____ THE _____ MATHEMATICS ___ ________ EDUCATOR _____ Volume 14 Number 1 Spring 2004 MATHEMATICS EDUCATION STUDENT ASSOCIATION THE UNIVERSITY OF GEORGIA Editorial Staff A Note from the Editor Editor Dear TME readers, Holly Garrett Anthony Along with the editorial team, I present the first of two issues to be produced during my brief tenure as editor of Volume 14 of The Mathematics Educator. This issue showcases the work of both Associate Editors veteran and budding scholars in mathematics education. The articles range in topic and thus invite all Ginger Rhodes those vested in mathematics education to read on. Margaret Sloan Both David Stinson and Amy Hackenberg direct our attention toward equity and social justice in Erik Tillema mathematics education. Stinson discusses the “gatekeeping” status of mathematics, offers theoretical perspectives he believes can change this, and motivates mathematics educators at all levels to rethink Publication their roles in empowering students. Hackenberg’s review of Burton’s edited book, Which Way Social Stephen Bismarck Justice in Mathematics Education? is both critical and engaging. She artfully draws connections across Laurel Bleich chapters and applauds the picture of social justice painted by the diversity of voices therein. Dennis Hembree Two invited pieces, one by Chandra Orrill and the other by Sybilla Beckmann, ask mathematics Advisors educators to step outside themselves and reexamine features of PhD programs and elementary Denise S. Mewborn textbooks. Orrill’s title question invites mathematics educators to consider what we value in classroom Nicholas Oppong teaching, how we engage in and write about research on or with teachers, and what features of a PhD program can inform teacher education. Beckmann looks abroad to highlight simple diagrams used in James W. Wilson Singapore elementary texts—that facilitate the development of students’ algebraic reasoning and problem solving skills—and suggests that such representations are worthy of attention in the U. S. MESA Officers Finally, Bharath Sriraman and Melissa Freiberg offer insights into the creativity of 2004-2005 mathematicians and the organization of rich experiences for preservice elementary teachers, respectively. Sriraman builds on creativity theory in his research to characterize the creative practices President of five well-published mathematicians in the production of mathematics. Freiberg reminds us of the Zelha Tunç-Pekkan daily challenge of mathematics educators—providing preservice teachers rich classroom Vice-President experiences—and details the organization, coordination, and evaluation of Family Math Fun Nights in Natasha Brewley elementary schools. It has been my goal thus far to entice you to read what follows, but I now want to focus your Secretary attention on the work of TME. I invite and encourage TME readers to support our journal by getting Amy J. Hackenberg involved. Please consider submitting manuscripts, reviewing articles, and writing abstracts for previously published articles. It is through the efforts put forth by us all that TME continues to thrive. Treasurer Last I would like to comment that publication of Volume 14 Number 1 has been a rewarding Ginger Rhodes process—at times challenging—but always worthwhile. I have grown as an editor, writer, and scholar. NCTM I appreciate the opportunity to work with authors and editors and look forward to continued work this Representative Fall. I extend my thanks to all of the people who make TME possible: reviewers, authors, peers, Angel Abney faculty, and especially, the editors. Undergraduate Holly Garrett Anthony Representatives 105 Aderhold Hall [email protected] The University of Georgia www.ugamesa.org Erin Bernstein Athens, GA 30602-7124 Erin Cain Jessica Ivey About the cover Cover artwork by Thomas E. Ricks. Fractal Worms I of the Seahorse Valley in the Mandelbrot Set, 2004. For questions or comments, contact: [email protected] Benoit Mandelbrot was the pioneer of fractal mathematics, and the famous Mandelbrot set is his namesake. Based on a simple iterative equation applied to the complex number plane, the Mandelbrot set provides an infinitely intricate and varied landscape for exploration. Visual images of the set and surrounding points are made by assigning a color to each point in the complex plane based on how fast the iterative equation’s value “escapes” toward infinity. The points that constitute the actual Mandelbrot set, customarily colored black, are points producing a finite value. The Mandelbrot set is a fractal structure, and one can see self-similar forms within the larger set. Using computing software, anyone can delve within this intricate world and discover views never seen before. Modern computing power acts as a