Recently, there has been an increasing interest in collegiate mathematics education, especially teaching and learning calculus (e.g., Oehrtman, Carlson, & Thompson, 2008; Speer, Smith, & Horvath, 2010). Of many calculus concepts, the derivative is known as a difficult concept for students to understand because it involves various concepts such as ratio, limit, and function (e.g., Thompson, 1994) and it can be represented in multiple ways (e.g., Zandieh, 2000).This study explored and compared university calculus instructors' and students' discourse on the derivative with the lens of a communicational approach to cognition (Sfard, 2008). Specifically, it examined how they describe a) the concept of derivative, b) the relationships between a function, the derivative function, and the derivative at a point, and c) the derivative of a function as another function. The data for this study were collected from the three calculus classes during six weeks of derivative lessons at a large Midwestern university. The study used mixed methods including classroom observations, student survey, and student and instructor interviews. Surveys were scored and used for selecting the interview participants. Classes and interviews were videotaped and transcribed. Transcripts of the instructors' and students' discourse were coded for discussion topic, including the derivative of a function (f '(x)), the derivative at a point (f '(a)), the relationships among f (x), f '(x), and f '(a), and f '(x) as a function. Analysis of instructors' discourse shows that they explicitly addressed what f '(a) and f '(x) represent in relation to f(x) by stating them as the slope of f(x) or describing the behavior of f(x) based on their signs. They addressed the relationship between f(x) and f '(x) with differentiation rules, and the relationship between f(x) and f '(a) by addressing the property of the derivative function at a point where f(x) has extreme values. How f '(x) and f '(a) are related was also explicitly addressed with the substitution method. However, the instructors did not explicitly addressed the relationship between f '(x) and f '(a) in most cases. First, they tended to use the word "derivative" without specifying it as the derivative function or the derivative at a point. This use of the word was mostly identified when they used f '(a) as a representative of f '(x) on an interval, which might have confused students not only about what the word, "derivative" referred to but also what f '(a) and f '(x) represent in terms of f(x). Second, they also did not specifically address that f '(a) is a number, a value of f '(x) that is a function. Although instructors stated once or twice the aspect that f '(x) is a function, they used this aspect mostly without mentioning. Analysis of students' discourses showed that they explained more consistently and correctly the aspects of the derivative addressed explicitly in their classrooms than those addressed implicitly. In other words, they explained the relationships between f(x) and f '(a), and between f(x) and f '(x) better than the relationship between f '(x) and f '(a), and f '(x) as a function. Though most students performed well in problems involving the first two relationships, some of them showed a mixed notion of the derivative as a point-specific value and a function on an interval without distinguishing these two aspects; the most common incorrect description of the derivative was the tangent line at a point, which was a point-specific object but a function. The results of this study indicate that when instructors were clear and unambiguous about mathematical aspects of the derivative, students' thinking on these aspects were closer to what is considered as true in mathematical community. This suggests that using exact mathematical terms and discussing the mathematical aspects that students have trouble explaining or using may provide better opportunity for students to learn the concept of a derivative.
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