Capturing Growth in Teacher Mathematical Knowledge: An Inquiry into Algebraic Reasoning

1. Capturing Growth in Teacher Mathematical Knowledge An Inquiry into Elementary and Middle School Teacher Understanding of Algebraic Reasoning and Relationships The Association of Mathematics Teacher Educators Eleventh Annual Conference 26 January 2007 Dr. DeAnn Huinker, Lee Ann Pruske & Melissa Hedges The Milwaukee Mathematics Partnership University of Wisconsin - Milwaukee www.mmp.uwm.edu • This material is based upon work supported by the National Science Foundation Grant No. EHR-0314898.

2. Session Goals • Contribute to the discussions around defining and measuring the specialized mathematical knowledge needed for teaching. • Share and examine performance assessments that look more closely at growth in the mathematical knowledge targeted on algebra.

3. What distinguishes mathematical knowledge from the specialized knowledge needed for teaching mathematics?

4. Common vs. Specialized Mathematical Knowledge Encompasses • “Common” knowledge of mathematics that any well-educated adult should have. • “Specialized” to the work of teaching and that only teachers need to know. Source: Ball, D.L. & Bass, H. (2005). Who knows mathematics well enough to teach third grade? American Educator.

5. Mathematical Knowledge for Teaching (MKT) Some interesting dilemmas… • Why do we “move the decimal point” when we multiply decimals by ten? • Is zero even or odd? • For fractions, why is 0/12 = 0 and 12/0 undefined? • How is 7 x 0 different from 0 x 7? • 35 x 25 ≠ (30 x 20) + (5 x 5) Why? • Is a rectangle a square or is a square a rectangle? Why?

6. Capturing Growth in Teacher Mathematical Knowledge

7. Setting • Content Strand: Algebraic Reasoning and Relationships • Pretest: September 2005 • School Year: Monthly sessions (~20 hours) • Posttest: June 2006 • 120 Classroom teachers: Kindergarten - Eighth Grade

8. Algebraic Relationships Expressions, Equations, and Inequalities Generalized Properties Sub-skill Areas a x b = b x a Patterns, Relations, and Functions – 25= 37

9. Items • Measure mathematics that teachers use in teaching, not just what they teach. • Orient the items around problems or tasks that all teachers might face in teaching math. • MMP performance assessments to give insight into depth of teacher knowledge developed around monthly seminars.

10. Teacher Growth in Mathematical Knowledge for Teaching (MKT) Gain = 0.296 t = 5.584 p = 0.000

11. Complete the following: A) Draw a sketch of a rectangle to represent the problem 46 x 37. Partition and label the rectangle to show the four partial products. B) Make connections from your partial product strategy (in part A) to the traditional multiplication algorithm, explaining how they are related. C) Make connections from your partial products strategy (Part A) to the problem (4x + 6) * (3x + 6), explaining how they are related.

12. Reflect and Discuss • What is the “pure” mathematical knowledge you employed while completing this task? • What mathematical knowledge embedded in this task might be accessed during the teaching of this concept? • Is this knowledge the same?

13. Performance Assessment Gain additional insights into our teachers’ abilities to: • Make solid connections between the area model of multiplication and the distributive property. • Understand and explain connections between the standard algorithm and use of the distributive property for multiplication. • Generalize use of the distributive property.

14. Examining Teacher Work As you reflect on teacher work samples consider the following: • Is the mathematics correct? Are mathematical symbols used with care? • Are the connections between representations clear? • Are explanations mathematically correct and understandable?

15. Performance Activity Results • 16% (9/56) proficient, good explanations and connections. • 50% (28/56) getting there, good procedural skills, limited explanations. • 34% (19/56) did not accurately or completely solve the tasks.

16. Next Steps . . . Next steps… • Do teachers’ scores predict that they teach with mathematical skill, or that their students learn more, or better? • How might we connect teachers’ scores to student achievement data? • More open-ended items to show reasoning

17. Knowing mathematics for teaching includes knowing and being able to do the mathematics that we would want any competent adult to know. But knowing mathematics for teaching also requires more, and this “more” is not merely skill in teaching the material. Ball, D.L. (2003). What mathematical knowledge is needed for teaching mathematics? Secretary’s Summit on Mathematics, U.S. Department of Education, February 6, 2003; Washington, D.C. Available at http://www.ed.gov/inits/mathscience.

18. Mathematical knowledge for teaching must be serviceable for the mathematical work that teaching entails, for offering clear explanations, to posing good problems to students, to mapping across alternative models, to examining instructional materials with a keen and critical mathematical eye, to modifying or correcting inaccurate or incorrect expositions. Ball, D.L. (2003). What mathematical knowledge is needed for teaching mathematics? prepared for the Secretary’s Summit on Mathematics, U.S. Department of Education, February 6, 2003; Washington, D.C. Available at http://www.ed.gov/inits/mathscience. (p. 8)

19. Knowing Mathematics for Teaching Demands depth and detail that goes well beyond what is needed to carry out the algorithm • Use instructional materials wisely • Assess student progress • Make sound judgment about presentation, emphasis, and sequencing often fluently and with little time • Size up a typical wrong answer • Offer clear mathematical explanations • Use mathematical symbols with care • Possess a specialized fluency with math language • Pose good problems and tasks • Introduce representations that highlight mathematical meaning of selected tasks