Nature and culture of finger counting: diversity and representational effects of an embodied cognitive tool

Studies like the one conducted by Domahs et al. (2010, in Cognition) corroborate that finger counting habits affect how numbers are processed, and legitimize the assumption that this effect is culturally modulated. The degree of cultural diversity in finger counting, however, has been grossly underestimated in the field at large, which, in turn, has restricted research questions and designs. In this paper, we demonstrate that fingers as a tool for counting are not only naturally available, but are also-and crucially so-culturally encoded. To substantiate this, we outline the variability in finger counting and illustrate each of its types with instances from the literature. We argue that the different types of finger counting all constitute distinct representational systems, and we use their properties-dimensionality, dimensional representation, base and sub-base values, extendibility and extent, sign count, and regularity-to devise a typology of such systems. This allows us to explore representational effects, that is, the cognitive implications these properties may have, for instance, for the efficiency of information encoding and representation, ease of learning and mastering the system, or memory retrieval and cognitive load. We then highlight the ambivalent consequences arising from structural inconsistencies between finger counting and other modes of number representation like verbal or notational systems, and we discuss how this informs questions on the evolution and development of counting systems. Based on these analyses, we suggest some directions for future research in the field of embodied cognition that would profit substantially from taking into account the cultural diversity in finger counting.     

Infants recognize counting as numerically relevant

Children do not understand the meanings of count words like “two” and “three” until the preschool years. But even before knowing the meanings of these individual words, might they recognize that counting is “about” the dimension of number? Here in five experiments, we asked whether infants already associate counting with quantities. We measured 14‐ and 18‐month olds’ ability to remember different numbers of hidden objects that either were or were not counted by an experimenter before hiding. As in previous research, we found that infants failed to differentiate four hidden objects from two when the objects were not counted—suggesting an upper limit on the number of individual objects they could represent in working memory. However, infants succeeded when the objects were simply counted aloud before hiding. We found that counting also helped infants differentiate four hidden objects from six (a 2:3 ratio), but not three hidden objects from four (a 3:4 ratio), suggesting that counting helped infants represent the arrays’ approximate cardinalities. Hence counting directs infants’ attention to numerical aspects of the world, showing that they recognize counting as numerically relevant years before acquiring the meanings of number words.     

Mastery of the logic of natural numbers is not the result of mastery of counting: evidence from late counters

To master the natural number system, children must understand both the concepts that number words capture and the counting procedure by which they are applied. These two types of knowledge develop in childhood, but their connection is poorly understood. Here we explore the relationship between the mastery of counting and the mastery of exact numerical equality (one central aspect of natural number) in the Tsimane’, a farming‐foraging group whose children master counting at a delayed age and with higher variability than do children in industrialized societies. By taking advantage of this variation, we can better understand how counting and exact equality relate to each other, while controlling for age and education. We find that the Tsimane’ come to understand exact equality at later and variable ages. This understanding correlates with their mastery of number words and counting, controlling for age and education. However, some children who have mastered counting lack an understanding of exact equality, and some children who have not mastered counting have achieved this understanding. These results suggest that understanding of counting and of natural number concepts are at least partially distinct achievements, and that both draw on inputs and resources whose distribution and availability differ across cultures.     

Is the Chinese number-naming system transparent? Evidence from Chinese-English bilingual children.

Chinese-speaking children have been shown to have an advantage over English-speaking children in a variety of mathematical areas, including counting. One possible explanation for the advantage in counting is that the Chinese number-naming system is relatively transparent, compared to English, in that number names typically are directly indicative of base-10 structure (e.g., 12 is named "ten-two" rather than "twelve"). To determine whether the transparency of the Chinese number-naming system influences counting in bilingual children, we tested 25 Chinese-English bilingual children between the ages of 3 and 5 years, both in English and in Chinese. Children were asked to count as high as they could (abstract counting) and also to count objects in small, medium, and large arrays (object counting). No evidence was found for transparency or for transfer from one language to the other. Instead, relative proficiency in the two languages influenced counting skill. These results are discussed in terms of linguistic and cultural variables that might account for cross-linguistic differences in counting.     

