How do you help develop mathematical curiosity?

A recent New York Times article talks about how to fall in love with math. Related to this issue is how to develop mathematical curiousity in your students as a math teacher. In no particular order, these are some of my suggestions.

1. Build a strong positive relationship with your students. They will follow you farther into the unknown if they believe in you and trust you.
    
2. Give students opportunities to explore mathematical ideas for themselves without a predefined goal, except that which your students may define themselves.
    
3. Introduce your students to mathematical mysteries; such as the fact that there are as many fractions as whole numbers, but too many decimal numbers to count, or that there are shapes with infinitely long perimeters, but finite areas.
    
4. Ask questions you cannot answer.
    
5. Give problems which are easy to state but which either have no solution or the solution is not yet known to anyone.
    
6. Be mathematically curious yourself and demonstrate this curiosity to your students. Do mathematics yourself and make the mathematics you discover public.
    
7. Let your students do most of the thinking in your class. Too often we do important thinking for students, and if one is not thinking, one cannot be engaged (obedience without thinking is compliance, not engagement).
    
8. Open the black box of problem solving and give students problem solving heuristics they can use themselves.
    
9. Don’t grade everything. Leave as much as you can as activities which are worth doing because they are interesting, not because someone will judge their performance.
    
10. Develop learning mathematics as a social activity. As the African proverb goes, "If you want to go fast, go alone. If you want to go far, go together." Let students explore things on their own, or in small groups, but do some mathematics together in ways which respect every student’s contribution.
     
11. Help students learn some of the history of mathematics and the social contexts under which it was developed.
     
12. Teach mathematics as a narrative, rather than as a series of disconnected facts. Too often children experience learning as a series of sitcoms. Math should be more like an epic journey.