Making masks for Maui: Keeping the macro task in mind

New Zealand primary school children in technology lessons often design and create an artifact in response to a scenario that relates to their interests and experiences. Usually the task is undertaken over several days. In this paper we draw on data generated within the INSiTE study, a three-year study exploring the nature of effective student-teacher interactions around science and technology ideas. The teacher in this paper planned for her children to create a mask for their forthcoming school production: 'How Maui found the secret of fire'. As the children worked on the macro task, that of designing and making a mask, meso and micro tasks emerged. The teacher assisted the children to identity and resolve these, hearing in mind that the ultimate aim was their successful participation in the school production. When teachers assist children to maintain a focus on the overall or macro task goals their artifact fulfils the specifications of the scenario and children's technology understandings and skills are fostered

Product structures for Legendrian contact homology

Legendrian contact homology (LCH) is a powerful non-classical invariant of Legendrian knots. Linearization makes the LCH computationally tractable at the expense of discarding nonlinear (and non-commutative) information. To recover some of the nonlinear information while preserving computability, we introduce invariant cup and Massey products – and, more generally, an A∞ structure – on the linearized LCH. We apply the products and A∞ structure in three ways: to find infinite families of Legendrian knots that are not isotopic to their Legendrian mirrors, to reinterpret the duality theorem of the fourth author in terms of the cup product, and to recover higher-order linearizations of the LCH

Circles Minimize most Knot Energies

We define a new class of knot energies (known as renormalization energies) and prove that a broad class of these energies are uniquely minimized by the round circle. Most of O'Hara's knot energies belong to this class. This proves two conjectures of O'Hara and of Freedman, He, and Wang. We also find energies not minimized by a round circle. The proof is based on a theorem of G. Luko on average chord lengths of closed curves.Comment: 15 pages with 3 figures. See also http://www.math.sc.edu/~howard