ERP研究反映感数与计数的不同脑机制

对于较小数目的快速准确认知的“感数”现象一直是一个引人兴趣的问题,这一感数过程到底本质是什么,它与计数过程究竟是同一种加工还是分属两种不同类别,对于这个问题多年来一直存在着争论。本实验的目的在于研究感数加工的本质,运用ERP手段来探索其与计数的不同机制;并且以分心物变量为指标,研究在有分心物呈现时,感数与计数过程将会有哪些变化以及其潜在的神经机制。对14名正常青年人记录感数与计数加工过程中的事件相关脑电位（ERP）。刺激图片由位于屏幕中心的靶(白色矩形)或者靶+分心物(白色圆形)组成,靶的数目分为感数（1－3个靶）及计数（4－6）;而分心物的数目有三种水平：零（无分心物）,与靶的数目相同及两倍于靶的数目。被试计算图片上白色矩形的个数并对所得数目进行奇偶判断,最后根据奇偶性用左右手进行按键反应。行为结果表明,感数与计数的反应时在靶数目之间有显著性差异,靶数目相同时,分心物越多,则被试所需要的反应时就越长。ERP测量表明P1波幅随着靶数目的增加而增大,随着分心物数目的增加而增加;N1波潜伏期随着靶数目的增加和分心物数目增加而减小,N1波幅随着靶数目和分心物的增加而增加;P3波幅随着靶数目的增加而减小,在某些记录点具有靶效应和干扰效应。研究结果提示感数加工具有明显的分心物效应,而计数加工则不然,支持感数与计数分属两种不同功能加工过程的观点,感数加工更易受到分心物出现的干扰。     

COUNTING YOUR BLESSINGS: SACRED NUMBERS AND THE STRUCTURE OF REALITY

Abstract. Although numerical systems have been regarded as static models of a symbolic system and treated as mythological behavior, it is postulated that these systems are more profitably analyzed as dynamic models, better understood as ritual behavior. As ritual, numerical systems, limited in number and expressive of rhythmicity, contribute to the biogenetic structuralist's notion of "equilibration" between the central nervous system and the environment.
The relationship between concrete and abstract numeration is also examined, showing that counting behavior, requiring asymmetrical use of the hands, may contribute to understanding the relationships between handedness and brain hemisphericity, as well as enumeration and memorization.     

SYSTEMS THEORY AND RESEARCH IN THE STUDY OF ORGANIZATIONAL COMMUNICATION: THE CORRESPONDENCE PROBLEM

The principle of correspondence specifies that a necessary condition for the acquistion of valid scientific knowledge is that correctly applied analytic techniques be inherently capable of revealing the theoretical properties under examination. The problem raised in the present paper is whether traditional analytic techniques are capable of providing evidence for or against systems hypotheses, no matter how well they are employed. Traditional univariate as well as multivariate and structural equation analytic techniques are examined to determine the extent to which they are capable of revealing two selected properties: feedback and holism. The general problem of correspondence is discussed and recommendations are offered for correspondence analysis for (1) other conceptualizations of systems, (2) other systems properties, and (3) other analytical techniques.     

Nonword naming: further exploration of the pseudohomophone effect in terms of orthographic neighborhood size, graphemic changes, spelling-sound consistency, and reader accuracy.

Five experiments examined nonword pronunciation. As reported by McCann and Besner (1987), accurate, regular pronunciations increased as the number of orthographic neighbors (N) increased. Adults read pseudohomophones (nonwords that sound like a word) more accurately than other nonwords only when the nonwords were low n, shared the consonants with the words on which they were based, and overall accuracy was lower. Children showed a pseudohomophone advantage even when N was high. Adults pronounced nonwords comprised of inconsistent endings (with existing regular and irregular pronunciations) in an irregular fashion when this resulted in a word; this applied to relatively high-N items.     

中国幼儿数数过程信息加工活动研究

探讨中国幼儿数数发展特点。实验1探讨3－5岁幼儿数数的终结点分布状况,结果表明,最大的密集终结点是“19”。实验2进一步考察幼儿在其数数范围内与范围外的起点进行数数的成绩,结果表明,数数范围为11－19的幼儿就开始有部分人可以在其数数范围外的起点进行数数,此表明他们开始应用了数列规则。据此可以认为,中国幼儿的数数活动同样包括联结学习与数列规则学习两种信息加工活动,但中国幼儿数列规则的学习活动在“11－19”数数过程开始。     

The relationship between non‐verbal systems of number and counting development: a neural signatures approach

Two non‐verbal cognitive systems, an approximate number system (ANS) for extracting the numerosity of a set and a parallel individuation (PI) system for distinguishing between individual items, are hypothesized to be foundational to symbolic number and mathematics abilities. However, the exact role of each remains unclear and highly debated. Here we used an individual differences approach to test for a relationship between the spontaneously evoked brain signatures (using event‐related potentials) of PI and the ANS and initial development of symbolic number concepts in preschool children as displayed by counting. We observed that individual differences in the neural signatures of the PI system, but not the ANS, explained a unique portion of variance in counting proficiency after extensively controlling for general cognitive factors. These results suggest that differences in early attentional processing of objects between children are related to higher‐level symbolic number concept development.     

