Proceedings of the Day Conference held at the University of Cambridge on 17 Nov 2012
01 A functional taxonomy of multiple representations: A tool for analysing Technological Pedagogical Content Knowledge
Hatice Akkoç* and Mehmet Fatih Ozmantar
Marmara University and Gaziantep University
This study investigates the development of prospective mathematics teachers’ use of multiple representations during teaching in technology-rich environments. Forty prospective teachers took part in a teacher preparation programme which aims to develop technological pedagogical content knowledge (TPCK). As part of this programme, prospective teachers participated in workshops during which the TPCK framework was introduced focusing on function and derivative concepts. Various components of TPCK were considered. This study investigates one particular component of TPCK: knowledge of using multiple representations of a particular topic with technology. The content we focus on in this paper is the ‘concept of radian measure’. Two out of forty prospective teachers introduced the concept of radian measure as part of their micro-teaching activities. The data obtained from semi-structured interviews, videos of prospective teachers’ lessons, their lessons plans and teaching notes was analysed to investigate prospective teachers’ knowledge of representations and of connections established among representations using technological tools such as Cabri Geometry software. We use the framework of ‘functional taxonomy of multiple representations’ which differentiates three main functions that multiple representations serve in learning situations: to complement, constrain and construct. We discuss the educational implications of the study in designing and conducting teacher preparation programmes related to the successful integration of technology in teaching mathematics.
02 Coverage of topics during a mathematics pedagogy module for undergraduate pre-service primary teachers
Yahya Al Zahrani and Keith Jones*
University of Southampton
Recently, research on teacher preparation has begun examining the opportunities to learn that pre-service teachers have of the different forms of knowledge thought to be necessary for effective teaching. This paper reports on one component of a wider study of undergraduate pre-service specialist primary mathematics teacher preparation: the pre-service teachers’ opportunities to learn about the primary school mathematics curriculum during a final-year undergraduate module on mathematics pedagogy (MPM). Using data from observations of the complete teaching of this module at two university colleges in Saudi Arabia, the findings indicate that while the pre-service teachers had some opportunity to learn about teaching aspects of the primary school geometry curriculum, they had little or no opportunity to learn about teaching topics related to the algebra taught in the upper primary school years. The main reason for this discrepancy was that while the MPM contained some sessions on primary school geometry, there were no sessions explicitly related to primary school algebra even though the current version of the relevant primary school curriculum now includes some algebra for Grades 5 and 6 (pupils aged 10-12).
03 Rethinking partnership in initial teacher education and developing professional identities for a new subject specialist team which includes a joint school-university appointment: A case study in mathematics
Rosa Archer, Siân Morgan and Sue Pope*
University of Manchester
In a time of rapid and extensive change in initial teacher education policy, a new team of mathematics educators is establishing at the University of Manchester. How does a new team of mathematics educators (some with experience of other institutions) establish itself and ensure that previous strengths and successes are maintained and developed? One member of the team is a joint school-university appointment. What are the affordances of a joint school-university appointment? What are the personal challenges for the appointee and colleagues working with the appointee â€“ in school and in university? Evidence for the paper is through personal reflective accounts, focus group discussions with school and university colleagues, an anonymous questionnaire of student teachers and their course outcomes. The outcomes of this early experience have implications for the developing practice of the University of Manchester PGCE mathematics team and the way in which university and school based colleagues work together to optimise learning for beginning teachers, as new models of ITE are adopted within a well-established partnership. These implications may provide areas for consideration by institutions rethinking partnership in initial teacher education.
Michal Ayalon* and Ruhama Even
Weizmann Institute of Science, Israel
This study compares students’ opportunities to engage in argumentative activity between two classes taught by the same teacher and across two topics in beginning algebra: forming and investigating algebraic expressions and equivalence of algebraic expressions. The study comprises two case studies, in which each teacher taught two 7th grade classes. All four classes used the same textbook. Analysis of classroom videotapes revealed that the opportunities to engage in argumentative activity related to forming and investigating algebraic expressions were similar in each teacher’s two classes. By contrast, substantial differences were found between one teacher’s classes with regard to the opportunities to engage in argumentative activity related to equivalence of algebraic expressions. The discussion highlights the contribution of the topic, the teacher, and the class to shaping argumentative activity.
