Teachers of mathematics face a particular tension, which the authors call the planning paradox. If teachers plan from objectives, the tasks they set are likely to be unrewarding for the pupils and mathematically impoverished. Planning from tasks may increase pupils' engagement but their activity is likely to be unfocused and learning difficult to assess. By seeking inspiration from research in the areas of curriculum design, the nature of authenticity in the classroom and the use of tools, and by looking retrospectively at the design of computer-based tasks that have underpinned their research for many years, the authors recognise a theme of purposeful activity, leading to a planned appreciation of utilities for certain mathematical concepts. The authors propose utility as a third dimension of understanding, which can be linked to purpose in the effective design of tasks. The article concludes with a set of heuristics to guide such planning.

This study explores the impact that co-designing a virtual manipulative, Fractions Lab, had on teachers' professional development. Tapping into an existing community of practice of mathematics specialist teachers, the study identifies how a cooperative enquiry approach utilising workshops and school-based visits challenged 23 competent primary school teachers' technological, pedagogical and content knowledge of fraction equivalence, addition and subtraction. Verbal and written data from the workshops alongside observations and interviews from the school visits were analysed using the technological, pedagogical and content knowledge (TPACK) framework. The findings show that the assumptions of even experienced teachers were challenged when Fractions Lab was shared as an artefact on which they were asked to co-design and subsequently interact with, using it in subsequent phases of the cooperative inquiry process. Two original aspects of successful co-design of virtual manipulatives through communities of practice are identified and offered to others: (1) careful bootstrapping of the first design iteration that gathers intelligence about the content area and the technological affordances and constraints available; and (2) involvement of highly motivated teachers who perceive themselves as agents of change in the domain area.

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Abstract. This paper reports on our progress on a systematic approach to operationalizing support in Fractions Lab -an exploratory learning environment for learning fractions in primary education. In particular, we focus on the question of what feedback to provide and consider in detail the implementation of feedback according to two dimensions: the purpose of the feedback, depending on the task-specific needs of the student, and the level of feedback, depending on the cognitive needs of the student. We present early findings from our design-based research that includes Wizard-of-Oz studies of the intelligent feedback system and student perception questionnaires.

Abstract. We describe a Wizard-of-Oz study that investigates the impact of different types of feedback on students' affective states. Our results indicate the importance of matching carefully the affective state with appropriate feedback in order to help students transition into more positive states. For example when students were confused affect boosts and specific instruction seem to be effective in helping students to be in flow again. We discuss this and other effective ways to and implications for the development of our system and the field in general.