视觉形状知觉在近似数量系统和计算流畅性关系中的作用

已有大量研究揭示了近似数量系统与计算流畅性的相关关系, 但缺少对二者关系原因的系统检验与论证。视觉形状知觉假设有别于传统的数量领域特异性解释, 认为对形状的快速知觉是近似数量系统与计算流畅性的共同认知机制, 即视觉形状的快速知觉能力可以解释二者之间的相关关系。近似数量系统和计算流畅性在加工过程中依赖对形状的快速知觉, 二者在加工过程中都涉及了复杂视觉刺激的快速处理。视觉形状知觉假设得到了一系列研究结果的支持, 但局限在视觉形状知觉与二者关系的探讨上, 视觉形状知觉在二者关系中作用的加工机制仍不清楚。未来研究需要结合多种研究方法和技术, 多角度深入探讨视觉形状知觉在二者关系中作用的认知与脑机制, 并将研究结果应用于数学课堂教学和计算困难的干预中。     

个体近似数量系统与其数学能力之间的关系：发展研究的证据

近似数量系统在个体数学能力的发展中起着重要的作用, 二者之间的关系受到年龄因素的影响。主要表现为, 随着年龄的增长, 相关程度逐渐减弱, 二者之间关系的作用机制可能由基数知识中介转变为多种中介变量的共同作用。未来可采用更严格的实验设计和多种研究方法考察各年龄段儿童近似数量系统与不同数学能力之间关系的发展趋势、因果方向、关键转折点和潜在机制, 以更好地理解近似数量系统在个体数学能力发展中所起的作用。     

近似数量系统与数学能力的关系：一项元分析

近似数量系统(ANS)与数学能力的关系是国际认知与心理研究的经典问题，大量研究进行了探索，但仍存在争议。为了揭示二者关系及其方向和调节机制，本研究对来自55项研究、59个独立样本、12661名被试的242个效应值进行了元分析。结果发现，ANS与数学能力之间存在稳定的中相关，二者关系是双向的，受ANS测量指标、数学能力内容和年龄的调节。研究结果为深入的纵向研究、实验研究和干预研究奠定了基础，为数学教育提供了实证依据和视角。     

人类的近似数量系统

近似数量系统(Approximate Number System, ANS)指个体在不需要依赖于计算和数量符号的情况下, 对一组数量进行近似表征的系统。通过总结近十年来研究者们在ANS的遗传和神经基础、干预训练等方面取得的新进展, 指出未来应综合运用各种认知神经科学研究手段立足于ANS的基因和脑生理基础研究, 进一步揭示ANS的本质和内在发生发展机制, 并将有关研究发现运用到教育教学中, 对数学困难儿童进行干预, 以提高其数学能力和适应社会的能力。     

数学焦虑影响儿童的数量表征:认知抑制的调节作用

为了考察数学焦虑对儿童数量表征表现的可能影响及认知抑制的潜在调节作用,选取70名小学三年级儿童（高焦虑组36人,低焦虑组34人）为被试,在对抑制条件进行操控的情况下,要求其完成符号、非符号数量表征任务。结果发现,被试在两种数量表征任务中均出现距离效应,与符号数量比较任务相比,高焦虑组在非符号数量比较任务中的正确率显著低于低焦虑组,且高焦虑组表现出了更强的距离效应。鉴于非符号数量比较任务更能反映出个体近似数量系统（ANS）的敏锐性,上述结果意味着高数学焦虑儿童的数量表征更不精确,其在相对复杂问题上较差的表现或许源于基本数量能力缺陷。本研究还发现认知抑制能够调节数学焦虑对个体非符号数量表征的影响,抑制条件下高低焦虑组儿童在正确率指标上的差异大于非抑制条件,抑制条件的设置提高了个体对工作记忆资源的需求,此时焦虑情绪对认知资源的消耗会造成任务所需资源的不足,进而削弱高焦虑个体的认知效用。     

小学生近似数量系统敏锐度的发展趋势及其与数学能力的关系：抑制控制的中介作用

采用测量法和问卷法考察了172名小学生近似数量系统敏锐度的发展,以及抑制控制在近似数量系统敏锐度与数学能力关系中的中介效应。结果表明：(1)随着年龄增长,小学生的近似数量系统敏锐度逐渐提高;(2)近似数量系统敏锐度(负相关序列中的韦伯系数)和抑制控制均能显著正向预测小学生的数学能力;(3)抑制控制在小学生近似数量系统敏锐度(负相关序列中的韦伯系数)与数学能力的关系中起部分中介作用。     

非正式学习环境中幼儿的自发数量聚焦

大量从“能力”角度来探讨个体早期数学发展的研究仅专注于儿童早期的数学知识和技能, 而忽略了可能与儿童早期数学能力相关的方面。近年来, 数学认知发展研究者逐渐开始关注自发数量聚焦(Spontaneous focusing on numerosity, SFON), 即个体对环境中数量相关信息自发注意的认知倾向。SFON被视为其他数学聚焦倾向概念的基础, 但与自发数字聚焦(SAN)、自发阿拉伯数字聚焦(SFONS)和自发数量关系聚焦(SFOR)之间的联系仍需进一步明晰。作为数学表现的领域特异预测因子, 个体SFON的发展轨迹和作用机制是当前研究的重点所在。未来可从数学动机、数学焦虑等角度进一步探讨SFON的作用机制, 改良与创新SFON测量工具, 同时需深入推进家庭和学校场合的SFON干预研究。     

