Proceedings of the Day Conference held at the Sheffield Hallam University on 02 Jun 2013
Rosa Archer, Sue Pope, Alice Onion and Geoff Wake
The University of Manchester, University of Manchester, King’s College London and University of Nottingham
The working group met at BSRLM to share experiences of using lesson study in various ways in various contexts in schools, higher education (HE), initial teacher education (ITE) and continuing professional development (CPD) in mathematics education. Lesson study was established in Japan as an important and effective enquiry based approach to professional development. The use of lesson study in CPD and ITE is described in this short paper.
Michal Ayalon1, Stephen Lerman2 and Anne Watson1
The University of Oxford1, London South Bank University2
Despite a plethora of research about misconceptions and the teaching of functions, little is known about the overall growth of students’ understanding of functions throughout schooling. We aim to map the development of students’ understanding of concepts which contribute to understanding functions throughout the school in two different curriculum systems: in the UK and in Israel. The research uses a survey instrument that was developed in collaboration with a group of teachers.
Alison Clark-Wilson, Celia Hoyles and Richard Noss
London Knowledge Lab, Institute of Education
This paper builds on the outcomes of the Cornerstone Maths pilot study, a USA/UK collaboration, which is now in a phase of scaling to over 100 schools. We describe the vision for the project and its iterative design, both informed by a twenty-year history of research on dynamic digital technologies. The resulting intervention builds on our understanding of some of the constraints to the widespread use of dynamic digital technologies by pupils in mathematics, which relate to accessibility, teacher development, curriculum alignment and the need to support the instrumentation process for teachers. The accompanying research agenda is concerned with evaluating models for scaling that are mindful of the ‘grain size’ of analysis and the necessary re-alignment of the design principles of the innovation to take account of implementation imperatives.
University of Nottingham, UK.
Classroom mathematics and the mathematics of ‘real life’ or work often appear to be unrelated from a student viewpoint and tasks that are intended to be realistic rarely resemble the real tasks that might actually occur in life. Research highlights the difficulties of crossing the boundaries between the classroom and the world outside, referring to the problems of transferability and the situated nature of learning. In this session qualitative data from discussions with post-16 students about a range of contextualised functional mathematics tasks will provide some insight into their perceptions of relevance and relationship to life. The results indicate ways in which they make personal connections to the context, the activity or the mathematical content at different levels or reject tasks because of the contradictions they present. Their judgements lead to interesting distinctions between tasks that they believe remain firmly situated in a mathematics classroom and ones that may belong in a ‘real life’ situation. This research involved vocational students in colleges but has implications for much wider consideration across mathematics teaching.
Susan K. Forsythe
School of Education, University of Leicester
Students working in Dynamic Geometry Environments interact with geometric figures by dragging constituent objects on the computer screen. A number of researchers have described different dragging modalities and linked them to cognitive activity. This paper draws on data from recordings of students working with a dynamic figure based on fixed length perpendicular diagonals. The diagonals can be dragged in the figure thus generating a number of quadrilaterals and triangles. Two new dragging strategies have been observed in use by students within the context of working with the dynamic figure. Refinement dragging is used when students check and review side and angle properties of shapes they have generated. Dragging maintaining symmetry is used when students drag so that one diagonal bisects the other generating what could be termed a ‘dragging family’ of shapes. This paper describes these dragging strategies and relates them to the Van Hiele levels of reasoning. The students’ innate sense of symmetry also emerged as an important aspect of how they conceptualise 2D shapes.
06 Exploring the challenges for trainee teachers in using a Realistic Mathematics Education (RME) approach to the teaching of fractions
Sue Hough and Paul Dickinson
Manchester Metropolitan University
We report on the second part of a study into the subject knowledge of Secondary Mathematics trainee teachers enrolled on a Subject Knowledge Enhancement (SKE) course prior to their PGCE. In the first part of the study, trainees revealed a predominantly procedural knowledge of fractions. Most used the procedure as the authority over their answers, and few were able to make sense of the fractions or represent the fractions pictorially. The trainees then studied the teaching of fractions, examining alternative learning trajectories based on Realistic Mathematics Education (RME), after which they taught the topic in schools. We focus here on the challenges faced by trainees attempting to adopt a classroom approach that did not concur with the nature of their knowledge of fractions or with their own experience of learning fractions. Many trainees were able to adopt some of the features of RME, including appreciating the value of visualising fractions and the important role of discussion. However, the need for the trainee to have knowledge of a learning trajectory through fractions that is not dominated by procedures, and a belief in this trajectory, emerged as critical features.
