Designing a Cryptography Game to Assess Student Understanding With Respect to a Learning Progression for Mathematical Functions

• In Event: Designing Game-Based Assessment Around Learning Progressions

Abstract

At the college level, most students have not internalized a formal understanding of the definition of a mathematical function (e.g., Sfard, 1992; Vinner & Dreyfus, 1989). Historically, the development of the function concept has been described as a “three-centuries-long struggle for reification” (Sfard, 1992, p. 42). Because the formal definition of function is abstract, Sfard (1992), Davis (as cited in Kieran, 1993), and Vinner and Dreyfus (1989) recommended preliminary instruction prior to introducing the formal definition. Sfard proposed introducing an operational (i.e., rule-based) definition first, and Davis (as cited in Kieran) suggested emphasizing the nature of dependency. Consistent with this recommendation, and in the interest of making the content engaging, we are designing a game intended to support and assess students’ understanding of the operational definition of function, and to lay the groundwork for a schema that captures the more formal set-theoretic definition.

The central competencies assessed by the game include algebraic evaluation and algebraic representation (the latter being a form of mathematical modeling), as applied to working with mathematical functions. This focus is informed by a model of mathematical competency (Graf, 2009; Graf, Harris, Marquez, Fife, & Redman, 2010), which includes both cross-cutting concepts and content-specific procedures. The competency model was developed for the Cognitively Based Assessment of, for, and as Learning (CBAL) research initiative (Bennett, 2010; Bennett & Gitomer, 2009), which is intended to extend traditional assessment systems by providing evidence with respect to student competency that can inform educational policy, instructional support, and learning opportunities. CBAL assessments include technology-enhanced, innovative task types, and we are now exploring the use of games. The context for the game is cryptography; in particular, students will learn simple substitution ciphers and the properties of the functions that underlie them. Students will encrypt messages using ciphers (the evaluation component), and “break” ciphers of opposing teams (the modeling component), learning about the nature of the underlying functions in the process.

Learning progressions (e.g., Corcoran, Mosher, & Rogat, 2009 for a review), or learning trajectories as they are known in mathematics (e.g., Daro, Mosher, & Corcoran, 2011), are proposed models for how competencies develop as students learn. The game discussed in this paper is linked to a learning progression for functions (Arieli-Attali, Wylie, & Bauer, 2012). Ciphers of greater complexity correspond to higher levels of the learning progression; as students advance in the game they will encounter increasingly complex ciphers. It is hypothesized that the methods students use to break the ciphers will reveal evidence about their level of understanding with respect to the progression; for example, a student at an advanced level might use a general method to break a cipher based on a minimum number of clues, while a student at an earlier level may use an informal method such as guess and check and require more clues. This presentation will describe the competencies that will be assessed by the game, as well as how the game mechanics will provide evidence of students’ understanding with respect to a learning progression for functions.

Authors

• Edith Aurora Graf, ETS

• Paul Gestwicki, Ball State University

• Meirav Arieli-Attali, ETS