PME-NA coverFrameworks that Support Research and Learning
The 27th Annual Meeting of
PME-NA North American Chapter of the International Group
for the Psychology of Mathematics Education
Preface to the 2005 PME-NA Proceedings
It is with excitement that we at Virginia Tech have brought technological advancement to the 27th annual meeting of PME-NA. For the first time in the history of PME-NA, the conference Proceedings are entirely electronic. Upon registration in Roanoke, conference participants received this CD-ROM containing the Proceedings – instead of the paper volumes traditionally distributed at PME-NA meetings. An extensive program booklet, including abstracts for each session, was distributed at the conference so that participants could make informed decisions about session attendance. In addition, participants were able to browse binders containing complete printed copies of the Proceedings during the meeting. An online set of Proceedings1 was also available prior to the conference.
Users of this CD-ROM have the ability to access conference reports by viewing and selecting from (1) the daily conference schedules, (2) a list of presenters and their sessions, and (3) a list of session topics and themes. Links allow users to read, print, or download individual reports (.pdf files). It is our hope that the electronic Proceedings (both CD-ROM and online) offer PME-NA participants, and the larger mathematics education community, flexibility and easy access to all reports from the PME-NA meeting in Roanoke.
The contents of the Proceedings relate to the PME-NA 27 theme, "Frameworks that Support Research and Learning." The Proceedings include plenary reports by John Mason and Denise Mewborn, 12 working group and discussion group reports, 99 research reports, 55 short oral reports, and 41 poster descriptions. Proposals for working groups, discussion groups, research reports, short orals, and posters were submitted electronically to All Academic's online system. Proposals were reviewed by 2-3 reviewers and, based on the peer reviews, acceptance decisions were made by the editors and the Virginia Tech Planning Committee. Full papers were submitted electronically and edited for uniform formatting and style throughout the Proceedings.
The Proceedings of PME-NA 27 are dedicated to the memory of James Kaput, an internationally renowned mathematics education leader who died tragically on July 31, 2005. Jim's personal and professional contributions are described in a commemorative paper that begins the Proceedings. In remembrance of Jim, his name appears in all of his accepted sessions and papers in the PME-NA program and Proceedings. Jim's participation in this and future PME-NA meetings will be greatly missed.
The quality of these Proceedings has been enhanced greatly by the contributions and support of many people and organizations. The voluntary PME-NA peer reviewers and the 2004-2005 PME-NA Steering Committee helped to shape a high quality and intellectually demanding conference program. The staff at All Academic provided excellent support throughout the review process, paper submission, and the development of the program and Proceedings. We thank the Virginia Tech School of Education, Department of Mathematics and College of Science, and Continuing and Professional Education at Virginia Tech for the many resources that enabled the development of these Proceedings. Finally, enormous gratitude is owed to the Virginia Tech Planning Committee, particularly to those graduate students who devoted themselves for over a year to the many preparations for the PME-NA conference and the production of these Proceedings.
The Editors
Melvin (Skip) Wilson Jesse (Jay) L. M. Wilkins
Stephanie L. Behm
1 http://convention2.allacademic.com/index.php?cmd=pmena_guest
Reviewers for the 2005 PME-NA Proceedings Dor Abrahamson Sergei Abramovich Keith Adolphson Silvia Alatorre Lillie Albert Karen Allen Alice Alston Rebecca Ambrose Mette Andresen Peter Appelbaum Fran Arbaugh Nancy Ares Elizabeth Baker David Barker Michele Baron Jeffrey Barrett Stephanie Behm Babette Benken Robert Q. Berry, III James Beyers Maria Blanton Irene Bloom Katharine Borgen Beth Bos Michael Bossé Michelle Bower Townsend Brian Cathy Bruce Lecretia Buckley Sylvia Bulgar Lupita Carmona Alison Castro Laurie Cavey Jennifer Chauvot Jeffrey Choppin Michelle Cirillo Matthew Clark Jo Clay Olson Linda Condron AnnaMarie Conner Doug Corey Darryl Corey Jose Luis Cortina Bettina Dahl Soendergaard C.E. Davis Sarah Davis A. J. (Sandy) Dawson Ana Dias Jaguthsing Dindyal Barbara Dougherty Corey Drake Maria Droujkova Michael Todd Edwards Amy Ellis James Epperson Evrim Erbilgin Diana B. Erchick Axelle Faughn David Feikes Rosa Ferreira Halcyon Foster Avikam Gazit Cristina Gomez Tracy Goodson-Espy Karen Graham Randall Groth Lourdes Guerrero
