1. Summarizing “Teachers need to learn concepts.” –Mike Maxon “Teachers need to be able to connect and see mathematics in their everyday lives.” –Personal communication with a SDSU Prof. “There are a lot of holes in the dam.” –Larry Sowder

2. Content Conceptual understanding Problem solving Think mathematically Make conjectures Communicate math ideas Concrete to the abstract Reason Deductively

3. Content part 2 Make connections - “the connections are essential” –Mariam Clifford –Concepts and procedures –Math ideas to students’ lives –Longitudinal Coherence –Representations Equations, Tables, Graphs, Diagrams, Contexts –Content Knowledge in math and content appropriate for students Abstract Algebra and M.S. Algebra

4. Disposition What is math? –Is math a fixed field of facts and procedures or is there more to it than that? Like mathematics Developing stick-to-itiveness

5. Implementation Proceduralizing good problems Percentages of how often types of problems were presented.

6. Implementation Proceduralizing good problems Percentages of How Problems are Worked on During the Lesson.

7. Implementation Good use of manipulatives Using technology in the classroom Getting students to communicate about math ideas Modeling the way we want them to teach –Being explicit about the pedagogy we are using Covering content –“If you cover the content then students can’t see the math.” – Kate Masarik

8. College Course vs. Professional Development What content or ideas are students not ready to grapple with until they are in the classroom?