**01 Characterising patterning of teacher-learner interactions in mathematics classrooms.**

Julian Brown

*University of Bristol*

Deliberate development of our practices, in and out of the classroom, is supported by explicit awareness of possibilities in the moment, expanding possibilities for action. My role as a mathematics teacher educator involves working alongside teachers as they work on their own awarenesses. I am interested in how classroom observations might be used as a mechanism to follow and support development of the awareness of awareness. This report makes use of observations of sequences of lessons taught by two different, experienced teachers of mathematics to consider possibilities for characterising particular teacher-class environments and, hence, to identify shifts in these environments. Consideration is given to how such mechanisms might act as tools for development of in-the-moment awareness for practising teachers.

Libby Chancer

*Swansea University*

Within the current Welsh educational system, ‘real life’ problems have become a focus within Mathematics lessons. Pupils are given a worded or visual problem which often needs to be translated into a mathematical form to solve. The OECD refer to this process as mathematizing. It is this process of mathematizing that often causes difficulties for pupils. Pupils have to abstract during the translation, meaning they have to locate and express the mathematical structure within the problem. Only after abstraction is the problem in a form which is able to be manipulated and solved. This research is the 1 st iteration in a design based research project that covers four intervention lessons. This project aims to develop specific pedagogies which support pupils to focus on mathematical structure and think abstractly and algebraically. This paper seeks to justify and evaluate intervention tasks and suggests possible changes for the next cycle of intervention.

**03 Mathematics education and financial literacy: A valued but vulnerable symbiosis**

Jennie Golding

*UCL Institute of Education*

In England the development of financial literacy and related applications features in several aspects of the intended curriculum, including, but not primarily, mathematics. We argue that the development of key mathematical concepts and of financial literacy have historically often been symbiotic and this inter-dependency is reflected in many curriculum materials. This paper draws on cross-phase classroom-close evidence collected as part of a larger study, to report changes to the preparedness of some learners for engagement with related concepts. Subsequent student and teacher interviews suggested such changes were due to both changing patterns of family life and moves towards virtual, rather than physical, currency in many communities. We discuss the significant implications for organisation of both mathematics and financial literacy curricula, and suggest some ways forward.

**04 An online course promoting wider access to university mathematics**

Richard Gratwick, George Kinnear, Anna K. Wood

*University of Edinburgh*

In 2018, a new year 1 course, “Fundamentals of Algebra and Calculus” (FAC), was introduced at the University of Edinburgh to provide better support for incoming students with a range of mathematical backgrounds. The course is delivered online, interleaving textbook-style exposition with videos of worked examples, interactive applets, and practice questions implemented in the STACK assessment system. The design of the course incorporates aspects of educational theory such as specifications grading and the use of computer-aided assessment. We present statistical evidence of FAC removing an apparent attainment gap, and initial findings on students’ opinions on the design of the course.

Rachel Helme

*University of Bristol*

The concept of ‘mathematical identity’ is described as saying and doing in the context of mathematics, however the interpretation of another’s identity work is problematic due to the subjectivities of the observer. In foregrounding the participant’s own voice, the opportunity is afforded to consider the work of identity from within the lived experience. This discussion focuses on the analysis of the narrative data of Darren, a participant from a study into the experiences of low attaining students in mathematics. Observations noted a contrast emerging between Darren’s ability to correctly calculate an answer and his reluctance to record his workings. Introducing the Listening Guide as a method of analysis revealed the presence of two co-existing voices, the bravado of the ‘action’ voice contrasted with the more poignant voice of ‘struggle’, providing an insight into his internal quandary as he strives to explain his mathematical thinking.

Jenni Ingram, Vicky Neale, Natsuno Funada, and Kyla Smith

*University of Oxford*

Undergraduates studying highly mathematical subjects at university level often have a long history of success in mathematics, particularly in school-level mathematics. We report on the interim findings of a study of undergraduates who all achieved the highest possible grades in their school examinations but whose performance in university mathematics examinations reveal attainment gaps that are not predictable by prior attainment. Focusing particularly on attitudinal factors such as mindset and beliefs about ways of working with mathematics we examine students’ perceptions of what enables or prevents them from being successful with their mathematics at university level using a mixture of questionnaires and interviews.

