1. “Mathematician” Meets “Math Educator”: The Experience of Co-Teaching A Math Inquiry Course April 23, 2009 University of New Hampshire Juliana Belding, Harvard University

2. Outline The Course: Purpose and Structure Co-teaching: Challenges and Benefits of Collaboration Student Experiences: From Frustration to Exhilaration

3. “The real voyage of discovery consists not in seeking new landscapes, but in having new eyes.” - Marcel Proust

4. Overview of the Course “Researching Mathematics As Math Educators” University of Maryland, Spring 2008 Students: Graduate students in Math Ed (also Physics Ed and Mathematics) Instructors: Eden Badertscher (Math Ed), J.B.

5. Goals of the Course Main Goal: Students develop ability and desire to ask and investigate their own mathematical questions Other Goals: Mathematical communication Appreciation for process/struggle of math

6. Structure of Course: Part I Individual Projects –Students choose/develop own question –Weekly project journals (instructor and peer feedback) –Studio Times (1 hr., 5x during semester) –Final written project Some Examples Moving a Couch Through a Doorway Finding Geometric Proofs of Trig Identities Iterating Rational Functions, eg: f(x) = 1/(x-1)

7. Structure of Course: Part II In-Class Investigations –Weekly 2 hr. classes –Three topics (last 4-5 weeks each) –Small groups, informal presentations –In-class journals

8. In-Class Investigations “Wrestling with…” 1: Rational Numbers Farey Sequences, Representation of Rationals F5: 0, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1 2: Geometry Definitions of A Parabola, Taxi-cab Geometry 3: The Real Numbers Cardinality, Representation of Reals

9. The Creative Process in Mathematics “Mathematics has two faces. Presented in finished form, mathematics appears as a purely demonstrative [deductive] science, but mathematics in the making is a sort of experimental science. A correctly written mathematical paper is supposed to contain strict demonstrations only, but the creative work of the mathematician resembles the creative work of the naturalist; observation, analogy, and conjectural generalizations, or mere guesses play an essential role in both.” -Polya, 1952

10. Mathematical Themes Through the creative process, students develop need for and appreciation of Definitions (their role, principled choices, sense- making) What constitutes a proof? Multiple viewpoints (geometric, algebraic, etc.) Precise mathematical language and notation

11. “What-If-Not” From The Art of Problem Posing, Brown and Walter, 1983 Given a mathematical object/situation/problem: List attributes Ask “what-if-not” (tweak attributes) Formulate new questions

12. Example: A Parabola Definition: The locus of points equidistant from a line (directrix) and a point not on the line (focus) Attributes… –Points are equidistant –Directrix is a line –Focus is not on the line –Euclidean distance

13. Example: A Parabola Definition: The locus of points equidistant from a line (directrix) and a point not on the line (focus) What-if-not… –Not Equidistant (1/2 as far, 2x as far) –Directrix is another object (circle, a parabola) –Focus is on the line (degenerate conics) –Non-Euclidean distance (Taxi-cab geometry)

14. Collaborative Teaching Share values of each discipline with each other Model values in classroom Alternate roles in the classroom Share work of designing investigations

15. Instructors’ Roles In Class We both acted as… participants (“having new eyes”) translators (model communication) facilitators (restart, regroup and recap)

16. Instructors’ Role Outside Class Collaboratively and individually, reframe questions (mathematical) restructure groups (interpersonal) according to students’ interest, facility with formal math, working style and new questions that arise…

17. In working with a math educator… More transparent classroom norms (eg: flags) Investigations more accessible, open-ended (eg: less rush to formalize, prove) Better integrate with students’ coursework in education (eg: readings) Better support students with feedback (eg: projects) More genuinely model process of inquiry: (excitement, frustration, confusion)

18. In working with a mathematician… Benefit from knowledge of field (awareness of dead-ends, fruitful avenues) Better highlight over-arching math themes in projects and investigations (and connect to students’ previous math) More genuinely model process of formalizing math: (refining questions, conjectures, notation, proof, etc.)

19. What challenges for a Mathematician? Being the know-it-all “math person” When to hold back, when to contribute Some topics too familiar (on surface) De-emphasis on rigor/proof and content

20. What benefits to a Mathematician? Chance to share/explore math outside curriculum/department More freedom to ask questions, make guesses (than in traditional “expert” role) Exposure to more experimental, intuitive, “elementary” approaches to math Refresher for research (formulating new questions, trying multiple approaches)

21. The Students’ Experience Freedom of exploration/“Playfulness” Increased confidence in math & validity of own questions Improved communication of mathematics (through journals/project write-ups) Frustration with open-endedness/lack of final answer

22. Freedom of Exploration “It made me understand what it means to explore math as opposed to learn math and solve problems” (Ellen) “I’m left with having learned something and with this`anything is possible’ feeling.” (Jackie) “I don’t stop when I have an answer because I want to find out more. I truly believe that there is more than one way to “do” a problem. I’ll be able to believe that when I go back to the classroom.” (Susan)

23. Increased confidence in math “I feel much more comfortable with the idea of formulating a problem and moving forth with investigations of my own.” (John) “I think this class just restored my confidence that given the right opportunity I can understand abstract concepts.” (Kaitlyn)

24. Improved communication “For a little stretch, my partner did not give me much helpful feedback… however the fact that she felt lost by what I was writing taught me that I needed to write differently…Once I did that, she gave me some of the most useful ideas (I.e. tree diagrams).” (Larry)

25. Why Teach Such a Course? Enriches cross-communication of Mathematicians and Math Educators Informs Teaching: Eg: “Curiousity Questions” in Calculus Informs Scholarly Learning and Research

26. Why Take Such a Course? All learners (educators in particular) benefit from hands-on experience with... how math is created (what do mathematicians do?) how to communicate about math across disciplines (eg: Hy Bass) how to learn math outside a classroom or textbook This is a chance to do this through collaborative exploration of mathematics!

27. Thanks to… UNH University of Maryland, Center for Math Ed. and Math Dept. CSSM Institute, Educational Development Center, Newton, MA http://cssm.edc.org/AboutCSSM.html Workshop: June 10-13 2009

28. Other Readings Habits of Mind: An Organizing Principle of Mathematics Curricula Cuoco, Goldenberg and Mark. 1996 The Roles of The Aesthetic in Mathematical Inquiry Sinclair, 2004 Learning to learn Mathematics: Voices of doctoral students in mathematics education. In M. Strutchens & W. Gary Martin (eds.) The Learning of Mathematics. 69th Yearbook of the National Council of Teachers of Mathematics. NCTM: Reston, VA.

29. Appendix: Getting Into the Problems How did we set up each of the investigations?

30. Rationals

31. Geometry For each definition, 1. locus of points equidistant from focus and directrix 2. locus made from cutting plane parallel to side of cone 3. equation of form y = ax^2 + bx + c Generate attributes, what-if-nots and questions.

32. Real Numbers Read Zeno’s Paradox Read Hotel Infinity Generate, categorize and refine questions about real numbers