In this paper we investigate how to efficiently empower teachers to implement and orchestrate a mathematical learning activity supported by digital technologies. The particular learning activity in this study is intended to facilitate learners’ transition from the Pythagorean Theorem to the distance formula and the equation of a circle. The activity comprises structured and guided inquiries involving laptops with GeoGebra and traditional resources. It has been tested with 38 upper secondary students and two mathematics teachers. Our results indicate that a singular discussion with the teachers, based on the researcher’s prospective analysis of the activity with main focus on threshold constructs and self-regulating skills, suffices to support the teachers’ implementation and orchestration of the activity.

The unfolding of an affordance, as an opportunity for action, during a learning activity requires the learner to interact with mediating artifacts. The design of a learning activity involves appropriating affordances and embedding them in the activity in such a manner that the learner is invited to unfold the affordances, through interaction with their mediating artifacts in accordance with pre-defined hypothetical learning trajectories. In this paper, the notion of affordances is used explicitly in the discussion of two previous research efforts. We argue that the notion of affordances, which was tacitly used in these efforts and aligns well with the methodology of scenario-based design,may beused as an instrument for the collaborative design of innovative mathematical learning activities.

Designing curriculum-based learning activities involves identifying and implementing affordances that provide opportunities for students to engage in actions directed at achieving specific learning objectives. These affordances may be mediated by information and communication technologies (ICT) or traditional resources, peers or teachers, or other contextual features of the learning environment. The learning objectives guide the designer's choice of affordances, whose deployment on specific artifacts is based on assessing the available artifacts' mediating capabilities. Such a design approach puts high demands on the designer's technological, pedagogical and content knowledge. In this paper, we discuss how we have addressed these demands by adopting a flexible co-design approach that invites the creative blending of mobile technologies and augmented reality with traditional resources, for the purpose of designing innovative mathematical learning activities with high relevance for teachers' practice and the mathematics curriculum.

In this paper we describe and reflect on the design of a mathematical learning activity developed in collaboration between teachers, researchers and technical developers. By making use of augmented reality (AR) as a technology supporting augmentation of a real-world projection with computergene- rated images, we have designed an activity that promotes unique action and learning trajectories. These trajectories require the learners to engage in interactive-constructive actions that involve and stimulate the development of their self-regulatory skills by inviting them to vary and coordinate across the contextual affordances of the technologies and the physical resources in the classroom. Our learning activity is designed as a collaborative guided inquiry, implemented in a regular classroom and involved mathematical problem solving in relation to the geometric concept of scale. In order to successfully complete the activity, the learners are challenged to coordinate affordances from three distinct referential contexts by involving physical and virtual artifacts. In the design process, we identify critical aspects of the activity and embed affordances for corresponding scaffolding actions which turn out to play a crucial role when the activity is implemented with a group of four 15-year-old students. Although the AR technology has served us well in developing this particular activity, this specific technology appears to have limited applicability in mathematics education beyond geometry. We recommend that future research efforts move beyond AR and consider the broader context of embodied design with tangible user interfaces, that have recently shown great potential for the design of innovative activities for the learning of mathematics.

Flera års forsknings- och utvecklingsarbete har resulterat i en handfull lärandeaktiviteter för elever i årskurs 4-9, som i undersökande aktiviteter använder ny teknik i form av datorer, projektorer, interaktiva skrivtavlor, webbkameror och mobiltelefoner. Tekniken kan låta oss möta meningsfull matematik på nya sätt i nya sammanhang och kan göra det möjligt att sammanfläta undersökande utomhusaktiviteter i små grupper med inomhusaktiviteter i helklass. Vi ger flera konkreta exempel på hur tekniken kan stimulera och stödja kommunikation och representation av ett matematiskt innehåll.

Guided by the notion of design research we develop a learning activity for 12 year old students, who are asked to coordinate themselves physically in terms of distances with respect to two given points in an outdoor setting. The outdoor activity, as well as its continuation into the mathematics classroom, involves mobile software applica-tions specifically developed to support this activity. In this paper, we argue that the design of innovative learning activities is enhanced by the coordination of expertise and knowledge from several research domains, whose collaboration is facilitated by using affordances for representation and communication as design instruments. We present a case where ancient Greek mathematics, modern psychology and techno-logical affordances guide the design of an innovative spatial coordination activity.

Mathematical tasks have always been and still are a major object of concern in the mathematics classroom. Teachers’ daily work includes planning, implementing, and evaluating the mathematical tasks that student engage in during class by making use of material and immaterial artefacts. We argue that teachers’ mathematical knowledge has to include instruments for controlling how these artefacts become involved when students engage in solving mathematical tasks. We propose to meet this demand by coordinating the matching notions of affordances (for planning) and objects of activity (for evaluation). We briefly illustrate how these notions can be used as analytical instruments in a fashion that connects to what teachers already do in their daily work.

We design and evaluate a curriculum-based mathematical learning activity involving secondary students' geometrical constructions, mathematical modeling and algebraic validation of hypotheses based on hands-on explorations with the interactive geometry application GeoGebra available on individual laptops. We argue that guided inquiries in a technology-enhanced learning environment that invites blending of interactive technologies and traditional resources may be an efficient means for developing self-regulatory skills.