Weaving Research & Practice to Encourage Mathematical Thinking & Classroom Discourse     50th Annual Pennsylvania Council of Teachers of Mathematics Meeting Pittsburgh, Pennsylvania March 16th, 2001 Presenter: Roya Salehi (roya@mathforum.org)

 Goals of the Session: Provide a hands-on opportunity to examine the cylinder problem Overview the videopaper, Encouraging Mathematical Thinking (http://mathforum.org/brap/wrap/) Discuss the BRAP project as a model for Professional Development 1) Initial Experiment: Form two cylinders from a rectangular piece of paper, one by joining the long sides, one by joining the short sides. Which of these cylinders will have greater volume, or will they hold the same amount? See it in action Was your prediction correct? Do the two cylinders hold the same amount? Why or why not? Can we explain why they don't? 2) Second Experiment: Are there any other cylinders that we can make from this same sheet of paper?" Which of them would hold the most? Write down your predictions." How many other cylinders could we make from a rectangle with these same dimensions? Do you see any pattern that relates the size of the cylinder and the amounts they hold? Do your results confirm what we discovered with our physical models? 3) Why we chose the Cylinder problem? -The lesson involves multiple stages of investigation: prediction, testing, rejection or extension of hypotheses, discovering and exploring the underlying mathematics, and making generalizations and proving results. -It involves a number of mathematical concepts and problem-solving approaches that can lead to rich discourse in the classroom. Students interpret a graph in Judith Koenig's class Reasoning for predictions in Susan Stein's class -Students are engaged by the mathematical demonstration afforded by this investigation, the results of which seem counterintuitive to many of them. Aha moments -The lesson also offers multiple entry points for a variety of levels, from elementary school through calculus. -The problem has real life applications.Jon Basden's real life example 4) Predictions Explaining predictions is an important part of the process It eliminates thoughtless guessing and helps students think about what might happen and the mathematical relations inherent in the situation; Our experience suggests that the initial struggle students go through in formulating their reasons for a prediction -- whether naive or correct -- enriches their experience and motivates them to continue. PoW Analysis 5) Useful links: