1. Research Experiences in Mathematics for Mathematics Educators Dan Chazan and Sarah Sword January 12, 2006

2. A course for mathematics education Ph.D. students Inside a College of Education Mathematically diverse students: former teachers from elementary, middle, and high school mathematics and science Two semesters Co-taught

3. Purpose of the Course Our goal was to give students strategies to support lifelong learning of mathematics, particularly of mathematics related to their professional work.

4. Research Projects One project per semester –Three weeks: develop the question –Six weeks: research –Three weeks: editorial board work –Three weeks: revise and resubmit Weekly project journals shared with peers Editorial Boards - “Peer Review Process” Studio time - work and presentations

5. In-Class Explorations

6. Exploration topics included: Real numbers Continued fractions Means and inequalities Probability and measure

7. What structures supported the in-class explorations? Study of topics began either with questions from K-12 curriculum or with students’ questions Brown and Walter’s “what-if-not” strategy Journal time at the end of every class period Crystallizations

8. Interactions One student did a project on arithmetic base n…

9. Interactions One student did a project on arithmetic base n… During an in-class exploration of real numbers, another student characterized rational numbers with terminating decimal expansions… and was inspired by his classmate’s work to carry this out in bases other than ten….

10. Interactions One student did a project on arithmetic base n… During an in-class exploration of real numbers, another student characterized rational numbers with terminating decimal expansions… and was inspired by his classmate’s work to carry this out in bases other than ten…. Which inspired a third student to use other bases to help her think about why 0.9999999…. = 1.

11. All of which led to… the instructors being able to use the Cantor Set to think about measure theory and countability (because we could use ternary notation)

12. A student’s reflection What seems so central here is that the mathematics we studied mattered to us. …This had the potential to awaken the mathematical mind and to allow mathematics to be intrinsically meaningful. There are so many questions that have not been answered yet, and so many that have not even been asked.