Proceedings of the Day Conference held at King’s College, London on 01 Mar 2014
01 ‘It doesn’t have to be like this’: Women mathematics teachers’ experiences of professional learning
Sheffield Hallam University
Despite the range of possibilities for mathematics teacher professional learning and the reported success of individual initiatives, the overall picture in England appears to be one of restricted access to opportunity together with a lack of appropriate support for individuals. This study explores women mathematics teachers’ experiences of professional learning throughout their careers, focusing on the ways in which their learning is supported. Four in-depth life histories were elicited through semi-structured interviews in the form of guided conversations, supplemented by time-lines of mathematics and of professional learning. The teachers’ narratives reveal that much professional learning is informal, with teachers accorded limited agency and support. Spaces to discuss mathematics learning and teaching are constrained, with teachers appearing isolated within school environments. Where opportunities for collaborative professional learning exist, women participate actively in the wider mathematics education community. Analysis of the narratives suggests that teachers’ agency over their professional learning needs to increase, creating spaces for women to collaborate on mathematics-focused professional learning.
02 Exploring Prospective Mathematics Teachers’ School Placement Induction through Communities of Practice
Hatice Akkoç1, Mehmet Ali Balkanlioğlu2, and Sibel Yeşildere-İmre3
1Department of Secondary Mathematics Education, Marmara University, Turkey
2Department of Sociology, Marmara University, Turkey
3Department of Elementary Mathematics Education, Dokuz Eylül University, Turkey
Communities of practice is one of the most common interdisciplinary terms which is mainly used by anthropologists, sociologists and educationalists. This paper aims to analyse the induction experiences of prospective mathematics teachers during their school placements through the lens of communities of practice. The main research question was concerned with how they perceive what constitutes the practice of community. For that purpose, the research was designed as a qualitative cross-sectional study. A convenient sample of four prospective mathematics teachers was selected. Data collection consisted of face-to-face interviews. Interview transcriptions were analysed using content analysis method. Findings indicated that participants’ observations of professional interaction among colleagues in the school are concerned with teachers’ subject knowledge, cooperation among colleagues, assessment and being a mathematics teacher in a private school.
03 Responding to students’ contributions in the mathematics classroom: the case of Saudi trainee primary teachers.
Bader Aldalan and Tim Rowland
University of East Anglia
This ongoing doctoral study focuses on five case studies of Saudi primary trainee mathematics teachers of grades five and six (11 and 12-year-olds). It aims to identify and explore the relationship between the trainee teachers’ subject matter and pedagogical content knowledge, and their response to their students’ contributions in mathematics lessons. The teachers were observed and videotaped while teaching to identify how they responded to students’ contributions. This was followed by individual semi-structured interviews in which selected episodes were discussed in order to investigate the teachers’ rationales for their actions. In this paper we introduce preliminary analyses informed by Rowland and colleagues’ Knowledge Quartet of an episode from one lesson.
04 Hidden variation in school performance tables: the difficulties in identifying mathematics departments that are effective for disadvantaged students.
Fay Baldry1, Jenni Ingram2, Andrea Pitt3, Victoria Elliott4
1University of Leicester,2University of Oxford,3University of Warwick,4University of York
The achievement gap between disadvantaged students and others has received considerable attention from politicians and the press. School performance tables, which are used to rank schools, now include measures of this gap. Results from this study show that using alternative measures of the achievement gap would have a considerable impact on the position schools hold in these tables. Combining measures has allowed schools to be identified that are differentially effective for disadvantaged students in mathematics.
Alison Borthwick, Micky Harcourt-Heath, and Rose Keating
This paper reports on the findings from a sample of 985 Year 8 pupils who were given age-appropriate calculation questions and examines the range of strategies they used. The introduction of the National Numeracy Strategy (NNS) Framework (DfEE, 1999) brought together research and recommendations which sought to improve primary age children’s calculation strategies (e.g. Plunkett, 1979; Thompson, 1999). The Year 7, 8, 9 Framework (DfEE, 2001) for Key Stage 3 was the conduit for sharing this with secondary schools. Fourteen years later, just as the mathematics curriculum in the UK has been revised again, this study considers how pupils in Y8 respond to questions for each of the four operations. While over the last decade a range of strategies have been promoted, and recent research (e.g. Nunes, Bryant and Watson, 2009) has focused in particular on methods of calculation, our study suggests that Y8 pupils still struggle to select an efficient and effective strategy.
Jeremy Burke, Eva Jablonka, Chris Olley
King’s College London
With a focus on ‘translation’, we will discuss elements of a language of description that captures a range of strategies and criteria deployed in setting up a mathematical model and demonstrate its usefulness for analysing and evaluating school mathematical modelling activities. We see this as part of a larger project of exploring mathematical modelling and its recontextualisation in school mathematics.
