Phase Transitions and superuniversality in the dynamics of a self-driven particle

We study an active random walker model in which a particle's motion is determined by a self-generated field. The field encodes information about the particle's path history. This leads to either self-attractive or self-repelling behavior. For self-repelling behavior, we find a phase transition in the dynamics: when the coupling between the field and the walker exceeds a critical value, the particle's behavior changes from renormalized diffusion to one characterized by a diverging diffusion coefficient. The dynamical behavior for all cases is surprisingly independent of dimension and of the noise amplitude.Comment: 14 pages, 4 figure

Trade and piece : tradition and the Kissinger-Nixon foreign policy of the United States

'...without expansion of international trade, based upon fair dealing and equal treatment for all, there can be no stability and security either within or among nations... (Furthermore) the withdrawal by a nation (or group of nations) from orderly trade with the rest of the world inevitably leads...to preparations for (a military or trade) war and a provocative attitude toward other nations.'peer-reviewe

Assessment to support the development of problem-solving goals in mathematics curricula 5-16

Current mathematics education policy perspectives in England, as well as in many other developed nations, privilege problem solving as a key 21st century skill (e.g. DfE, 2014; Kaur, 2014). Importantly, there is among the mathematics, mathematics education and end-user communities widespread embrace of problem solving as a centrally valued mathematical activity (e.g. ACME, 2011). However, English teachers and assessors often have limited experience of teaching for/assessing genuine problem solving, and performance in mathematics in England is high-stakes (Ofsted, 2012). Problem solving is therefore unlikely to be widely developed in classrooms unless summatively assessed at key points including GCSE, the standard English external assessment at age 16. A coherent curriculum system (Schmidt and Prawat, 2006) whereby intended curriculum, assessments, resources and teacher development are aligned and consistent, is key to supporting principled enactment (Golding, 2017). We report on a set of longitudinal efficacy studies which, among other intentions, evaluate the impact on teachers and students of a leading curriculum and assessment provider’s support for, and assessment of, problem solving for the 2014 Mathematics National Curriculum for 5-16 year olds (DfE, 2014). These are highly influential in England because they are widely adopted: the provider’s GCSE assessments at 16, for example, accounted for about two-thirds of all cohort entries in 2017. Key theoretical constructs used are those of performativity (Ball, 1994) and curriculum coherence (Schmidt and Prawat, 2006). Teacher interviews (n=452), student focus groups (n=172), classroom observations (n=101) and student survey responses (n~3300) over two early years of curriculum enactment show teachers and students perceive the approaches adopted in related curriculum materials and in the first set of provider GCSE examination papers not only employ highly valid assessment of, and approaches to, mathematical problem solving, but support that with provision of free surround materials specifically designed to build up students' ability to demonstrate related skills in summative timed assessments. However, we also evidence early and emerging constraints on both assessment and classroom enactment of problem solving in the curriculum: teacher skills and knowledge for related teaching across the range of students, teacher time and opportunity to harness the (additional or included) professional development opportunities provided with the resources, perceptions of superficial interpretations of ‘problem solving’ in national assessments at age 11, and pressures on schools and GCSE assessors to adopt enactments of ‘problem solving’ that are of limited validity, or for a subset of students only. Teachers attribute this to a) the challenges associated with defining an agreed meaning for mathematical problem solving and b) perceived in-school tensions between validly enacting that and meeting high-stakes performance measures. We discuss some implications

A pandemic summer: Impact on teaching and learning for mastery in Power Maths primary schools

We report on findings from a 2019-2021 study of use and impact of Power Maths, a ‘mastery’-oriented primary (R-year 6) resource. The study follows 40 classes of 2019-20 Power Maths-using year 1, 3 and 5 children and their teachers over two years, exploring teacher/pupil use and impact on learning. We report initial high-level findings. Summer 2020 study data serendipitously enabled us to understand aspects of teachers’ work over the pandemic period. Teachers reported particular challenges in addressing new areas requiring conceptual development, and inability to effectively develop children’s mathematical language or reasoning, or to monitor deep progress in mathematics learning. However, some children’s learning benefited from small group in-school provision, and others’ from more contextualised and less time-constrained ‘home schooling’. Tentatively, children returning to school often showed initially slow, but accelerating, recovery from confidence and learning loss

Transforming triangulations on non planar-surfaces

We consider whether any two triangulations of a polygon or a point set on a non-planar surface with a given metric can be transformed into each other by a sequence of edge flips. The answer is negative in general with some remarkable exceptions, such as polygons on the cylinder, and on the flat torus, and certain configurations of points on the cylinder.Comment: 19 pages, 17 figures. This version has been accepted in the SIAM Journal on Discrete Mathematics. Keywords: Graph of triangulations, triangulations on surfaces, triangulations of polygons, edge fli

‘It’s been worth the effort’: Primary school teachers learning to teach mathematics remotely during the pandemic

We draw on Spring 2021 findings from a 2019-2022 study of impact and use of a ‘mastery’-oriented primary (ages 4-11) mathematics resource, ‘Power Maths’, in England. The study follows 40 classes of primary children and their teachers, in 20 schools, over two years. Our findings span the return into school from the early 2021 lockdown period, comparing and contrasting teachers’ approaches across the two pandemic-related lockdown periods, the first in March-June 2020. Most teachers developed a significantly wider range of, and confidence in, remote learning practices. They came to expect more, and active, new learning, rather than aiming just to consolidate prior knowledge. Many developed active selection of the most appropriate topics for home learning, substantially increased ‘live’ teaching, and found ways to more proactively monitor work. Despite this, some challenges persisted: providing effective formative assessment and insecure knowledge of parental support and of gaps in children’s learning

Covid 19: Impacts on teaching, learning and progression for A Levels in Mathematics.

We report on ways that teaching and learning for mathematics A Levels, studied pre-university in England (by students aged 16-18), was disrupted by Covid19 in Summer 2020. Findings are contextualised within teacher and student accounts of the aspirational and time-pressured nature of these reformed qualifications. We explore the nature of engagement with mathematics achieved by year 12 and 13 students during lockdown, and the preparedness of 2019-20 year 11 and 13 students for progression into A Level/Higher Education respectively. Our findings derive from the third year of a four-year study (2017/18 to 2020/21) exploring enactment and impact of reformed mathematics A Levels, and efficacy of associated Pearson resources and assessments. Research tools were adapted to focus on impacts of Covid19. We present a snapshot (March to July 2020), of teachers and students looking to the future in a time of uncertainty and rapid change

‘Hard to focus, difficult to learn’: Covid19 Impacts on teaching, learning and progression for A Levels in Mathematics

We explore year 13 (age 17-18) student accounts of how Covid19 has impacted their learning for pre-university mathematics qualifications in England. Findings derive from the final year of a four-year study (2017/18 to 2020/21) exploring enactment and impact of reformed mathematics ‘A Levels’, and efficacy of associated Pearson resources and assessments. Research tools were adapted to focus on impacts of Covid19. In this cohort’s first year of A level (2019/20), teaching and learning was severely disrupted. Teachers anticipated significant, wide-ranging learning gaps as students progressed to year 13. Using data from Autumn 2020 and Spring 2021 we analyse student accounts of how continued disruptions to teaching and learning have impacted them. Variable access to teachers, barriers to collaborative work, and challenges of remote or reduced contact working have resulted in reduced depth and breadth of learning. Additionally, many students reported negative impacts on mathematical confidence and wider mental health

Teaching and learning for ‘moving goal-posts’: reformed A Levels in mathematics

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