Timothy Gutmann Erhan Haciomeroglu Guney Haciomeroglu Amy Hackenberg Wendy Hageman Smith Jean Hallagan Michael Hardy Lynn Hart Beverly Hartter Beth Herbel-Eisenmann Abbe Herzig Margret Hjalmarson Amanda Hoffmann Karen Hollebrands Ilana Horn Verónica Hoyos Pao-sheng Hsu Markus Hähkiöniemi Santiago Inzunza Debra Johanning Jason Johnson Guzman Jose Debra Junk Pier A. Junor-Clarke Ann Kajander Signe Kastberg Sibel Kazak Paul Kehle Garrett Kenehan Margaret Kidd Thomas Kieren Ok-Kyeong Kim David Kirshner Robert Klein Donna Kotsopoulos Nayoung Kwon Andrea Lachance Teruni Lamberg Keith Leatham Sarah Ledford Hea-Jin Lee Hollylynne Stohl Lee Jacqueline Leonard Peter Liljedahl Louis Lim Kien Lim Gwendolyn Lloyd LouAnn Lovin Michael Lutz Kathleen Lynch-Davis Azita Manouchehri Dragana Martinovic Peter McCarthy Raven McCrory Douglas McDougall Jean McGehee Michael Meagher David Meel Miguel Mercado Denise Mewborn Aki Murata Immaculate Namukasa Susan Nickerson Clara Nosegbe-Okoka Michael Oehrtman Rosa Paez Nikita Patterson
Barba Patton Erkki Pehkonen Louise Poirier Neil Portnoy Jeanne Rast Stacy Reeder Araceli Reyes Ginger Rhodes Robin Rider Laurie Riggs Olgamary Rivera-Marrero John Ross Joanne Rossi Becker Amy Roth McDuffie Behnaz Rouhani Elena Ruiz Ana Isabel Sacristan Roberta Schorr Jennifer Seymour Daniel Siebert Elaine Simmt Stephanie Z. Smith Natasha Speer Laura Spielman Bharath Sriraman Heidi Staebler Megan Staples Jon Star Brenda Strassfeld Sharon Strickland Andreas Stylianides Gabriel Stylianides Lynn Tarlow James Tarr Mourat Tchoshanov Anne Teppo Christine Thomas Patrick Thompson Tony Thompson Guenter Toerner Maria Trigueros Mary Truxaw Zelha Tunc-Pekkan Andrew Tyminski Elizabeth Uptegrove Juliana Utley Veronica Vargas-Alejo Jana Visnovska David Wagner Lisa Warner Keith Weber Dorothy White Tobin White Joy Whitenack Jesse 'Jay' Wilkins Skip Wilson Terry Wood Paul Yu Ismail Zembat Qing Zhao
Lloyd, G. M., Wilson, M., Wilkins, J. L. M., & Behm, S. L. (Eds.). (2005). Proceedings of the 27th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education.
JAMES J. KAPUT – HIS CONTRIBUTIONS OF INTELLECT AND CHARACTER
Jeremy Roschelle Center for Technology in Learning, SRI International
jeremy.roschelle@sri.com
August 5, 2005
Jim Kaput liked to tell me stories about his huge lung capacity. He could take in more air than anyone else. As a child, he surprised and delighted others by swimming underwater for longer than anyone imagined possible. Now, like many things, this seems like a metaphor. He lived life more fully than most of us. He took more in, he worked harder, and he gave more to others.
Jim could frame a vision in a phrase. “Democratizing access to the mathematics of change” was the mission I shared with him for twelve years. It's the flag that those who loved him will carry onwards.
Jim’s personal mission was to bring much more powerful and meaningful mathematics to many more people. He was a theorist of democratization. In his view, access to concepts was a function of representation and pedagogy. By transforming the way concepts were represented and methods of instruction, many more people would be able to gain access to difficult but important ideas. He wanted to accomplish in mathematics something like the democratization of literacy that began with the printing press. Before the printing press, scribes were thought of a very special people. Who could imagine that someday everyone would read and write? Surely many people thought, "I'm just not good at reading." Jim believed in a future in which everyone would be able to access the mathematical jewels of their cultural heritage -- the jewels that lay beyond the "basics" of shopkeeper arithmetic. He believed it was the responsibility of an advanced civilization to make powerful mathematics learnable, meaningful, and useful.
