Seeing the Quality of Mathematical Knowledge in Instruction

1. Seeing the Quality of Mathematical Knowledge in Instruction TNE Observation Protocol Meeting Academy for Educational Development University of Wisconsin-Milwaukee October 17-19, 2007 Learning Mathematics for Teaching ProjectUniversity of Michigan Deborah L. Ball, Hyman Bass, Merrie Blunk, Katie Brach, Charalambos Charalambous, Carolyn Dean, Seán Delaney, Imani Masters Goffney, Heather Hill, Jennifer Lewis, Geoffrey Phelps, Laurie Sleep, Mark H. Thames, and Deborah Zopf

2. “Mathematical Knowledge for Teaching” • Essential for high-quality mathematics teaching • Learned in a variety of settings: school, university, teaching • Includes common content knowledge • Comprises other domains of mathematics content particular to the work of teaching

3. Common content knowledge: Can you simplify these radicals? √48 √63 √99 √150 Mathematical knowledge for teaching: Mr. Squire is designing a lesson on simplifying radical expressions. He wants to pick one example from the previous day’s homework problems to review at the beginning of today’s class. His goal is to select a problem that will lead to a good discussion about different solution strategies for simplifying radicals. Which of the following is best for setting up a good discussion about strategies for simplifying radical expressions? √48 √63 √99 √150 An example of MKT

4. Why does MKT matter? • MKT is associated with children’s achievement; therefore we want to know: • What are the domains of MKT? • How do teachers acquire MKT? • What is the relationship between instructional practices and MKT? • What professional development experiences contribute to MKT?

5. Measuring MKT • Paper-and-pencil multiple-choice items • Develops the construct of MKT • Provides data about professional development

6. Validating MKT measures Does achievement on paper-and-pencil measures matter in instruction? We videotaped lessons of a subset of teachers to validate. • 10 teachers • 9 videotaped lessons each, over one year • Interviews If the MKT measures are valid, then teachers’ performance in practice would correspond to their achievement on the paper-and-pencil test.

7. Coding Videotapes of Mathematics Teaching • Videocodes cover these domains: • Mathematical content and instructional format • Teacher’s use of mathematical language, representations, explanations • Teacher’s mathematical interaction with student productions • Teacher’s use of mathematics to teach equitably • Videocodes are neutral regarding teaching style (“reform-oriented,” “didactic,” etc.)

8. Sample: Representations

9. An excerpt of code

11. A teacher models • Red “pies” represent negative numbers • Green “pies” represent positive numbers

12. Coding Glossary • To code, first decide whether the mathematical element is present (P) or not present (NP). If present, then: • Mark appropriate (A) if the teacher’s use of the element was, for the most part, mathematically accurate and appropriate—it did not distort the mathematics. • Mark inappropriate (I) if the teacher’s use of the element distorted the mathematics or was inappropriate for the grade level. • If not present, then: • Mark appropriate (A) if absence of the element seems appropriate. • Mark inappropriate (I) if absence of the element seems problematic—i.e., the element should have happened.

13. Use of videocodes • Our findings: • inter-rater reliability of .7 to .85 • teachers’ clinical practice scores and test scores correlation of .75 or higher • Others’ potential use of codes would require: • Shared understandings of the codes • Users with MKT

14. Continuing challenges • How do we appraise teaching for equity? • Some essential qualities difficult to nail down (e.g., “scaffolding”) • High qualifications for coders • Gap between quality teaching as envisioned by the field and quality teaching that produces results in schools