Manahel Alafaleq and Lianghuo Fan
University of Southampton, UK
Assessment is an essential process for gathering information about students’ learning and achievement. This process should be integrated with learning and teaching to establish ways for teachers to understand their students’ learning and make informed decision about their instruction. In this paper, our focus is on a new approach to mathematics assessment in Saudi Arabia, which has been implemented recently. The new assessment approach is essentially a criterion-referenced assessment which aims to support students’ learning rather than measuring their progress solely. It is employed more as part of the students’ learning process. We explain why the new assessment approach is introduced, what it is, and how teachers deal with it. Moreover, we also discuss the challenges and implications of implementing new assessment approaches to mathematics teachers, educators and policy makers.
The Programme of International Student Assessment (PISA) and Trends in International Mathematics and Science Study (TIMSS) create much international interest in those countries perceived as high achieving. One such system, rarely acknowledged, is Flanders, the Dutch-speaking region of Belgium. In this paper I present the results of focused analyses of four sequences of video-taped mathematics lessons taught to students aged 10 to 14 years. These confirmed a mathematics education tradition drawing on two well-known curricular movements. The first presents mathematics as a Bourbakian set of interconnected concepts. The second exploits realistic problems in its presentation of mathematics.
Michal Ayalon, Steve Lerman* and Anne Watson
University of Oxford; *London South Bank University
We focus on a ‘typical’ task in which students have to give a functional generalisation in algebraic form of a growing sequence of spatial structures. We analyse the contribution of this task to a coherent knowledge of functions. 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 in two different curriculum systems: the UK and Israel. The research uses a survey instrument that was developed in collaboration with a group of teachers and the task for this paper is one of six that span several routes to understanding functions. Our data appears to contradict some other studies as well as to suggest conjectures about how students’ willingness to use covariational reasoning depending to some extent on task features.
04 Using a digital tool to improve students’ algebraic expertise in the Netherlands: crises, feedback and fading
University of Southampton
Enhancing ways of developing students’ algebraic expertise remains an important focus for research. This paper reports on a design research study which involved a digital intervention for 17-18 year old students, implemented in nine schools in the Netherlands (N=324). For the intervention, algebra tasks for the conceptual and procedural components of algebraic expertise were placed in a sequence based on three design principles: (i) ‘crisis’ items that intentionally questioned the use of standard algorithms, (ii) feedback provided by the digital system, and (iii) the ‘fading’ of feedback during the sequence to increase transfer. Data collected included results from student pre- and post-tests, questionnaires, and scores and log files of their digital work. Results from the study show that the intervention was effective in improving algebraic expertise, and that the aforementioned design principles have merit. This paper reports on the effects and illustrates the design principles through a case example. The intervention shows a significant effect in improving algebraic expertise. It shows that well-thought-out design principles augment learning. The paper fits in a broader discussion on how to integrate algebraic expertise and ICT use in the classroom through the use of educational design.
Department of Education and Professional Studies, King’s College London
This paper concerns a philosophical, but highly practical, issue arising at the interface between mathematics and education, and I claim that mathematics education offers insights where mathematical philosophy has ground to a halt. More specifically it concerns the two related but distinct concepts of how-many-ness (alias quotity) and rank order, whose separate identities are traditionally obscured by the language of ‘number’. (Sometimes they are called ‘cardinal number’ and ‘ordinal number’. More often they are wrapped up together as ‘natural number’.) The teacher of young children has the advantage over the philosopher that she works with people before they have acquired all the prejudices of their native language, and we shall build on the analysis of counting by Gelman and Gallistel (1978), concluding that ‘natural number’ as normally conceived is something of an illusion, for only quotity has the properties expected of ‘number’, while rank order is a mere quality.
Melise Camargo and Kenneth Ruthven
University of Cambridge, Faculty of Education
Classroom-based assessment has been a matter of concern and discussion in academia, especially in recent years. Many studies have been conducted, particularly about the implementation of formative assessment. Although it has been heralded as an important practice, there is still little research about this subject related to mathematics education, particularly in Brazil. Aiming to seek information about the types of approach that secondary-school mathematics teachers in Brazil have been taking in their classrooms, survey research was conducted via an e-questionnaire. The teachers were asked, among other aspects, about the frequency with which they apply and the importance they give to specific assessment methods or procedures. The results from the quantitative analysis show that tests and homework assignments are the methods most commonly used by mathematics teachers, whereas self- and peer-assessment are still not common practice.
Ann Marie Casserly, Pamela Moffett and Bairbre Tiernan
St Angela’s College, Sligo; Stranmills University College, Belfast; St Angela’s College, Sligo
Teacher-facilitated “math talk” in the early years significantly increases children’s growth in understanding of mathematical concepts (Klibanoff et al., 2006). Although young children may have a beginning understanding of early number concepts, they often lack the language to communicate their ideas. Teacher modelling and fostering of mathematical language throughout the day and across various subject areas, allows children to articulate their ideas and communicate their understanding. Encouraging “math talk” in young children as they explain, question and discuss their strategies is important. The teacher plays a significant role in guiding children to make connections, to recognise how their thinking relates to key mathematical number concepts and to make further conjectures and generalisations. This paper will outline the theoretical perspectives underpinning the development of a resource of key vocabulary and teaching and learning strategies for teachers to support their planning and teaching in early number.
