Beginning calculus students\u27 understanding of the derivative: Three case studies

This exploratory study analyzed the development of beginning calculus students\u27 understanding of the derivative in relation to their initial understandings of the three highly interconnected mathematical content strands of Variable, Function, and Rate. The research framework defined critical elements of understanding of these content strands, based on existing research and a rational task analysis. Twelve students participated in task-based clinical interviews to establish initial profiles of understanding based on the research framework. Each student\u27s developing understanding of the derivative was documented in eight subsequent weekly interviews. Three students with different initial profiles were chosen for full case study analysis. The following major findings emerged from this study: (1) There was a correspondence between students\u27 initial understandings of the three content strands and their understanding of the derivative: strong initial understanding in all three content strands supported a strong understanding of the derivative; while weak rate understanding impeded it. (2) Rate understanding appeared to play the largest role in determining how a student made sense of the derivative. (3) Speed and motion situations presented special obstacles to student understanding, confirming results of other studies both in mathematics education and physics education. (4) Some students regarded graphs as simple pictorial displays of individual data points, rather suggesting lack of understanding of covariation. (5) Strong initial content understanding provided a good foundation for developing a strong understanding of the derivative, but was not sufficient to support learning how to apply calculus; the ability to create and use symbolic representations was needed. (6) Algebraic manipulation skill and use of memorized formulas and procedures created the illusion (both for instructor and student) of learning. (7) Powerful representational ways of thinking about change enabled students to conceptualize variation and covariation, to define a function symbolically, to clearly distinguish a function from its rate of change, and to recognize and articulate pattern in rate of change. (8) Mature rate understanding involves a long-term learning process in which many mathematical ideas need to be connected