Generative number concepts in children

正如成语所说，自然数就像一二三一样简单。然而，作为一个孩子，了解他们并不容易。人们需要几年的正规和非正规教育才能理解数字的含义，每个数字与另一个数字的关系，以及数字是无限的。这个项目调查了孩子们是如何理解数字是生成的，也就是说，一个新的数字可以无限地产生。一个新的理论框架，整合了心理学，语言学和历史的思想，认知发展实验将评估该理论框架。该项目的成果预计将解决人类数学思维的基本能力来自哪里，更广泛地说，扩展我们对人类如何“无限利用有限手段”的知识，这是尚未回答的最深刻的认知科学问题之一。从这个项目中获得的基础知识也有望激发基于数字习得自然机制的STEM教育的新教学方法。数字习得研究中的一个关键问题是儿童如何学习一组规则允许无限期地生成新数字。现有的心理学理论建议儿童通过使用正式的数学原理来学习生成数字概念，这使得难以产生可证伪的假设并根据经验进行测试。这个项目建议从抽象的数学角度出发，而是提出了一个经验上可检验和可证伪的假设，从广泛的文献的跨学科分析。具体来说，它提出并测试了中心假设，即生成数概念是通过学习由数字词和阿拉伯数字的组合结构实现的量的多项式表示来获得的。这一假设将在4至8岁的儿童中使用新的认知实验进行测试。该项目的成果预计将建立一个新的心理上合理的和可证伪的理论，关于数字概念和他们的收购。这个奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

As the idiom goes, the natural numbers are as easy as one-two-three. However, learning about them as a child is not easy. It takes several years of formal and informal education to understand what number words mean, how each number relates to another, and that numbers are infinite. This project investigates how children come to understand that numbers are generative, that is, a new number can be generated endlessly. A new theoretical framework that integrates ideas from psychology, linguistics, and history is proposed, and cognitive developmental experiments will evaluate that theoretical framework. The outcomes of this project are expected to address where the fundamental human capacity for mathematical thinking comes from and, more broadly, to expand our knowledge about how humans make "infinite use of finite means," one of the deepest cognitive science questions yet to be answered. The foundational knowledge gained from this project is also expected to inspire new pedagogical approaches to STEM education that are based on the natural mechanisms of number acquisition.A key question in number acquisition research is how children learn that a set of rules allows the generation of new numbers indefinitely. Existing psychological theories propose children learn generative number concepts by using a formal mathematical principle, making it difficult to generate falsifiable hypotheses and test them empirically. This project suggests a departure from the abstract mathematical perspective and instead proposes an empirically testable and falsifiable hypothesis developed from an interdisciplinary analysis of the broad literature. Specifically, it proposes and tests the central hypothesis that generative number concepts are acquired by learning the polynomial representation of quantity enabled by the combinatorial structure of number words and Arabic numerals. This hypothesis will be tested using novel cognitive experiments in children between four and eight years of age. The outcomes of this project are expected to establish a novel psychologically plausible and falsifiable theory about number concepts and their acquisition.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.