Question 1: Important mathematics (but what's that?). Doing mathematics. Internalizing mathematics. Talking mathematics. Physical models for abstract things, abstract models for physical things. Mathematics as a natural part of the world. Imagination and intelligence. Intelligent use of machines.

All this for both college-bound and non-college bound.

The hard questions: defining "important mathematics," "intelligent use of machines" needs mathematicians, math ed folks, teachers talking to each other and working together to begin to delineate parameters (this stuff will never be settled). My own bar pretty high in most ways.

Question 2: Algebra, geometry (along the lines of EDC's [Education Development Center] Connected Geometry, some of the BBN [Bolt, Beranek, and Newman] intuitive higher geometry stuff) , functions and their graphs (including trig functions), basic probability and statistics, lots of mucking about with applications (e.g. surveys, modelling).

Question 3: When kids talked mathematics intelligently, as naturally as they talk anything else, whenever mathematics is appropriate. When kids want to talk mathematics.

Question 4: No different from any mathematics teacher, K through graduate school: mathematically, to have experienced creating mathematics, to have internalized mathematics (deep, not superficial knowledge), to have learned much more than they will ever teach, and to not stop learning; pedagogically, to draw on their experience doing mathematics, on the experiences of their colleagues, and on good pedagogical research to understand what goes on when people learn mathematics and to be flexible in their approaches, and to not stop observing themselves and their students and learning from colleagues and research.

Question 5: The proof that the real numbers are uncountable, encountered in 7th grade. No kidding.