Proceedings of the Day Conference held at the Institute of Education, London on 11 Mar 2011
Contents
01 Extending Valsiner’s zone theory to theorise student-teacher development
Mohammed Abdul Hussain, John Monaghan, John Threlfall, School of Education, University of Leeds
This paper sketches an extension of Valsiner’s ‘zone theory’ to theorise student-teacher development in inquiry classrooms. The paper is structured as follows. We begin with the empirical classroom study that grounds the theoretical extension we propose. We then provide the basic ideas of Valsiner’s zone theory, the Zone of Free Movement (ZFM) and the Zone of Promoted Actions (ZPA) followed by illustrative results. The final section presents the substance of the paper, that over the course of the study there were transformations of students’ ZFM/ZPA, of teachers’ ZFM/ZPA and these transformations were interrelated.
02 Family mathematics/numeracy: identifying the impact of supporting parents in developing their children’s mathematical skills
Jackie Ashton, Graham Griffiths, David Kaye, Beth Kelly and Daian Marsh, LLU+, London South Bank University
For a number of years, parents have been encouraged to become involved in their children’s learning. This has led to ‘family learning’ provision of various types being developed and funded. There have been a number of studies looking at parental involvement in their children’s learning, though less with a focus on the perspective of the parents (although see Abreu and Cline 2005). The researchers have started a small scale, pilot investigation in the impact of the provision on parents in supporting their children. Previous authors (McMullen and Abreu 2010) have noted that such parental support means that parents are engaging in some aspect of teaching. The study involves interviewing parents about their motivations for learning, their views on their ability to support their child’s learning, and the extent to which the courses involved have assisted this process. The data collected so far indicates heterogeneity in motivations although some possible categories are emerging which may assist planning for such programmes.
03 A classification of questions from Irish and Turkish high-stakes examinations
Tugba Aysel, Ann O’Shea, Department of Mathematics and Statistics, NUI Maynooth, Kildare, Ireland and Sinead Breen, Department of Mathematics, St. Patrick’s College, Drumcondra, Ireland
In both Turkey and Ireland entrance to third level education is determined by performance on a high-stakes examination at the end of second level education. However, the examination systems in Ireland and Turkey are quite different from each other. In order to compare the examinations, we attempted to classify the types of questions asked in 2009 and 2010. We used various classification systems including the Levels of Cognitive Demand Framework developed by the QUASAR Project (Smith & Stein 1998). We will report on the use of these frameworks, and the results obtained for the Turkish and Irish mathematics examinations.
04 Inducting young children into mathematical ways of working in Hungary
Jenni Back, Centre for Innovation in Mathematics Teaching, University of Plymouth
This paper outlines some initial findings from a small qualitative study exploring the ways in which teachers and kindergarten practitioners induct young children into mathematical ways of working in the last year of kindergarten and the first year of formal schooling. It presents a ‘telling case’ (Mitchell 1984) of one lesson as an example of the way in which mathematics is presented to a class in the first few weeks of formal schooling in Hungary. The mathematics involved and the ways in which the teacher works with the children are described to illustrate the centrality of mathematics in the approach and the care with which it is presented and developed with the children.
05 Consulting pupils about mathematics: a straightforward questionnaire?
Patti Barber and Jenny Houssart, Institute of Education, London
We reflect on experiences of working with the staff of a primary school to ascertain pupil views of mathematics. Our focus is on methodological issues. We consider the process of building on existing questionnaires to develop one appropriate for a particular school, identifying how discussions with school staff illuminated school practices. We discuss how pupils responded to the questionnaire, considering whether we can learn from the questions they found difficult to answer. A key observation is that researchers and teachers are likely to approach pupil consultation in different ways, that are likely to impact on findings.
06 A representational approach to developing primary ITT students’ confidence in their mathematics
Patrick Barmby, David Bolden and Tony Harries, School of Education, Durham University
Representations of mathematical concepts play an important role in the understanding of learners (Greeno and Hall 1997), and also in the pedagogical processes involved in developing that understanding (Leinhardt et al. 1991; Brophy 1991). In this paper, we report on work with a cohort of pre-service primary teachers, with the aim of developing their understanding of mathematics and their confidence in their subject knowledge and their teaching of mathematics. This was attempted through the introduction and use of representations associated with mathematical concepts covered in primary schools. We present the results of attitude measures and qualitative questionnaire comments in identifying whether and how the use of representations supported pre-service teachers’ confidence in teaching mathematics.
07 Children’s perceptions of, and attitudes towards, their mathematics lessons
Alison Borthwick
Among the reasons attributed to the crisis in mathematics education, disaffection with pupils remains high. While there are studies that investigate this pupil disaffection at secondary school, there are few that consult younger children in order to ascertain their views of mathematics. The research study examines this issue by using drawings as the primary source of data collection, followed by interviews. It offers a view of how some children perceive their mathematics lessons and what this could mean for the future of the subject.
08 The use of mathematical tasks to develop mathematical thinking skills in undergraduate calculus courses — a pilot study
Sinéad Breen and Ann O’Shea, CASTeL, St Patrick’s College Drumcondra; National University of Ireland, Maynooth
Mathematical thinking is difficult to define precisely but most authors agree that the following are important aspects of it: conjecturing, reasoning and proving, making connections, abstraction, generalization and specialization. In order to develop mathematically, it is necessary for learners of mathematics not only to master new mathematical content but also to develop these skills. However, undergraduate courses in Mathematics tend to be described in terms of the mathematical content and techniques students should master and theorems they should be able to prove. It would appear from such descriptions that students are expected to pick up the skills of (advanced) mathematical thinking as a by-product. Moreover, recent studies have shown that many sets of mathematical tasks produced for students at the secondary-tertiary transition emphasize lower level skills, such as memorization and the routine application of algorithms or procedures. In this paper we will consider some suggestions from the literature as to how mathematical thinking might be specifically fostered in students, through the use of different types of mathematical tasks. Efforts were made to interpret these recommendations in the context of a first undergraduate course in Calculus, on which large numbers of students may be enrolled. This itself constrains to some extent the activities in which the teachers and learners can engage. The tasks referred to here are set as homework problems on which students may work individually or collaboratively. We will report preliminary feedback from the students with whom such tasks were trialled, describing the students’ reactions to these types of tasks and their understanding of the purposes of the tasks.
09 Gattegno’s ‘powers of the mind’ in the primary mathematics curriculum: outcomes from a NCETM project in collaboration with ‘5x5x5=Creativity’
Alf Coles, University of Bristol Graduate School of Education
In this paper I report on the outcomes of a ‘Mathematics Knowledge Network’ (MKN) project, aimed at developing rich tasks in the primary curriculum. The work was funded by the National Council for Excellence in Teaching Mathematics (NCETM) and carried out in collaboration with the Arts based charity “5x5x5=Creativity”. The approach to the project was informed by Gattegno’s ideas on the ‘powers of the mind’. I report on the answers to three questions: can KS2 students access rich tasks designed for KS3, using their ‘powers of the mind’?; what is the potential for KS2 students’ capacities to use and apply mathematics?; what is the potential for collaboration between mathematics and arts-based education?
10 The application of lesson study across mathematics and mathematics education departments in an Irish third-level institution
Dolores Corcoran and Maurice O’Reilly, with Sinéad Breen, Therese Dooley, and Miriam Ryan, CASTeL, St Patrick’s College, Dublin City University
This presentation reports preliminary findings arising from a research project, which embodied cross-disciplinary collaboration into the teaching and learning of Mathematics. The project involved the use of a form of Japanese lesson study by colleagues from the Education Department and the Mathematics Department of a College of Education and Humanities in the Republic of Ireland. Five colleagues worked together to explore the goals of teaching two research lessons; the first of which was part of a module in the history of mathematics for BA students, and the second, a lesson in mathematics education for BEd (Primary) students. Following ethical clearance, the research lessons were videotaped using both a static camcorder focused on the teacher and a roving camera to record student participation. The research lessons were also observed in situ by the remaining participants of the lesson study group. Both research lessons were later transcribed. In this presentation we will report on our initial findings from the different perspectives of preparing, teaching, observing and reviewing the first research lesson. The potential for conducting lesson study in a cross-disciplinary fashion will be discussed.
Extra Where has all the beauty gone?
Martin Griffiths, University of Manchester
Bertrand Russell famously talked of mathematics as possessing an “austere beauty”. It would seem though that the capacity to appreciate the aesthetic aspects of our field is not necessarily the preserve of the mathematical elite. Indeed, a number of educators believe that such considerations have, in conjunction with various cognitive factors, the potential to play a significant role with respect to the student learning of mathematics in the classroom. We consider here the notion of the mathematical aesthetic within this context, drawing on the work of a number of key thinkers in this area. Our preliminary explorations focus on a number of lesson observations, and the intention at this stage is merely to ascertain whether or not aesthetic considerations are playing any part in students’ mathematical development in the classroom. We provide a brief discussion of our findings thus far, highlighting potential issues and dichotomies that would appear to arise as a consequence of the current climate of test-score-driven schooling.
11 Exploring children’s interest in seeing themselves on video: metacognition and didactics in mathematics using ‘Photobooth’.
Rose Griffiths, University of Leicester
This paper examines the process of interviewing five children aged 7 to 11 doing arithmetic, to begin to explore the benefits and limitations of using video of children in three main areas: the benefits to the researcher of making a video record of an interview; the use of visually stimulated recall; and the potential for the teacher, especially with children whose experience is one of failure in mathematics, to show the child they have made progress and thus to influence their future learning.