Tandi Clausen-May and Remegious Baale
National Curriculum Development Centre, Kyambogo, Kampala, Uganda
The Ugandan secondary school mathematics curriculum was established in colonial times to serve a small, select minority of academic high-achievers, and it is delivered with a ‘dominant pattern of expository, whole-class teaching’ (Centre for Global Development through Education 2011). But following the introduction, in 1997 and 2007, of Universal Primary and Secondary Education Policies the curriculum has become increasingly irrelevant and inaccessible to the majority of learners. How can mathematics education be reformed to make it more appropriate to the modern Ugandan context, with large, mixed-ability classes and very limited resources? In an effort to increase access and improve learners’ performance in the first four years of secondary education, the Ugandan National Curriculum Development Centre is developing a new mathematics curriculum with a range of materials that encourage alternative teaching and learning strategies using low-cost, locally-available resources. These have been trialled in urban, peri-urban and rural secondary schools, where lessons have been observed and learners’ work has been collected and analysed. The trials indicate that learners may be more willing to adopt a new approach when the tasks are novel and unfamiliar. When they are associated with established mathematical knowledge and techniques, pedagogical change may be more difficult to establish.
Fiona Cockerham and Rob Timlin
Faculty of Education, Manchester Metropolitan University
In this paper we discuss the difficulties and tensions currently facing our initial teacher training (ITT) programme in secondary mathematics at Manchester Metropolitan University, and outline the pilot we have trialled to try to address some of these issues. We have called the schools we have worked with for this pilot ‘University schools’. This model of teacher training is currently in its third year of development in 10 schools; it has been evaluated throughout this period via interviews with student teachers, weekly feedback from student teachers in the form of reflections, lesson observations, discussion with MMU staff involved in the programme, and discussions with teachers in the pilot schools. The paper outlines our findings and then draws conclusions about the success of this training model.
10 The case of the square root: Ambiguous treatment and pedagogical implications for prospective mathematics teachers
Institute of Education, University of London
I report on a small-scale study rooted in the UK context that was conducted with eight volunteers from a cohort of PGCE secondary mathematics students (participants). The participants’ own understanding of the square root concept and use of the associated symbol were explored and the findings revealed that they may not possess adequate subject knowledge about and for teaching this concept. Access to instructional materials, mainly textbooks and discussions with other more experienced teachers were identified as the main external sources consulted by the participants in order to refresh their knowledge of the square root concept. During this study, those participants who became aware of the shortcomings of their conceptual understanding of the square root felt, at first, uncomfortable with modifying their personal knowledge and their long held beliefs about this concept. Group discussion helped most of the participants become aware of connections between their more advanced knowledge of mathematics and the square root concept. Such awareness empowered the participants to clarify this concept for themselves and critically scrutinise the (re)sources available. A tension between employing their modified knowledge about the square root and adherence to the widely accepted view about this topic in school mathematics has also been identified.
Similar to other curricula, the Swedish mathematics curriculum emphasises problem solving both as an end in itself and as a means to becoming a competent citizen. Thus, a goal for mathematics is the creation of a problem-solving citizen. In this paper, I explore how critical discourse analysis, and parts of social activity theory can be used to operationalise Bernstein’s pedagogic device in relation to the construction of the problem-solving citizen. It is proposed that critical discourse analysis can be used for a linguistic analysis of official documents, like the curriculum whilst social activity theory’s different domains, mainly public and esoteric, can be used to analyse the national tests. These tests assess students’ problem solving both as a means and an end in two different senior secondary programmes, an academic orientated one and a vocational one. In this exploration, two examples are given to show how these methodological tools can be used.
University of Nottingham, UK
This study of recent school-leavers in Further Education explores students’ opinions of relevance and how these are influenced by their experiences of different mathematics curricula in school and college. These vocational students are taught mathematics as a functional ‘tool for life’ rather than a discipline of rules. Perceptions of relevance are influenced by personal goals and interests (Ernest, 2004) and may depend on whether students identify a value for the qualification, a practical usefulness or some transferable skills (Sealey and Noyes, 2010). These can provide reasons for studying mathematics but, in this study, students who encountered mathematics as a ‘tool for life’ engaged in learning experiences that connected with their personal life experience. This changed their conceptual view of mathematics and added a different perspective to their views of relevance. The research is part of a larger study of the student experience of functional mathematics in colleges but this paper will focus on qualitative data from student focus groups and lesson observations. Transcriptions were coded and compared to identify common themes in student experiences. The results suggest how teaching mathematics as a ‘tool for life’ can influence perceptions of relevance and effect some positive changes in student attitudes.
13 Problem solving tasks in mathematics classrooms: An investigation into teachers’ use of guidance materials
University of Nottingham
This paper reports on a design research study undertaken as part of the Mathematics Assessment Project (MAP). The project aims to support teachers in implementing a new curriculum in US schools, through the use of formative assessment lessons (FALs) designed by the MAP team. Here we report results of our research into teachers’ use of the accompanying guidance materials as they implement problem-solving FALs, drawing on detailed case studies of lessons from a sample of UK teachers. Although we observe much variation in the ways in which teachers use the guidance, both when in and out of the classroom, we identify the provision of a ‘Common issues’ table, outlining likely responses from students together with advice of potential ways to respond, as one of the most valued and used aspects of the guidance materials provided.