University of Bristol
How do we learn mathematical concepts? How can we learn them fast? In this paper, I offer a lighting on the work of Gattegno and Davis and suggest that one common feature was a linking of mathematical concepts and symbols to relations (e.g., relations between physical objects, or actions performed on the objects). Visible and tangible resources are used to support the awareness of relationships between symbols, rather than offer a meaning for symbols. I suggest a distinction between an ‘absolute’ and a ‘relational’representation of mathematical concepts (I am endebteed to Tim Rowland for suggesting these labels at a BSRLM Conference).
08 Analysing two group-tasks leading to a collaborative classroom practice with Engeström’s activity theory
Umeå University, Sweden; University of Oxford, UK
Two teachers, Olaf and Knut, conducted two group tasks in succession, early in the academic year at a gymnasium or upper secondary school in Norway. In doing so, they steered classroom practice away from traditional instruction, with Olaf alone as the teacher, to cooperative learning in small groups with guidance from both. While the first group-task titled When Together initiated cooperative learning by students in small groups, the second titled How Heavy initiated student groups to build upon group cooperation and work with other groups in a collaborative classroom practice. It was Olaf and Knut’s intention to have their students cooperate in small groups at all times and collaborate with students from other groups on occasion. A few months into the year, Olaf and Knut’s students’ groups had the opportunity to discuss rules of cooperation whereupon their collaborative classroom practice became the norm. Using examples of students’ attempts at both group-tasks, I portray Olaf and Knut’s initiation of such a practice. Using Engeström’s activity model, I shed light on how students’ participation was transformed to meet with their intentions.
Peter Gates and Shafia Abdul Rahman
University of Nottingham and Universiti Sains Malaysia
There is concern internationally that socio-economic class and ethnicity remain the most significant predictors of outcomes in mathematics; performance is often largely dependent on family income and level of parental education. Consequently, the influence of pupils’ socio-economic backgrounds remains a major challenge to those of us in the field concerned with achieving an equitable education. However, the ways in which socio-economic factors play out in different parts of the world subject to different political systems and structures remains unclear. In this paper, we present an analysis of mathematics achievement in Penang to offer a localised perspective on the ways in which socio-economic status and ethnicity affect achievement.
Institute of Education, London
I report on an initial study, which aims to ascertain teachers’ opinions and practice relating to the place of real-world equity issues in the mathematics classroom and identify any barriers they perceive. Although academics have examined ways to implement a culture of critical mathematics education, it has also been suggested that there is little evidence of real world problems being addressed in the mathematics classroom. The National Curriculum states that mathematics is ‘for understanding the world, the ability to reason mathematically, and a sense of enjoyment and curiosity’ which contrasts starkly with a ‘back to basics’ curriculum in which Citizenship at Key Stages 3 and 4 has been disapplied. Initial interviews were held to investigate secondary mathematics teachers’ beliefs in relation to classroom mathematics, critically addressing real equity world issues.
School of Education, University of South Wales, Newport
Investigations were carried out at a Further Education college in Wales to determine the extent to which students are prepared for the numeracy demands of employment. Surveys were carried out with teaching staff and students of courses including science, computing, business studies, and construction. It was found that a majority of students would require additional job-specific training after entering the workplace, but that further help could be provided in college by developing the wider transferable skills of numeracy, including mathematical methods, problem-solving, data collection and processing including the use of electronic devices, and effective communication of mathematical results. Students showed the greatest motivation when undertaking authentic real-world problem-solving tasks, particularly when personally involved in the design of a project and collecting their own data for analysis.
Jenni Ingram1, Fay Baldry2 and Andrea Pitt3
1University of Oxford,2University of Leicester,3University of Warwick
In this paper we will explore the role of errors in both the teaching and the learning of mathematics. Analysis of classroom interactions shows that mathematics teachers are implicitly treating errors as something to avoid despite commenting on the positive role they have in the learning of mathematics. Similarly, the students themselves treat errors as something to be avoided. This leads us to consider, and therefore explore, what possible roles errors may have in the learning and teaching of mathematics.
Mercer University, Atlanta, Georgia, USA
Released in 2010 by the Common Core State Standards Initiative (CCSSI), the Common Core State Standards for Mathematics have been adopted by the majority of states in the USA and are expected to be fully implemented by the end of 2014. With this in mind, teacher education programs must provide training in the new standards immediately, especially in those states that have already adopted the standards. Otherwise, new mathematics teachers may enter the profession with little to no exposure or understanding of the standards that they will be expected to teach. The Common Core State Standards consist of two major components: (1) Standards for Mathematical Content and (2) Standards for Mathematical Practice. This paper will focus on the Standards for Mathematical Practice, which are devoted to the nurturing of problem-solving skills, critical thinking abilities, and mathematical habits of mind. This paper also presents ideas for future research endeavours generated by a group of mathematics educators attending the March 2014 BSRLM Conference.
