Journal of Mathematical Behavior 27 (2008) 246–249

Contents lists available at ScienceDirect

The Journal of Mathematical Behavior

journa l homepage: www.e lsev ier .com/ locate / jmathb

Multipurpose Professional Growth Sequence: The catwalk problem asa paradigmatic example

Chris RasmussenSan Diego State University, Department of Mathematics and Statistics, 5500 Campanile Drive, San Diego, CA 92182-7720, United States

a r t i c l e i n f o

Article history:Available online 17 March 2009

Keywords:ModelingRepresentationTeacher learningTask design

a b s t r a c t

An important concern in mathematics teacher education is how to create learning oppor-tunities for prospective and practicing teachers that make a difference in their professionalgrowth as educators. The first purpose of this article is to describe one way of working withprospective and practicing teachers in a graduate mathematics education course that holdspromise for positively influencing the way teachers think about mathematics, about studentlearning, and about mathematics teaching. Specifically, I use the “catwalk” task as an exam-ple of how a single problem can serve as the basis for a coherent sequence of professionallearning experiences. A second purpose of this article is to provide background informationthat contextualizes the subsequent two articles, each of which details the positive influenceof the catwalk task sequence on the authors’ professional growth.

© 2009 Elsevier Inc. All rights reserved.

Speiser and Walter (1994, 1996), Speiser, Walter, and Maher (2003), in a series of compelling case studies with high schoolstudents, undergraduate calculus students, and university professors from diverse disciplines, explore a surprising range ofpossibilities for exploring, knowing, and making sense of change and motion mathematically. Each of these case studiesbegins with a set of 24 time-lapse photographs by Eadweard Muybridge (Muybridge, 1887/1957). For the photos shown inFig. 1, one simply asks: How fast is the cat moving at frame 10? At frame 20? As Speiser’s and Walter’s work makes clear, theseseemingly simple questions can give rise to deep practical and philosophical reflections, reflections that can inform our workwith students in addition to our own understandings of motion and modeling.

Inspired by this prior work with this set of photos, I was intrigued to explore ways that this task might serve as anopportunity to develop a coherent sequence of professional learning experiences for practicing high school and communitycollege teachers enrolled in a masters degree program in mathematics education. Similar to the goals that Zaslavsky (2008)detailed for general teacher education, my broad goals in designing this sequence of experiences with the catwalk taskwere to foster the development of more knowledgeable, analytic, insightful, observant, and reflective professionals. Howone might actually use tasks to make progress on these perhaps ambitious goals has been a central and continuing concern,if not a dilemma, for teacher educators (Watson & Mason, 2007). In the paragraphs that follow, I argue that the catwalk task,configured as a Multipurpose Professional Growth Sequence (MPGS), can be a vehicle to achieve at least some of these broad,ambitious purposes. The two articles that follow are reports from practicing teachers on their personal and professionalexperiences with the catwalk task.

An MPGS, as the name implies, is a sequence of experiences, centered on the same problematic situation, that can servethe purpose of addressing multiple professional growth goals. In particular, I have used the catwalk task with teachers in amathematics education masters program

E-mail address: chrisraz@sciences.sdsu.edu.

0732-3123/$ – see front matter © 2009 Elsevier Inc. All rights reserved.doi:10.1016/j.jmathb.2009.01.002

C. Rasmussen / Journal of Mathematical Behavior 27 (2008) 246–249 247

Fig. 1. Muybridge’s catwalk photos. The time interval between successive photographs is 0.031 s. The distance between lines in the background grid is 5 cm.

• to deepen and enrich their mathematical insights into important mathematical ideas,• to model learning environments where participants share and discuss their mathematical thinking,• as a springboard to read and critically discuss research articles and other artifacts that feature student reasoning,• as an opportunity to engage their own students with the same important and challenging mathematics, and• as an occasion to reflect on, analyze, and write about their interactions with their students and the reasoning and products

produced by their students.

The catwalk task, based on the photographs shown in Fig. 1, provides a common foundation for all of these activities, toform a coherent collection of professional learning experiences.