14 Using context and models at Higher Level GCSE: adapting Realistic Mathematics Education (RME) for the UK curriculum
Paul Dickinson, Steve Gough and Sue Hough
Manchester Metropolitan University
Since 2003, staff at Manchester Metropolitan University (MMU) have been involved in a number of projects related to Realistic Mathematics Education (RME). This originally involved trialing materials with 11-14 year olds and then, in collaboration with the Freudenthal Institute, writing materials for Foundation level GCSE. In 2012, these materials were published by Hodder Education as a series of books called Making Sense of Maths. Classroom trials of the original materials showed an increased willingness of students to discuss and engage with their mathematics, and to attempt to ‘make sense’ of what they were doing rather than simply to try to remember taught procedures. The results from the trials led, in 2009, to a further project designing materials for Higher level GCSE. The focus here is on the development of these materials, and how we have interpreted the original design principles of RME for UK schools. In particular, we focus on the use of context, the notion of progressive formalisation and the use of models. We provide excerpts from the materials that exemplify these principles and discuss the issues for teachers attempting to integrate this approach into an examination driven curriculum.
Liping Ding, Keith Jones*, Birgit Pepin** and Svein Arne Sikko**
Sör-Tröndelag University College, Norway and Shanghai Soong Ching Ling School, China; * University of Southampton, UK; ** Sör-Tröndelag University College, Norway
We report on one component of a study of school-based teacher professional development (TPD) in Shanghai, China. Here we focus on an experienced primary teacher who is teaching the topic of angle measurement to 10 year-olds. Using data from the teacher’s original lesson plans, her modified lesson plans, together with an expert teacher’s advice and the teacher’s reflections on her lesson design, we illustrate how the support of an expert teacher enabled the teacher to improve her instructional practice. This was by supporting her in thinking explicitly about the traditional classroom practice with which she was familiar and in building her ‘wisdom of practice’ within the context of instructional reform taking place in China.
Education School, University of Southampton
This report leads from findings of an earlier research project in which pre-service mathematics and science teachers identified areas where they felt they needed greater support during their training year. In September 2011 a closed group was set up on the social network site Facebook as a support mechanism for a cohort of pre-service secondary mathematics teachers. The rationale was that the Facebook group, which was established as a ‘secret staffroom’ would provide a common, yet secure area where the pre-service teachers could share ideas and resources as well as engage in discussion about their progress on the course. Analysis of the data collected from the interactions that took place identifies the frequency and ways in which the social network site has been used during their training year and first year of teaching. These findings are contrasted with the results of a series of attitudinal questionnaires completed by the participants at various stages of these two years. The results indicate that Facebook can be used to support pre-service teachers; however there remains a challenge in using social networking to support teachers in their first year of teaching and beyond.
17 Acquisition of mathematical skills in trigonometrical concepts through project based learning in junior secondary schools in Calabar municipality of Cross River State, Nigeria.
Cecilia O. Ekwueme, Esther Ekon and Anne N. Meremikwu
Department of Curriculum and Teaching, Faculty of Education, University of Calabar, Calabar, CRS, Nigeria
This paper focuses on the interactive development of junior secondary three (JS3) students’ knowledge and performance in Pythagoras theorem using project-based learning skills (collaboration and critical thinking). The purpose therefore, is to acquaint the students with the necessary mathematical skills that enrich their knowledge of formation and solution of Pythagorean triple triangles. A non-equivalent pre-test post-test quasi-experimental design was used on a sample of 280 JS3 students from two private and two public schools to ascertain the knowledge level and cognitive achievement before and after exposing them to project-based learning strategy. Students were also interviewed. Special lesson plans were developed and the teachers were trained on the use of this strategy. Mean scores, standard deviations and dependent t-tests were used to analyse the data. It was discovered that students’ performance was enhanced and they were able to construct and solve Pythagorean triple triangles easily and quickly. Recommendations were made based on these findings.
18 The Role of Sample Pupil Responses in Problem-Solving lessons: Perspectives from a Design Researcher and Two Teachers
Sheila Evans, Nicola Mullins and Lucy Waring
University of Nottingham, The Joseph Whitaker School, Rainworth, Toot Hill School, Nottingham
The benefits of learning mathematics by comparing, reflecting on and discussing multiple approaches to a problem are well-known (Silver, 2005). However, teachers using non-routine problem-solving tasks designed to encourage multiple approaches face challenges: understanding how pupils make sense of the problem, especially when pupils use unique or unanticipated approaches and helping pupils make connections between disparate approaches and aligning these with lesson goals. In an attempt to address such challenges an extensive set of problem solving lessons were developed. Each lesson includes a range of sample solution-methods that expose pupils to multiple perspectives. A detailed teacher guide supports each lesson. This paper focuses on the use of these sample solution-methods. It explores their development from initial design to final versions. We analyse the varied interpretations and use made of sample solution-methods, in both US classrooms and by two UK teachers, and reflect on how these interpretations align with the designers’ intention.
