1. Developing a Framework for Mathematical Knowledge for Teaching at the Secondary Level The Association of Mathematics Teacher Educators (AMTE) Eleventh Annual Conference Irvine, CA January 27, 2007 Mid Atlantic Center for Mathematics Teaching and Learning Center for Proficiency in Teaching Mathematics

2. Situations Research Group Glen Blume Brad Findell M. Kathleen Heid Jeremy Kilpatrick Jim Wilson Pat Wilson Rose Mary Zbiek Bob Allen Sarah Donaldson Ryan Fox Heather Godine Shiv Karunakaran Evan McClintock Eileen Murray Pawel Nazarewicz Erik Tillema

3. Collaboration of Two CLT’s to Identify and Characterize the Mathematical Knowledge for Teaching at the Secondary Level 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

4. Our collaborative work is addressing: the mathematical knowledge the ways of thinking about mathematics that proficient secondary mathematics teachers understand.

5. How to get teachers acquainted with secondary mathematics in ways that are useful in their teaching. How to help secondary mathematics teachers connect collegiate mathematics with the mathematics of practice The Problems

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, making this mathematics for 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 Commentaries - providing rationale and extension Mathematical Foci - created from a mathematical perspective

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. 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.

11. 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.

12. 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 they 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.

13. Example of a Situation: Inverse Trig Functions Commentary The problem seems centered on knowing about the entity of inverse. Connections can be made to the notion of inverse from abstract algebra. When we think about inverses, we need to think about the operation and the elements on which the operation is defined. The selection of foci is made to emphasize the difference between an inverse for the operations of multiplication and composition of functions. The foci contrast how the multiplicative inverse invalidates the properties for an inverse element for the operation of composition. The contrasts will be illustrated in a variety of approaches: graphical, numerical, and verbal.

14. Example of a Situation: Inverse Trig Functions Mathematical Focus 1 [What does it mean to be an inverse?] The problem seems centered on knowing about the mathematical entity of inverse. An inverse requires two elements: the operation and the elements on which the operation is defined. csc(x) is an inverse of sin(x), but not an inverse function for sin(x). For any value of x such that csc(x) ≠ 0, the number csc(x) is the multiplicative inverse for the number, sin(x); multiplication is the operation in this case and values of the sin and csc functions are the elements on which the operation is defined. Since we are looking for an inverse function, the operation is composition and functions are the elements on which the operation is defined.

15. Example of a Situation: Inverse Trig Functions Mathematical Focus 2 [Are these three functions really inverses of sine, cosine, and secant?] Suppose cosecant and sine are inverse functions. A reflection of the graph of y = csc(x) in the line y = x would be the graph of y = sin(x). Figure 1 shows, on one coordinate system graphs of the sine function, the line given by y = x, the cosecant function, and the reflection in y = x of the cosecant function. Because the reflection and the sine function graph do not coincide, sine and cosecant are not inverse functions. The reflection in the line given by y = x of one function and the graph of an inverse function coincide because the domain and range of a function are the range and domain, respectively, of the inverse function.

16. Example of a Situation: Inverse Trig Functions Mathematical Focus 3 [For what mathematical reason might one think the latter three functions are inverses of the former three functions?] The notation f -1 is often used to show the inverse of f in function notation. When working with rational numbers, f -1 is used to represent the reciprocal of f. If people think about the “inverse of sine” as sin -1, they might use to represent the inverse of sine.

17. 1. Adding Radicals A mathematics teacher, Mr. Fernandez, is bothered by his ninth grade algebra students ’ responses to a recent quiz on radicals, specifically a question about square roots in which the students added and and got. Samples of Prompts for the MAC-CPTM Situations Project

18. 2. Exponents In an Algebra II class, the teacher wrote the following on the board: x m. x n = x 5. The students had justt finished reviewing the rules for exponents. The teacher asked the students to make a list of values for m and n that made the statement true. After a few minutes, one student asked, “ Can we write them all down? I keep thinking of more. ”

19. Mathematical Lenses Mathematical Objects Big Mathematical Ideas Mathematical Activities of Teachers

20. 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.

21. 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; Begins with a mix of curriculum content and practice and uses each to inform the other; and Accounts for overarching mathematical ideas that cut across curricular boundaries and carry into collegiate mathematics while staying connected to practice.

22. 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.

23. 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.

24. 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.

25. Looking at the Inverse Trig Function Mathematical Foci through an Activities 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. The Problems How to get teachers acquainted with secondary mathematics in ways that are useful in their teaching. How to help secondary mathematics teachers connect collegiate math with the math of practice

27. 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

28. The Mathematics of Your Courses Where do you see objects, big ideas, and mathematical activities in your courses? Which mathematical lenses (these or others) or combination of mathematical lenses influence your courses?

29. What insights can you now offer regarding the problems we posed? How to get teachers acquainted with secondary mathematics in ways that are useful in their teaching. How to help secondary mathematics teachers connect collegiate math with the math of practice

30. 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.