用绳子和大头针解决几何问题

IF 1.6 4区 心理学 Q3 PSYCHOLOGY, EXPERIMENTAL
C. Freksa, T. Barkowsky, Zoe Falomir, J. V. D. Ven
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引用次数: 6

摘要

如果空间和抽象问题可以可视化和/或物理操作,人类解决这些问题会更容易。我们从认知的角度分析了几何问题解决的领域,并确定了在问题解决过程中相互作用的几个层次的领域抽象。我们讨论了空间构型的物理表现,它们的操作,以及它们对理解问题解决过程的感知的作用。我们提出了构造几何经典解题曲目的扩展,以更直接地解决某些问题,而不是在指南针和直线范式下。具体地说,我们引入弦和针作为有益的隐喻,以推广建设性几何方法。我们从空间问题解决的经典问题来说明“弦和针”范式。讨论了三个案例研究:(i)椭圆结构问题的串销解决方案;(ii)最短路径问题;(三)角三分问题。与正式解决方案进行了比较。分析了“圆规直尺”范式与“绳针”范式的异同。构造几何和描绘几何的特点和局限性以及对计算方法的影响进行了讨论。弦针域比罗盘直尺域更普遍,限制更少。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Geometric problem solving with strings and pins
ABSTRACT Humans solve spatial and abstract problems more easily if these can be visualized and/or physically manipulated. We analyze the domain of geometric problem solving from a cognitive perspective and identify several levels of domain abstraction that interact in the problem solving process. We discuss the roles of physical manifestations of spatial configurations, their manipulation, and their perception for understanding problem solving processes. We propose an extension of the classical problem solving repertoire of constructive geometry to approach certain problems more directly than under the compass-and-straightedge paradigm. Specifically, we introduce strings and pins as helpful metaphors for a generalization of the constructive geometry approach. We present classical problems from spatial problem solving to illustrate the ‘strings and pins’ paradigm. Three case studies are discussed: strings-and-pins solutions to (i) the ellipse construction problem; (ii) the shortest path problem; and (iii) the angle trisection problem. Comparisons to formal solutions are drawn. Differences and similarities between the compass-and-straightedge paradigm and the strings-and-pins paradigm are analyzed. Features and limitations of constructive and depictive geometry as well as implications for computational approaches are discussed. The strings-and-pins domain is shown to be more general and less restrictive than the compass-and-straightedge domain.
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来源期刊
Spatial Cognition and Computation
Spatial Cognition and Computation PSYCHOLOGY, EXPERIMENTAL-
CiteScore
4.40
自引率
5.30%
发文量
10
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