For example, I first used the catwalk task with masters students for mathematical purposes. In particular, I wanted mystudents to revisit the concept of rate of change from a (possibly) new and deeper perspective. I wanted them to them tocreate models to present their analyses of the catwalk photos, and in the process rethink what models and functions are.As students went about their work on the catwalk task, I used this as an occasion to foster the social norms that studentsexplain their reasoning, present their work to others in some visible form, listen to others’ reasoning, comment on alternativeapproaches, etc. As such, I used the catwalk task for socialization purposes.

After working on the task themselves, students critically read and discussed selected research articles that emphasizedcase studies of the catwalk task (e.g., Speiser & Walter, 1994, 1996; Speiser et al., 2003; Speiser & Walter, 2008). Studentsalso watched and discussed video footage of high school students working on the catwalk task. This video was featured inthe “Private Universe in Mathematics” series, produced by the Harvard-Smithsonian Center for Astrophysics (2000). Theseexperiences offered opportunities for students to become more analytic and reflective, as professionals. Students were latergiven the choice of using the catwalk task for their final course project. In particular, they would work with a group of highschool or community college students on the catwalk task, document this work, and write a final report about the evolutionof their students’ work and thinking in relation to this task.

Finally, I would like to comment on how the catwalk task has contributed to my own professional growth as a teachereducator. Certainly the many creative and innovative presentations that my students have developed to capture their analysesof the cat’s motion has enlarged my own thinking about rate of change, function, and modeling. The two articles that followgive a glimpse of the creativity and insight of these students. Moreover, reflecting on the collection of experiences centeredon the catwalk photos has caused me to notice more clearly the tremendous potential of problems like the catwalk task.Attaching the label MPGS to the series of experiences centered in the catwalk task might encourage other mathematicseducation researchers to develop further MPGS. Indeed, I feel that the community of mathematics teacher educators, as wellas the professionals we serve, could benefit greatly from such efforts.

248 C. Rasmussen / Journal of Mathematical Behavior 27 (2008) 246–249

A fairly common approach in teacher education is to use a mathematics problem first with teachers, and then invite theteachers to use this same problem (or a suitable variant) with their students. The main feature that distinguishes MPGS fromthis fairly common use of problems in teacher education is that an MPGS includes a collection of published research studiesand other artifacts that can be used for critical analysis, reflection, and hence further professional growth. Moreover, MPGSare intended for teachers, in turn, to share their work with others publicly. The two articles that follow offer rich examplesof such sharing.

Next, I would like to identify particular characteristics of the catwalk task that could make it a paradigmatic MPGS. Toidentify such characteristics might help enable others to develop further MPGS. These characteristics necessarily refer bothto the task itself and to available resources that the research field has created.

• There is a clear, unambiguous question. “How fast is the cat moving in frame 10 and in frame 20?” This makes the taskreadily understandable.

• The task affords a range of points of view from which to understand the problem situation, so that the task becomesaccessible to a wide range of students. Such points of view, in this case, include the photographs, personal experiencein seeing cats move, the possibility to recreate the motion for oneself, a certain familiarity with the relevant informalmathematical concepts (such as speed and motion), and readily accessible measurements that can be used for computationsand comparisons.

• The task brings includes strikingly contrasting cases. Specifically, the degree of certainty that one might reasonably haveabout the cat’s speed turns out to be quite different at frame 20 than it might be at frame 10. Such contrasting casesencourage (and perhaps demand) diverse and innovative methods and ways to approach and to approach the problem atthe outset, and diverse and innovative ways to present subsequent analyses.

• The task focuses attention on important (and often challenging) mathematical concepts.• The task has an interesting and indeed authentic historical connection. In the case at hand, more than a century ago,

Muybridge made a similar set of photos of a walking horse to help settle a controversy (involving Leland Stanford, amongothers) that all four hooves would never leave the ground at the same time.

• Artifacts to trace others’ engagement with the task are readily available. In the present case, the catwalk task has beenused in a number of different research studies, and reports of these studies are accessible to students. Further, the catwalktask is also featured in a widely distributed video workshop series, explicitly designed for the professional developmentof teachers.