Alison Borthwick* and Micky Harcourt-Heath
In 2006, 2008 and 2010 we collected and analysed answers from a Year 5 QCA test paper to explore the range of calculation strategies used by a sample of approximately 1000 Year 5 children. Once again in 2012 we have repeated this research using the same group of 22 schools. This paper explores the findings from the 2012 data, including case studies. It examines the range of strategies used by the children. We conclude by considering if and how the use of particular calculation strategies has impacted on the overall results and we ask if this shows greater clarity about which strategies lead children to success.
06 Relentless consistency: Analysing a mathematics prospective teacher education course through Fullan’s six secrets of change
University of Bristol, Graduate School of Education
In the leadership of change literature, Michael Fullan’s work is influential. He has developed theories about the process of working rather than the content of that process. The work of a mathematics teacher educator could be seen as leading change for a group of prospective teachers. This paper aims to use Fullan’s ‘six secrets of change’ to analyse the structure of the mathematics education aspects of the one-year University of Bristol Post-graduate Certificate of Education (PGCE) course, to gain insight into both practices that illustrate Fullan’s ‘secrets’ and possible developments to the course given aspects of the secrets not in evidence. Fullan’s idea of ‘relentless consistency’seems to fit with the way the prospective teachers evaluate strengths of the course.
Maria Chionidou-Moskofogloua*, Georgia Liarakoua, Efstathios Stefosa, Zoi Moskofogloub
aUniversity of the Aegean- Rhodes Greece, bUniversity College London
A game called Euro-Axio-Polis was constructed by students of the Aegean University aiming to promote teaching and learning on mathematics and sustainability for 6th grade pupils. 40 students played Euro-Axio-Polis and Monopoly to investigate differences between the two games, and wrote five key words that characterized each game. Also 19 sixth grade pupils played the Euro-Axio-Polis game during students’ teaching practice and wrote five key words about the game. The research results suggest that Monopoly reflects capitalist economic terms and social values while Euro-Axio-Polis reflects social values associated with sustainable development such as solidarity and equity. Pupils were more likely than students to make reference to socio-political issues such as parliament, education, democracy, elections and political power. As far as mathematics is concerned, most students and half of 6th grade pupils recall the mathematical concepts percentages and interest rates while they played Euro-Axio-Polis.
Institute of Education, University of London
Following on from my observations of the inconsistencies and misuse of the radical symbol amongst pupils, undergraduates, teachers and some authors of school textbooks, I became interested in those decisions that teachers take when confronted with inaccurate or ambiguous representations of the square root concept and its associated symbol notation. The impact that the ambiguous treatment of this mathematical concept and its associated symbol notation has on a number of PGCE students’ conceptual understanding and pedagogical affinity will be discussed.
Sue Cronin and Denise Hardwick
Liverpool Hope University
We share an emerging pedagogy for Initial Teacher Education (ITE) mathematics tutors who are seeking new ways to work with student teachers in what Zeichner (2010) defines as hybrid spaces. In terms of Initial Teacher Education, hybrid spaces are those spaces which are formed to ‘bring together school and university based teacher educators and practitioners and academic knowledge in new ways to enhance the learning of prospective teachers’ (92). For the last three years the PGCE secondary mathematics programme in the authors’ university has included a Saturated Learning Project (SLP). This has involved taking all of the secondary mathematics students into school one morning for each of ten weeks to work with groups of pupils in a shared communal space, supported by class teachers and university tutor. The project has now been extended to the PGCE primary course with ten student teachers specialising in mathematics. They also worked over a number of weeks with a group of Y6 pupils. The experiences in such hybrid spaces enriched and extended students’ practical and pedagogical knowledge by facilitating understanding of theories about teaching and learning mathematics in a real, shared context. This new pedagogical approach is strengthening school-university partnership and improving learning experiences for both student teachers and their pupils.
