Measuring Algebraic Sophistication: Instrumentation and Results David Kirshner Beth Chance Louisiana State University California Polytechnic University dkirsh@lsu.edu bchance@calpoly.edu Because number sense “resists the precise forms of definition we have come to associate with the setting of specified objectives for schooling” (Resnick, 1989), it retains an ineffable quality that makes it difficult to observe and measure. Arcavi (1994) has extended number sense to the algebraic realm under the rubric “symbol sense.” Building on Arcavi’s work, this paper grapples with how to understand and measure students’ algebraic sophistication (conceived in the spirit of, but somewhat more broadly than, symbol sense). We report on an instrument developed to measure algebraic sophistication for 131 Calculus II students at an elite private university, exploring the subconstructs identified for algebraic sophistication, and the degree of sophistication found for these students. Increasingly, mathematics educators are turning away from skills and concepts as the only goals of mathematics instruction to embrace “sociomathematical norms” (Yackel & Cobb, 1996), “habits of mind” (Cuoco, Goldenberg, & Mark, 1996), or “mathematical dispositions” (Kirshner, 2002, 2004) as additional important accomplishments for students. Norms, habits of mind, and dispositions, in this sense, aren’t limited to affective tendencies such as “enjoyment of math” or “persistence in problem solving,” but include valued ways of thinking and of approaching mathematical activity. “Number sense” is a wonderful exemplar of the attractions and challenges of such interests. For, while highly valued in our community, number sense is not reducible to a discrete listing of skills and concepts, and hence “resists the precise forms of definition we have come to associate with the setting of specified objectives for schooling”(Resnick, 1989, p. 37). In the algebraic sphere, Arcavi (1994) introduced “symbol sense” as a concomitant to “number sense” in the arithmetic realm. Arcavi has done an excellent job of setting forth concrete examples of some of the constituent aspects of symbol sense. However, as with number sense, symbol sense retains and ad hoc character making it difficult to address in curriculum, and difficult to measure. Our study takes up the challenge of symbol sense under the rubric of algebraic sophistication–a label that signals a somewhat broadened interest from that pursued in Arcavi (1994). As with number sense and symbol sense, we assume that algebraic sophistication must retain a measure of the ineffable, the uncodifiable. Nevertheless, it is important for our community to continue to grapple with its many facets, to find ways to understand it more fully, and to learn how to identify its presence and absence with respect particular programs of instruction. We report, here, on an instrument used in evaluating an innovative Calculus II curriculum at an elite private university in the mid-western United States. The instrument was designed to