Many studies have shown that the strategies used in making judgments of chance are subject to systematic bias. Concerning chance and randomness, little is known about the relationship between the external structuring resources, made available for example in a pedagogic environment, and the construction of new internal resources. In this study I used a novel approach in which young children articulated their meanings for chance through their attempts to “mend” possibly broken computer-based stochastic gadgets. I describe the interplay between informal intuitions and computer-based resources as the children constructed new internal resources for making sense of the total of 2 spinners and 2 dice.

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.

Previous research has not supported the validity of Olson's “curvilinear hypothesis.” Convergent validity studies of the FACES I and II have indicated that the “cohesion” and “adaptability” dimensions have a linear relation with healthy family functioning. It is not clear, however, whether these findings are due to the measurement of the FACES' constructs or to the curvilinear theory. This study tested the “curvilinear hypothesis” in terms of these theoretical and measurement issues. The results indicate that FACES II and III do not adequately operation‐alize the “curvilinear hyothesis.” However, an item consistent with the “curvilinear hypothesis” utilizing a bipolar item response format did adequately opera‐tionalize the theory. The use of a bipolar item response format is suggested in the measurement of “cohesion” and “adaptability.”

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