Nottingham Trent University
The purpose of this study was to discover the individual and distinct ways in which each student teacher understands fractions and their strategies for working with them. A phenomenographical approach was adopted in order to provide insight into each student teacher’s subject knowledge of fractions. This study involved detailed scrutiny of six self-selected small groups, which enabled a range of rich and honestly reflective data to be collected. Groups undertook two collaborative tasks involving the sequencing of fractions by magnitude, followed by reflective interviews. Each group also undertook a diagnostic interview, considering a range of questions, which they had ordered in terms of their perceived difficulty. A constructivist perspective was adopted giving students the opportunity to reconstruct their own understanding of fractions through the explanation and discussion of their existing ideas. A range of successful strategies was demonstrated, especially the use of mathematical anchors and the use of residual or gap thinking as a means of comparison. Improper fractions and reunitising were the main difficulties cited by many in the group. A common assumption was that there was a particular ‘correct’ method to be adopted. The study helps to identify misconceptions that can be addressed within teacher training.
School of Education, University of Nottingham
It is now almost 40 years since Skemp’s (1976) seminal division of understanding into ‘instrumental’ and ‘relational’ categories, yet the current political direction of mathematics education in the UK is decidedly towards the traditional teaching of ‘standard algorithms’ (DfE, 2013). In this research paper, I draw on a lively staffroom discussion about different approaches to the teaching of quadratic equations, in which one method used was derided as ‘a trick’. From this, I discuss reasons why certain mathematical processes are often regarded as inherently and irretrievably ‘procedural’. Informed by recent theoretical interpretations of procedural and conceptual learning in mathematics, which increasingly stress their intertwining and iterative relationship (Star, 2005; Baroody, Feil and Johnson, 2007; Star, 2007; Kieran, 2013), I make a case that stigmatising particular methods and censoring their use may deny students valuable opportunities to make sense of mathematics. I argue instead that encouraging students to take a critical stance regarding the details and the value of the procedures that they encounter can cultivate in them a deeper awareness of mathematical connections and a more empowered sense of ownership over their mathematics.
University of Leicester
In several countries, there is concern about the low levels of educational attainment achieved by many children in public care. This paper outlines some of the reasons why looked after children’s average attainment in mathematics is poor. Using the case of Ronan, aged 8, I examine the experience that this child’s schools offered him in mathematics, as he moved from a school designated ‘satisfactory’ to one acclaimed as ‘outstanding’. Whilst his experience in most areas of school life improved, his mathematics lessons became less effective. I explore the mathematics he did at home with his foster carers, and note that there was little co-ordination between school and family in mathematics.
22 Improving students’ understanding of algebra and multiplicative reasoning: Did the ICCAMS intervention work?
Jeremy Hodgen, Rob Coe*, Margaret Brown and Dietmar Küchemann
King’s College London, Durham University*
In this paper we report on the intervention phase of an ESRC-funded project, Increasing Competence and Confidence in Algebra and Multiplicative Structures (ICCAMS). The intervention was designed to enable teachers to use formative assessment in mathematics classrooms by evaluating what students already knew, then adapting their teaching to students’ learning needs. A key feature was the use of models and representations, such as the Cartesian graph, both to help students better understand mathematical ideas and to help teachers appreciate students’ difficulties. Twenty-two teachers and their Year 8 classes from 11 schools took part in the intervention during 2010/11. Pre- and post-tests in algebra, decimals and ratio were administered to the students of these classes, and compared to a control group of students matched from the ICCAMS national longitudinal survey (using propensity score matching). The students in the intervention group made greater progress than the matched control.
23 Trajectory into mathematics teaching via an alternate route: A survey of graduates from Mathematics Enhancement Courses
Sarmin Hossain1, Jill Adler2, John Clarke3, Rosa Archer4 and Mary Stevenson5
1Brunel University London, 2King’s College London, 2University of the Witwatersrand, Johannesburg, 3University of East London, 4University of Manchester, 5Liverpool Hope University
We report survey data collated from past Mathematics Enhancement Course (MEC) students. The survey is part of a larger project involving three UK institutions offering the MEC, a UK-based government initiative to address the shortage of secondary mathematics-teachers: whereby non-mathematics graduates enter the teaching profession via a subject enhancement course. We have reported qualitative aspects of this route (Adler et al., 2013; Hossain et al., 2013). Here we discuss the survey conducted to ascertain MEC graduate’s experiences of appointment, retention and progression, and our interpretation of this as to whether the MEC with its particular focus on understanding mathematics in-depth is ‘fit for purpose’. We asked ex-MEC students from three institutions to respond to an on line survey and 118 participated. Findings include that a large majority (100/118) secured and retained teaching posts, with some progressing in their positions in schools. Our study of subject enhancement courses as an alternative route into teaching emerges at a critical juncture given the cultural and political scepticism concerning such routes and their longevity in the current UK education climate.