Embodied numerosity: Implicit hand-based representations influence symbolic number processing across cultures

In recent years, a strong functional relationship between finger counting and number processing has been suggested. Developmental studies have shown specific effects of the structure of the individual finger counting system on arithmetic abilities. Moreover, the orientation of the mental quantity representation (“number line”) seems to be influenced by finger counting habits. However, it is unclear whether the structure of finger counting systems still influences symbolic number processing in educated adults.In the present transcultural study, we pursued this question by examining finger-based sub-base-five effects in an Arabic number comparison task with three different groups of participants (German deaf signers, German and Chinese hearing adults). We observed sub-base-five effects in all groups, but particularly so for both German groups who use an explicit sub-base-five system in their finger counting habits. It is concluded that bodily experiences – namely finger counting – influence the structure of the abstract mental number representations even in adults. Thus, the present findings support the general idea that even seemingly abstract cognition may at least partially be rooted in our bodily experiences.     

Why preschoolers are reluctant to count spontaneously

Two experiments are reported which examine children's counting and its role in reasoning about the relative numerosity of two arrays. In the first experiment, children's number judgements were compared under different conditions designed to evaluate the importance of three different cues to number—length and density of rows, small number perception and counting. Children were found to count very rarely unless specifically asked to do so. Experiment 2 investigated some possible reasons why children who count readily in some situations are reluctant to count spontaneously in this number judgement task. Spontaneous counting in 4-year-olds increased in one condition only: when they were given feedback as to the correctness of their previous judgements. This feedback showed that basing judgements on number as counted was always correct whereas length and density judgements were only sometimes correct. Preschoolers' preference for length as a cue to number may therefore be due to their belief that length is a more reliable cue than counting, rather than to their ignorance about the link between counting and numerical reasoning.     

感数——它能告诉我们什么?

研究者在列举任务中发现,被试对3个以内项目的报告既快又准确（一般称之为“感数”）,而对3个以上项目的报告既慢又容易出错误（“计数”）,由此他们提出感数和计数属于两种不同性质的加工过程,一系列行为数据的反应时和正确率指标支持了这一假设。此外,最近的脑成像、电生理研究还发现,感数和计数在对注意的需求上同样存在着分离——感数无需注意,只有计数过程才需注意的参与。在此基础上,研究者从不同角度提出了一些解释感数现象的理论     

Preschool children master the logic of number word meanings

Although children take over a year to learn the meanings of the first three number words, they eventually master the logic of counting and the meanings of all the words in their count list. Here, we ask whether children's knowledge applies to number words beyond those they have mastered: Does a child who can only count to 20 infer that number words above 'twenty' refer to exact cardinal values? Three experiments provide evidence for this understanding in preschool children. Before beginning formal education or gaining counting skill, children possess a productive symbolic system for representing number.     

汉语母语者英语动词三种屈折变化形式加工机制的差异

采用ERP技术考察了汉语母语者,英语动词三种屈折变化形式加工机制的差异。结果发现,英语动词进行式的屈折变化形式- ing的错误使用诱发了P600成分;完成式的屈折变化形式- ed的错误使用诱发了N400成分;完成式中没有明显屈折变化标识的不规则词的错误使用没有诱发典型的ERP成分。研究结果表明：对于缺乏英语使用环境、母语为汉语的晚期英语学习者来说,屈折变化形式- ing,完成式规则动词的变化形式- ed和没有明显形态变化标识的不规则动词的变化形式具有不同的加工机制,前者可能处于由陈述性记忆系统向程序性记忆系统加工的转变过程中,而后两者更多地由陈述性系统进行加工。这种加工机制的差异可能与动词屈折变化规则的复杂程度有关。     

An association between understanding cardinality and analog magnitude representations in preschoolers