核心知识系统及其对相关研究的启示

核心知识系统出现于人类个体发展和种系发展的早期,在人类复杂认知能力的发生发展中起着建构模块的作用。该文以表征物体和数量关系的两类核心知识系统的性质及二者在儿童的“数”概念形成和成人的数学思维中所起的作用为例,综述了以灵长类动物、婴儿、儿童和成人为被试的研究证据,并进而提出了核心知识系统给相关研究带来的若干启示。     

数学焦虑个体近似数量加工的神经机制:一项EEG研究

近似数量加工是对大数目物体数量在不依赖逐个数数前提下的估计。行为学研究提示高数学焦虑人群近似数量加工能力下降, 但神经机制未明。本研究探讨高数学焦虑个体近似数量加工的神经机制, 比较高低数学焦虑脑电活动的差异:(1)行为上无显著组间差异; (2)高数学焦虑组的P2p成分波幅增加; (3) δ频段ERS及β频段ERD无显著数量比例效应, 而低数学焦虑组在上述指标的数量比例效应显著。本研究为高数学焦虑人群近似数量加工能力下降提供了电生理学的证据。     

语言与数量认知关系的新认识

数量认知研究近年有长足发展。文章从新近提出的独立于语言的两个数量表征核心系统,语言与精确数量运算,语言与算术事实的储存,语言对儿童早期数概念发展的影响,语言与数量认知关系的最新脑科学证据,以及语言在数量认知模型中的角色等方面,介绍和评述了人类存在依赖和不依赖语言的两级数量能力的新认识。对于是否还存在其它不依赖语言的理解数量的系统,以及这些非语言数量表征系统的认知机制,文章认为有待进一步研究     

Preschool acuity of the approximate number system correlates with school math ability

Previous research shows a correlation between individual differences in people's school math abilities and the accuracy with which they rapidly and nonverbally approximate how many items are in a scene. This finding is surprising because the Approximate Number System (ANS) underlying numerical estimation is shared with infants and with non-human animals who never acquire formal mathematics. However, it remains unclear whether the link between individual differences in math ability and the ANS depends on formal mathematics instruction. Earlier studies demonstrating this link tested participants only after they had received many years of mathematics education, or assessed participants' ANS acuity using tasks that required additional symbolic or arithmetic processing similar to that required in standardized math tests. To ask whether the ANS and math ability are linked early in life, we measured the ANS acuity of 200 3- to 5-year-old children using a task that did not also require symbol use or arithmetic calculation. We also measured children's math ability and vocabulary size prior to the onset of formal math instruction. We found that children's ANS acuity correlated with their math ability, even when age and verbal skills were controlled for. These findings provide evidence for a relationship between the primitive sense of number and math ability starting early in life.     

Bidirectional,Longitudinal Associations Between Math Ability and Approximate Number System Precision in Childhood

Research suggests that individual differences in math abilities correlate with approximate representations of quantity that are supported by a primitive Approximate Number System (ANS). However, relatively little research has addressed the direction of this association in early childhood. Here we examined the development of the ANS and math ability longitudinally in 3- to 5-year-old children. Children were observed at three time points roughly six months apart; they completed a nonsymbolic numerical comparison task that measured ANS precision and a standardized math assessment. A series of cross-lagged panel models was then estimated to explore the associations between ANS precision and math ability over time. Bidirectional associations between ANS precision and math ability emerged: Early ANS precision was related to children’s later math skills, and early math ability also significantly predicted children’s later ANS precision. Evidence for mutual enhancement over time between the ANS and symbolic math ability adds to our growing understanding of the ANS and how the ANS and math knowledge interact.     

Number sense at 12 months predicts 4-year-olds’ maths skills

Preverbal infants spontaneously represent the number of objects in collections. Is this ‘sense of number’ (also referred to as Approximate Number System, ANS) part of the cognitive foundations of mathematical skills? Multiple studies reported a correlation between the ANS and mathematical achievement in children. However, some have suggested that such correlation might be mediated by general-purpose inhibitory skills. We addressed the question using a longitudinal approach: we tested the ANS of 60 12 months old infants and, when they were 4 years old (final N = 40), their symbolic math achievement as well as general intelligence and inhibitory skills. Results showed that the ANS at 12 months is a specific predictor of later maths skills independent from general intelligence or inhibitory skills. The correlation between ANS and maths persists when both abilities are measured at four years. These results confirm that the ANS has an early, specific and longstanding relation with mathematical abilities in childhood.

