1. Putting the Focus on Mathematics:Content-based Professional Development for Grade 5-12 Teachers Steve Benson Education Development Center sbenson@edc.org MAA Session on Professional Development Programs for K-12 Teachers, ISan Antonio, TX; January 12, 2006

2. Motivation I’m still not sure why I had to learn about rings and fields and other such topics to be a high school math teacher. — A veteran high school teacher

3. Messages from the mathematics community Over the past 15 years, two refrains have echoed through the discourse about teachers’ knowledge of mathematics: (1) that U.S. teachers mathematical knowledge is weak(2) that the mathematical knowledge needed for teaching is different from that needed by mathematicians. — Mathematical Proficiency for All Students: Toward a Strategic Research and Development Program in Mathematics Education (RAND, 2001) The mathematical knowledge needed by teachers at all levels is substantial, yet quite different from that required by students pursuing other mathematics-related professions. . . . Collegecourses developing this knowledge should make connections between the mathematics being studied and mathematics prospective teachers will teach. — The Mathematical Education of Teachers (CBMS, 2001)

4. Messages from the mathematics community • Teachers need several different kinds of mathematical knowledge: • Knowledge of the whole domain • Deep, flexible knowledge about curriculum goals and about the important ideas that are central to their grade level • Knowledge about the challenges students are likely to encounter in learning these ideas • Knowledge about how students’ understanding can be assessed — Principles and Standards for School Mathematics (NCTM, 2001)

5. Our assumptions • Existing and emerging 6–12 curricula contain many problems that serve as starting points for deep mathematical exploration. • To teach exploratory mathematics, one must be willing to engage in exploratory mathematics oneself. • If you see yourself as a mathematical explorer, you’re more likely to view your students as explorers, too. • Immersion in content alone is not enough; the choice of content is crucial.

6. Knowledge of Mathematics for Teaching • Not everything a teacher needs to know ends up on the chalkboard. • — Mark Saul • The ability “to think deeply about simple things” (A. Ross)What’s really behind the geometry of multiplying complex numbers? • The ability to create activities that uncover central habits of mindWhat do 53/2and 5 mean?

7. Knowledge of Mathematics for Teaching (cont’d) • The ability to see underlying connections and themes • Connections • Linear Algebra brings coherence to secondary geometry • Number Theory sheds light on what otherwise seem like curiosities in arithmetic • Abstract Algebra provides the tools needed to transition from arithmetic with integers to arithmetic in other systems. • Analysis provides a framework for separating the substance from the clutter in precalculus • Mathematical Statistics has the potential for helping teachers integrate statistics and data analysis into the rest of their program

8. Knowledge of Mathematics for Teaching (cont’d) • The ability to see underlying connections and themes • Themes • Algebra: extension, representation, decomposition • Analysis: extension by continuity, completion • Number Theory: reduction, localization

9. Knowledge of Mathematics for Teaching (cont’d) • The “mining” of student ideas • The class was using calculators and estimation to get decimal • approximations to . One student, Marla, looked at how you do out long multiplication and realized that none of these • decimals would ever work because if you square a finite (non-integer) decimal, there’ll be a digit to the right of the decimal • point, so you can’t ever get an integer. So, Marla had the start • of a proof that can’t be represented by a terminating decimal. • But where does she go from here? • — Adapted from “A Dialogue About Teaching” in What’s • Happening in Math Class? (Teachers College Press, 1996).

10. Responses to the call for connections Ways to Think About Mathematics: Activities and Investigations forGrade 6-12 Teachers; Benson, Addington, Arshavsky, Cuoco, Goldenberg, Karnowski; Corwin Press, 2004. Mathematical Connections: A Companion for Teachers and Others;Cuoco, MAA, 2005. Mathematics for High School Teachers - An Advanced Perspective;Usiskin, Peressini, Marchisotto, Stanley; Prentice Hall, 2003. Seeing the Connections: Promoting Profound Understanding of Secondary Mathematics; Benson, Cuoco, Graham, Greenes, Grundmeier, Portnoy (in preparation)

11. Foci of our materials • On problem solving and reflecting on methods of solution. • On mathematical ways of thinking. • On making connections among mathematical ideas; connections to current and emerging 6-12 curricula; connections to methods of effective teaching; and cross-grade connections. • On analyzing problems.

