Software design is increasingly being recognized as an important factor in student learning when interacting with interactive geometry software (IGS). A categorization of the operations possible within an IGS is used to identify and analyse design decisions made in a number of current IGS programs: Cabri II Plus, Cabri 3D, Cinderella, GeoGebra and Geometer's Sketchpad. The analysis, in the context of exploring the area of a circle, is focused on construction, dragging, and alternative spatial and semantic views. A wide diversity of both design issues and individual design decisions was identified, illustrating both the scarcity of research in this area and a number of inevitable tensions, such as between functionality and complexity, and between static and dynamic geometry, related to which future research questions might be posed.

Constructionism, best known as the framework for action underpinning Seymour Papert's work with Logo, has stressed the importance of engaging students in creating their own products. Noss and Hoyles have argued that such activity enables students to participate increasingly in a web of connections to further their activity. Ainley and Pratt have elaborated that learning is best facilitated when the student is engaged in a purposeful activity that leads to appreciation of the power of mathematical ideas. Constructionism gives prominence to how the learner's logical reasoning and emotion-driven reasons for engagement are inseparable. We argue that the dependence of constructionism upon the orienting framework of constructivism fails to provide sufficient theoretical underpinning for these ideas. We therefore propose an alternative orienting framework, in which learning takes place through initiation into the space of reasons, such that a person's thoughts, actions and feelings are increasingly open to critique and justification. We argue that knowing as responsiveness to reasons encompasses not only the powerful ideas of mathematics and disciplinary knowledge of modes of enquiry but also the extralogical, such as in feelings of the aesthetic, control, excitement, elegance and efficiency. We discuss the implication that mathematics educators deeply consider the learner's reasons for purposeful activity and design settings in which these reasons can be made public and open to critique.Keywords Space of reasons . Constructivism . Constructionism . Purpose and utility .

This chapter focuses on digital technologies and geometry education, a combination of topics that provides a suitable avenue for analysing closely the issues and challenges involved in designing and utilizing digital technologies for learning mathematics. In revealing these issues and challenges, the chapter examines the design of digital technologies and related forms of learning activities for a range of geometries, including Euclidean and co-ordinate geometries in two and three dimensions, and non-Euclidean geometries such as spherical, hyperbolic and fractal geometry. This analysis reveals the decisions that designers take when designing for different geometries on the flat computer screen. Such decisions are not only about the geometry but also about the learner in terms of supporting their perceptions of what are the key features of geometry.

Mathematics-specific anxiety is anxiety that impedes mathematical thinking and progress, and creates distress for many learners, or at the least a tendency to avoid mathematical thinking. Such anxiety is prevalent. The importance of mathematics to economic recovery is well-established; in order to meet the need for mathematics, the high levels of mathematics anxiety that stand in the way of individual mathematical progress should be addressed. Using a case study involving an adult learner, we use Self-Determination Theory to explain why mathematical resilience is a concept which can work against anxiety and for a positive stance towards mathematics. Work on mathematical resilience demonstrates that well-informed, subject-specific interventions can help people manage emotions, including anxiety, and improve progress and uptake in mathematics. We illustrate ways in which the focus of Self-Determination Theory on meeting basic psychological needs (autonomy, competence and relatedness), to enhance wellbeing and prevent harm, provides grounding for much good practice in mathematics education and specifically for work in mathematical resilience. The tools of mathematical resilience go beyond what is currently proposed in SDT research. We illustrate ways in which these tools can specifically facilitate learners' emotion regulation, which we propose is integral to mathematical learning competence, leading to greater mathematical wellbeing, learning, and release from mathematics anxiety.

The aim of this discussion group was to put contemporary philosophy to work (cf. Cobb, 2007). Inferentialism is an example of contemporary philosophy (Brandom, 2000) that increasingly receives interest in mathematics and statistics education. It can be considered an orienting framework that provides epistemological foundations for conceptualizing and analyzing knowledge, learning, communication, and reasoning in the fields of mathematics and statistics. Inferentialism avoids a representationalist perspective on knowledge and learning by focusing on reasoning and inferences (Bakker & Derry, 2011). The Discussion Group (DG) brought together researchers who are interested in the role and use of inferentialism or other contemporary philosophies in mathematics and statistics education. It gave the attendants the opportunity to share perspectives, to question, to discuss, and to make joint efforts in answering the posed key issues. The DG format at ICME provided the opportunity to discuss the significance and the restrictions of the perspective of inferentialism and other contemporary philosophies on the learning and teaching of mathematics and statistics. The discussion was initiated by several talks: Arthur Bakker (Utrecht) introduced inferentialism as a semantic theory and Maike Schindler (Örebro) gave an overview on researchers presently working with inferentialism in mathematics and statistics education. Paul Ernest (Exeter) talked about meaning in mathematics and mathematics education and anti-representationalism, and Dave Pratt (London) gave a talk on constructionism. Alexandra Thiel-Schneider (Dortmund) presented an empirical study using inferentialism and Luis Radford (Ontario) summarized the discussion elaborating on how inferentialism relates to existing theories in our domain. The participants experienced the discussion group as a fruitful gathering of researchers interested in philosophy in mathematics education; and of various perspectives on inferentialism and its possible use. The talks were welcomed as an input and promoter of discussion among all participants.