1. Mathematical Knowledge for Teaching Algebra: Validating an Assessment of Teacher Knowledge Robert E. Floden Raven McCrory Michigan State University January, 2007

2. KNOWING MATHEMATICS FOR TEACHING ALGBRA (KAT) PROJECT (NSF REC No. 0337595) Joan Ferrini-Mundy, Initial PI Robert Floden, Current PI Raven McCrory Mark Reckase Sharon Senk Karen Allen & Xuhui Li

3. GOALS OF KAT PROJECT Instrument Design (2004 - 2005) Develop theoretical constructs, items, test forms Assessment Validation (2006 - 2008) Do the forms measure distinct constructs? What differences in knowledge affect teacher responses? Sensitivity to Instruction (2007 - 2008) How do teacher preparation experiences affect responses?

4. Item Development August 2004 - October 2005 Constructs defined Item writing workshops with mathematicians, math educators, secondary teachers Additional Items written by KAT faculty & GAs Items reviewed by mathematicians Items edited by KAT staff

5. Mathematical Knowledge for Teaching Algebra (simplified for assessment design) Knowledge of school algebra algebra in middle and high school Advanced mathematical knowledge related college math, e.g. calculus, abstract algebra Teaching knowledge knowledge of typical errors, canonical uses of school math, curriculum trajectories, etc.

7. Sample Item: Identify an Exponential Function (School Knowledge) Which of the following situations can be modeled using an exponential function? i. The height h of a ball t seconds after it is thrown into the air. ii. The population P of a community after t years with an increase of n people annually. iii. The value V of a car after t years if it depreciates d% per year. A. i only B. ii only C. iii only D. i and ii only E. ii and iii only

8. Pilot Test Results: Identify Exponential Function Number of casesDifficulty Pre-service teachers 1380.297 In-service teachers 2870.324 Total4310.313

9. Sample Item: Number Systems (Advanced Knowledge) For which of the following sets S is the following statement true? For all a and b in S, if ab = 0, then either a = 0 or b = 0. i. the set of real numbers i. the set of complex numbers iii. the set of integers mod 6 iv. the set of integers mod 5 v. the set of 2x2 matrices with real number entries A. i onlyD. i, ii, iii and iv only B. i and ii onlyE. i, ii, iii, iv, and v C. i, ii and iv only

10. Pilot Test Results: Properties of Number Systems (Advanced Knowledge) Number of casesDifficulty Pre-service teachers 860.151 In-service teachers 960.177 Total1860.161

11. Sample Item: Identifying Student’s Error in Solving a Linear Equation (Teaching Knowledge) A student solved the equation 3(n - 7) = 4 - n and obtained the solution n = 2.75. What might the student have done wrong?

12. Pilot Test Results: Identify Student’s Error Number of casesDifficulty Pre-service teachers 970.760 In-service teachers 140.946 Total1150.787

13. Validation Study Design Recruit sample with varying combinations of constructs Combination of factor analysis and cluster analysis to identify empirical sub-scales Analysis of clustered items to interpret clusters Interviews with test-takers to understand reasoning behind correct and incorrect responses

14. Number of Participants in Validation Studies Pre-service teachers In-service teachers 327 492 Math majors Non-math major 535 283 Total822 Career stage 818 Major

16. Next Steps Complete interpretation of clusters Examine results for special samples (e.g., very high math knowledge) Interview study of new sample of teachers Measure change in assessment results in selected teacher preparation programs

17. Questions for the Future Will our assessment reveal differences in the effects among approaches to mathematics teacher education (pre-service or inservice)? Will programs vary in their relative impact on the three components?: Knowledge of school algebra Advanced mathematical knowledge Teaching knowledge Will differences in assessment results predict differences in teaching effectiveness?

18. Discussion What information from our project would be useful to you in designing courses for teachers? How can we best make our results useful to teacher educators? What do you hope our project can tell you?

19. Contacts for KAT Project Information Xuhui Li, Project Manager,lixuh@msu.edu Robert E. Floden, PI, floden@msu.edu Raven McCrory, Co-PI,mccrory@msu.edu