1. Mathematical Knowledge for Teaching at the Secondary Level Using Scenarios to Develop a Framework American Mathematical Society 2008 Fall Central Section Meeting October 18, 2008 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 Maureen Grady Svetlana Konnova

3. Situations Project Mathematical knowledge for teaching at the secondary level (MKTS) What is it? Why are we interested in it? Recommendations for preparing mathematics teachers Strategy for investigating MKTS Examples of situations Time to create a Situation of your own Uses of implications for collegiate education of teachers

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. Recommendations on Formal Mathematics Background for Secondary Teachers 2001 CBMS “Prospective high school teachers of mathematics should be required to complete the equivalent of an undergraduate major in mathematics, that includes a 6-hour capstone course connecting their college mathematics with high school mathematics.” (MET Report)

8. 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

9. 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.

10. 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.

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

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

13. 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 Italicized statement - captures the essence of the focus

14. Example of a Situation: Inverse Trig Functions Prompt Three prospective teachers 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 high school students first encounter what the prospective teachers called “the three basic trig functions”: sine, cosine, and tangent. The prospective teachers indicated in their plan that students next would work with “the inverse functions,” identified as secant, cosecant, and cotangent.

15. Example of a Situation: Inverse Trig Functions Commentary The foci draw on the general concept of inverse and its multiple uses in school mathematics. Key ideas related to the inverse are the operation involved, the set of elements on which the operation is defined, and the identity element given this operation and set of elements. The crux of the issue raised by the prompt lies in the use of the term inverse with both functions and operations.

16. Example of a Situation: Inverse Trig Functions Mathematical Focus 1 An inverse requires three entities: a set with a binary operation, and an identity element given this operation and set of elements. Mathematical Focus 2 Although the inverse under multiplication is not the same as the inverse under function composition, the same notation, the superscript -1, is used for both. Mathematical Focus 3 When functions are graphed in an xy- coordinate system with y as a function of x, these graphs are reflections of their inverses’ graphs (under composition) in the line y = x.

17. Mathematical Lenses Mathematical Objects Big Mathematical Ideas Mathematical Processes Mathematical Work of Teachers

18. Mathematical Lens: Mathematical Objects A “mathematical-objects” approach Centers on mathematical objects, properties of those objects, representations of those objects, operations on those objects, and relationships among objects; Starts with school curriculum; and Addresses the larger mathematical structure of school mathematics.

19. Mathematical Lens: Big Mathematical Ideas A “big-mathematical-ideas” approach Centers on big ideas or overarching themes in secondary school mathematics; Examples: ideas about equivalence, variable, linearity, unit of measure, randomness;and Accounts for overarching mathematical ideas that cut across curricular boundaries and carry into collegiate mathematics while staying connected to practice.

20. Mathematical Lens: Mathematical Processes 1 A “mathematical processes” approach Focuses on actions involved in doing mathematics Examples: representing, justifying, defining Processes operate on objects 1 Zbiek et al.

21. Mathematical Lens: Mathematical Activities of Teachers A “mathematical-activities” approach Partitions or structures the range of mathematical activities in which teachers engage Examples: defining a mathematical object, giving a concrete example of an abstraction, formulating a problem, introducing an analogy, or explaining or justifying a procedure. May also draw on the mathematical processes that cut across areas of school mathematics.

22. Looking at the Inverse Trig Function Mathematical Foci through an Object Lens Focus 1: Inverse Focus 2: Relationship between graphs of inverse functions Focus 3: A conventional symbolic representation of “the inverse of ” f is f -1. The exponent or superscript -1 has several different meanings, not all of which are related to inverse in the same way.

23. Looking at the Inverse Trig Function Mathematical Foci through an Big Ideas Lens Focus 1: Two elements of a set are inverses under a given binary operation defined on that set when the two elements used with the operation in either order yield the identity element of the set. Focus 2: Equivalent Functions/ Domain and Range: Two functions are equivalent only if they have the same domain and the same range. Focus 3: The same mathematical notation can represent related but different mathematical objects.

24. Looking at the Inverse Trig Function Mathematical Foci through a Mathematical Processes Lens Focus 1:Conceptualizing the concept of inverse Focus 2:Interpreting a symbolic representation Focus 3:Representing graphically and justifying that representation

25. Looking at the Inverse Trig Function Mathematical Foci through an Work of Teaching Lens Focus 1: Appealing to definition to refute a claim Focus 2: Using a different representation to explain a relationship Focus 3: Explaining a convention

26. Situation 40: Powers Prompt During an Algebra I lesson on exponents, the teacher asked the students to calculate positive integer powers of 2. A student asked the teacher, we ’ ve found 2 2 and 2 3. What about 2 2.5 ?

27. Each group -- develop assigned prompt into a Situation (sketch out two mathematical foci) Present some ideas for your prompts (prepare a transparency for your presentation)

28. The Problem How to get teachers acquainted with secondary mathematics in ways that are useful in their teaching.

29. Engaging with Mathematical Lenses Big Ideas Lens: Two UGA courses (Concepts in Secondary School Mathematics and Connections in Secondary School Mathematics) based on secondary mathematics from an advanced standpoint Two PSU courses (Foundations of Secondary Mathematics: Functions and Data Analysis) based on fundamental ideas related to function and data analysis in secondary mathematics Mathematical Activities Lens: PSU course (Understanding secondary mathematics in classroom situations) UGA course (Teaching and Learning Secondary School Mathematics) with a major component on Situations

30. How could you use Situations or some of the ideas from today’s session in your mathematics class or mathematics education class?

31. This presentation is based upon work supported by the Center for Proficiency in Teaching Mathematics and the National Science Foundation under Grant No. 0119790 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.