24 How working on mathematics impacts primary teaching: Mathematics Specialist Teachers make the connections
Jenny Houssart and Caroline Hilton
Institute of Education, London
We draw on analysis of assignments by primary teachers as part of the assessment for the Mathematics Specialist Teachers programme (MaST). In the assignment teachers are asked to work on some mathematics themselves, write up the mathematical part of their work then write about how this experience has impacted on their practice as a primary teacher. We focus first on case studies of teachers who included algebraic work in the first part of their assignments and look at what they say about the connections between this and their practice as primary teachers. Connections are made in a range of ways, but an overall finding is that teachers tended to focus more on the process of doing mathematics and the consequences this had for their practice rather than knowing mathematics. A further theme was feelings about mathematics, entailing positive consequences for practice, even where the initial feelings included negative dimensions. Examination of assignments on other aspects of mathematics confirms the presence of these three themes. Across all the assignments there was strong evidence that this experience of doing mathematics impacted positively on how teachers worked mathematically with their primary classes.
25 Classroom environment variables and mathematics achievement of junior secondary school students in Cross River State, Nigeria
Irem Egwu Igiri*, Anne Ndidi Meremikwu, Emmanuel E. Ekuri and Alice E. Asim
Primary Education Studies Department, Federal College of Education, Obudu, Cross River State, Nigeria*, Faculty of Education, University of Calabar, Calabar, Cross River State, Nigeria.
The study was designed to investigate the influence of classroom environment variables on mathematics achievement of junior secondary schools students in Cross River State, Nigeria. It was a survey research involving 1200 Junior Secondary 2 (JS2) students from 48 secondary schools in Cross River State. A valid and reliable achievement test and questionnaire were used for data collection, while multiple regression analysis technique was used to analyse the data. The research findings indicated that: there is a significant individual and combined prediction effect of the classroom environment variables on students’ mathematics achievement; twenty-seven (27) out of the thirty-six (36) paths in the hypothesised recursive model are significant at .05 probability level. Based on the findings, it was concluded that classroom environment variables significantly influence students’ mathematics achievement in secondary school. It was recommended among others that teachers and school counsellors should educate the students on the need to establish a good relationship with teachers and themselves in the classroom in order to improve their performance. Also, classrooms should be furnished and equipped to enhance effective teaching and learning.
King’s College London
This paper attempts to map a range of modes of mathematical reasoning employed in classrooms from Germany, Hong Kong and the United States taught by experienced teachers locally judged to be competent. Reasoning here is used as an umbrella term for modes of justification within a range of strategies that aim at making discursively available some elements of mathematical practice. The significance of this analysis consists in the attempt of describing modes of reasoning in a way that accommodates the diversity of mathematical topics, achievement levels, curriculum traditions and culturally sanctioned modes of interaction, rather than in the outcome of the comparison itself.
Marie Joubert and John Larsen
University of Nottingham and The Trinity Catholic School, Nottingham
It is well recognised that professional development research often struggles to demonstrate that changes in a teacher’s practice are as a result of a professional development initiative (e.g. Guskey, 2007). One reason is that teachers are influenced by a ‘patchwork’ of learning opportunities and it is sometimes impossible to pick out how, and to what extent, each opportunity may have contributed to these changes. The research reported here was a joint effort between a mathematics teacher and a researcher. The paper, which draws on ‘research conversations’ between the authors, explores what counts as professional development for this teacher and describes his patchwork of professional development in terms of the processes in which he engages within his professional practice: exploring, experimenting and reflecting; mainly within the context of teaching but also more widely. We argue that critical reflection is crucially important within professional development, but so too is appropriate action to carry exploring and experimenting through to real development.
Sibel Kazak, Rupert Wegerif and Taro Fujita
Graduate School of Education, University of Exeter
Research has shown that pupils and many adults have intuitions about probability that are often at odds with accepted probability theory. Drawing on the literature on probabilistic reasoning, effective pedagogical approaches and the use of technology tools, our aim is to examine the relationship between students’ talk together, their use of TinkerPlots software and the development of their reasoning about uncertain outcomes. In this paper we report on findings from the first iteration of a design study conducted in an afterschool club for Year 7 students in Exeter. More specifically we describe the trajectory of two students making conjectures about the fairness of some games involving combined events, testing and revising their initial theories based on simulation data. Our analysis shows that these students’ use of dialogic talk in combination with the technology leads to a shift from intuitive reasoning to probabilistic reasoning.
29 Networking theories of society and cognitive science: An analytical approach to the social in school mathematics
Institute of Education, London
Debate about the interplay between social and individual aspects of mathematics teaching and learning remains at the cutting edge of theoretical understanding of mathematics education research. In trying to make sense of the insights of these divergent perspectives I ask: How is it that social reality exists? What are the merits and limitations of considering the students in our classrooms as only collections of individual minds, in contrast with perspectives that posit the primacy of the social in determining the identity of mathematics learners? Can each be accorded its relative legitimacy in a rigorous and rational manner? Recent developments in analytical social theory may have the potential to address this issue productively. This paper covers the conflict between social-constructivist and socio-cultural perspectives in the literature and the critical role of inter-subjectivity in communicating mathematics through interaction. The paper concludes by drawing on Searle’s notion of collective intentionality to address the networking and complementary use of theories based in cognitive science and critical theory and the interplay of the individual and social in school mathematics.
