This paper explores two types of lever potentials of CAS and a tool for designing such activities. In the first activity, CAS is used for students to study the concept of equations. In the second activity, CAS is used to strengthen the relation between traditional algebraic paper-andpencil manipulations of equations and the theory that the solution of the equation must remain the same. As the design tool for CAS-based activities that capitalise on the potential of CAS, the notion of praxeology from the Anthropological Theory of Didactical is employed.

The presentation of TSG57 at ICME-14 outlined that mathematics education is a scientific field in which many theoretical cultures coexist, and that “this diversity can be regarded as richness but it also challenges research as well as communication and cooperation in the field”. It was added that “how the scientific community can cope with this diversity with scientific integrity remains an open question”. In this paper, we propose to contribute to the reflection on this challenging issue by considering research on technology-based teaching and learning. First, we present a brief overview of the theoretical landscape in this area of research, highlighting its diversity. We then introduce two conceptual tools that have proven their effectiveness in addressing issues of theoretical diversity: the scale of networking strategies and the concept of research praxeology, before focusing on two case studies. These regard the instrumental approach and the documentational approach to didactics, the emergence and development of which illustrate well the global dynamic of the field towards increasing theoretical diversity, the questions raised by this dynamic and the insightful efforts made to deal with it.

The paper offers a theoretical reflection on the role of a priori analysis in theory. For each theory the a priori analysis has a role of reference while comparing with the a posteriori analysis. Different theories offer different a priori analyses. The different a priori analyses result from the different priorities of the theories with regard to the focus of analysis. In the a priori analyses these priorities and their differences are made explicit. This study demonstrates the important role of the a priori analysis in the different theories and the benefit of comparing a priori analyses for networking the theories.

Despite some recent progress, ethics remains a peripheral topic in mathematics education research. My intention in the first part of the article is to show that ethics is omnipresent in the mathematics classroom. However, ethics does not always operate in the same way. To better grasp the nature of ethics, in the second part of the article, I discuss two ethical systems that have been influential in Western thought (Hobbes’s and Kant’s). These systems have largely informed the understanding of the mathematics classroom, even if, more often than not, they remain implicit. Then, I move to a short discussion of ethics in postmodern times and try to pinpoint what is at stake in ethics. The previous theoretical considerations pave the way to approach, in the last part of the article, ethics from an educational viewpoint. I argue that, in educational contexts, such as the school, ethics appear framed by the way in which we understand teaching and learning. I end the article with an outline of the communitarian oriented relational ethics articulated in the theory of objectification—a communitarian ethics whose practice features responsibility, commitment, and care.

This study aims to identify the theoretical underpinnings of researchers’ work on networking theories in terms of the Anthropological Theory of the Didactic. The study is based on the concept of research praxeology, which allows us to characterise researchers’ practice and knowledge of networking. Three research examples, conducted in the Japanese educational context, are used to illustrate a means to characterise the four praxeological elements (type of tasks, technique, technology, and theory). The results imply that the notion of theoretical grain sizes (grand, intermediate, and local levels) can be used to deepen our understanding of researchers’ work on networking theories. Based on our results, different characteristics of networking theories are discussed according to the cultural issues and specificities of the illustrative examples.

How digital feedback supports teaching/learning with a digital tool is not yet well understood. In a networking of theories approach, Activity Theory and the Instrumental Approach are combined to investigate the role of digital feedback for the teaching/learning of integers with the MAL-system, a multimodal algebra learning system. The MAL-system is designed as a multi- touch tangible user interface with feedback functions allowing students to mathematically operate with (negative) numbers represented as virtual tiles. We explore the role of digital feedback by a multi-case study at the grade five level, in which we conducted experimental task-based interviews of four student pairs, each supported by a tutor. Findings show that (digital) feedback mediates the teaching/learning activity in a supportive way. Reflection on the way the two theories are combined reveals that they can be regarded as locally integrated into a layered model enriching the describing of the transformation of teaching/learning mediated by digital feedback.