CAREER: Becoming Euclid: Characterizing the geometric intuitions that support formal learning in mathematics

只有人类才有能力进行形式化数学运算，就像欧几里得元素中的几何那样。这个几何学的论点和证明要求人们想象这样的点是如此之小，以至于没有维度，而线延伸到如此之远，以至于没有尽头。然而，日常生活中经历的点和线是有维度的，是有限的。人类独有的抽象点和抽象线的想法从何而来？人类的推理在多大程度上是理想化和抽象化的，在多大程度上仍然植根于日常的、物理的有限？这个项目将通过探索人类几何能力的三个相互关联的认知方面来解决这些问题：1)心理模拟和基于规则的推理在儿童和成人关于三角形的基本判断中的作用；2)允许婴儿、儿童和成人将线识别为两点之间最短距离的条件；以及3)在物理环境(例如，重力)中描述几何对儿童和成人关于空间中两个位置之间最快路径的判断的影响。这一建议将描述对日常生活中的场景和对象的特性的早期感知敏感性如何可能形成人类对不可感知事物的理解的基础。这种理解是人类许多成就的核心，也是学校学习的主要目标之一。这个项目的前景是，在揭示人类在面临困难和新颖的几何问题时所依赖的基本和普遍的空间直觉时，该领域将能够更好地开发基于这些直觉的优势和补充的教学方法。该项目由EHR核心研究(ECR)项目授予纽约大学的职业奖项支持，该项目支持推进STEM学习基础研究文献的工作。在三个系列的实验中，该项目将在积极的教育环境中进行研究，并为未来利用我们基本和普遍的空间直觉的教育干预奠定基础。系列1将描述人们如何直观地对三角形的性质进行推理。它们的性质是通过心理模拟和想象来评估的，还是通过抽象的规则来评估的？系列2将探索人们对台词的看法。在什么情况下，线被认为是两点之间的最短路径？这种认知与日常体验有什么关系，比如通过开发从一个地方导航到另一个地方？系列3将探索在有重力的物理环境中有效路径的几何直觉。实验将在实验室和国家数学博物馆(MoMath)进行。MoMath还将是一个广泛传播该项目发现的网站，包括一个将认知科学家、教育工作者和公众聚集在一起的谈话系列。因此，这个项目将通过基础科学研究和在博物馆的推广接触到成千上万的个人。这个奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Only humans are capable of doing formal mathematics, like the geometry in Euclid's Elements. The arguments and proofs of this geometry require one to imagine points so small they have no dimension and lines that extend so far they never end. And yet the points and lines experienced in everyday life have dimension and are finite. Where do the uniquely human ideas of abstract points and lines come from? To what extent does human reasoning work with idealizations and abstractions and to what extent does it remain rooted in everyday, physical finitudes? This project will address such questions by exploring three interrelated cognitive aspects of human geometric aptitude: 1) the role of mental simulation and rule-based reasoning in children's and adults' basic judgments about triangles; 2) the conditions that allow infants, children, and adults to identify lines as the shortest distance between two points; and 3) the effects of couching geometry in a physical context (e.g., gravity) on children's and adults' judgments about the fastest path between two locations in space. This proposal will characterize how early emerging perceptual sensitivities to the properties of the scenes and objects of everyday life might form the foundation of human understanding of the imperceptible. This understanding lies at the heart of much human achievement and also constitutes one of the main aims of school learning. The promise of this project is that in uncovering the basic and universal spatial intuitions humans rely on when confronted with difficult and novel geometry problems, the field will better be able to develop pedagogies that build on the strengths and supplement the limitations of these intuitions. The project is supported by a CAREER award to New York University by the EHR Core Research (ECR) program, which supports work that advances the fundamental research literature on STEM learning.In three series of experiments, the project will both conduct research in active educational settings and lay the foundation for future educational interventions that harness our basic and universal spatial intuitions. Series 1 will characterize how people reason intuitively about the properties of triangles. Are their properties evaluated through mental simulation and imagery or through abstract rules? Series 2 will explore how people think about lines. Under what conditions are lines recognized as the shortest path between two points? How does that recognition relate to everyday experiences, like navigating from one place to another, through development? Series 3 will probe geometric intuitions about efficient paths in physical contexts with gravity. The experiments will take place in the laboratory and in the National Museum of Mathematics (MoMath). MoMath will also be a site for general dissemination of the project's findings, including a talk series that will bring together cognitive scientists, educators, and the public. This project will thus reach thousands of individuals through basic science research and through outreach at the museum.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.