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。需要几年的正规和非正规教育才能理解数字词的含义，每个数字如何与另一个数字相关，以及数字是无限的。这个项目调查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休止地产生。本文提出了一个整合了心理学、语言学和历史学思想的新理论框架，认知发展实验将对该理论框架进行评估。这个项目的成果预计将解决人类基本的数学思维能力来自哪里，更广泛地说，扩大我们对人类如何“无限使用有限手段”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启发基于数字习得的自然机制的STEM教育的新的教学方法。数字习得研究的一个关键问题是儿童如何学习一套规则允许无限期地产生新的数字。现有的心理学理论认为，儿童通过使用正式的数学原理来学习生成性数字概念，这使得生成可证伪的假设并对其进行实证检验变得困难。这个项目建议背离抽象的数学观点，相反，提出了一个通过对广泛文献的跨学科分析而形成的经验性和可证伪性的假设。具体地说，它提出并检验了中心假设，即生成的数字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而获得的。这一假设将通过对四到八岁的儿童进行新颖的认知实验来验证。这个项目的结果有望建立一个新的心理学上可信的和可证伪的关于数字概念及其习得的理论。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。

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.