10 From failure to functionality: a study of the experience of vocational students with functional mathematics in Further Education
University of Nottingham, UK.
Many students who undertake vocational courses in Further Education colleges in England enter post-compulsory education as mathematical ‘failures’ at GCSE level but their experience in college has the potential to change not just their attainment, but also their future attitude and ‘functionality’ with mathematics in employment and society. This paper outlines the early stages of a mixed methods study to identify the main influences on the student experience and their effects on the aspirational trajectory from ‘failure’ to ‘functionality’.
Paul Dickinson* and Sue Hough
Manchester Metropolitan University
At Manchester Metropolitan University, approximately eighty students each year qualify to become teachers of secondary mathematics. Of these, roughly half do not have a mathematics degree, but have studied on a Subject Knowledge Enhancement (SKE) course. This research study is concerned not with the pure mathematical knowledge of such trainees, but with the nature of their knowledge. Asking them relatively routine questions on fractions showed almost all trainees reaching for a known procedure to answer the questions. Furthermore, when asked how they knew they were correct, most trainees used the procedure as the authority for this. The trainees then studied the teaching of fractions, after which they taught the topic in school. This paper focusses on the first part of the study, which analyses the trainees’ own knowledge of fractions. A later paper will report on the classroom work of the trainees.
Ceneida Fernándeza*, Alf Colesb, Laurinda Brownb
aUniversity of Alicante (Spain); bUniversity of Bristol
In this paper, we report on our analysis of four transcripts of teacher meetings that took place over the academic year 2011-12. These meetings took place in the context of a project looking into tackling underachievement in primary mathematics through a focus on creativity. We bring the idea of growth indicators (Jacobs, Lamb and Philipp 2010) within the framework of noticing (Mason 2002) in order to analyse shifts in teacher discourse. There is evidence of growth but we conclude by discussing the complexity of teacher change and problems with any set of indicators.
Thurston Community College/University of Cambridge
Recent developments in mathematics education place an unprecedented emphasis on the role of discourse in developing students’ conceptual understanding, with a corresponding de-emphasis on the use of ‘telling’ : the stating of facts and demonstration of procedures. This action research study investigated teacher-student dialogue during one-to-one interactions in my post-16 mathematics classroom. The participants were four A-level students. Data sources included clinical interviews, student feedback interviews and an analytical log; and the data were coded using a framework of scaffolding categories drawn largely from current research literature. The findings suggest that, although I utilised more ‘telling’ than ‘questioning’ interventions, often these ‘telling’ actions served useful and necessary functions. They also indicate that my scaffolding skills developed as a result of the process of critical analysis; and that the scaffolding strategies valued by my students were those that they felt best promoted their independence. The study concludes by suggesting that context is a crucially important factor in addressing the dilemma of whether or not to tell.
14 Using scenes of dialogue about mathematics with adult numeracy learners â€“ what it might tell us.
Institute of Education, University of London
The study concerns the use of prepared dialogue scenes involving mathematics with groups of adult learners. It is intended to consider how we might characterise discussion following the reading of scenes of dialogue. The article outlines some examples of scenes and the response from the use of these in an early exploratory phase with some adult learners intending to become teaching assistants. A discussion of the scenes and responses leads to some conclusion about the characteristics of more appropriate scenes for the main study.
Guðný Helga Gunnarsdóttir*, Jónína Vala Kristinsdóttir and Guðbjörg Pálsdóttir
University of Iceland â€“ School of Education
Icelandic student teachers’ professional development starts at the onset of their initial teacher education. We have studied our teaching as teacher educators with a focus on the development of learning communities and reflective practices that are considered important elements of effective professional development. Our studies have given us some guidelines to work with and strengthened our beliefs on the importance of collaboration and discussions.