30 The use of alternative double number lines as models of ratio tasks and as models for ratio relations and scaling
Dietmar Küchemann, Jeremy Hodgen and Margaret Brown
King’s College London
In this paper we draw on ICCAMS project materials that used the double number line (DNL) to develop secondary school students’ understanding of multiplicative reasoning. In particular, we look at the use of a DNL, and its alternative version, as a model of ratio tasks, as a model for developing an understanding of ratio relations, and finally (but only briefly) as a model for developing the notion of multiplication as scaling.
31 From the physical classroom to the online classroom – providing tuition, revision and professional development in 16-19 education
Mathematics in Education and Industry
In recent years Mathematics in Education and Industry (MEI) has undertaken extensive work to develop techniques for utilising online classroom technologies to deliver tuition, revision and professional development in 16-19 education. Interaction using these technologies has involved tens of thousands of students and many hundreds of teachers. In this paper a short background section on online learning will be presented, before discussion of how MEI has evolved its work to move from the ‘physical’ classroom for tuition, revision and professional development, to that of using online technology. An overview of feedback from online participants is then given. Techniques and strategies for utilising the technology are then discussed before final conclusions of the work are made.
32 What makes a claim an acceptable mathematical argument in the secondary classroom? A preliminary analysis of teachers’ warrants in the context of an Algebra Task
Elena Nardi, Irene Biza and Steven Watson*
University of East Anglia, *University of Cambridge
The study we report builds on previous research conducted by Nardi, Biza and colleagues, which examined mathematics teachers’ considerations of what makes a claim an acceptable mathematical argument in the secondary classroom. We identify teachers’ considerations in their written responses to tasks and then in semi-structured interviews that probe these written responses. Here we present data from six teachers and one such task, a (GCSE-level) Algebra Task. The tasks we invite the teachers to engage with, of which the Algebra Task is one, are structured as follows: a mathematical problem that students are likely to encounter in typical secondary mathematics lessons; fictional student responses to the problem (grounded on student responses found by relevant research as typical); and, an invitation to teachers to solve the problem, consider the purposes of its use in the lesson, reflect on the student responses and describe the feedback they would provide to the students. So far, we have proposed a theoretical tool for analysing mathematics teachers’ warrants for the preferences they express in their written responses and the interviews. The tool is based on our adaptation of Toulmin’s model of argumentation in which we classify teachers’ warrants according to pedagogical, epistemological and institutional considerations.
33 Lesson study and Project Maths: A Professional Development Intervention for Mathematics Teachers Engaging in a New Curriculum
Aoibhinn Ní Shúilleabháin
Trinity College Dublin
Since 2010 there has been a phased introduction of a new post-primary mathematics curriculum in Ireland entitled ‘Project Maths’. This new curriculum places a greater emphasis on problem solving and on an investigative approach for students. This implies not only changes in the curriculum content, but also changes to teaching and learning approaches within the classroom. This research aims to provide teachers with a school-based professional development structure through which they can engage with the curriculum and attempt new teaching and learning strategies. This structure involves mathematics teachers engaging in lesson study as a professional development intervention and is investigated in two schools (phase 1 and phase 2 of Project Maths). Teachers engage in lesson study cycles repeated throughout the academic year and the research questions how effective an approach this may be in encouraging teachers to engage with and implement a new centralised mathematics curriculum. The research also investigates how effective an approach this may be in developing teachers’ pedagogical practices. In this paper, initial findings will be discussed from teacher research meetings and interviews.
Craig Pournara and Jill Adler
University of Witwatersrand, Johannesburg; University of Witwatersrand and Kings College London
Learning mathematics-for-teaching (MfT) involves revisiting school mathematics and learning new mathematics. The notion of revisiting is operationalised and exemplified within pre-service secondary mathematics teacher education, specifically a financial mathematics course. A framework for MfT was developed consisting of nine interrelated aspects of teachers’ mathematical knowledge, and provided an analytic tool for exploring opportunities to learn MfT within the course. These opportunities are exemplified through a revisiting task where pre-service teachers (PSTs) were required to make sense of learners’ responses to a compound growth task. The learners’ responses provided a springboard for learning other aspects of MfT of compound growth such as essential features, modelling and applications, and knowledge of context. The decompressed nature of the learners’ responses opened opportunities for PSTs to reconsider their own knowledge of compound growth, of the mathematics that underpins the formula, the process to obtain the formula and thus the way in which the formula models compound growth in different contexts.
35 Lesson study as a Zone of Professional Development in secondary mathematics ITE: From reflection to reflection-and-imagination
Darinka Radovic, Rosa Archer, David Leask, Sian Morgan, Sue Pope and Julian Williams
Manchester Institute of Education, University of Manchester
We here add to the sparse literature on the use of Lesson Study (LS) in initial teacher education (ITE), reporting how LS can mediate development of reflective practice (RP). A cohort of 50 student-teachers, all secondary mathematics postgraduates, were involved in lesson study subgroups: planning, teaching-and-observing, analysing and reflecting on observations, and re-teaching. Each study group worked on their own LS in selected schools with teachers/school based mentors that had some previous experience of lesson study. We tutors/researchers/authors observed the planning, teaching and post-lesson analyses, and one of us interviewed selected participants. Two main findings were: (i) the significance of imagination for reflective practice, here prompted by the focus on improving and ‘re-teaching’ the lesson; and (ii) the importance of the ITE-peer group and its relations with more powerful others (mentors and tutors) to development. We conclude that LS may complement and even ‘lead’ the development of reflective practices of student teachers, providing a Zone of Professional Development.