Jim believed deeply in democratization and acted on his beliefs with passion. For decades, he taught a class at the University of Massachusetts, Dartmouth for academically disadvantaged freshman preparing for technical majors. Many accomplished professors want to work with only the best students. Jim, in contrast, dedicated his thoughtful class preparation to students who were underprepared. He constantly sought to improve his course, striving to help these students go on to further coursework in science and engineering. Often when I called, he would start the conversation by telling me excitedly about the innovation he planned for the next day's class.
When Jim was with students, he would coax thinking out of them. He celebrated each little step a student made as a major advance. “Notice what is going on here!” he would say, drawing the class’s attention to the student's idea and embellishing it, making it more precise, more mathematical, more fruitful for later growth. “Ordinary kids can do extraordinary things,” was one of his core beliefs.
Likewise, Jim’s research projects always committed to do research in schools that had very little. To him, it would be meaningless to show that technology could help elite students; he wanted to prove technology's potential for helping the least advantaged to master mathematical concepts. His signature technology was SimCalc, an approach to introducing the mathematical ideas of rate and accumulation through more intensive use of computer-based graphs and
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animation, used alongside conventional tables and algebraic symbols. Many people think of SimCalc as a piece of software. There were many pieces of SimCalc software, and these were always works in progress. There were also many curricular units to accompany the software, and these too were always in a state of perpetual improvement. SimCalc was an idea that Jim was constantly refining with the best and brightest team he could inspire to join him.
I still have the original video that Jim created to communicate his vision of SimCalc. It’s a sign of how he worked that there are almost no features of that video are part of today's MathWorlds software. SimCalc was really a process of constant iterative improvement towards the goal of democratizing access to the mathematics of change. Jim's teams proposed, debated, tested and implemented new features all the time. Gradually, in the crucible of classroom experience, his teams separated the wheat from the chaff. Some half-baked ideas died quickly; some transformed slowly, but in the end Jim never held onto a feature or idea that didn't prove out in the classroom. And he was always open to yet more powerful capabilities of technology that would require rethinking everything all over again but might lead to a quantum leap. The most recent instance of this was his passionate work on taming classroom networks to become instruments of a participatory, engaged, emergent mathematical experience for all his students. Jim’s vision will only become a concrete thing when every student has in hand a powerful combination of representation and communications that enables him or her to participate meaningfully in expressing, constructing, modeling, and analyzing concepts using the mathematics of change.
One pervasive character of Jim’s work was a drive to scale. Once he accomplished something, he always raised the stakes. He wanted to take it to the next level on the path to massive impact. And so SimCalc went from studying a few students, to studying a few teachers, to studying teachers in a few regions of the country, to statewide studies scale up among many teachers in Texas. He also pushed hard to get elements of his designs on calculators, thereby bringing his mathematical representations into widespread use.
Jim acted locally in his region; he acted nationally to influence key reports and standards documents; he acted globally wherever mathematics educators met. A distinguished National Science Foundation (NSF) program officer once challenged the field to imagine how it could fruitfully spend $1 billion to make an impact on math and science education through technology. The assembled room of 100 distinguished educators collectively hid under their chairs mumbling, "we're not ready. We haven't done enough research yet." Not Jim. He lept out his chair and proclaimed, "Wait a minute! I think I could use at least half of it!"
Jim always took the long view. He refused to work on fads or political imperatives designed for short-term impact. He reminded anyone who would listen that 100 years ago only 3 percent of students studied algebra: today we expect algebra for all. But that is only today's problem, Jim would remind us. Schooling is only slowly catching up to 19th-century mathematics. Jim was addressing the problem of how to teach everyone 21st-century mathematics. He wanted to transform the current curriculum, in which calculus is icing on the layer cake of mathematics education into a continuous strand of mathematical concepts that are introduced to students throughout the math curriculum as they advance from grade to grade. He believed that we live in a time of change and every student could benefits from an understanding of the mathematics of change. Because he had a long-range view, Jim refused to be pinned down to research questions that could be answered in a year or two. “You can test this little fraction of my ideas,” he’d say, “but there isn’t any way to test the whole thing.” I always teased him that his epitaph should say, “we will know if he was right in 50 years.” Indeed, it would take 50 years to fully test the scope
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of Jim’s vision for improving mathematics education, which stretched from elementary grades though university education.
