ers to national policy makers, agrees on the importance of connecting mathematics — to science, to social science, to art, to everyday life, and to other parts of mathematics. Accordingly, the prototypes should develop links with science, with the visual arts, and with the language arts.

  • Thoughtful approaches: Insofar as possible, the tasks should promote "higher-order" thinking. Just as the verbs explore, justify, represent, solve, construct, discuss, use, investigate, describe, develop, and predict are used in the Standards to convey "active physical and mental involvement of children" in learning mathematics, they are appropriate to seek in assessment activities as well. Further, given a choice between cognitive complexity and simplicity, the focus of these tasks should be on the former.

  • Mathematical communication: The tasks should emphasize the importance of communicating results — not simply isolated answers, but mathematical representations and chains of reasoning. Children should have opportunities to work in groups to explain their thinking to others, and to write explanations of their approaches.

  • Rich opportunities: The tasks should let children solve problems via a variety of creative strategies; demonstrate talents (artistic, spatial, verbal) beyond those normally associated with numerical mathematics; invent mathematics that (to them) is new; recognize opportunities to use and apply mathematics; and show what they can do (as opposed to what they cannot do).

  • Improved instruction: The tasks should have the potential for influencing instruction positively, both in content and in pedagogy. Teachers who use these tasks should become better teachers as a result of the experience; children who participate should emerge with increased self-confidence and heightened expectations for future mathematics courses.

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