**07 Centres for Excellence in Mathematics: Research trials focusing on specific teaching approaches**

Marie Joubert ^{1} , Marc North ^{1} , Geoff Wake ^{1} , Diane Dalby ^{1} and Shobhna Fletcher ^{2}

^{1} University of Nottingham, ^{2} Education and Training Foundation

The government in England aims to improve the teaching of mathematics up to level 2 in post-16 educational settings (colleges of Further Education and Sixth Form Centres) through twenty-one Centres for Excellence in Mathematics in England. One strand of the work of the Centres for Excellence is to participate in national trials which aim to research how, and how well, specific teaching approaches work. This paper introduces the research trials, outlining their design and considering the pilot implementation, which took place between October 2019 and March 2020. The paper provides some emerging findings and recommendations for a wider roll-out of the trials.

Stephen Lee ^{1} , Matthew Walker ^{2} and Suzanne Straw ^{2}

^{1} Mathematics in Education and Industry, ^{2} National Foundation for Educational Research

Mathematics in Education and Industry’s government funded Advanced Mathematics Support Programme (AMSP) has been in place since May 2018. During that period, the National Foundation for Educational Research (NFER) has conducted an independent evaluation of the programme. Key aspects of the evaluation include a national school/college survey into level 3 mathematics, teacher interviews and student focus groups.

Schools/colleges have engaged in various AMSP support activities including teacher professional development and student enrichment and support, but they have faced key barriers such as releasing teachers/students from school/college and the cost or availability of teacher cover. Lack of support from senior leadership was not seen as an issue by many.

This paper draws on analysis from one of the largest responses (717 schools/colleges) to a survey into post-16 mathematics since the GCSE/A level curriculum changes took place in 2015. It also considers feedback from in-depth case studies across two touchpoints.

**09 ‘I want to help others’: Why female A-Level mathematics students reject undergraduate mathematics**

Ems Lord

*University of Cambridge*

Although university mathematics departments are increasingly maximising the potential of tracking undergraduate applications, there is a dearth of data regarding A- Level mathematics students who choose not to apply to those institutions. This pilot case study focused on a mixed group of Y13 (17- and 18-year-old) A-Level mathematics students (N=18) attending an urban secondary school. The study was

conducted after the closing date for their university applications. Using mind maps, the students were asked to share their reasons for continuing their studies to undergraduate

level, as well as their choice of course and institution. The findings indicated several gender differences in their decision-making, including the perceived connection between their university course and their desire to help others. The possible implications of these findings are considered for their potential to inform future, larger-scale studies of interest to both schools and university outreach departments.

Kate Mackrell ^{1} and Sue Johnston-Wilder ^{2}

^{1} University College London, ^{2} Warwick University

One approach to the problem of mathematics anxiety, that of developing mathematical resilience (MR) focuses on enabling learners to remain in the growth zone, where learners experience challenge and manage any threat. This approach, involving the use of three tools (the growth zone model, hand model of the brain and the relaxation response) has been successful in small-scale studies. We show here how the theory and practice of MR can be grounded in self-determination theory (SDT) (Deci & Ryan, 2000), with connections to SDT concepts of: autonomous motivation; the basic psychological needs of autonomy, competence and relatedness; and emotion regulation. Extensive research evidence has indicated that the satisfaction of basic psychological needs leads to well-being and that frustration of these needs leads to ill-being, indicating the potential of SDT to support research and practice in the specific area of ill-being known as mathematics anxiety.

Njaru Mbogo

Mathematical resilience defined as a positive adaptive stance to mathematics allows children to function optimally in a mathematics lesson. Therefore, children experiencing mathematics anxiety require mathematical resilience to aid optimal mathematics functioning in a mathematics lesson. Interestingly, mathematics anxiety research focuses on maladaptive responses to learning mathematics. Against this background this proposed PhD study contends that the mathematical resilience grade three children with mathematics anxiety require to aid optimal mathematics functioning in a mathematics lesson is a consequence of various attributes including daily classroom instructional strategy, the nature of mathematics itself and, pervasive beliefs about mathematics ability being fixed. This proposed PhD study intends to determine why grade three children aged eight years are experiencing mathematical resilience deficits in Kenya.

