This article proposes a reframing of the purposes of mathematics education for the twenty-first century that combines apparently divergent philosophical approaches, arguing that the consequent empowerment should as a matter of individual equity be available to all young people (as well as of benefit to wider society). It suggests that the global mathematics attainment 'spotlight', and the English policy context in particular, offer both opportunities and constraints for the development of such a high-quality mathematics education. The article also discusses the challenging implications for the curriculum, and for the nature of teacher expertise, particularly subject-specific expertise, that is needed.

Mathematics teachers across the Western world are faced with an expectation that they make significant change to their teaching, but repeated attempts have shown little embedded success. This paper draws on a longitudinal study of two apparently well-placed English mathematics departments attempting to make change aligned with both policy and internationally-valued 'good practice'. It suggests deep teacher change draws on a wide range of both social and affective characteristics, as well as sophisticated professional skills and knowledge. The study supports a construct of '(mathematics) teacher capacity for change' at both individual and group levels within teachers' 'personal domains' , synthesising the range of characteristics apparently needed by teachers in times of change. In particular, it argues for the development of dispositions for collaborative learning and of other learning-supportive affects. Such an approach has the potential to place teachers in a better position to respond to demanding aspirations.

Online education is widely heralded as cost-effective and convenient, but there is a recognised need to research appropriate pedagogies for synchronous online (SOL) teaching. Prior research suggests that course design, including formative assessment, is critical in achieving accessible and flexible learning episodes. We harness a well-tested theoretical framework to analyse the development of pedagogies for synchronous online mathematics teacher education in a course designed to induct early career teachers into pre-university teaching. The lenses of 'teaching-', 'social-', 'learner-', and 'cognitivepresence' reveal opportunities for improving provision. We show participants' learning of mathematics content knowledge can then appear comparable to that achieved face-to-face, but it is more challenging to design for the growth of mathematics pedagogical knowledge in a synchronous online environment. Many digital tools work well in this medium, but the use of physical tools requires significant development before teacher engagement compares with that achieved face-to-face. However, some approaches developed to address challenges within a synchronous online course appear to have benefits also for face-to-face learning.

We report on the professional learning of a small purposive sample of teachers beginning to teach an English post-16 mathematics course centred around contextualised problem solving. We describe their accounts of change, their intentions and justification for those, and how development was supported. The demands of the course require 'boundary crossing' between contexts and mathematics that was novel for these teachers and their students, and learning to teach for such approaches is known to be demanding. By drawing on peer support and their own robust subject knowledge, though, these teachers were able to accommodate change to classroom authority and expertise, attention to students' relationship with mathematics, longerterm planning sequences, and growth in their own management of uncertainty within the classroom.

This paper considers the policy 'roles' adopted by teachers enacting policy in a department. It draws on a longitudinal study of two secondary mathematics departments endeavouring to make deep change aligned with a demanding curriculum policy. The study validates aspects of an existing typology, demonstrates the existence of a variety of policy 'critics' and adds a category of 'survivor' teacher, showing the role can have considerable impact on enactment. The paper argues for an extension to a group construct of 'policy role', here at a department level, and shows that as teachers struggle to marry the constraints of the range of policies to which they are subject with the time and effort needed to maintain deeply espoused professional values, an adopted group role can serve either to support or constrain individual teacher efforts.