Foundations of Numerical Cognition: From Perceptual Quantities to Abstract Numbers / 数感发展的基础：从感知数量到抽象数字

第 1 章：数感发展的基础：从感知数量到抽象数字

Chapter 1: Foundations of Numerical Cognition: From Perceptual Quantities to Abstract Numbers

引言：理解数字认知发展的核心问题

How do young children develop the cognitive foundations for understanding numerical concepts, and what early experiences support this development? This question lies at the heart of early mathematics education research and practice. The development of numerical cognition represents one of the most remarkable cognitive achievements of early childhood, yet its mechanisms remain subjects of ongoing scientific investigation. Understanding how children transition from innate, non-verbal number sense to sophisticated symbolic mathematical understanding provides the essential groundwork for designing effective early mathematics curricula and interventions.

理解幼儿如何发展理解数字概念的认知基础，以及哪些早期经验支持这种发展——这一核心问题处于早期数学教育研究与实践的核心位置。数感的发展代表了幼儿期最显著的认知成就之一，然而其机制仍然是持续科学研究的课题。理解儿童如何从先天的、非语言的数感过渡到复杂的符号数学理解，为设计有效的早期数学课程和干预措施提供了必要的基础。

The human capacity for numerical thinking emerges remarkably early in development. Within the first year of life, infants demonstrate sensitivity to changes in numerical quantity, suggesting that some form of numerical processing is present before formal education or even language acquisition. By the age of two or three, children engage spontaneously with numerical concepts during everyday activities—counting fingers, comparing amounts of snacks, or noting how many children are present in a group. These early numerical encounters are not merely coincidental; they form the foundation upon which formal mathematical understanding will eventually be constructed.

人类的数思维能力在发展中出现得非常早。在生命的头一年，婴儿就表现出对数量变化的敏感性，这表明某种形式的数字处理在正规教育甚至语言习得之前就已存在。到两三岁时，儿童在日常活动中自发地与数字概念互动——数手指、比较零食的数量，或注意到一组孩子中有多少人。这些早期的数字接触并非偶然；它们构成了最终构建正式数学理解的基础。

This chapter examines the cognitive mechanisms underlying early numerical development, analyzing both the innate perceptual capacities that children bring to numerical tasks and the learned processes that transform these capacities into abstract numerical understanding. The analysis proceeds from basic perceptual quantity discrimination through more complex symbolic number processing, examining how these different levels of numerical cognition interact and build upon one another. Throughout this exploration, we maintain a dual focus: understanding the theoretical foundations of numerical cognition while simultaneously considering the practical implications for educators and caregivers who seek to support young children's mathematical development.

本章考察了早期数感发展的认知机制，分析了儿童在进行数字任务时所具备的先天的感知能力以及将这些能力转化为抽象数字理解的后天学习过程。分析从基本的感觉数量辨别开始，逐步深入到更复杂的符号数字处理，探讨这些不同层次的数感如何相互作用和相互构建。在整个探索过程中，我们保持双重关注：既要理解数感的理论基础，同时考虑教育工作者和照护者在支持幼儿数学发展方面的实践启示。

第一节：近似数量系统及其在前语言期数认知中的作用

Section 1: The Approximate Number System and Its Role in Pre-verbal Numerical Cognition

1.1 ANS的理论基础与核心机制

The Approximate Number System (ANS) represents a foundational cognitive capacity that supports non-verbal numerical judgments. First systematically described in the cognitive science literature of the 1980s and 1990s, the ANS refers to an inherent ability to estimate and compare numerical quantities without reliance on symbolic representations or verbal counting. This system operates continuously throughout the lifespan, providing humans with an intuitive sense of numerical magnitude that guides behavior in numerous contexts.

近似数量系统（ANS）是支持非语言数字判断的基础认知能力。该系统最早在20世纪80年代和90年代的认知科学文献中被系统描述，ANS指的是一种无需依赖符号表示或口头计数就能估计和比较数量大小的内在能力。这个系统在人的一生中持续运作，为人类提供了一种直觉性的数量感，指导着各种情境中的行为。

The neural substrate of the ANS has been identified through neuroimaging studies revealing activation in the intraparietal sulcus (IPS) and surrounding parietal regions during numerical magnitude tasks. Research conducted by Dehaene and colleagues established that these parietal regions form part of a "number sense circuit" that responds to numerical information regardless of the format in which it is presented—whether as arrays of dots, sequences of tones, or collections of objects. This neural architecture suggests that the ANS is not merely a behavioral capacity but reflects specialized neural organization for numerical processing.