Evi Papadaki and Irene Biza

*University of East Anglia*

This report introduces part of a larger study on secondary teachers’ mathematical and pedagogical discourses that are significant to the coherence of mathematical ideas and practices across educational levels. The study draws on the literature related to what ismostly called Horizon Content Knowledge and specifically on the theoretical construct of the Discourse at the Mathematical Horizon. The aim of this report is to propose and exemplify an analytical approach that conceptualises and identifies the characteristics of this discourse in a lesson observation and an interview with one newly qualified mathematics teacher. The proposed analytical approach illustrates the teacher’s actions of interpreting and giving meaning to students’ unexpected ideas and how these actions can lead in the identification of discursive patterns of how the teacher goes beyond the content of a specific teaching situation.

**13 Teaching and learning for ‘moving goal-posts’: Reformed A Levels in mathematics**

Ben Redmond ^{1} , Jennie Golding ^{2} and Grace Grima ^{1}

^{1} Pearson, ^{2} UCL Institute of Education

Reformed English pre-university mathematics ‘A levels’ feature enhanced content and renewed focus on mathematical reasoning and problem solving. Related assessments, at scale from 2019, had negligible piloting, and preparation time for resources and teaching was pressured, with teachers/assessors typically having little experience of teaching/assessing for the renewed foci. We used an institutional ethnographic lens to study the first 3 years’ enactment from the leading awarding organisation, and impact on students’ learning, affect and pathways. We followed students and teachers in a fairly representative sample of 46 classes, drawing on termly data collection. Initial ‘specimen assessments’ were largely considered valid and accessible; however, we evidence insecurity due to perceptions of ‘moving goal posts’. Early final assessments were perceived as significantly more demanding than predecessor comparators and of limited reliability for many students. We analyse contribution to knowledge around introduction of curriculum aspirations at this level and discuss ways to address identified challenges.

**14 Estimation: An inadequately operationalised national curriculum competence**

Judy Sayers ^{1} , Jöran Petersson ^{2} , Eva Rosenqvist ^{2} and Paul Andrews ^{3}

^{1} Leeds University, UK, ^{2} Malmö University, Sweden, ^{3} Stockholm University, Sweden

Research has highlighted the importance of estimation, in various forms, as both an essential life-skill and a significant underpinning of other forms of mathematical learning. It has also highlighted a lack of opportunities for learners to acquire estimational competence. In this paper, we present a review of the literature that identified four forms of estimation. These are measurement, computational, quantity (or numerosity) and number line estimation. In addition to summarising the characteristics and significance of each form of estimation, we examine critically the estimation-related expectations of the English national curriculum for primary mathematics to highlight a problematic lack of opportunity.

**15 Beginning mathematics and science teachers’ understanding of randomness**

Victoria Wong and Jenni Ingram

*University of Oxford, UK*

An understanding of randomness is essential for understanding many aspects of both the school mathematics and science curricula. Yet research has shown that many people find randomness difficult to perceive and argue about, with many holding a number of different and contradictory views about the nature of randomness. This study explores beginning mathematics and science teachers’ understanding of randomness. A questionnaire was used with a single cohort of mathematics (n=28) and science (n=30) students on an initial teacher education course (secondary) to explore their understanding of and reasoning about randomness and random events. Results suggest that mathematics and science beginning teachers conceptualise and argue about randomness in different ways. Further that these different conceptualisations affect how they respond to common classroom tasks involving randomness.

**16 Report from the Critical Mathematics Education Working Group meeting**

Pete Wright

*UCL Institute of Education*

BSRLM’s CME Working Group met for the fifth time for a discussion prompted by the question: ‘What are the implications of Bourdieu’s ideas for the mathematics classroom?’ The meeting provided an opportunity for the 18 delegates attending to discuss how researchers, educators and teachers might draw on Bourdieu’s ideas,

including ‘cultural capital’, ‘symbolic violence’ and ‘reflexive sociology’, to inform their practice. A series of prompts was used to facilitate discussion and six themes emerged from the responses: the contested nature of the term ‘cultural capital’; the relevance of Bourdieu’s theories to mathematics teaching; the extent to which the reproductive function of school mathematics is intentional; the relevance of Bourdieu’s ideas to an analysis of global inequalities; making sense of the self-perpetuation of conventional approaches to teaching mathematics; and possibilities offered by Bourdieu’s analysis for challenging the exploitative nature of school mathematics.