Investigating Approximate Number System Computation in Children

这个项目调查了儿童在学校接受正式数学训练之前，在没有符号或形式符号的情况下进行算术计算的能力。近似数系统(ANS)是一种认知系统，从婴儿期起就可以操作，它允许人类对没有语言或符号的项目集进行近似量化。研究表明，ANS可能支持加法和减法等算术运算，因此ANS可以作为学习正式数学的桥梁。然而，人工神经网络的算术能力以及这种能力是如何发展的还没有被很好地理解。这个项目通过系统地检查ANS的计算能力来填补这一知识空白，该项目使用了一系列旨在评估幼儿在学校学习正式算术规则之前的实验。该项目解决了关于人工神经网络早期认知架构及其在算术计算中的作用的理论辩论。该项目将确定教育工作者可以利用儿童符号前的数学直觉来帮助他们学习正规数学的潜在方法。这对STEM(科学、技术、工程和数学)教育有影响。这个项目将检查在ANS表示上执行的计算与真实算术计算的平行程度。将审查ANS计算能力的发展。符号算术计算遵循一套函数规则。例如，像5+6这样的算术计算的结果是一个新的、独立的量，它与推导它的量一样精确，并且可以在进一步的计算中操作和使用。这些函数规则使符号算术运算既强大又灵活。本项目旨在利用ANS来识别非符号算术的功能规则。在一系列实验中，4至6岁的儿童将被要求解决具有未知加数的非符号问题(例如，5+_=11)。这项任务要求儿童进行类似算术的计算，在脑海中记住两个独立的ANS表示(例如，大约五个和大约11个)，并对它们进行计算(例如，从大约11个减去大约五个)以得出解决方案。每个实验都旨在考察非符号算术的功能规则的不同组成部分，包括ANS计算的解决方案的表征结构和精度，以及这些解决方案可以在进一步计算中使用的程度。工作记忆容量、符号数学成绩和ANS表征精度的其他测量被用来阐明这些认知系统对ANS的计算能力和发展的贡献。数据将使用传统的零假设显著性检验、贝叶斯因子分析和Logistic回归相结合的方法进行分析。因此，该项目将通过揭示ANS的计算体系结构和早期儿童时期这种体系结构的发展来补充已知的ANS代表性结构。该奖项反映了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。

This project investigates children's capacity for arithmetic computation without symbols or formal notation, before they encounter formal mathematics training in school. The Approximate Number System (ANS), a cognitive system which is operational from early infancy onward, allows humans to approximately quantify sets of items without language or symbols. Research suggests that the ANS could potentially support arithmetic operations, such as addition and subtraction, and that the ANS could therefore serve as a bridge to learning formal mathematics. However, the arithmetic capacity of the ANS, and how this capacity develops, is not well understood. This project fills that knowledge gap by systematically examining the computational capacity of the ANS using a series of experiments designed to assess young children before they learn the formal rules of arithmetic in school. The project addresses theoretical debates about the early cognitive architecture of the ANS and its role in arithmetic computation. The project will identify potential ways that educators could leverage children's pre-symbolic mathematical intuitions in order to help them learn formal mathematics. This has implications for STEM (Science, Technology, Engineering, and Mathematics) education. This project will examine the degree to which computations performed over ANS representations parallel true arithmetic computations. The development of the computational capacity of the ANS will be examined. Symbolic arithmetic computation obeys a set of functional rules. For example, the result of an arithmetic computation like 5+6 is a new, independent quantity that is just as precise as the quantities it was derived from, and which can be manipulated and used in further computations. These functional rules make symbolic arithmetic computation both powerful and flexible. This project aims to identify the functional rules of non-symbolic arithmetic with the ANS. In a series of experiments, four to six year-old children will be asked to solve non-symbolic problems with unknown addends (e.g., 5+__=11). This task requires children to perform arithmetic-like computation, holding two separate ANS representations in mind (e.g. approximately five and approximately 11) and performing a computation over them (e.g. subtracting approximately five from approximately 11) to derive a solution. Each experiment is aimed at examining different components of the functional rules of non-symbolic arithmetic, including the representational structure and precision of the solutions to ANS computations and the extent to which these solutions can be used in further computations. Additional measures of working memory capacity, symbolic math performance, and ANS representational precision are used to elucidate contributions of these cognitive systems to the computational capacity and development of the ANS. Data will be analyzed using a combination of traditional null hypothesis significance testing, Bayes factor analysis, and logistic regression. The project will therefore compliment what is known about the representational structure of the ANS by shedding light on its computational architecture and the development of this architecture in early childhood.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.