14 Adopting Goldin et al.’s student ‘engagement structures’ for investigating teacher ‘engagement structures’: Some preliminary analyses
University of East Anglia, School of Education
As part of PhD research into mathematics teachers’ emotional engagement, I am exploring means of addressing the complexity of classroom interactions, whilst incorporating teacher beliefs. Goldin, Epstein, Schorr, and Warner (2011) propose ‘engagement structures’ which are suggested as a theoretical model for framing analysis of the complex nature of affect for students of mathematics. In this paper, I examine whether this ‘engagement structures’ construct can be appropriately and usefully adapted for secondary mathematics teachers. I then discuss the fit of emerging affective characteristics to each of the strands outlined by Goldin et al. (2011). The teacher data includes a pre-observation career story; videoed observations of lessons; and post-observation discussion of video extracts where the teacher recalls emotions. If transference proves useful, then linking teacher and student ‘engagement structures’ could support detailed examination of classroom interactions.
In this paper, I report early findings on characteristics of university mathematics teaching in small group tutorials, and in particular, I focus on a lecturer’s use of generic examples in tutoring. The characterization draws on data from single tutorials of 26 lecturers and on data from a systematic study of tutorials of 3 of the 26 lecturers for more than one semester. A teaching episode has been selected from this data as a paradigmatic case to illuminate the lecturer’s use of genericity across examples, supported by data from interviews in which her underlying considerations emerge.
16 Remarks on (dis)connectivities across institutional, sociocultural and discursive approaches to research in university mathematics education
Elena Nardi1, Irene Biza1, Alejandro González-Martín2, Ghislaine Gueudet3 and Carl Winsløw4
1University of East Anglia, UK;2Université de Montréal, Canada;3Université Européenne de Bretagne, France;4University of Copenhagen, Denmark
In this paper we present the work generated during and after a Working Group session at the BSRLM Conference of March 1st, 2014 entitled Institutional, sociocultural and discursive approaches to research in (University) mathematics education: (Dis)connectivities, challenges and potentialities. In the session, we organised a discussion based on highlights from a Special Issue (SI) for Research in Mathematics Education (entitled Institutional, sociocultural and discursive approaches to research in university mathematics education, 16(2)) which we had just finished writing and editing, together with 20 other colleagues from 11 countries. The approaches covered by the SI papers are: Anthropological Theory of the Didactic; Theory of Didactic Situations; Instrumental and Documentational Approaches; Communities of Practice and Inquiry; and, Theory of Commognition. The papers present recent cutting edge research on several aspects of university mathematics education: institutional practices, analysis of teaching sequences, teacher practices and perspectives, mathematical and pedagogical discourses, resources and communities of practice. In the WG session we invited participants to generate university mathematics education research questions in a small group discussion, and then address these to the whole group in order to discuss how different issues could be dealt with by the different approaches covered by the SI. Our overall aim was to explore how these approaches may offer complementary, overlapping and in some cases diverging or even incommensurable points of departure for dealing with such questions. The participant small groups generated the following list of questions: (1) How can issues of equity and gender be explored by the frameworks presented in the SI? (2) What are the praxis and logos in different courses (e.g. in pure and applied mathematics)? (3) What are the distinct differences of the didactic contract in different courses (including those other disciplines with a strong mathematical component)? (4) What communication practices can we discern in students’ writing? In this paper, we present short answers from each framework to a (slightly amended version of) one of the research questions asked by the WG participants, namely (4). To this purpose, we first outline how we developed a more detailed question based on (4), which, for the purposes of this paper, will act as a common Research Question. We then use this as a platform on which to illustrate the potentialities of the frameworks presented in the SI. We conclude with a few thoughts on ways forward of this work.
Andy Noyes1, Jeff Evans2, David Pepper3, Jeremy Hodgen3
1University of Nottingham,2Middlesex University,3Kings College London
This working group has been meeting for about a year at BSRLM day conferences. We focus on two surveys discussed at the March 2014 conference: PIAAC (aka Survey of Adult Skills) and PISA. Jeff Evans outlines some basic points to look for in the methods used in educational surveys and illustrates these in relation to the PIAAC adult skills survey results. He argues that we need to distinguish three aspects of validity: construct validity, internal validity, and external validity. David Pepper and Jeremy Hodgen outline the OECD validation of PISA and argue that it is not sufficient for the proposed high stakes use of the assessment. They focus on PISA’s assessment of student confidence in mathematics.