12. Ways to Think About Mathematics: Activities and Investigations for Grade 6-12 Teachers isavailable from Corwin Press. A Facilitator’s Guide and Supplementary CD (including solutions and additional activities) are also available. More information athttp://www2.edc.org/wttam

13. Ways to Think About Mathematics: Activities and Investigations for Grade 6-12 Teachers I. What is Mathematical Investigation? Problem solving and problem posing You’ve got a conjecture-now what? Do it yourself You know the answer? Prove it. Discerning what is; predicting what might be II. Dissections and Area Be a mathematical cut-up Making assumptions, checking procedures Thinking about area Areas of nonpolygonal area Transformations and area III. Linearity and Proportional Reasoning Mix it up Filling in the gaps Guess my rule Functions of two variables From cups to vectors IV. Pythagoras and Cousins What would Pythagoras do? Puzzling out some proofs Pythagoras’s second cousins Pythagorean triples (and cousins) More classroom cousins V. Pascal's Revenge: Combinatorial Algebra Trains of thought Getting there Trains and paths and triangles, oh my! Binomial theorem connection Supercalifragilisticgeneratingfunctionology VI. Problems for the Classroom (+ solutions) VII. Answers to Selected Problems http://www2.edc.org/wttam

14. Seeing the Connections:Promoting Profound Understanding of Secondary Mathematics A collaborative curriculum project from Education Development Center University of New Hampshire Stony Brook University Funded by NSF DUE-0231342 Steve Benson sbenson@edc.org Karen Graham karen.graham@unh.edu Al Cuoco acuoco@edc.org Neil Portnoy nportnoy@math.sunysb.edu http://www2.edc.org/connect

15. The Seeing the Connections materials The Seeing the Connections materials are the “offspring” of three NSF-funded proof-of-concept projects: Making the Connections: Higher Algebra to School Mathematics(DUE-9950722) Carole Greenes, PI, BU Al Cuoco, Co-PI, EDC Carol Findell, BU Emma Previato, BU Making Mathematical Connections in Programs for Prospective Teachers (DUE-9981029) Karen Graham, PI, UNH Neil Portnoy, CSU, Chico* Todd Grundmeier, UNH‡ Gateways to Advanced Mathematical Thinking (DUE-9450731) Al Cuoco PI, EDC Wayne Harvey, Co-PI, EDC * Now at Stony Brook University ‡ Now at California State University San Luis Obispo

16. The Seeing the Connections project is producing curriculum modules for use in mathematics courses that help preservice teachers develop a knowledge ofmathematics for teaching. • The StC curriculum will help secondary teachers develop important mathematical knowledge and skills required in their future careers: • designing effective lessons • emphasizing certain ideas over others • connecting ideas across the grades • understanding germs of insight in students' questions • placing precollege topics in the broader mathematical landscape. • The project staff, combining extensive expertise in curriculum development, undergraduate and secondary teaching, teacher preparation and professional development, and education research, are creating and making widely available (in paper and electronic formats) a library ofmaterials that can be used in a wide range of preservice and inservice environments.

17. Making Mathematical Connections in Programs for Prospective Teachers Making Mathematical Connections in Programs for Prospective Teachers, developed a series of activities that provide prospective teachers with the opportunity to make connections between two mathematical areas (transformational geometry and liner algebra) and school and university mathematics. In addition, there is a series of 3 pedagogical activities that the prospective teachers explore within the context of the developing mathematical understandings above. These activities involve the prospective teachers in the analysis of pre-college mathematical curricula and tasks, the analysis of classroom observations conducted in middle school and/or high school classrooms, and the development, implementation, and evaluation of a class activity focused on transformational geometry.

