Understanding Qualitative Calculus: A Structural Synthesis of Learning Research |
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Authors: | Walter M Stroup |
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Institution: | (1) The Center for Generative Learning, University of Utah, 1705 E. Campus Center Drive, Rm 142, Salt Lake City, UT 84112-9256, USA |
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Abstract: | More than a decade of research and innovation in using computer-based graphing and simulation environments has encouraged
many of us in the research community to believe important dimensions of calculus-related reasoning can be successfully understood
by young learners. This paper attempts to address what kinds of calculus-related insights seem to typify this early form of
calculus reasoning. The phrase “qualitative calculus” is introduced to frame the analysis of this “other” calculus. The learning
of qualitative calculus is the focus of the synthesis. The central claim is that qualitative calculus is a cognitive structure
in its own right and that qualitative calculus develops or evolves in ways that seem to fit with important general features
of Piaget's analyses of the development of operational thought. In particular, the intensification of rate and two kinds of
reversibility between what are called “how much” (amount) and “how fast” (rate) quantities are what interactively, and collectively,characterize
and help to define understanding qualitative calculus. Although sharing a family resemblance with traditional expectations
of what it might mean to learn calculus, qualitative calculus does not build from ratio- or proportion-based ideas of slope
as they are typically associated with defining rate. The paper does close, however, with a discussion of how understanding
qualitative calculus can support and link to the rate-related literature of slope, ratio and proportion. Additionally, curricular
connections and implications are discussed throughout to help illustrate and explore the significance of learning qualitative
calculus.
This revised version was published online in July 2006 with corrections to the Cover Date. |
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Keywords: | calculus computer simulation modeling Piaget qualitative calculus rate |
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