King’s College, London
The processes for developing, sharing, and using models – idealised, partial, purposeful descriptions – in the workplace and everyday life are reviewed. They seem to be intuitive. But we are reluctant to replace our out-of-date models, leaving this to the next generation. We live in an era of the unprecedented rate of change with the corresponding need for model-making on an increasing scale and so the possibility of enhancing, in secondary education, the intuitive modelling skills of the next generation should be investigated. As a first step, a controlled experiment in learning mathematical model-making at secondary level, by means of one-day investigative workshops is proposed.
19 Coaching: What do primary teachers perceive as the effective elements of a specialist-coaching approach when developing their classroom practice in mathematics?
This paper reports on a small-scale study, based in a primary school in south-east England, that sought to provide insight into specialist coaching as a model for teacher professional learning by researching its effectiveness in a sustained mathematics development project. The project took the form of regular specialist coaching sessions for every teacher combined with whole-school training, both of which were delivered by the consultant who later became the researcher and author of this paper. The principal driver for the improvement of teaching and learning was the outcome of an Ofsted inspection, with the school being given ‘Notice to Improve’. Informed by the research literature on the specialist coaching approach, coaching as a model of Continuing Professional Development, primary teachers’ attitudes towards and views of mathematics and Professional Learning Communities, a case-study approach, focused on three teachers was adopted and semi-structured interviews used to collect data along with documentary evidence from the Ofsted reports. The picture that emerged is complex. The analysis resulted in a set of guidelines for the specialist coach and for schools. The possible potential of the approach and its subsequent ability to support teacher professional learning in the twenty-first century became apparent.
20 Exploring academic achievement in mathematics and attitudes towards mathematics: The role of Bourdieu’s elusive habitus.
School of Sport and Education, Brunel University, London, UK
This paper reports on the quantitative part of my doctoral study carried out in three state-funded secondary schools in England. Using Bourdieu’s trilogy of habitus, embodied cultural capital and field, I have explored the interaction of social class, gender, ethnicity and attitudes towards mathematics, and their impact on mathematical achievement. The findings suggest that although cultural capital correlates positively with mathematical achievement, social class and habitus, which has been operationalised as mathematics-related career aspirations, have a more significant impact on mathematical achievement.
The national numeracy strategy (NNS) (DfEE, 1999) in England promoted mental calculation skills, built on a strong ‘number sense’, developed throughout primary schooling. The new national curriculum (DfE, 2013) places emphasis on formal algorithms for fluency in the calculation. At a time of transition, this paper explores three contrasting theoretical perspectives on number sense: cognitive psychology situated cognition and Bourdieusian social theory. It is suggested that cognitive theories dominate the teaching literature, while limited attention has been paid to social perspectives in this area. From this position, it is proposed that number sense acts as a gatekeeper to wider mathematical opportunity.
22 Exercises in mathematical imagining: setting out a teaching instrument that evokes imaginings and utilises visualisations in secondary school mathematics
University of Applied Sciences Northwestern Switzerland, School for Teacher Education
This paper sets out a teaching instrument which could be referred to as ‘mental imagery exercises in mathematics’ or ‘exercises in mathematical visualisation’. Originally developed in the context of the teaching principles of the German-speaking countries, and then based on theories of the German mathematics education literature, the paper will reframe this task design and conclude that the term ‘exercises in mathematical imagining’ is most appropriate. The fact that imagining, i.e. mentally forming and manipulating images, is personal means that this approach enables students to experience mathematics as commencing in their own minds. Teachers are also simultaneously able to gain insight into their students’ thought processes through this approach. In contrast with mental arithmetic, this task design does not focus on training a particular ability. Rather it promotes a specific heuristic strategy, within which students first conceive of and imagine a mathematical topic, and then construct and explore visualisations in order to understand their implications. This paper should be read as a case study of the researcher’s day-to-day teaching. It makes explicit a practice that has for many years produced positive results in terms of students developing mathematical understanding.
23 Planning for active participation in mathematics: promoting democratic practices in mathematics classrooms.
Peter Winbourne1 and Suman Ghosh2
1London South Bank University, 2Institute of Education, London
We believe students’ participation in school activities should be democratic and that this can best be achieved by planning for ‘authentic’ mathematical activity, which is characterised by the way in which students and their teachers work together mathematically. Students have the opportunity ‘to pose their own problems, make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs’ and to engage in mathematics as ‘art of explanation’ (Lockhart, 2008, p.5). We make use of ‘Big Ideas’ as a tool for shifting the object of activity in the mathematics classroom to participation in the authentic mathematical activity. This report draws on data from an EU-sponsored research project, ‘Awareness of Big Ideas in Mathematics Classrooms’, and a small-scale follow-up project with Secondary Mathematics PCGE students at London South Bank University.