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Posing and Pursuing One’s Own Questions: Experiences of Graduate Students in Math Education and Math

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##### Posing and Pursuing One’s Own Questions: Experiences of Graduate Students in Math Education and Math

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**Posing and Pursuing One’s Own Questions:Experiences of**Graduate Students in Math Education and Math Eden M. Badertscher, Institute for Learning at University of Pittsburgh Juliana Belding, Harvard University**Overview 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 • Who: Graduate students in Math Ed (also Physics Ed and Mathematics)**Structure of Course**• In-Class Investigations • Weekly 2 hr. classes • Three topics (last 4-5 weeks each) • Small groups, informal presentations • Individual Projects • Student-designed, Outside of class • Studio Times (1 hr., 5 during semester) • Final written project • Journals (in-class and weekly project updates)**In-Class Investigations: Wrestling with…**• 1: Rational Numbers Farey Sequences, Representation of Rationals • 2: Geometry Definitions of A Parabola, Taxi-cab Geometry • 3: The Real Numbers Cardinality, Representation of Reals**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**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**“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**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)**Other Resources**• Habits of Mind: An Organizing Principle of Mathematics Curricula, Cuoco, Goldenberg and Mark. 1996 • The Roles of The Aesthetic in Mathematical Inquiry, Sinclair, 2004**Instructors’ Role**• In class: • participant (“having new eyes”) • translator (model communication) • facilitator (restart, regroup and recap) • Outside of class: • reframe questions • restructure groups according to interest, facility with formal math, working style and new questions that arise…**The Students’ Experience**“It made me understand what it means to explore math as opposed to learn math and solve problems” (from student’s reflection) • Freedom of exploration/“Playfulness” • Increased confidence in math & validity of own questions • Improved communication of mathematics (through journals/project write-ups)**What can a Mathematician (in training) gain?**• Challenges: • Being the know-it-all “math guy” • When to hold back, when to contribute • Some topics already familiar • At times, less rigorous than used to**What can a Mathematician (in training) gain?**• Benefits: • More freedom to ask questions, make guesses (than in graduate math classes) • Exposure to more experimental, intuitive, “naïve” approaches to math • Practice for graduate research (formulating a thesis question, trying multiple approaches)**Why?**All learners (educators in particular) need 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**For More Information**• CSSM Institute at Educational Development Center, Newton, MA http://cssm.edc.org/AboutCSSM.html • 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.**Appendix: Getting Into the Problems**• How did we set up each of the investigations?**Geometry**For each definition, locus of points equidistant from focus and directrix locus made from cutting plane parallel to side of cone equation of form y = ax^2 + bx + c Generate attributes, what-if-nots and questions.**Real Numbers**Read Zeno’s Paradox Read Hotel Infinity Generate, categorize and refine questions about real numbers**Individual Projects**• Key Elements: • Students choose/developown question • Instructors and peers give weeklyfeedback • Structured communication (oral and written) Some Examples • Moving a Couch Through a Doorway • Finding Geometric Proofs of Trig Identities • Iterating Rational Functions