16 Engaging students with pre-recorded ‘live’ reflections on problem-solving: potential applications for ‘Livescribe’ pen technology
Faculty of Education and Theology, York St John University
Building on the author’s PhD work with part time postgraduate (PGCE) primary student teachers, this paper considers the potential application of Livescribe pen technology to facilitate/support reflection on collaborative mathematical problem solving, allowing opportunities for participants to engage in ‘live’ reflection on their ‘free’ problem solving performance in order to elicit reasoning/effective strategies and thereby inform their future practice. With recorded (group) thinking aloud, followed and supplemented by a stimulated recall/task-based interview opportunity and associated problem solving/talk framework, participants are encouraged to articulate their problem solving strategies, experiences and understanding with the benefit of potentially reduced influence from the researcher. The risk of think-aloud protocols impacting negatively on problem solving performance is arguably reduced by the use of a technology that allows the ‘replay’ of participants’ workings/jottings alongside their verbal contributions.
Institute of Education, University of London
In university mathematics education courses, messages about the pedagogy and content of teaching mathematics are conveyed to student teachers. During the teaching practicum, mentor teachers also have their own set of messages about mathematics teaching. My research investigates the messages conveyed to student teachers and the ways student teachers acknowledge these messages and incorporate them into their teaching using the notion of recontextualisation. The use of information and communication technology (ICT) in teaching mathematics is generally viewed positively in the university and by mentor teachers. In this paper I share some data and analysis of the messages about ICT, and how one student teacher recontextualises these messages into his own teaching of mathematics.
Loughborough University, Mathematics Education Centre
Research into the teaching of mathematics to engineering students to promote their conceptual understanding (Jaworski and Matthews 2011) has shown the problematic nature of planning for and identifying understanding. I review the project briefly and introduce the idea of competencies from the Danish project, KOM (translated as Competencies and Mathematical Learning). Through the medium of designing inquiry-based tasks for students and use of the competency framework for analysis of tasks, I consider the relevance of such a competency-based analysis and its usefulness (or otherwise) for recognising student understanding. This leads to important questions for further research of a developmental nature.
Dr Cecilia Kilhamn
Faculty of Education, University of Gothenburg, Sweden
Swedish students have not been successful in solving geometrical pattern tasks in the TIMSS study and as a result it has been introduced as explicit core content in the National Syllabus (Lgr11) for grades 1-6. Analysis of video recordings of three student groups working with a task taken from TIMSS07 showed that students’ initial approach to the task was often unsuccessful. In this situation it was then a call for justification that led them on, for example through questioning why a solution was correct or what the answer meant. The call for justification came from the teacher, from other students or from a student’s wish to understand. An implication of this study is that students would benefit from incorporating justification as an essential part of their problem solving process.
Oslo and Akershus University College of Applied sciences
In this paper I take as a starting point social inequalities and pupils’ different learning possibilities as a result of their social background, and consider mathematics on three levels: The level of Discourse, which primarily encompasses cultural relations and communities of meanings in school; the level of genre which concerns recognizable common cultural texts and the frames of reference which support their understanding, and finally, the level of paradigmatic and syntagmatic modes of thought which are necessary for learning within mathematics. My argument is that in order to decrease the school’s reinforcement of social inequalities, teaching should be based on meta-awareness rather than acquisition through pupils’ activities.
21 Stimulating an increase in the uptake of Further Mathematics through a multifaceted approach : Evaluation of the Further Mathematics Support Programme.
Stephen Lee* and Jeff Searle
Mathematics in Education and Industry and Durham University
Over recent years there has been a marked increase in the number of students studying A-level Further Mathematics in England. In 2012 12,688 students sat the qualification, with the numbers having more than doubled from 5,627 in 2005 (Joint Council for Qualifications figures). The increase has been evident despite the common perception that Further Mathematics is a difficult subject. The work of Mathematics in Education and Industry’s (MEI) government-funded Further Mathematics Support Programme (FMSP) has been highly influential in stimulating this increase through not only enabling all students who wish to study Further Mathematics to have access to tuition, but also through supporting teachers and students in schools and colleges in a variety of ways. An external evaluation of the FMSP has been undertaken by the Centre for Evaluation and Monitoring at Durham University. This paper reports on aspects of the evaluation and how these relate to the multifaceted approach taken by the FMSP to increase participation in Further Mathematics, including: innovative tuition models, enrichment events, extensive provision for teachers to undertake professional development and also an insight into direct attempts by the FMSP to engage with schools and colleges who have not traditionally offered the subject.
