1. Mathematical Knowledge for Teaching at the Secondary Level Examples from the Situations Project Mathematics Education Research Colloquium November 16, 2007 Mid Atlantic Center for Mathematics Teaching and Learning Center for Proficiency in Teaching Mathematics

2. Situations Research Group Jeremy Kilpatrick Jim Wilson Pat Wilson Glen Blume M. Kathleen Heid Rose Mary Zbiek Bob Allen Sarah Donaldson Kelly Edenfield Ryan Fox Brian Gleason Eric Gold Sharon O’Kelley Heather Godine Shiv Karunakaran

3. Situations Project Mathematical knowledge for teaching at the secondary level (MKTS) Recommendations for preparing mathematics teachers Strategy for investigating MKT Examples of situations Future directions

4. Mathematical Knowledge for Teaching Mathematics at the Secondary Level Why is MKT difficult to define? What is the difference between MKT and mathematical knowledge? MKT and pedagogical content knowledge? Is MKT at the secondary level distinct from MKT at the elementary level?

5. Recommendations on Formal Mathematics Background for Secondary Teachers 1911 ICTM: “dealing critically with the field of elementary mathematics from the higher standpoint” 1935 MAA: “calculus, Euclidean geometry, theory of equations, and a history of mathematics course” 1959 NCTM: 24 semester hours of mathematics courses

6. Recommendations on Formal Mathematics Background for Secondary Teachers 1991 MAA’s Committee on the Mathematical Education of Teachers (COMET): “the equivalent of a major in mathematics, but one quite different from that currently in place at most institutions” 2000 NCATE: “know the content of their field (a major or the substantial equivalent of a major)” Compiled in Ferrini-Mundy and Findell

7. MET Report Recommendations Knowledge of the mathematical understandings and skills of elementary and middle school students Knowledge of the post-secondary mathematics (collegiate, vocational or work). Ability to continue growth of mathematical knowledge and its teaching.

8. MET Report Recommendations properties Understanding of the properties of the natural, integer, rational, real, and complex number systems. algebraic structures underlie rules for operations Understanding of the ways that basic ideas of number theory and algebraic structures underlie rules for operations on expressions, equations, and inequalities. algebra to model and reason about real-world situations Understanding and skill in using algebra to model and reason about real-world situations. problem solving and proof Ability to use algebraic reasoning effectively for problem solving and proof in number theory, geometry, discrete mathematics, and statistics. use graphing calculators, computer algebra systems, and spreadsheets to explore algebraic ideas and algebraic representations Understanding of ways to use graphing calculators, computer algebra systems, and spreadsheets to explore algebraic ideas and algebraic representations of information, and in solving problems.

9. Strategy for Learning about Mathematical Knowledge for Teaching at the Secondary Level Begin with practice Identify mathematical ideas and ways of thinking about mathematics that could be useful to secondary mathematics teachers Use what we learn to build a way to think about Mathematical Knowledge for Teaching at the Secondary level

10. Begin with Practice We draw from events that have been witnessed in practice. Practice includes but is not limited to classroom work with students. Events were in high schools or universities. Events were related to secondary level mathematics. We write brief prompts that describe mathematical events from practice.

11. Identify Mathematical Ideas and Ways of Thinking about Mathematics Given an event from practice (Prompt), We describe the mathematical ideas that could be useful to a teacher in that situation We are not trying to decide what a teacher should do! We write a Situation which includes the practice-based Prompt and a set of Foci and Commentary.

12. Then we… Argue and rewrite Debate and rewrite Defend and rewrite Rethink and rewrite...

13. Mathematical Knowledge for Teaching at the Secondary Level Mathematics Classroom Create Situations Mathematics Knowledge for Teaching

14. Situations We are in the process of writing a set of practice-based situations that will help us to identify mathematical knowledge for teaching at the secondary level. Each Situation consists of: Prompt - generated from practice Commentaries - providing rationale and extension Mathematical Foci - created from a mathematical perspective

15. Prompts A prompt describes an opportunity for teaching mathematics E.g., a student’s question, an error, an extension of an idea, the intersection of two ideas, or an ambiguous idea. A teacher who is proficient can recognize this opportunity and build upon it.

16. Commentaries The first commentary offers a rationale for each focus and emphasizes the importance of the mathematics that is addressed in the foci. The second commentary offers mathematical extensions and deals with connections across foci and with other topics.

17. Mathematical Foci The mathematical knowledge that teachers could productively use at critical mathematical junctures in their teaching. Foci describe the mathematical knowledge that might inform a teacher’s actions, but they do not describe or suggest specific pedagogical actions.

18. Website of Situations Situations project webpage

19. Where are we going? Teacher preparation Methods courses Shadow courses Capstone courses Professional development Reflection on practice Situation study Theory building Research Projects Dissertations Assessment

20. This presentation is based upon work supported by the Center for Proficiency in Teaching Mathematics and the National Science Foundation under Grant No. ESI 0227586. and the Mid-Atlantic Center for Mathematics Teaching and Learning under Grant Nos. 0083429 and 0426253. Any opinions, findings, and conclusions or recommendations expressed in this presentation are those of the presenter(s) and do not necessarily reflect the views of the National Science Foundation.