School of Education, University of Southampton, UK
Diagrams are generally taken to be an integral component of doing and understanding mathematics. In the teaching and learning of geometry, the use of diagrams is not only because of the nature of geometrical objects, but also because a diagram is often a particularly effective problem representation that enables complex geometric processes and structures to be represented holistically. At the same time, learners can be misled by diagrams. This brief paper provides some results from research on the affordances and limitations of diagrams in the teaching and learning of geometry. The paper concludes by suggesting some ideas for future research.
Liverpool Hope University (University of California, Los Angeles from September 2013)
Teacher collaboration, and teacher professional development within this context has become an area of interest in recent years. In particular, mathematics teacher education has seen the rise of collaboration as an effective school-based professional development activity, where in-service teachers plan, observe and reflect on lessons together. This paper presents some of the findings from a year-long case study research project which investigated the professional development of mathematics teachers in cross-phase and cross-school collaborations. The findings show that although the teachers were encouraged to jointly plan lessons and peer observe, they were reluctant to do so. This paper explores their reasons for choosing to work alone and the implications this has for collaborative development.
Nicola Trubridge and Ted Graham
This article considers the various dichotomies between types of mathematical understanding. It concludes that whilst the different categorisation is useful; it is the interplay and connections between these types of understanding that are more beneficial to student learning. Theories are drawn from a wide literature base to consider what this might look like in the secondary mathematics classroom, and we propose the Collaborative Connected Classroom Model.
St. Patrick’s College, Dublin City University
The purpose of my research is to explore to what extent children in Irish primary schools are developing skills in algebraic reasoning. In addressing the challenge faced by students encountering algebra in secondary school, one recommendation is to commence algebra instruction early in a child’s schooling (Kaput, 2008; Carpenter and Levi, 1999; Mason, 2008; among others). Underpinning my research will be a framework of growth points, which outlines a possible developmental pathway in algebraic reasoning for primary school. In this paper, I present five growth points in algebraic reasoning with interim trajectories which may be supportive in exploring children’s emerging understanding.
University of Leeds
Two concerns for UK government are: the shortage of mathematics teachers and the poor mathematics results of school-leavers. In order to increase the supply of teachers, the government have sponsored ‘ subject knowledge enhancement’ (SKE) courses to graduates from numerate disciplines to enable those without mathematics degrees to train as mathematics teachers. Additionally, policy makers and researchers have seen defining and codifying the subject knowledge required by teachers as central to improving student learning. The University of Michigan have developed measures to test teachers’ ‘ mathematical knowledge for teaching’. These measures have primarily been used within the US but have been adapted for use in a number of other countries (Ireland, Norway, Ghana, Indonesia and Korea). For this study, the measures were utilised with a sample of trainee secondary mathematics teachers in England. In contrast to existing studies, an alternative approach to selecting measures was employed. The selection process and preliminary results from administering the measures in a pre- (teacher-training course) questionnaire will be discussed. In particular, the responses of mathematics graduates and SKE students will be compared.
Kat Hall and Rosie Everley
Balby Carr Community Sports and Science College and Wickersley School and Sports College
Kat and Rosie, two teacher delegates, were invited to respond to the discussion in Rebecca Warburton’s session on Mathematical Knowledge for Teaching.
Jehad Alshwaikh1 and Candia Morgan2
1Birzeit University, Palestine 2Institute of Education, University of London, UK
The project reported here aims to produce an analysis of Palestinian school mathematics textbooks, focusing on the nature of the mathematics portrayed in the texts and of the mathematical activity expected of students. The academic collaboration enabled by the project includes comparison with a sample of English textbooks. We share some results of an early analysis of two extracts of textbooks and discuss the nature of the mathematics that students are expected to experience.