36 Towards a model of professional development for mathematics teachers integrating new technology into their teaching practice
University of Leeds
This paper focuses on the professional development journeys of mathematics teachers when integrating new technology into their teaching practice. The research is part of a year-long innovative cooperative intervention in a cross-phase and cross-school setting with transcribed interviews, teacher meetings and on-line communication used for data collection. The Interconnected Model of Professional Growth was used to analyse these journeys and one teacher’s journey is illustrated here. The analysis indicates that all four teachers’ development journeys exhibited similar cycles of practical experimentation and reflection that led to long-term changes in teacher practice. A model for how mathematics teachers’ integrate technology into their teaching is proposed.
37 Researching children’s ‘self’ constructs and their success at solving word problems: a pilot study
Joan Rigg, Anesa Hosein* and Sarantos Psycharis**
Liverpool Hope University, *University of Surrey, **Faculty of Pedagogical and Technological Education – Athens Greece
Few studies have investigated self-constructs on primary school age children’s achievement in mathematics. However, studies on secondary and tertiary levels suggest that academic achievement is influenced by a person’s self-efficacy/self-confidence, (a belief in their own ability), social comparison, (comparing own performance with others), self-concept, (perceived ability in a particular area) and attitude towards mathematics. In preparation for a research study in the following academic year, twenty-one year 5 pupils took part in a pilot study in which they completed a psychometric inventory which measured social comparison, self-concept and attitude. Additionally pupils rated their confidence in solving a range of mathematical word problems prior to solving them. Analysis suggests that these self-constructs influence primary age pupils’ academic performance.
38 Development and evaluation of a partially-automated approach to the assessment of undergraduate mathematics
Nottingham Trent University
This research explored assessment and e-assessment in undergraduate mathematics and proposed a novel, partially-automated approach, in which assessment is set via computer but completed and marked offline. This potentially offers: reduced efficiency of marking but increased validity compared with examination, via deeper and more open-ended questions; increased reliability compared with coursework, by reduction of plagiarism through individualised questions; increased efficiency for setting questions compared with e-assessment, as there is no need to second-guess the limitations of user input and automated marking. Implementation was in a final year module intended to develop students’ graduate skills, including group work and real-world problem-solving. Individual work alongside a group project aimed to assess individual contribution to learning outcomes. The deeper, open-ended nature of the task did not suit timed examination conditions or automated marking, but the similarity of the individual and group tasks meant the risk of plagiarism was high. Evaluation took three forms: a second-marker experiment, to test reliability and assess validity; student feedback, to examine student views particularly about plagiarism and individualised assessment; and, comparison of marks, to investigate plagiarism. This paper will discuss the development and evaluation of this assessment approach in an undergraduate mathematics context.
The notion of an autonomous teacher has long been accepted as an important characteristic of a good mathematics teacher. In this paper, the beliefs and practices of six primary teachers, all of whom are construed against various criteria as autonomous, were examined. Data, which derived from multiple interviews with and observations of each teacher, were analysed using constant comparison and yielded three themes related to the interaction of belief and practice that separated the six teachers into two equal groups. The results show that autonomy depends on one’s perception, therefore challenging the sufficiency of the construct as a characteristic of a good mathematics teacher.
Malmö University, Sweden
Recent research shows that the dominant practice in mathematics education in Sweden involves students working individually from a textbook. However, to read mathematical texts involves comprehending the global meaning from the page and this requires specific reading skills. In this study, the reading strategies of six 10-year old students, with different levels of mathematical achievement, are identified. The analysis is based on Palinscar and Brown’s reciprocal activities prediction, clarification and summarisation and Halliday’s Systemic Functional Linguistics (SFL). In this small study, high achieving students more often described that they used appropriate reading strategies.
University of Washington
Many studies have examined counting skills in young children with language-related disabilities but few studies have examined counting skills in older children with these disabilities. In this study I examined the counting skills of fifteen 9 to10-year-old students with dyslexia. During an individual clinical interview students worked on an object counting task, counting by tens tasks, and word problems. Video analysis of these tasks revealed that twelve students made errors on the counting tasks and that these counting difficulties impacted the students’ abilities to complete the word problems accurately. Even in upper primary school students with dyslexia have difficulties with counting and these difficulties with counting impact their abilities to accurately solve more complex mathematical problems.
St. Patrick’s College, Dublin City University
The Irish Primary Mathematics Curriculum consists of five content strands, namely Number, Algebra, Data, Measure and Shape and Space (Government of Ireland, 1999). Children engage with material from all strands throughout their primary education. Thus the Irish education system fulfills the widespread recommendations in research of commencing algebra early (e.g. Kaput, 2008; Carpenter and Levi, 2000; Cai and Knuth, 2011). However, national and international studies of student attainment suggest that many children in Irish schools may not be developing robust skills in algebraic reasoning (Eivers et al., 2010; OECD, 2009; Eivers and Clerkin, 2011). In my research I plan to investigate to what extent children in Irish primary schools are developing skills in algebraic reasoning. In order to do so, an assessment instrument must be developed which will facilitate an exploration of children’s thinking as their skills develop. The clinical interview is an instrument which allows access to children’s emergent thinking (Ginsburg, 1997; Piaget, 1929) and in this paper I discuss the design of a clinical interview with specific relevance to children’s skills in algebraic reasoning.