The preschool years are a time of great advances in children’s numerical thinking, most notably as they master verbal counting. The present research assessed the relation between analog magnitude representations and cardinal number knowledge in preschool-aged children to ask two questions: (1) Is there a relationship between acuity in the analog magnitude system and cardinality proficiency? (2) Can evidence of the analog magnitude system be found within mappings of number words children have not successfully mastered? To address the first question, Study 1 asked three- to five-year-old children to discriminate side-by-side dot arrays with varying differences in numerical ratio, as well as to complete an assessment of cardinality. Consistent with the analog magnitude system, children became less accurate at discriminating dot arrays as the ratio between the two numbers approached one. Further, contrary to prior work with preschoolers, a significant correlation was found between cardinal number knowledge and non-symbolic numerical discrimination. Study 2 aimed to look for evidence of the analog magnitude system in mappings to the words in preschoolers’ verbal counting list. Based on a modified give-a-number task ( [Wynn, 1990] and [Wynn, 1992] ), three- to five-year-old children were asked to give quantities between 1 and 10 as many times as possible in order to assess analog magnitude variability within their developing cardinality understanding. In this task, even children who have not yet induced the cardinality principle showed signs of analog representations in their understanding of the verbal count list. Implications for the contribution of analog magnitude representations towards mastery of the verbal count list are discussed in light of the present work.     

The Power of 2: How an Apparently Irregular Numeration System Facilitates Mental Arithmetic

Mangarevan traditionally contained two numeration systems: a general one, which was highly regular, decimal, and extraordinarily extensive; and a specific one, which was restricted to specific objects, based on diverging counting units, and interspersed with binary steps. While most of these characteristics are shared by numeration systems in related languages in Oceania, the binary steps are unique. To account for these characteristics, this article draws on—and tries to integrate—insights from anthropology, archeology, linguistics, psychology, and cognitive science more generally. The analysis of mental arithmetic with these systems reveals that both types of systems entailed cognitive advantages and served important functions in the cultural context of their application. How these findings speak to more general questions revolving around the theoretical models and evolutionary trajectory of numerical cognition will be discussed in the 6 .     

The mathematics of sharing / Las matemáticas que realizan los niños al compartir

Abstract

The purpose of sharing is to construct equivalent sets, making it an ideal context for analysing important quantitative concepts such as counting, equivalence and cardinality. Two studies analysed how four- and five-year-olds shared blocks in equal sharing and reciprocity conditions and their number inferences about one set after counting the other. The researcher asked children to share double and single blocks between two characters. They succeeded more in building equivalent shares in an equal sharing than reciprocity condition. Most children who shared correctly also made appropriate number inferences. To examine whether perceptual cues helped children share the blocks, a second study used Canadian $1 and $2 coins. A double block is twice the size of a single, whereas there is no visual cue about the value relation between coins because they are the same size. Unexpectedly, children shared equally well with blocks and coins, and most children made number inferences.     

Evidence for counting in insects

Here we investigate the counting ability in honeybees by training them to receive a food reward after they have passed a specific number of landmarks. The distance to the food reward is varied frequently and randomly, whilst keeping the number of intervening landmarks constant. Thus, the bees cannot identify the food reward in terms of its distance from the hive. We find that bees can count up to four objects, when they are encountered sequentially during flight. Furthermore, bees trained in this way are able count novel objects, which they have never previously encountered, thus demonstrating that they are capable of object-independent counting. A further experiment reveals that the counting ability that the bees display in our experiments is primarily sequential in nature. It appears that bees can navigate to food sources by maintaining a running count of prominent landmarks that are passed en route, provided this number does not exceed four.     

Attaching meaning to the number words: contributions of the object tracking and approximate number systems

Children's understanding of the quantities represented by number words (i.e., cardinality) is a surprisingly protracted but foundational step in their learning of formal mathematics. The development of cardinal knowledge is related to one or two core, inherent systems – the approximate number system (ANS) and the object tracking system (OTS) – but whether these systems act alone, in concert, or antagonistically is debated. Longitudinal assessments of 198 preschool children on OTS, ANS, and cardinality tasks enabled testing of two single‐mechanism (ANS‐only and OTS‐only) and two dual‐mechanism models, controlling for intelligence, executive functions, preliteracy skills, and demographic factors. Measures of both OTS and ANS predicted cardinal knowledge in concert early in the school year, inconsistent with single‐mechanism models. The ANS but not the OTS predicted cardinal knowledge later in the school year as well the acquisition of the cardinal principle, a critical shift in cardinal understanding. The results support a Merge model, whereby both systems initially contribute to children's early mapping of number words to cardinal value, but the role of the OTS diminishes over time while that of the ANS continues to support cardinal knowledge as children come to understand the counting principles.