1. UNIVERSITY OF LOUISVILLE Assessing the Mathematics Knowledge of Teachers William S. Bush University of Louisville North Carolina Association of Mathematics Teacher Educators Chapel Hill, North Carolina April 21, 2007

2. University of Louisville 2 Purpose of Session To analyze the types of mathematics knowledge necessary for teaching To analyze the types of mathematics knowledge necessary for teaching To review assessment items that assess these types of knowledge To review assessment items that assess these types of knowledge To discuss strategies for helping teachers build these types of knowledge To discuss strategies for helping teachers build these types of knowledge

3. University of Louisville 3 Think, Pair, Share What kinds of mathematics knowledge do teachers of mathematics need to do their work effectively? Discuss in groups of three or four.

4. University of Louisville 4 What Others Say Lee Shulman (Educational Researcher, February 1986) Lee Shulman (Educational Researcher, February 1986) Content Knowledge Content Knowledge –Subject matter content knowledge –Pedagogical content knowledge –Curricular knowledge

5. University of Louisville 5 Subject Matter Content Knowledge The amount and organization of knowledge per se in the mind of the teacher... Teachers must not only be capable of defining the accepted truths in a domain. They must also be able to explain what a particular proposition is deemed warranted, why it is worth knowing, and how it relates to other propositions, both within the discipline and without, both in theory and in practice.

6. University of Louisville 6 Pedagogical Content Knowledge Includes, for the most regularly taught topics in one’s subject area, the most useful forms of representation of those ideas, the most powerful analogies, illustrations, examples, explanations, and demonstrations... the ways of representing and formulating the subject that makes it comprehensible to others... what makes the learning of topics easy or difficult, the conceptions and preconceptions students of different ages and backgrounds bring with them to the learning of those most frequently taught topics and lessons.

7. University of Louisville 7 Curricular Knowledge The full range of programs designed for the teaching of particular subjects and topics at a given level, the variety of instructional materials available in relation to these programs, and the set of characteristics that serve as both indications and contraindications for the use of particular curriculum or program materials in particular circumstances.

8. University of Louisville 8 What Others Say (Continued) Ball and Others—Univ. of Michigan Mathematics knowledge for teaching requires knowing in detail the topics and ideas that are fundamental to the school curriculum, and beyond. This detail involves a kind of unpacking that is often difficult to produce... It requires knowing how mathematical topics are connected, and how particular ideas anticipate later ones... [It] also involves using tools and skills for reasoning about mathematical ideas, representations, and solutions, as well as knowing what constitutes adequate proof... [It] also requires fluency and care with mathematical language and notation... [It] a Mathematics knowledge for teaching requires knowing in detail the topics and ideas that are fundamental to the school curriculum, and beyond. This detail involves a kind of unpacking that is often difficult to produce... It requires knowing how mathematical topics are connected, and how particular ideas anticipate later ones... [It] also involves using tools and skills for reasoning about mathematical ideas, representations, and solutions, as well as knowing what constitutes adequate proof... [It] also requires fluency and care with mathematical language and notation... [It] also depends on a broad familiarity with applications of mathematics.

9. University of Louisville 9 Mathematics Knowledge about Instruction Design mathematically accurate explanations Design mathematically accurate explanations Use mathematically appropriate and comprehensible definitions Use mathematically appropriate and comprehensible definitions Represent ideas using a variety of models Represent ideas using a variety of models Respond productively to mathematical questions Respond productively to mathematical questions Explain mathematical terms and concepts Explain mathematical terms and concepts Use representations accurately in the classroom Use representations accurately in the classroom Use appropriate technology Use appropriate technology Help students abstract/generalize mathematical ideas Help students abstract/generalize mathematical ideas Assist students in building mathematical structures Assist students in building mathematical structures

10. University of Louisville 10 Mathematics Knowledge About Student Learning Pose good mathematical questions and problems that are productive for student learning Pose good mathematical questions and problems that are productive for student learning Interpret and make mathematical and pedagogical judgments about students ideas and work Interpret and make mathematical and pedagogical judgments about students ideas and work Assess students’ mathematics learning Assess students’ mathematics learning Interpret student statements/solutions and provide remedies Interpret student statements/solutions and provide remedies Provide students examples of mathematical concepts, algorithms, or proofs Provide students examples of mathematical concepts, algorithms, or proofs Know when student reasoning is valid and assist them with errors in reasoning Know when student reasoning is valid and assist them with errors in reasoning

11. University of Louisville 11 Mathematics Knowledge About Curricula Make judgments about the mathematical quality of instructional materials and modifying as necessary Make judgments about the mathematical quality of instructional materials and modifying as necessary Judge and correct textbook treatments of particular topics Judge and correct textbook treatments of particular topics Connect mathematical ideas within and across other mathematical topics Connect mathematical ideas within and across other mathematical topics

12. University of Louisville 12 What the Research Says Hill, H. C., Rowan, B., & Ball, D. L. (2005, Summer). Eff Hill, H. C., Rowan, B., & Ball, D. L. (2005, Summer). Effects of teachers’ mathematical knowledge for teaching on student achievement. American Educational Research Journal, 42(2), 371- 406. Teacher and student data collected in 115 elementary schools in 2001-02 through 2003-04. Teacher and student data collected in 115 elementary schools in 2001-02 through 2003-04. Mathematics achievement data and parent interview data collected on 1 st and 3 rd grade students. Mathematics achievement data and parent interview data collected on 1 st and 3 rd grade students. Teacher questionnaires, including mathematics knowledge for teaching, collected on teachers Teacher questionnaires, including mathematics knowledge for teaching, collected on teachers Teachers’ mathematical knowledge for teaching positively predicted student gains in mathematics. Teachers’ mathematical knowledge for teaching positively predicted student gains in mathematics.

13. University of Louisville 13 Think, Pair, Share Review the following sets of assessment items. Which type of mathematics knowledge for teaching is being assessed? Which type of mathematics knowledge for teaching is being assessed? What underlying mathematics knowledge is needed to respond to the item? What underlying mathematics knowledge is needed to respond to the item?

14. University of Louisville 14 Questions for Discussion Where can preservice and practicing teachers learn these types of mathematical knowledge? Where can preservice and practicing teachers learn these types of mathematical knowledge? What strategies can we use to help teachers gain this knowledge? What strategies can we use to help teachers gain this knowledge?

15. University of Louisville 15 Closing Comments Mathematics knowledge for teaching is based on an underlying knowledge of mathematics. Mathematics knowledge for teaching is based on an underlying knowledge of mathematics. Tasks that focus on or assess mathematics knowledge for teachers: Tasks that focus on or assess mathematics knowledge for teachers:  can uncover teachers’ misconceptions about mathematics.  are inherently more relevant and motivating for teachers.  Can and should be used in methods and mathematics courses for teachers.

16. University of Louisville 16 For More Information about Teacher Assessments University of Louisville DTAMS University of Louisville DTAMS William S. Bush bill.bush@louisville.edu William S. Bush bill.bush@louisville.edubill.bush@louisville.edu Web http://louisville.edu/edu/crmstd/ Web http://louisville.edu/edu/crmstd/http://louisville.edu/edu/crmstd/http://louisville.edu/edu/crmstd/ University of Michigan Teacher Assessments University of Michigan Teacher Assessments Heather Hill hhill@umich.edu Heather Hill hhill@umich.eduhhill@umich.edu Web http://www.sii.soe.umich.edu/ Web http://www.sii.soe.umich.edu/http://www.sii.soe.umich.edu/