Proceedings of the Day Conference held at the University of Birmingham, November 2006.
Kim Beswick, University of Tasmania
This paper begins with a brief discussion of existing understandings of the nature of beliefs system, and of complexity science. The extent to which belief systems might be complex is then considered and possible implications of thinking of beliefs systems in this way are discussed with a particular focus on how insights gained from a complexity science perspective might inform attempts to influence the development of the beliefs of pre-service and in-service teachers of mathematics.
Camilla Gilmore, Learning Sciences Research Institute, University of Nottingham
Children’s knowledge of arithmetic is made up of many components (e.g. conceptual understanding, procedural skill, factual knowledge) and is therefore difficult to capture with a single measure. Profiles of understanding map children’s performance across tasks and can reveal differences among children in the relationship between components of arithmetical skill. A re-analysis of several studies examining children’s conceptual and procedural knowledge of addition and subtraction revealed subgroups of children with different profiles of understanding. These subgroups may represent different routes to the development of arithmetic knowledge and so it may be important to consider profiles of development as well as profiles of understanding.
Rose Griffiths, School of Education, University of Leicester
Tony Brown, Una Handley and Susan Darby, Manchester Metropolitan University
This small-scale study sent a film crew to twelve families with children under five and asked them to show us some of the things they did to help their children learn to count, to use in a DVD for other parents. The film clips provided examples of children aged 2, 3 and 4 engaged in counting with a parent, sibling or on their own. The clips were analysed to see what the parents considered important, and how they supported their children’s learning. Parents’ and carers’ support for their children’s learning included aspects that are perhaps not seen as often in a classroom setting (including individual attention for sustained periods). There was also evidence of wide differences between families in children’s and parents’ favoured activities.
Tansy Hardy, Sheffield Hallam University
I offer an exploration of both what is often named ‘identity’ and of what it means to be confident in learning maths. I discuss the theoretical model offered by ‘subjectivity’ and discuss how this can give a better analytical frame. Through an analysis of teacher guidance material together with extracts from interviews with pre-service student teachers I explore the discursive practices of teachers and learners and the effect of this teaching. I discuss how shaking up constructs of ‘confidence’ and ‘competence’ and that of a ‘good learner’ highlights the key of being willing and able to participate in classrooms in particular validated ways.
Tony Harries and Patrick Barmby, Durham University
In this paper, we examine the importance of representations, in particular with respect to the understanding of multiplication by primary school pupils. The first half of the paper examines the theoretical background to representations in mathematics. The second half of the paper examines some preliminary work that we have carried out, examining Year 4 and Year 6 pupils’ use of the array representation for multiplication calculations. Using a novel methodological approach of recording children’s workings on a computer, we observed that the array representation can be a powerful tool for supporting work in multiplication. At the same time, we also observed pupils who were unable to access the mathematical meanings of the representation. We must be aware of such difficulties when developing the use of the array as a tool for mathematical understanding.
We examine the written responses of fifteen students (aged about 141/2 years) to a homework task and their responses to the same task in a subsequent lesson. Students were asked to make observations about the sum of three consecutive numbers and to explain why they thought their observations were true, thereby giving students the opportunity to engage in structural reasoning. The teaching sequence had four phases designed to allow students to make, share and develop their observations and reasoning, and we found a clear improvement in the quality of students’ responses. As far as students’ reasoning is concerned, this suggests limitations may stem at least in part from a lack of familiarity with the nature of mathematical reasoning.
William O. Lacefield, III, Tift College of Education, Mercer University, Atlanta, Georgia, USA
Children’s literature, when used appropriately and creatively, serves as an impetus for exciting teaching and learning, rich instructional tasks, and valuable assessment opportunities. Fiction and non-fiction books, as well as poetry and other forms of literature, have long been respected in reading and language arts classrooms. However, today’s teachers are encouraged to integrate literature throughout the content areas of the early childhood curriculum. Ideas for using children’s literature in mathematics are limited only by teachers’ imaginations and creativity.
John Mason, Open University
Participants were prompted to consider core issues concerning CPD, then presented with a window on what might be possible via the NCETM CPD FW website. The session ended with questions and issues centred on the PedMaPedia (the component which it is hoped will develop into a wikipedia for mathematics related pedagogy).
Julie-Ann Edwards, University of Southampton
Sue Pope, St Martins College, Lancaster
This paper is a report from the ITEmaths working group on Masters level work in PGCE courses. This BSRLM working group arose as a result of a session on Masters level work at the AMET conference in September, 2006, during which some issues arose which were left unresolved due to a lack of discussion time. The paper examines a) the reasons why Masters level PGCE courses have, more recently, become a priority for some institutions; b) some models which have resulted from the changes; and c) some of the positive outcomes already experienced and some of the issues raised.
Anne Watson, University of Oxford
My task is to analyse a collection of classroom videos in a way which draws attention to the range of mathematical possibilities available to teachers in the public tasks, questions and prompts of mathematics classrooms. To do this I have explored the use of several frameworks which identify different intended learning outcomes, different ‘levels’ of thought and different aspects of mathematical thinking. None of these pick up the subtle variations I see in the videos.
Paul Wilson, College of St Mark and St John, Plymouth
This is an account of research conducted before and after the implementation of the National Key Stage 3 Strategy, exploring Year 8 pupils’ affective responses to mathematics. The results suggest that the teaching styles recommended for the Strategy did not have an adverse affect on pupils’ attitudes. The responses highlight some issues concerning social acknowledgement of the personal importance of mathematics and how teachers’ difficulties may have an adverse affect on pupils’ feelings about mathematics.
Marcus Witt, Graduate School of Education, University of Bristol
Studies looking at the connections between working memory and children’s mathematical achievement are often limited by the general nature of the measures of either working memory or mathematics. In this study, thirty-five children in Year 5 (9 to 10 years of age) were given tasks designed to measure their phonological working memory, visual working memory, inhibitory skills and their mastery of addition and multiplication facts. The findings suggest that different components of the working memory model (Baddeley and Hitch, 1974; Baddeley, 1996) may be involved in these two different mathematical operations.
Peggy Woods and Nick Pratt, University of Plymouth
This article reports on a small scale study in which PGCE primary students were asked to record and report their experiences of mathematical problem solving during school placements. We comment on these experiences, but also on how the involvement in the project was important in helping students to construct new ideas about effective maths teaching more generally.