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.
Tobin White has written an interesting article about cryptography and problem solving. The article is entitled Encrypted objects and decryption processes: problem-solving with functions in a learning environment based on cryptography, and the article was published online in Educational Studies in Mathematics on Thursday. Those of you who don’t have a subscription to this journal will be interested to know that the article is an Open Access article, and it is therefore available to all! (Direct link to pdf download) Here is the abstract of the article: