Distributed by Design: On the Promises and Pitfalls of Collaborative Learning with Multiple Representations

Abstract

This article presents a designed learning environment intended to engage students in learning about the relationships among multiple representations as they work together on a shared task. Over the course of several extended problem-solving sessions, groups developed several successive alignments of participants and representations as they learned to solve increasingly difficult tasks. Our findings highlight the emergent and often unexpected meanings that learners established for representational tools as their groups reorganized into increasingly effective problem-solving ensembles. Our findings echo those of prior research regarding learners' considerable competence and creativity in interpreting and applying distributed representational tools, as well as the careful coordination among learners involved in establishing and acting on those interpretations. Challenges in this design space include instances in our data where students capitalized on connections among representations without really trying to understand those connections, temporarily undermined the distributed character of the representations, and worked more efficiently by reducing the number of participants actively involved in breaking codes. Our findings indicate that managing these challenges requires presenting groups with regular opportunities to reconsider and reorganize their roles, and to experiment with different meanings and uses of flexible tools in the context of tasks with carefully sequenced levels of difficulty.

ACKNOWLEDGMENTS

We would like to acknowledge the pioneering work of the Middle School Math through Applications Project in creating the Codes Inc. unit upon which the handheld wireless application Code Breaker was based and to thank Shelley Goldman for her design contributions to the Code Breaker system. Thanks also to John Murray, Nicolai Scheele, and Wolfgang Effelsberg for their programming of Code Breaker. Hewlett Packard provided a generous 2003 hardware grant to Roy Pea in support of this project from their Applied Mobility Technology Solutions in Learning Environments Grant Initiative. We also appreciate funding for this work in its early stages by the Wallenberg Global Learning Network, as well as more recent support for Tobin White through National Science Foundation (NSF) Early Career Development Award DRL-0747536 and for Roy Pea through LIFE Center NSF Grant SBE-0354453. We dedicate our article to the memory and leadership of Jim Kaput (1942–2005), whose seminal publications and talks on learning with multiple representations have been so foundational to the field of technology-enhanced mathematics education. Finally, we gratefully acknowledge Dor Abrahamson, Lee Martin, Jeremy Roschelle, and several anonymous reviewers for providing insightful comments on earlier drafts.

Notes

¹Code It! was an adaptation of the “Codes Inc.” curriculum unit originally developed at the Institute for Research in Learning by Goldman, Greeno, and colleagues in the Middle School Math through Applications Project to provide a slice of the real world chosen to engage young people in mathematics (Greeno et al., 1999), in this case having students role-play cryptologists designing and testing codes for clients. Their curriculum used code making and code breaking as a vehicle for teaching algebraic functions. For the Codes Inc. unit, students used software running on desktop personal computers to create and break codes and passed coded messages to each other via e-mail. When Roy Pea wished to develop a classroom network using handheld wireless computers for collaborative math learning, he sought out Goldman's collaboration to redesign the activities so as to support collaborating groups in competing with one another in creating and breaking codes (see Goldman, Pea, & Maldonado, 2004; Goldman, Pea, Maldonado, Martin, & White, 2004, for design rationale and preliminary accounts of this intervention). We thus adapted the Codes Inc. unit for small-group collaborative learning using handheld wireless PocketPCs. In this new Code Breaker system, handheld devices communicated with each other via a classroom-based server computer, rather than directly, and different representations of candidate functions were updated dynamically on the handheld displays of each group member, unlike the original Codes Inc. design.

²In addition to the polynomial mapping, some codes also featured an “offset.” Offsets shifted the relationship between the letters of the alphabet and their ordinal values, so that whereas an offset of 0 meant that A was associated with an input value of 1, an offset of 1 associated A with an input of 2, B with 3, and so on, including associating Z with 1. Codes featuring offsets were introduced approximately halfway through the 3-week series of decryption activities as part of an effort to steadily increase the difficulty of the tasks.

³For a more detailed recounting of the inverse function table, its bug, the insights it occasioned, and the reasons for its replacement, see White (2008).

⁴Hereafter, we use coded text to refer to this representation in the Code Breaker software and ciphertext to refer to the actual encrypted text message or the set of numerical characters it included.

⁵The X-column shown in Figure 6 was added after the removal of the buggy inverse function table and retained some of the same functionality, displaying a value whenever the inverse of the current candidate mapped a ciphertext value to an integer between 1 and 26.

⁶Topics on the pretest and an identical posttest included reading graphs and tables; evaluating arithmetic and simple algebraic expressions; solving linear equations; and defining vocabulary words related to functions, algebraic expressions, and graphs. The results of focus group students on these assessments are reported in White (2006).

⁷The groups each worked on some messages assigned by the teacher and some encrypted by other groups within parameters established by the teacher for a given session (e.g., only linear encoding functions). Consequently, the respective series of codes attempted by the two groups were similar though not identical.

⁸In fact, this function did not solve the code, because Tina had incorrectly assumed that it should be linear; after a hint to this effect from the teacher she quickly came up with the correct quadratic encoding function.

⁹See White (2008) for a detailed analysis of this approach.

¹⁰The group's fourth member, Reggie, was absent on this day.

¹¹In this sense, the relevant analog from Hutchins's analysis to classroom learning activity is not the routine of the fix cycle but rather the crisis prompted by a failed gyrocompass. Such disruptions of regular practice, so undesirable and potential hazardous in the workplace or in the wild, are precisely the moments to strive for in the design of learning environments.