18. Making Mathematical Connections in Programs for Prospective Teachers • 1- Isometries of the Plane • Discover the four basic isometries (rotation, reflection, translation, and glide). • Reinforce the place of definition in mathematics. Sharing definitions and the ensuing discourse is likely to bring out the importance of careful wording. • Identify similarity transformations. • Make connections between functions and geometric transformations. • 2- Rotations, Reflections, Translations, and Glides • Discover basic properties of various isometries. • Understand definitions and invariants of each isometry. • 3- Compositions • Discover that the class of isometries is preserved by composition. • View isometries as functions.

19. Making Mathematical Connections in Programs for Prospective Teachers • 4- Proof with Isometries • Be familiar with the use of isometries in proof. • Consider basic Euclidean postulates. • 5- The Human Vertices • Enable students to make connections (physically) between transformational geometry and linear algebra. • Linear transformations are functions. • Non-invertible transformations collapse R2 to R1or to {0}. • Sign of the determinant indicates orientation. • 6- Isometries and Linear Algebra • This activity is meant to bring closure to the mathematical ideas connecting transformational geometry and linear algebra by introducing the idea of a group structure.

20. Making the Connections: Higher Algebra to School Mathematics Making the Connections: Higher Algebra to School Mathematicswas a proof-of-concept project, funded by the National Science Foundation (DUE-9950722), which produced materials for use in courses for preservice mathematics teachers that make explicit connections between the mathematics they learn in college to the mathematics they will eventually teach. The content focus of this project was algebra and number theory with three main themes: Modular Arithmetic, Periods of RepeatingDecimals, and The Chinese Remainder Theorem.

21. Making the Connections: Higher Algebra to School Mathematics • Numbers, Systems, and Divisibility 1. Algebra as Structure 2. Modular Arithmetic 3. Making it a System 4. Decimals, Fractions, and Long Division 5. The Fundamental Theorem of Arithmetic 6. Interlude 7. Units, Orders, and Periods 8. The Chinese Remainder Theorem • Etude 10. Euler, Units, and Periods of Decimals 11. Irrational Numbers: An Introduction

22. Gateways to Advanced Mathematical Thinking Gateways to Advanced Mathematical Thinking was a dual curriculum development/research project funded by the National Science Foundation (DUE 9450731). The development component of the project built a model curriculum module for use with undergraduates, and particularly with preservice teachers, which motivates appreciation for mathematics, focuses on conceptual understanding without sacrificing formal techniques, and makes explicit connections to the high school curriculum. Topics include precalculus methods for solving — and/or approximating the solution of — optimization problems.

23. Gateways to Advanced Mathematical Thinking • Part1: Geometric Techniques • Minimizing Distance • Maximizing Area • Contour Lines • Part 2: Algebraic Techniques • Squares are never negative • The Arithmetic - Geometric Mean Inequality • Part 3: Graphical Techniques • The Box Problem

24. Seeing the Connectionsmaterials are available online Making Mathematical Connections in Programs for Prospective Teachers http://www2.edc.org/connect/mathconnlink.html Making the Connections: Higher Algebra to School Mathematics http://www2.edc.org/connect/connectionslink.html Gateways to Advanced Mathematical Thinking http://www2.edc.org/connect/gatewayslink.html Copies of slides and handouts will be available at http://www2.edc.org/cme/showcase.html All files are in PDF or Powerpoint format Questions? Problems? Send email to sbenson@edc.org

25. Where have these materials been used? Master of Science for Teachers program at the University of New Hampshire (1-unit content courses and supplement to core courses)http://www.math.unh.edu/~mathadm/mst/ Focus on Mathematics Mathematics Science Partnership(3-hour seminars, weekly study group content, 1-week all-day summer content institutes for grade 5-12 mathematics teachers) http://focusonmath.org MAA Minicourses at Mathfest and Joint Meetings (2005-2006)Next minicourse at Lexington Mathfest, August 2006. University, district, and school-based courses and professional developmentworkshops for inservice and preservice middle and high school teachers Interested in using any of these materials? Send me a note: sbenson@edc.org