12 I can be quite intuitive”: Teaching Assistants on how they support primary mathematics
Jenny Houssart, Institute of Education, London
This paper reports on initial work on Teaching Assistants’ (TAs) perceived contribution to Mathematics teaching in primary schools. Extracts are presented from interviews with three TAs who provide support to individuals with particular needs. The focus is on what interviewees say about the knowledge and understanding they bring to their work. The paper also identifies how they feel they acquired this knowledge. I show that they draw on pedagogic content knowledge and subject-specific knowledge of individuals. In discussing the source of this knowledge, interviewees value experience and use of initiative.
13 Economic activity and maths learning: project overview and initial results.
Tim Jay and Ulises Xolocotzin Eligio, Graduate School of Education, University of Bristol
We present the overview and initial results of a project that explores the links between the out-of-school economic activities of UK children (8 – 16) and their learning of mathematics. The aim is to inform and enhance children’s classroom mathematics experience to make it meaningful and engaging, especially for underachievers. Economic activity is interpreted and researched in a broad sense, covering activities around money (e.g., paid work, pocket money, gambling) but also other non-monetary transactions (e.g., swapping, collecting, giving gifts). The project methodology has three stages: A survey of children’ seconomic activities, Qualitative studies including diary studies and focus groups and task-based interviews. The initial results of the survey have both similarities and differences with previous studies of children’s money usage, and revealed a wide range of economic activities in which non-monetary goods are central. Moreover, some of these activities are likely to involve mathematics more complex than simple arithmetic. These findings are discussed in relation to the anticipated results of the project.
14 Imperative- and punctuative-operational conceptions of the equals sign
Ian Jones, Matthew Inglis, Loughborough University and Camilla Gilmore, University of Nottingham
At the British Society for Research into Learning Mathematics day conference held at the London Institute of Education in March 2011 we presented evidence for the existence of a substitutive conception of the equals sign. During the session, Jeremy Hodgen (Figure 1) questioned the use of active and passive items in the instrument, suggesting our results demonstrate not a substitutive conception but rather children’s preference for passive items. This is an astute observation and one we have investigated albeit within the context of operational rather than substitutive conceptions. Specifically, we hypothesised and tested whether Year 7 children (N=99) distinguished between imperative (active) and punctuative (passive) formulations of the operational conception. We found no difference, thereby refuting both our own hypothesis and Hodgen’s suggestion. In this paper, we present these previously unpublished findings.
15 Models and representations for the learning of multiplicative reasoning: Making sense using the Double Number Line
Dietmar Küchemann, Jeremy Hodgen and Margaret Brown, King’s College London
There has been a great deal of work on the didactical use of models, such as the Double Number Line, much of it focused on using models as a support for teaching (e.g., Van den Heuvel-Panhuizen 2003). However, less attention has been devoted to documenting the ways in which students make sense of and engage with such models in developing understandings of multiplicative reasoning. In this paper, we will discuss this issue drawing on data from the ESRC-funded Increasing Competence and Confidence in Algebra and Multiplicative Structures (ICCAMS) study. Drawing on lesson observations, we will examine the relationship between the Double Number Line and students’ informal methods. Our work suggests that, whilst the Double Number Line is a valuable pedagogic tool, the development of multiplicative reasoning is nevertheless a long term process.
16 ‘Ability’ in primary mathematics education: patterns and implications
Rachel Marks, Department of Education and Professional Studies, King’s College, London
Ability is a powerful ideology in the UK, underscoring many educational practices. We have extensive evidence pertaining to the impacts of these, particularly setting, in secondary mathematics, but there is relatively little research into the impacts in primary schools, despite an increase in ability-grouping practices at this level. This paper begins to address this gap, discussing some of the results from my doctoral study. It explores the pervasive nature of ability and the strength of young children’s convictions in innate ability. It also examines the role of assessment in perpetuating an ability ideology, suggesting that many of the implications seen in secondary education are also issues for primary mathematics.
17 Topologic and topographic features of parameters of functions and meaning transitions within a microworld–microidentity interaction
David Martín Santos Melgoza and Armando Landa Hernández, Autonomous University of Chapingo, Mexico
Two year 10 English students, one boy and one girl, worked together in a task developed using a GeoGebra software (Dynamic Geometrical System) to promote the development of ideas approaching to the notion of function. During the task, students’ voices and the computer screen were recorded to assess how this task promotes the developmental process of what Mason call mathematical being through a perceptually guided activity regarding to parameters notion.