通过神经影像学研究，ANS的神经基质已被确定——在数字量任务期间，显示出顶内沟（IPS）及周围顶叶区域的激活。Dehaene及其同事进行的研究证实，这些顶叶区域形成了一个"数感回路"的一部分，无论数字信息以何种形式呈现，该回路都会对其做出响应——无论是点阵阵列、音调序列还是物体集合。这种神经架构表明，ANS不仅仅是一种行为能力，而是反映了用于数字处理的专门神经组织。

The signature characteristic of ANS representations is their inherent imprecision. Unlike exact symbolic numbers, ANS representations exhibit a "ratio-dependent" discrimination threshold, meaning that two quantities can be reliably distinguished only when they differ by a sufficient ratio. Psychophysical research has formalized this relationship through the Weber fraction (w), which describes the proportional change in magnitude required for reliable discrimination. For adult humans, the Weber fraction for non-symbolic numerical comparisons typically ranges from approximately 0.10 to 0.15, indicating that adults can reliably distinguish 10 groups of 50 dots from 11 groups of 50 dots under optimal conditions.

ANS表征的显著特征是其固有的不精确性。与精确的符号数字不同，ANS表征表现出"比率依赖"的辨别阈值，这意味着只有当两个数量之间有足够大的比率差异时才能可靠地区分它们。心理物理学研究通过韦伯分数（w）正式描述了这种关系，韦伯分数描述了可靠辨别所需的成比例变化量。对于成人来说，非符号数字比较的韦伯分数通常在0.10到0.15之间，这表明在最佳条件下，成人可以可靠地区分10组50个点与11组50个点。

$$P(\text{correct}) = \Phi\left(\frac{|N_1 - N_2|}{w \cdot \sqrt{N_1^2 + N_2^2}}\right)$$

The equation above, derived from the popular neuronal accumulator model, captures how discrimination accuracy depends on the ratio between numerical magnitudes, with larger ratios producing higher accuracy and smaller ratios producing performance near chance levels. This ratio-dependent pattern distinguishes ANS processing from exact symbolic number processing, which permits precise discrimination regardless of numerical ratio.

上式来自流行的神经元累积器模型，捕捉了辨别准确率如何取决于数量之间的比率，较大的比率产生较高的准确率，较小的比率则产生接近随机水平的绩效。这种比率依赖模式将ANS处理与精确符号数字处理区分开来，后者允许无论数字比率如何都能进行精确辨别。

1.2 婴儿期近似数量判断的证据

The developmental origins of the ANS can be traced to infancy, with converging evidence from preferential looking paradigms, habituation procedures, and looking-time measures demonstrating numerical sensitivity within the first year of life. landmark studies by Karen Wynn and colleagues in the early 1990s established that five-month-old infants expect that 1 + 1 should equal 2, demonstrating surprise (measured through increased looking time) when a puppet is placed behind a screen and only one puppet emerges rather than two. These findings suggested that infants possess some form of numerical representation that can be manipulated through simple arithmetic operations.

ANS的发展起源可追溯到婴儿期，来自偏爱注视范式、习惯化程序和注视时间测量的多方面证据表明，在生命的第一年内就存在数字敏感性。Karen Wynn及其同事在1990年代初期进行的开创性研究证实，五个月大的婴儿期望1+1等于2，当玩偶被放到屏幕后面而只出现一个玩偶而非两个时，他们会表现出惊讶（通过增加注视时间来测量）。这些发现表明，婴儿具有某种形式的数字表征，可以通过简单的算术运算进行操作。

Subsequent research extended these findings by examining spontaneous numerical discrimination in pre-verbal infants. Studies using arrays of dots, arrays of sounds, or arrays of objects demonstrated that infants as young as six months can discriminate quantities when the ratio between them is sufficiently large (typically greater than 2:1). For instance, infants look reliably longer at displays showing 16 objects alternating with 8 objects compared to 16 objects alternating with 12 objects, indicating sensitivity to numerical ratios rather than continuous physical properties alone.

后续研究通过检查前语言期婴儿的自发数字辨别来扩展这些发现。使用点阵、声音阵列或物体阵列的研究表明，六个月大的婴儿在数量之间的比率足够大（通常大于2:1）时可以辨别数量。例如，婴儿在观看显示16个物体与8个物体交替的画面时，注视时间明显长于16个物体与12个物体交替的画面，这表明他们对数字比率敏感，而不仅仅是对连续物理属性敏感。

However, the interpretation of these findings has been subject to scholarly debate. Some researchers have argued that performance on habituation-dishabituation tasks may reflect sensitivity to cumulative surface area or other continuous quantities rather than genuine numerical representation. Rigorous control studies by Susan Hespos and colleagues addressed this concern by using novel paradig