Geoff Wake, Colin Foster and Malcolm Swan
University of Nottingham
This article reports on a study that has researched teacher professional learning in lesson study communities that enquired into how we might better support students develop skills in problem solving and mathematical modelling. A rationale for professional development of this type, both in in terms of its structure and focus, is presented followed by an illustrative description from the study of a typical research lesson and issues raised in the post-lesson discussion. This is used to provide insight into some of the key issues to consider in developing teacher knowledge for modelling and problem solving.
44 SKE courses and bursaries: examining government strategies to tackle mathematics teacher quantity and quality issues
University of Leeds
The shortage of secondary mathematics teachers is a concern for UK government. In order to increase the supply of teachers, the government has sponsored subject knowledge enhancement (SKE) courses for non-specialist graduates (‘SKE Policy’). Additionally, the government has offered financial incentives to attract high attaining graduates into teaching through differentiated bursaries (‘Bursary Policy’). Existing research into the efficacy of these measures is limited but studies on two individual teaching courses suggest that student teachers who have taken SKE courses do no worse in their teaching course and that those who achieve more highly in their first degree do not achieve significantly higher outcomes in their teaching course respectively. This study corroborated results of these smaller studies through testing over 100 secondary mathematics student teachers on a sample of ‘mathematical knowledge for teaching’ (MKT) items, designed by the University of Michigan. Results suggest that whilst the SKE Policy may be a good strategy in that there was no overall significant difference in MKT scores between SKE students and mathematics graduates, the Bursary Policy may be a flawed since there is no evidence that higher degree classes lead to greater success on PGCE courses.
45 The impact of professional development on the teaching of problem solving in mathematics: A Social Learning Theory perspective
University of Cambridge
Teachers’ professional development (PD) has been described as an ‘unsolved problem’, particularly where there is an expectation to change teaching practices from teacher-centred orthodoxies to more student-centred approaches. This paper considers PD that has been designed to support the teaching of problem solving in secondary classes. One of the main problems in PD design and research has been identified as the limitations of existing professional learning theory. To respond to this, the research reported in this paper is intended to contribute to the theorisation of PD. I will present a case study of one teacher – taken from a larger multiple case study – as they take part in a programme of school-based PD. Social learning theory (SLT) is used to analyse and explain the learning processes as a result of participation in the PD programme. This reveals that SLT provides a useful theoretical approach. As a result, I suggest the approach could be used more extensively in professional learning and PD, to understand, evaluate and develop programmes more effectively.
University of Bedfordshire
The purpose of my research is to explore first year undergraduate perceptions in learning mathematics and their identification of strategies to support them in this area. I work as a senior lecturer in mathematics education on a BA Applied Education Studies course where students consistently identify the mathematics education units as a source of anxiety. Having established a potential link between mathematics anxiety in teachers and trainee teachers, and the potential that this anxiety could be passed on to children in classrooms, my aim is to identify whether there may be strategies that could support adults in learning mathematics. To explore this area, I tracked a sample of first year undergraduate students through their initial mathematics education unit, establishing their perceptions before and after the teaching of this unit. Initial findings demonstrated that the students had negative perceptions about learning mathematics (twice as many negative as positive) and that this was reversed following the completion of the first mathematics education unit. A range of factors were identified as affecting how they felt about learning mathematics, including the role of the teacher, personal perceptions regarding learning mathematics and the role of discussion.
University of Southampton
This research paper reports findings from an exploratory multiple case-study investigating children’s cognitive representations of number, in particular during interactions with number line estimation tasks. The paper considers the cases of two Year 1 children (ages 5 and 6) and the findings from their participation in the pilot stage of this larger study. Children participated in individual task-based interviews, which were video-recorded and analysed using multimodal analysis. This methodology identified the cognitive representation of many structural aspects of natural number, and enabled a fine-grained differentiation of children’s strategies during number line estimation tasks. Both cases considered invite continued investigation of the connection between strategies and estimation results. The implications for our understanding of children’s cognitive representations of number, and the interpretation of number line estimation tasks, are discussed.
Institute of Education, University of London
There is growing consensus, amongst teachers, teacher educators and researchers, that a more engaging and relevant school mathematics curriculum is needed, with greater emphasis on problem-solving skills and the development of conceptual understanding. Too much focus on factual recall and procedural understanding has led to unacceptable levels of disengagement and disaffection amongst learners. This paper reports on initial findings from a project involving a group of teacher researchers who share a commitment to addressing the alienation of learners, raising their awareness of the nature of mathematics and the reason for learning it, developing student agency and an appreciation of how mathematics can be used to better understand the world around them. I make use of a participatory action research methodology to explore how being part of a research group can help teachers to begin to develop their classroom practice in ways which resonate with a commitment to teaching mathematics for social justice. I identify four themes emerging from an analysis of an initial series of semi-structured empathetic interviews with the teacher researchers that provide a useful theoretical framework for the development of the project and useful insight for others wishing to carry out research in a similar field.