Exploring Mathematical Knowledge for Teaching Secondary Level

1. Identifying Mathematical Knowledge for Teaching at the Secondary Level (6-12)from the Perspective of Practice • Joint NSF-CLT Conference on • Curriculum, Teaching & Mathematical Knowledge • University of Maryland • November 18, 2006 • Mid Atlantic Center for Mathematics Teaching and Learning • Center for Proficiency in Teaching Mathematics

2. Mid Atlantic Center for Mathematics Teaching and Learning Focus: The preparation of mathematics teachers The Center for Proficiency in Teaching Mathematics Focus: The preparation of those who teach mathematics to teachers Collaboration of Two CLT’s to Identify and Characterize the Mathematical Knowledge for Teaching at the Secondary Level

3. Our collaborative work is investigating: • mathematical knowledge • ways of thinking about mathematics • that proficient secondary mathematics teachers can use in their teaching.

4. Two Assumptions • Every day teachers need to integrate their knowledge in order to respond to students whose real questions about mathematics do not fall into neat compartments. • The structure and content of mathematics requirements for a mathematics major falls short of what is needed for secondary teachers.

5. Our work is centered on finding a balance between the complexity and reality of practice and the goal of creating a new vision. • We are trying to address mathematical knowledge for teaching by grounding it in practice.

6. Grounding our Work in Practice • We are drawing from events that have been witnessed in practice. • Practice has many faces, including but not limited to classroom work with students. • Situations come from and inform practice.

7. Working toward a Framework • We would like to build a framework of Mathematical Knowledge for Teaching at the Secondary level. • The framework could be used to guide: • Research • Curriculum in mathematics courses for teachers • Curriculum in mathematics education courses • Design of field experiences • Assessment

8. 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 • 3-6 Mathematical Foci - created from a mathematical perspective • Commentary - introduced and connected foci

9. 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.

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

11. Some issues in creating and identifying mathematical foci • Straying into pedagogy • Determining appropriate level of mathematical detail • Determining the appropriate level of mathematics • Determining what makes two foci different

12. Commentary • Commentaries were added as we worked on creating Foci for Prompts. • A commentary offers a reason for each carefully selected focus and points out the importance of the mathematics that was addressed across the set of foci.

13. Where are we in our work? • Collecting prompts from our experiences in schools and teacher preparation • Creating a variety of foci that illustrate useful mathematical knowledge • Working on characterizing and classifying domains of knowledge that are salient in the foci

14. We need your help • We think your experiences with developing, implementing, analyzing, studying, or assessing curricula and the use of curricula prepare you to: • See events that suggest rich prompts • Develop insights related to foci • Provide guidance in designing our framework • Provide helpful feedback to our process

15. Sample Prompt Three prospective teachers have planned a unit of trigonometry as part of their work in a methods course on the teaching and learning of secondary mathematics. They developed a plan in which students first encounter what they call the three basic functions: sine, cosine, and tangent. They indicated in their plan they would next have students work with the "inverse functions" apparently meaning the secant, cosecant, and cotangent functions.

16. Sample Mathematical Focus • What is an inverse? • The inverse is based upon set and operation • In this case trigonometric functions are the set and composition of functions is the operation.

17. Brainstorming What other foci can be developed to explore the mathematics that underlie this prompt? In other words, what other key mathematical ideas would you identify that directly relate to this prompt?

18. Other Foci We Developed • In a Cartesian coordinate plane, a graph of an inverse of a function is the graph of the function reflected over the line, y = x . • If a function is the inverse of another function, then the original input should be the output of the composition.

19. Commentary • Introducing and relating the various mathematical foci • Providing mathematical extensions

20. Analyzing the Situations • Identifying the mathematics: • Within a given situation • Across the set of situations • The goal of the analysis is to develop a framework, based in situations arising from practice, that characterizes mathematical knowledge for teaching at the secondary level

21. How might this analysis proceed? • How do we go from the situations to a framework for MKT at the secondary level? • What might such a framework look like?

22. Three Potential Lenses for Developing a Framework • Mathematical objects, representations, properties, and operations • Big Ideas • Mathematical activities in which teachers engage

23. Mathematical Objects, etc. • Mathematical objects • Properties of those objects • Relationships between objects • Representations of those objects • Relationships between representations • Operations on objects • Properties of those operations

24. Big Ideas in Secondary Mathematics • Equivalence • Variable • Linearity • Unit of measure • Randomness • …

25. Mathematical Activities in Which Teachers Engage • Defining a mathematical object • Giving a concrete example of an abstraction • Introducing an analogy • Explaining or justifying a procedure • Reasoning using cases

26. Defining Informal (characteristics) Formal (minimal set) Descriptive Constructive Justifying/Proving Non-proof justifying (validating/refuting) Proving (validating/refuting) Generalizing Recogning invariants in a number of instances Subclass to larger class Symbolic working Creating symbols Interpreting symbols Manipulating symbols Connect symbols to other representations Reasoning from symbols Processes

27. Mathematical Activity Producing general case Reasoning from def’n Using extreme cases Reasoning from graph Manipulating symbols Proof by region subdivision Interpret relationships transformationally Content Rate of change Gaussian elimination Linearity/non-linearity Distributions Mean as balance pt. Categories from the Situations

28. Some issues in developing a framework • 1. Grain size • 2. Direction • Given a situation, what processes are there? • Given certain processes, are they in situations? • 3. General framework vs. Specific framework • 4. Straying into pedagogy

29. Feedback on Lenses • Preferences among the three lenses • Mathematical objects • Big ideas • Teachers’ mathematical activities • How might this differ from the perspective of a mathematician, curriculum developer, teacher educator, researcher, …? • Alternative lenses

30. Generating Additional Prompts • NSF-funded curricula • Different mathematics • Different pedagogy • Who might best provide such prompts? • Teachers? • PD providers? • Curriculum writers? • Researchers whose data include interesting classroom instances?

31. What do we hope to get from you? • New prompts • Strategies for getting additional prompts • Who might be productive contacts? • Other strategies? • How might one use these prompts?