Investigating the relationship between the problem and the solver: Who decides what math gets used?

Tasks that are descriptive of a goal state and not prescriptive of the paths students must take to reach it inevitably generate spaces of possible interpretations of givens and goals, as well as possible paths from givens to goals, each featuring elements of a bounded space of mathematical concepts. When a sample comprised of students at elementary and post-baccalaureate levels of schooling was given one of these tasks, the solutions expressed rich and deep understandings of mathematical concepts that were common among groups at both levels of schooling. These findings are less supportive of the foundational metaphor of curriculum in which understandings serve to support the acquisition of more formal mathematics, and more supportive of the notion of a curriculum that "spirals" around central ideas that are revisited at multiple levels of schooling in order to provide learners with greater access to powerful ways of understanding mathematics.