• Journal of Educational Research in Mathematics
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• v.21 no.1
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• pp.33-55
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• 2011

This study explores the characteristics of calculus students' and instructors' discourses on the derivative using a communicational approach to cognition. The data were collected from surveys, classroom observations, and interviews. The results show that the instructors did not explicitly address some aspects of the derivative such as the relationship between the derivative function (f'(x)) and the derivative at a point (f'(a)), and f'(x) as a function, and that students incorrectly described or used these aspects for problem solving. It is also found that both implicitness in the instructors' discourse, and students' incorrect descriptions were closely related to their use of the word, "derivative" without specifying it as "the derivative function" or "the derivative at a point." Comparison between instructors' and students' discourses suggests that explicit discussion about the derivative including exact use of terms will help students see the relationship that f'(a) is a number, a point-specific value of f'(x) that is a function, and overcome their mixed and incorrect notion "the derivative" such as the tangent line at a point.

• School Mathematics
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• v.12 no.1
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• pp.1-15
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• 2010

The study presented in this paper, which serves as a pilot study for a future comprehensive project, was to investigate how students deal with the concepts of infinity and limit. Based on the communicational approach to cognition, according to which mathematics is a kind of discourse, we tried to identify the characteristics of students' discourse on the topics. Four American and four Korean students were interviewed in English on limits and infinity and their discourse was scrutinized with an eye to common characteristics as well as culture, age, and education-related differences.