-based facets of Initial Teacher Education (ITE) in England influence teacher quality in relation to both student teachers and early career teachers, working in secondary schools. Drawing on 34 interviews with school leaders, school mentors and ITE tutors, we critically explore the ways in which teacher quality was developed through key aspects of teachers’ pedagogy and practice during the pandemic crisis when schools were closed and teaching moved online. Our findings show that the pandemic crisis has highlighted the different facets of teacher quality which arguably disrupt narrow and prescriptive understandings of what constitutes “quality” in policy terms. Although there were many instances of challenge in the development of new and student teachers, our data also shows how ITE tutors, school mentors and leaders responded creatively to the crisis. Participants highlighted the opportunities afforded by the pandemic to develop diverse and innovative pedagogies and practice, enhance students’ subject knowledge, as well as overcome some of the challenges in other areas of pedagogy and practice. Furthermore, the study shows that teacher quality was not substantially reduced despite the challenges arising from the pandemic and concerns that pre-service teachers would not be ready and prepared for a career in the classroom

Collaborative identity development during a global pandemic: exploring teacher identity through the experiences of pre-service high school teachers in England

Since early 2020, COVID-19 has had a substantial impact on teacher education. We consider novel aspects of how pre-service teachers have collaboratively developed their professional identities during the pandemic. Drawing on findings from forty-five interviews with pre-service high school teachers working in England during September 2020 – June 2021, we share how collaborative identity development was central and occurred in a variety of spaces, communities and modes. Collaborative identity development featured in how pre-service teachers saw themselves making a positive contribution to society through education and, in strong subject connections. Reflection that is collaborative, personalised, iterative, and separate from notions of formal progression enables positive identity work. Notions of identity are absent from international policy initiatives in ITE (Initial Teacher Education). This case study provides insights for policy makers in and beyond England who aim to support teachers at the beginning of their career so that they are retained

Learning to ‘deal’: A microgenetic case study of a struggling student’s representational strategies for partitive division

International audienceThis paper focuses on the arithmetical understandings and behaviours of one fifteen-year old student with very low attainment in mathematics, as she worked on a sequence of scenario-based partitive division (sharing) tasks with individually-tailored verbal and visuospatial support. The student’s independent and co-created visuospatial representations of arithmetical structures, along with her verbal comments, were analysed qualitatively using a multimodal microgenetic approach. This paper focuses on three particular excerpts which illustrate the fundamentally componential nature of the concept and practice of division, some difficulties that may be experienced when modelling ‘sharing’ tasks, and the pedagogical importance of spatial structuring when a learner is moving between different kinds of representation

Spatial structuring, enumeration and errors of S.E.N. students working with 3D arrays

International audienceThe move from understanding and working with arithmetical structures in one dimension (i.e. additive) to two or more dimensions (i.e. multiplicative) requires a significant change in children’s thinking. This paper investigates the varied and developing strategies and understandings of young people struggling with that change, through a series of 3D array enumeration tasks. Participants relied heavily on counting-based strategies, and a new analytical framework is proposed with which to diagnose initial (mis-)conceptions and observe microprogressions on the path towards multiplicative understanding

Learning to ‘deal’:A microgenetic case study of a struggling student’s representational strategies for partitive division

International audienceThis paper focuses on the arithmetical understandings and behaviours of one fifteen-year old student with very low attainment in mathematics, as she worked on a sequence of scenario-based partitive division (sharing) tasks with individually-tailored verbal and visuospatial support. The student’s independent and co-created visuospatial representations of arithmetical structures, along with her verbal comments, were analysed qualitatively using a multimodal microgenetic approach. This paper focuses on three particular excerpts which illustrate the fundamentally componential nature of the concept and practice of division, some difficulties that may be experienced when modelling ‘sharing’ tasks, and the pedagogical importance of spatial structuring when a learner is moving between different kinds of representation