Encrypted objects and decryption processes: problem-solving with functions in a learning environment based on cryptography

Abstract

This paper introduces an applied problem-solving task, set in the context of cryptography and embedded in a network of computer-based tools. This designed learning environment engaged students in a series of collaborative problem-solving activities intended to introduce the topic of functions through a set of linked representations. In a classroom-based study, students were asked to imagine themselves as cryptanalysts, and to collaborate with the other members of their small group on a series of increasingly difficult problem-solving tasks over several sessions. These tasks involved decrypting text messages that had been encrypted using polynomial functions as substitution ciphers. Drawing on the distinction between viewing functions as processes and as objects, the paper presents a detailed analysis of two groups' developing fluency with regard to these tasks, and of the aspects of the function concept underlying their problem-solving approaches. Results of this study indicated that different levels of expertise with regard to the task environment reflected and required different aspects of functions, and thus represented distinct opportunities to engage those different aspects of the function concept.

Graphing in Groups: Learning About Lines in a Collaborative Classroom Network Environment

“…These designs align each student participant in a small group with one of a small number of components of a single shared mathematical object. In a previous cycle of design work along these lines, we used a local network of handheld computers to allow each member of a small group to examine different dynamically linked representations of a common function (White, 2006(White, , 2008(White, , 2009White & Pea, 2011). In our current designs for networked graphing calculators, teams of two or three students contribute and then transform expressions that form opposing sides of an algebraic equation, or move respective coordinate points in a Cartesian plane to jointly manipulate lines, parabolas, or quadrilaterals, or operate on individual terms to collectively construct polynomial expressions.…”