The huge upswell of feeling for Jim as people learned of his death was not, however, merely a reflection of his beliefs, his ideas, or his projects. Jim was there for so many people at crucial moments in their careers; he shared their personal hardships and help them at the junctures and gauntlets of their lives. Whether in the role of mentor, colleague or boss, he was unbelievably generous with his time. James Burke told me, "he had the most compassionate intelligence." Jim slaved over letters of recommendation to help junior faculty whom he believed were qualified for tenure. He helped colleagues reframe their ideas to achieve publication and develop their funding proposals.
A virtual college formed around Jim. He connected people and helped them work together successfully. This college had no formal organization, no web site, no e-mail list. But it was no less real. He brought people together, from local schools, national universities, and major technology companies. Jim worked through powerful connectivity as well as powerful representation of ideas.
Jim was committed to his family. He would always get home from his busy travel schedule by Friday night or early Saturday morning, to spend the weekend with his family and to read aloud to his disabled son. Despite offers from universities all over the country, he wanted to stay in Massachusetts, close to family and friends. He was legendary not just for his research but for his Super Bowl parties.
More than anyone I know, Jim maintained intense working relationships with people across time and space. His town was home base, but only a starting point. He did not allow his location to limit his work.
Jim was a character from toe to head. He perpetually wore running shoes and a yellow, orange or red shirt. He had an Abe Lincoln beard. But what I will miss most is his eyes. Jim had extraordinarily expressive eyes. When he heard an idea he liked, they would grow huge with excitement and radiate light. His eyes beamed enthusiasm for the contributions he devined in others' thoughts. I will miss that the most.
Jim was my mentor and made me feel incredibly special and important. When, as a green kid, I proposed that I join him full time in 1993, he acted as if I were doing him the greatest honor in the universe. As we worked together, he sang my praises to leaders at NSF, credited project accomplishments to me, and promoted me to co-Principal Investigator. We stayed up nights together despite 2600 miles between us, sometimes passing drafts coast to coast over the Internet to meet a deadline. As he did with so many other young researchers, he poured energy into my career development. As I progressed, the relationship shifted and he became more of a colleague. We worked as co-PIs on eight grants totalling more than $10 million over a time span of 12 years so far ( three more years remain of our largest project together). And yet, we did so by expanding the umbrella to embrace all the best people we each could bring to the mission, by intuitively anticipating what the project needed next, by agreeing on what quality meant, not how it was achieved. Our actual acts of overt coordination were surprisingly sparse. This was how Jim led.
Jim relished his upcoming retirement and had many dreams. Among them, he planned to build a tower in his backyard. He wanted to be higher than the trees, to see clearly out to the horizon, to the ocean.
Jim was an engaged visionary, a compassionate intellect, an inspirational poet of mathematics education reform. Although he will be missed by many, I believe his dreams will
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some day be realized: new curriculum coupled with new technology will someday enable many more ordinary people to accomplish extraordinary things through mathematics. Many people have asked what they can do. Jim’s family would like all contributions to go to a scholarship fund when established. If you wish to make a contribution now please make checks payable to UMDF and write "Kaput Scholarship" in the memo line. These should be forwarded to: UMass Dartmouth Foundation Foster Administration University of Massachusetts Dartmouth 285 Old Westport Road, N. Dartmouth, MA 02747-2300. USA
Lloyd, G. M., Wilson, M., Wilkins, J. L. M., & Behm, S. L. (Eds.). (2005). Proceedings of the 27th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education.
FRAMEWORKS FOR LEARNING, TEACHING AND RESEARCH: THEORY AND PRACTICE
John Mason The Open University
j.h.mason@open.ac.uk
I have chosen to reverse the order of the key words in the conference title, and to interpolate a third term in order to fit with my view of the role and functioning of frameworks. I begin by introducing a framework for learning in which systematic variation can be used to provoke learners into becoming aware of mathematical structure. Structural Variation Grids have evolved over several years and I indicate some of the history of their development. I then use some frameworks for teaching based on Ference Marton’s notion of variation, some based on George Polya’s descriptions of mathematical thinking, some based on Jerome Bruner’s three modes of re-presentation, and one based on my own work on the structure of attention, in order to provide theory-based justifications for pedagogical and didactic choices that the Structural Variation Grids afford. These frameworks can be used to enhance and enrich the learning potential of particular instances of grids, but also any other mathematical task in any mathematical topic. Like most frameworks for teaching, the ones I will use can be transformed into frameworks for learning through the process of scaffolding and fading (Brown et al 1989), itself a framework for teaching. In the final section I suggest why and how these frameworks work, and this includes a description of the methods used to justify the claims in this paper. My aim is to exemplify what I think is at the heart of learning and of being taught, at the heart of professional development, and indeed at the…