Understanding Qualitative Calculus: A Structural Synthesis of Learning Research
Understanding Qualitative Calculus: A Structural Synthesis of Learning Research
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理解定性微积分：学习研究的结构综合
DOI：
10.1023/a:1021147132127
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发表时间：
2002
期刊：
影响因子：
--
通讯作者：
W. Stroup
中科院分区：
文献类型：
--
作者：
W. Stroup
十多年来在使用基于计算机的图形和模拟环境方面的研究和创新鼓励我们研究界的许多人相信年轻学习者可以成功理解与微积分相关的推理的重要方面。本文试图探讨哪些与微积分相关的见解似乎代表了这种早期形式的微积分推理。引入“定性微积分”一词来框架对这种“其他”微积分的分析。定性微积分的学习是综合的重点。核心主张是，定性微积分本身就是一种认知结构，并且定性微积分的发展或演变方式似乎符合皮亚杰对运算思维发展的分析的重要一般特征。特别是，速率的强化以及所谓的“多少”（量）和“多快”（速率）量之间的两种可逆性，交互地、集体地表征和帮助定义理解定性微积分。尽管与学习微积分的传统期望有相似之处，但定性微积分并不是基于比率或比例的斜率概念建立的，因为它们通常与定义速率相关。然而，本文最后讨论了理解定性微积分如何支持和链接到斜率、比率和比例的速率相关文献。此外，整个过程中还讨论了课程联系和含义，以帮助说明和探索学习定性微积分的重要性。
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.