Research Highlights

• In the literature there is a lively debate about the correlation between the ANS and maths skills.
• We longitudinally tested a sample of 60 preverbal infants at 12 months and rested them at 4 years (final sample of 40 infants).
• The ANS tested at 12 months predicted later symbolic mathematical skills at 4 years, even when controlling for inhibition, general intelligence and perceptual skills.
• The ANS tested at 4 years remained linked with symbolic maths skills, confirming this early and longstanding relation in childhood.

     

Numerical ordering ability mediates the relation between number-sense and arithmetic competence

What predicts human mathematical competence? While detailed models of number representation in the brain have been developed, it remains to be seen exactly how basic number representations link to higher math abilities. We propose that representation of ordinal associations between numerical symbols is one important factor that underpins this link. We show that individual variability in symbolic number-ordering ability strongly predicts performance on complex mental-arithmetic tasks even when controlling for several competing factors, including approximate number acuity. Crucially, symbolic number-ordering ability fully mediates the previously reported relation between approximate number acuity and more complex mathematical skills, suggesting that symbolic number-ordering may be a stepping stone from approximate number representation to mathematical competence. These results are important for understanding how evolution has interacted with culture to generate complex representations of abstract numerical relationships. Moreover, the finding that symbolic number-ordering ability links approximate number acuity and complex math skills carries implications for designing math-education curricula and identifying reliable markers of math performance during schooling.     

Language counts: Early language mediates the relationship between parent education and children's math ability

Children's early math skills have been hailed as a powerful predictor of academic success. Disparities in socioeconomic context, however, also have dramatic consequences on children's learning. It is therefore critical to investigate both of these distinct contributors in order to better understand the early foundations of children's academic outcomes. This study tests an integrated model of children's developing math ability so as to (1) identify the specific skills and abilities most clearly linked to early math achievement and (2) measure the influence of children's socioeconomic context on each of these skills. We first evaluated the early vocabulary, number word knowledge (knower level), and Approximate Number System (ANS) acuity of a diverse group of preschoolers. Then, approximately 1 year later as they entered Kindergarten, we administered a test of early math achievement. We find that children's early language (general vocabulary and number word knowledge) fully mediates the relationship between parent education and math ability. Additionally, number word knowledge mediates the relationship between ANS acuity and early math. We argue that increased focus on number word knowledge, as well as general vocabulary, may help to minimize disparities in math ability as children enter kindergarten. We also highlight the role of parent education on children's learning and note that this may be an important locus for intervention.     

Intergenerational associations in numerical approximation and mathematical abilities

Although growing evidence suggests a link between children's math skills and their ability to estimate numerical quantities using the approximate number system (ANS), little is known about the sources underlying individual differences in ANS acuity and their relation with specific mathematical skills. To examine the role of intergenerational transmission of these abilities from parents to children, the current study assessed the ANS acuities and math abilities of 54 children (5–8 years old) and their parents, as well as parents' expectations about children's math skills. Children's ANS acuity positively correlated with their parents' ANS acuity, and children's math abilities were predicted by unique combinations of parents' ANS acuity and math ability depending on the specific math skill in question. These findings provide the first evidence of intergenerational transmission of an unlearned, non‐verbal numerical competence and are an important step toward understanding the multifaceted parental influences on children's math abilities.     

Children's expectations about training the approximate number system

Humans possess a developmentally precocious and evolutionarily ancient approximate number system (ANS) whose sensitivity correlates with uniquely human symbolic arithmetic skills. Recent studies suggest that ANS training improves symbolic arithmetic, but such studies may engender performance expectations in their participants that in turn produce the improvement. Here, we assessed 6‐ to 8‐year‐old children's expectations about the effects of numerical and non‐numerical magnitude training, as well as states of satiety and restfulness, in the context of a study linking children's ANS practice to their improved symbolic arithmetic. We found that children did not expect gains in symbolic arithmetic after exercising the ANS, although they did expect gains in ANS acuity after training on any magnitude task. Moreover, children expected gains in symbolic arithmetic after a good night's sleep and their favourite breakfast. Thus, children's improved symbolic arithmetic after ANS training cannot be explained by their expectations about that training.     

Differences in the acuity of the Approximate Number System in adults: The effect of mathematical ability

It is largely admitted that processing numerosity relies on an innate Approximate Number System (ANS), and recent research consistently observed a relationship between ANS acuity and mathematical ability in childhood. However, studies assessing this relationship in adults led to contradictory results. In this study, adults with different levels of mathematical expertise performed two tasks on the same pairs of dot collections, based either on numerosity comparison or on cumulative area comparison. Number of dots and cumulative area were congruent in half of the stimuli, and incongruent in the other half. The results showed that adults with higher mathematical ability obtained lower Weber fractions in the numerical condition than participants with lower mathematical ability. Further, adults with lower mathematical ability were more affected by the interference of the continuous dimension in the numerical comparison task, whereas conversely higher-expertise adults showed stronger interference of the numerical dimension in the continuous comparison task. Finally, ANS acuity correlated with arithmetic performance. Taken together, the data suggest that individual differences in ANS acuity subsist in adulthood, and that they are related to mathematical ability.