The main purpose of this section is to explore the design of and design with digital resources (DRs) for mathematics teaching and learning. Design of DRs involves teachers’ engagement in the process where DRs are the product of design. Design with DRs refers to cases where DRs facilitate the design-work offering an environment for teacher collaboration (e.g., communication platform) or an environment for designing tasks and lessons. The chapters of this section address both forms of design.

of teacher design in collaborative settings in the light of the new opportunities that digital resources (DRs) offer. Taking into account the growing research interest into the collective dimension of teachers’ design-work, our aim is to answer what are the forms, conditions, and products of teachers’ collective design-work with DRs as well as what and how the individual teacher learns by participating in collaborative work with DRs. We also aim to answer what theoretical and analytical perspectives are used by researchers to study teachers’ collective design-work with DRs. To answer these questions, we conducted a systematic literature study leading finally to 36 peer-reviewed publications. Our first thematic analysis resulted in two main themes (the process of teacher collaboration, the impact of teacher collaboration on teacher professional learning) and corresponding subthemes. The next step of our analysis focused on: the context, the product, the purpose, and the processes of the design-work; the theoretical and methodological approaches by which it was studied; and the main findings. The final synthesis indicates that (a) teachers’ collaborative design-work has usually positive learning outcomes for individual teachers, (b) the role of DRs in the collaboration depends on their affordances, and (c) the collective-individual interplay has been studied mainly by a focus on the effects of the collaboration on the individual teacher and not the other way around. Areas of further research are also discussed.

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.

This chapter reports on the work of Working Group 4 and focuses on the integration of digital resources into mathematics teaching and learning practices.There are five central sections, focusing on, instrumental genesis, instrumental orchestration, the documentational approach to didactics, digital resources andteacher education, and the design of learning environments with the use of digital resources. A range of constructs and theoretical approaches are covered in these five sections, and the opening section comments on construct validity and issues in “networking” theoretical frameworks. The chapter can be viewed as a literature review which surveys past and present (at the time of writing) scholarship with an eye to possible future research. The chapter is extensive in several dimensions: a large range of digital resources and applications are considered; the subjects using digital resources are not just teachers but also students, student teachers and student teacher educators. Issues raised in the sections include individual and collective use of resources, the adaptation of these resources for specific learning goals and to prepare (pre- and in-service) teachers for the use of digital resources.

This paper aims to analyze the structure and quality of prospective mathematics teachers’ (PMTs)’ argumentation when identifying and interpreting critical incidents from their initial field experiences. We use Toulmin’s model and recent elaborations of it to analyze the discussions that took place at the university where PMTs reflected on their recent classroom experiences. Our aim is to identify the structure of the argumentation and characterize the emerging warrants, backings, and rebuttals. Results indicate different argumentation structures and types of warrants, backings, and rebuttals in the process of PMTs’ interpretations of students’ mathematical activity. We discuss these findings from the perspective of noticing to identify shifts at the level of PMTs’ interpretations.

This chapter describes how novice and experienced mathematics teachers integrate authentic workplace contexts into mathematics teaching. This goal is inspired by the European MaSciL project and introduced to the teachers in the context of a masters programme in mathematics education. Under an Activity Theory perspective, we use the notions of activity system and boundary crossing to study the process of teachers’ professional learning. In particular, we analyse teachers’ boundary crossings between two activity systems: mathematics teaching and workplace. Results indicate that collaborative task design and reflection made teachers combine elements from the workplace into mathematics teaching. Different ways of linking reality and mathematics teaching were identified in the modelling process in which the students were asked to be engaged.