In accordance with Mason, who thinks that teacher can not do the learning for their learners, we assume that learning comes about from the activities students develop in a learning episode. Here we will discus the capabilities of the micro-world to direct students’ attention to some general topographic and topological features of the geometric enactment of algebraic expressions; and the meaning transitions of the micro-world elements experimented by students regarding to the specific manipulation type (dragging points, sliders, or typing).
18 The extent to which a primary maths teacher’s success in the classroom is dependent on subject knowledge.
Robert Newell, Institute of Education, London
This paper tracks 5 Primary PGCE Trainee Teachers through their Course: In particular, it considers their Subject Knowledge ( as measured through exam results and the PGCE mid- course Audit) and explores the extent of its significance in helping children to understand mathematical ideas and to make connections. It analyses the choices the trainees make prior to, and during, the 10 lessons observed. The Trainees reflections are heard in their post-lesson discussions and in their focus group discussion at the end of the course. This is evaluated in terms of the balance between Subject Knowledge and Pedagogic Content Knowledge alongside Generic Teaching Pedagogic knowledge. Consideration is given to the need to create teachers who help children to make connections in maths when many trainees have not experienced such teaching themselves and are often fearful of trying to teach in such a way.
19 Women’s stories of learning mathematics
Alice Onion, Department of Education and Professional Studies, Kings College London
In this session we looked at women from three generations of one family. All three women have no formal qualifications in mathematics and all left education at the minimum school leaving age. They were video recorded talking about their experiences of learning mathematics at school and their current levels of confidence. Given their different ages (82, 64 and 44) we might expect their stories to be different, but there are surprising similarities. I am at an early stage in this ‘grounded’ research, approaching the topic with no intended preconceptions of what I might discover.
20 Tablets are coming to a school near you
Peter Osmon, Department of Education and Professional Studies, King’s College London
Improving mathematics learning is a major educational challenge. It is predicted that schoolchildren across the developed world, will soon have personal Tablet computers with the potential to support learning. The scope for improvement in mathematics learning support is examined from several related viewpoints: previous contributions of Information Technology, including PC labs for mathematics classes; IT innovations children themselves adopt; an analogy between office work and classroom learning; individualised learning environments such as SMILE; and alternative classroom configurations. The potential of personal Tablet computers as a learner’s interactive textbook, notebook, test-paper and progression-record and as a teacher’s class management tool is outlined
21 What is ‘mathematical well-being’? What are the implications for policy and practice?
Tracy Part, London Metropolitan University
This project will attempt to investigate the ‘usefulness’ of the capabilities framework as a means to empower adult learners to identify and reflexively consider the mathematics that holds intrinsic value to them. The discourse terrain offered by the twin concepts of capabilities and well-being will be used to sketch a theoretical landscape to show how a learners approach to learning mathematics can either impinge or capitalise on the substantive opportunities that present for improved mathematical wellbeing.
22 Early entry in GCSE Mathematics
Sue Pope and Andy Noyes, Liverpool Hope University and Nottingham University
The change in school accountability measures at KS4 to include GCSEs in mathematics and English at grade C or above has led to increasing use of early entry to ensure that performance targets are met. We discuss the evidence around school entry practices from two surveys completed as part of the independent evaluation of the mathematics pathways project and discuss the need for a quantitative research study into the impact of early entry on participation and attainment.
23 Mathematics but and yet: Undergraduates narratives about decision making
Melissa Rodd, Institute of Education, University of London
Abstract: This paper draws on research from the ESRC-funded project Understanding Participation rates in post-16 Mathematics And Physics (UPMAP) and draws on the strand of this project that has interviewed undergraduates about their choice of university course. These interviews were conducted in a ‘narrative-style’ and their construction and analysis were informed by psychoanalytical theory and practice that acknowledges unconscious influences on decision-making. The focus here is on narratives from undergraduates who are reading mathematics with other subjects of study. The questions ‘why is this person studying mathematics?’ and ‘what is the role of the minor or joint subject?’ are both considered. The observation is made that while mathematics functions as a place where results are definite — a notion cited by many students in the study — the minor or joint subject functions as a place for fantasies or is used as defence.
24 Primary school teachers in Seychelles reporting on their impressions of a mathematics teaching reform
Justin Davis Valentin, PhD Student, King’s College London, UK
The Mathematics Lesson Structure reform in Seychelles is stimulating debates on the way mathematics teaching can be improved in the context of a small island developing state. Due to limited research on the reform, to date, its impact in schools has not been established empirically. Informal evidence suggests that teaching is gradually changing to accommodate the major ideas of the reform. A systematic inquiry is needed to develop an understanding of its impact on teaching and learning. This paper is a first step towards documenting the impact of the reform. Analyses carried out on data collected during the early years of implementation show that the teachers have been well sensitised to incorporate MLS in their practices even if they find some elements of the structure difficult to apply. The study suggests implications for in-service teacher education.