University of Oxford and Open University
I propose that exchange is a core idea underlying much of school mathematics. Alerted by young children struggling with the difference between coins as objects and coins as having value, I began to explore the action of exchanging one thing for another. If exchange is augmented to include substitution then it shows up everywhere, from counting to algebra, from money to currency, from ratio to algorithms and Turing machines.
John Monaghan and John Mason
University of Leeds; University of Oxford and Open University
About 10 years ago the Gibsons’ notion of ‘affordance’ was extended to cultural objectives underlying designed computer systems. Chiappini (2012) extends this idea to mathematics software. We critically, but respectfully, review these extensions â€“ does ‘cultural affordance’ add anything new to valuations of software for doing mathematics?
24 Doing the same mathematics? Exploring changes over time in students’ participation in mathematical discourse through responses to GCSE questions
Candia Morgana*, Sarah Tanga, Anna Sfardb
aInstitute of Education, University of London, UK; bUniversity of Haifa, Israel
The project ‘The Evolution of the Discourse of School Mathematics’ uses the lens of GCSE examinations to investigate changes over the last three decades in what is expected of students in England. We have identified differences in the discursive features of examination questions through this period and now seek to investigate how these differences may have affected the nature of student participation in mathematics discourse. Students have been tested using questions varying in characteristics typical of different points in time. We discuss the design of the test, and present some preliminary results.
King’s College, London
Throughout the last century the mathematics of the continuum underpinned the science and technology of the developed world. Today’s developed world is increasingly dominated by the artefacts and processes of information technology and it is discrete mathematics that underpins this technology. A finite state machine description of the behaviour of vending machines, in the form of state transition diagrams and state transition tables, is used as an example to demonstrate that modelling numerous artefacts of today’s everyday world would be within reach of many 15-19 year old learners if the curriculum were to give more emphasis to discrete mathematics.
26 Gendered styles of linguistic peer interaction and equity of participation in a small group investigating mathematics
Anna-Maija Partanen and Raimo Kaasila
Ã…bo Akademi University and University of Oulu, Finland
In a teaching experiment with two Finnish upper secondary classes, the basics of calculus were studied using an investigative approach and a small-group setting. As part of the ethnographic teacher research, the different styles of talking of the girls and boys in four groups were analyzed through application of the concept of sociolinguistic subcultures. This paper focuses on the interactions in one of the groups where two girls and a boy discuss mathematics. We show how the linguistic strategies typical of these boys prohibited the full potential of the contributions of the girls to be utilized in the collective construction of meaning in the group. Promoting democratic discussions in small groups may need attention in terms of gendered ways of interacting.
Beauty has been discussed since ancient times, but discussions of beauty within mathematics education are relatively limited. This lack of discussion is surprising given the importance of beauty within the practice of mathematics. This study explores one particular metaphor of beauty, that of beauty as fit, as a way to distinguish between proofs that are considered beautiful and those that are not. Several examples are examined, supported by empirical data of mathematicians and mathematics educators who judged and ranked different proofs in a seminar on mathematical beauty.
Sør-Trøndelag University College and Norwegian University of Science and Technology, Trondheim, Norway
This presentation is based on a study of 20 pupils, aged 9-10, in a Norwegian primary school. The pupils were exposed to two, rather different, classroom situations and in both situations the concept of fraction was central. The pupils were given tasks and in order to accomplish these tasks it was necessary to make sense of fractions in some way. An interesting observation is how the presence of different mediating artefacts influences the pupils’ meaning making.