microscope allowing extraordinary magnification of the set’s detail. The fanciful drawing Fractal Worms I is based on the structure of spirals residing in the commonly called “Seahorse Valley” of the Mandelbrot Set. Using a lightboard, Thomas Ricks drew the fractal worms on a sheet of art paper laid over a computer printout of the Seahorse spirals. With the light shining through both sheets of paper, he drew the various fractal worms following the general curve of the spirals. The printout was produced by a Mandelbrot set explorer software package called “Xaos”, developed by Jan Hubicka and Thomas Marsh and available at: http://xaos.theory.org/ This publication is supported by the College of Education at The University of Georgia. ____________THE________________ ___________ MATHEMATICS________ ______________ EDUCATOR ____________ An Official Publication of The Mathematics Education Student Association The University of Georgia Spring 2004 Volume 14 Number 1 Table of Contents 2 Guest Editorial… Do You Need a PhD to Teach K–8 Mathematics in Ways Respected by the Mathematics Education Community? CHANDRA HAWLEY ORRILL 8 Mathematics as “Gate-Keeper” (?): Three Theoretical Perspectives that Aim Toward Empowering All Children With a Key to the Gate DAVID W. STINSON 19 The Characteristics of Mathematical Creativity BHARATH SRIRAMAN 35 Getting Everyone Involved in Family Math MELISSA R. FREIBERG 42 In Focus… Solving Algebra and Other Story Problems with Simple Diagrams: a Method Demonstrated in Grade 4–6 Texts Used in Singapore SYBILLA BECKMANN 47 Book Review… Diverse Voices Call for Rethinking and Refining Notions of Equity AMY J. HACKENBERG 52 Upcoming Conferences 53 Subscription Form 54 Submissions Information © 2004 Mathematics Education Student Association. All Rights Reserved The Mathematics Educator 2004, Vol. 14, No. 1, 2–7 Guest Editorial… Do You Need a PhD to Teach K–8 Mathematics in Ways Respected by the Mathematics Education Community? Chandra Hawley Orrill The genesis of this editorial was a conversation and middle school mathematics in ways that about an article in which Ball (1991) provided mathematics educators would value?” After all, the descriptions of three teachers’ approaches to working Balls, Lamperts, and McClains1 in the literature offer with their students. In Ball’s article, teachers without high-quality mathematics instruction, attend to student PhDs in mathematics or mathematics education thinking, provide opportunities for knowledge struggled to engage their students in developing construction, and introduce students to a variety of meaningful concepts of mathematics. They could not tools they can use later (e.g., visual representations and provide multiple interpretations of problem solving strategies). Further, these researcher- concepts—particularly representations that provided teachers seem to have a gift for promoting student concrete explanations or tie-ins to the real world. They thinking and moving an entire class forward by demonstrated only stepwise approaches to doing scaffolding lessons, questioning students, and creating mathematics, clinging tightly to procedures and a classroom community where learners consider each algorithms, and provided no evidence that they had a other’s work critically and interact meaningfully. The deeper understanding of the mathematics. In stark reality, however, is that not all mathematics teachers contrast, the same Ball article offered a vignette of have PhDs and it is unlikely that most ever will.2 Lampert’s teaching that illustrated a rich mathematical In working through this question both with the experience for students. Lampert provided multiple graduate students with whom I work and in preparation perspectives, introduced multiple representations, and for this editorial, I have developed some ideas both demonstrated a deep understanding of both about researcher-teachers as a “special” group and mathematics and student learning throughout the about why having a PhD might matter. Based on my episode described. thoughts I would like to propose two conjectures about Given the number of articles in the literature researcher-teacher efforts. First, I conjecture that we painting the ‘typical’ mathematical experience as one should consider the way we think about researcher- that is impoverished, and the growing body of teachers versus research on/with teachers. Second, I literature written by PhD researcher-teachers, I propose that certain features of PhD programs can be wondered, “Do you need a PhD to teach elementary applied to teacher professional development and/or undergraduate education to support all teachers in Chandra Hawley Orrill is a Research Scientist in the Learning creating richer mathematics learning experiences for and Performance Support