• Teachers write reports of their experiences with the task to publicly share with others. Such sharing might take place, forexample, in conference presentations, journal articles, and local teacher training efforts.

MPGS, rather than handing research results to teachers, are intended to bring mathematics and mathematics educationresearch into a teacher’s lived experience, experience as a mathematics learner, as a teacher, and as a professional who sharestheir work with others. As Mason (1998) has emphasized,

I am convinced that failure is the only possible outcome for any approach to research in mathematics education inwhich researchers hand their results to curriculum developers and teachers, who are then expected to apply themin their practices. Education simply does not work that way, either in how other people can be influenced to try out‘ideas’, or in how new ideas are located and developed in the first place. (p. 375)

MPGS, in contrast, can encourage teachers to try out personal, original, ideas, in human settings that can honor the specificmathematical and human contexts in which such ideas took shape. The two papers that follow give concrete evidence thatthe catwalk task, viewed as an MPGS, can facilitate teachers’ professional development in ways that build from teachers’personal and collective intellectual strengths.

The authors of these two articles were students in the graduate course where the catwalk task was used as an MPGS. Eacharticle, based in part on its author’s final project, gives a retrospective reflection and synthesis of the authors’ experiences withthe catwalk task. Each has three sections. The first, Reflections on My Own Work, recounts the authors’ personal mathematicalwork on the catwalk task, and each such narrative describes personal struggles, hard-won progress, and success. The secondsection, Reflections on Students’ Work, reports on and synthesizes the experience of working with a group of students (in onecase high school students, in the other community college students) on the catwalk task. Surprising and creative studentpresentations of the cat’s motion have been detailed and described. In the third and final section, Looking Back (and Forward)on the Entire Experience, each author takes stock of the whole experience, as a mathematics learner, as a teacher, and as aprofessional.

Steven Case wrote the first article. Case, a recent graduate from the masters in mathematics education program, is a highschool mathematics teacher with 9 years of teaching experience, and he is currently the chair of his department. In his report,Case details his interactions with and the work of high school students on the catwalk task. Emiliano Vega and Shawn Hicks,both of whom are also recent graduates of the masters in math ed program, are coauthors on the second article. In addition toindividually detailing reflections on their own work and on the entire experience, they describe joint work with communitycollege calculus students.

C. Rasmussen / Journal of Mathematical Behavior 27 (2008) 246–249 249

References

Harvard-Smithsonian Center for Astrophysics, Science Media Group (Producer). (2000). Workshop 6: Possibilities of real-life problems. In M. Schneps(Executive producer), Private universe project in mathematics, [A six workshop television series]. Cambridge, MA: The Annenberg/CPB Channel.

Mason, J. (1998). Researching from the inside in mathematics education. In A. Sierpinska & J. Kilpatrick (Eds.), Mathematics education as a research domain:A search for identity, Book 2 (pp. 357–377). Dordrecht, The Netherlands: Kluwer.

Muybridge, E. (1887). Animals in motion. New York: Dover. (Reprint, 1957 ed.).Speiser, B., & Walter, C. (1994). Catwalk: First-semester calculus. Journal of Mathematical Behavior, 13, 135–152.Speiser, B., & Walter, C. (1996). Second Catwalk: Narrative, context, embodiment. Journal of Mathematical Behavior, 15, 351–371.Speiser, B., & Walter, C. (2008). Rethinking change. In M. Carlson & C. Rasmussen (Eds.), Making the connection: Research and teaching in undergraduate

mathematics education (pp. 15–25). Washington, DC: The Mathematical Association of America.Speiser, B., Walter, C., & Maher, C. (2003). Representing motion: An experiment in learning. Journal of Mathematical Behavior, 22, 1–35.Watson, A., & Mason, J. (2007). Taken-as-shared: A review of common assumptions about mathematical tasks in teacher education. Journal of Mathematics

Teacher Education, 10, 205–215.Zaslavsky, O. (2008). Meeting the challenges of mathematics teacher education through design and use of tasks that facilitate ts T. Wood (Series Ed.) and B.

Jaworski, International handbook of mathematics teacher education. (Vol. 4 Ed.). Rotterdam, The Netherlands: Sense Publishers.