This study explores teacher educators’ (TEs’) activity as they support mathematics and science teacher collaboration in co-designing and jointly implementing tasks. We view TEs’ activity through the lens of Activity Theory and expansive learning and draw evidence from data generated within the mascil project that linked mathematics and science teaching with workplace situations through inquiry-based teaching. We focus on five TEs’ actions and goals, use data from their professional development sessions with teachers and from the TEs’ interactions during their own meetings, and highlight the illuminating case of one teacher educator. We trace evidence indicating paths of actions followed by each Teacher Educator and look for indications of their professional learning. Our analysis reveals generic and content-focused actions. All TEs faced different kinds of contradictions and had difficulties handling them. In terms of professional learning, all TEs adapted their prior teacher education practices and appreciated the critical role of epistemological differences between the two disciplines.

The chapter focuses on the attempts of a group of mathematics teacher educators (MTEs) to support teachers in exploiting workplace situations in their mathematics teaching. We report on MTEs’ professional learning in the context of a European-funded project which brought together 18 partners from 13 countries. In Greece, 11 MTEs (academic researchers, teachers and mentors) with different research and teacher education experiences worked with thirteen groups of practising teachers who collaborated to plan, enact and reflect on lessons aligned to the aims of the project. The project provided substantial opportunities for challenging MTEs’ professional knowledge and teacher education practice. The analysis of the discussions during a series of meetings, where design and reflection on professional development activities took place, allowed for identifying and describing MTEs’ concerns and emerging tensions. Using the construct of boundary crossing we traced shifts in MTEs’ movements across different practices indicating an interplay of research, teacher education and mathematics teaching.

The chapter is on how social issues influence teachers’ use of digital resources in mathematics classrooms. The study is on an experienced, digitally competent, Danish teacher, Sofia, and one question is how her use of digital resources relates to her shifting professional identities. To address the question, a framework called Patterns of Participation, PoP, is used, one that draws on the notions of practice and figured worlds from social practice theory and of self and interaction from symbolic interactionism. Another question is whether PoP is helpful for understanding how Sofia contributes to classroom interaction when using digital resources. Sofia’s case was previously analysed with another framework, Structuring Features of Classroom Practice, which is developed to study teachers’ expertise and development in relation to digital resources. The PoP perspective supplements the previous and primarily descriptive account by providing explanations for how digital resources are used in Sofia’s classrooms, including a focus on procedures and a paucity of attention to conceptual understanding and mathematical reasoning. These explanations relate to Sofia’s identities, understood as her professional experiences of being, becoming and belonging. The PoP analysis, then, offers contextual interpretations and explanations of teachers' acts as related to broader social enterprises beyond classroom interactions.

The chapter is on how to analyse classroom situations and students’ evolving conceptualisation of function as covariation at upper secondary level in authentic modelling situations involving the use of digital tools. To address this aim we take a networking perspective to develop a framework by combining Connected Working Spaces and Abstraction in Context. We privilege authentic modelling tasks utilising the potential of different models and the use of digital environments offering integrated algebraic and geometrical representations of function. Another question is how the combination of the two frameworks can help to make sense of students’ evolutions in the path from physical context to algebra. The combined analyses based on the two frameworks allow a deeper look at students’ cognitive evolution as they experience functions in a plurality of settings: physical context, geometry, measures, algebra. Connected Working Spaces allows distinguishing these settings and their connections focusing on instrumental, semiotic and discursive dimensions and their coordination in students’ work. Abstraction in Context offers concepts and expected strategies and an account of knowledge construction within and between these settings allowing to make sense of students’ progress.