Dr Sashi Sharmaa*, Phil Doyleb, Viney Shandilc and Semisi Talakia’atuc
aThe University of Waikato; bThe University of Auckland; and cMarcellin College
Despite statistical literacy being relatively new in statistics education research, it needs special attention as attempts are being made to enhance the teaching, learning and assessing of this strand. It is important that teachers are aware of the challenges of teaching and assessing of this literacy. In this collaborative research study, two cycles of teaching experiments were carried out in two year 9 classes. The data set consisted of audio and video-recordings of classroom sessions, copies of students’ written work, audio recorded interviews conducted with students, and field notes of the classroom sessions. The results shed light on tools and techniques which the research team used to help students develop critical statistical literacy skills. The findings have implications for teaching and further research.
Graduate School of Education, University of Bristol, UK and Bath Spa University UK
This paper reports on changes in students’ perceptions of assessment during a Mathematics Subject Knowledge Enhancement Course (MEC). Students’ views were gathered pre- and post-MEC via an open-question questionnaire with semi-structured interviews for some. Pre- and post- MEC understanding of mathematics features highly in the students’ sense of progress, but few had experienced feedback prior to the MEC. Post-MEC feedback is viewed as the most useful aspect aiding their sense of progress.
Tracy L Weston*, Bodil Kleve, Tim Rowland
University of Alabama; Oslo and Akershus University College of Applied Sciences; University of East Anglia/University of Cambridge
This paper provides a brief overview of the work to date of an international research team that has worked together since Fall 2011. The team members are mathematics educators and researchers who use the Knowledge Quartet (Rowland et al. 2009) in their work as researchers as a framework by which to observe, code, comment on and/or evaluate primary and secondary mathematics teaching across various countries, curricula, and approaches to mathematics teaching. The countries represented on the team include the UK, Norway, Ireland, Italy, Cyprus, Turkey and the United States. The team has developed a Knowledge Quartet coding manual for researchers which is freely available for other researchers to use. This is a collection of primary and secondary vignettes that exemplify each of the 21 Knowledge Quartet (KQ) codes, with classroom episodes and commentaries provided for each code. This work provides increased clarity on what each of the KQ dimensions ‘look like’ in a classroom setting, and is helpful to researchers interested in analysing classroom teaching using the KQ. This paper provides an overview of the Knowledge Quartet, describes the working methods of the team and offers examples of classroom vignettes that exemplify two of the codes as an indication of what can be found on the coding manual website.
Zeynep Yildiza*, Hasan Unala, A. Sukru Ozdemirb
aDepartment of Elementary Education, Faculty of Education, Yildiz Technical University; bDepartment of Elementary Education, Faculty of Education, Marmara University
In this study, the purpose was to reveal thinking styles and different points of view of pre-service primary school teachers about the concept of ‘rotation’ in mathematics. The study was conducted with undergraduate students who are studying in the department of primary school teacher education. The subject of ‘rotation’ in this study has two sub-topics which are rotation of points around a point and rotation of shapes about a point in a coordinate plane. A test about rotation was applied to 44 students and then interviews were made with 5 students. Results of the study include an analysis of correct and incorrect answers of students.
Working Group Reports
Nichola Clarkea*, Maria Chionidou-Moskofogloub, Zoi Moskofoglouc, Alison Parrishd, Anna-Maija Partanene
aUniversity of Nottingham, UK; bUniversity of the Aegean-Rhodes Greece; cUniversity College, London; dWarwick University UK; eAbo Akademi University, Denmark.
The Sustainability in Mathematics Education Working Group discusses research on how to integrate learning about climate change and sustainable living with the learning of mathematics. In the third group meeting, participants from Denmark, Greece and the UK focused on the design of cross-curricular tasks for the simultaneous learning of mathematics and sustainability issues. We drew on examples of task design experiences from Greece and the UK.
Alf Coles* and Yvette Solomon
University of Bristol and Manchester Metropolitan University
In this paper, we report on the discussion and issues raised at the working group session at the day conference in Cambridge.