Using cardinality to compare quantities: the role of social-cognitive conflict in early numeracy

A key question in early number development is how 4- and 5-year-olds learn the roles that counting and cardinal numbers play when comparing quantities. Children who wrongly used length to identify numerosity were assigned to five experimental groups and trained to judge whether a puppet--who sometimes miscounted--created equivalent sets. Over three training sessions, children who were asked to compare sets after they were counted learned to base their judgments on cardinal numbers when the puppet counted accurately by being given feedback. However, only the groups who were also asked to explain either their own or the experimenter's reasoning made progress in identifying the puppet's miscounts. This ability to recognize the importance of counting accuracy for quantitative comparisons predicted whether children would spontaneously count to compare sets on a post-test. The importance of asking children to identify miscounts is discussed alongside the social factors that influence children's recognition of the relationship between procedural counting, cardinality and relative number.     

幼儿在集合比较中运用数数的实验研究

本研究中运用了两个实验探讨数数干预和测查条件对儿童在集合比较中运用数数的影响。干预对3岁儿童(M=3：9)没有影响。在平均年龄为4岁4个月时．干预组儿童比控制组儿童更倾向于用数数比较集合．自然组儿童也比传统组更倾向于用数数。许多4岁儿童在无干预时不用数数可能是因为,1)不知数数比视觉性比较更有效,或2)他们在集合比较中的数数极易受测查情景因素的影响。儿童在集合比较中的数数运用与他们的数数水平密切关联。     

Small Subitizing Range in People with Williams syndrome

Recent evidence suggests that the rapid apprehension of small numbers of objects-- often called subitizing-- engages a system which allows representation of up to 4 objects but is distinct from other aspects of numerical processing. We examined subitizing by studying people with Williams syndrome (WS), a genetic deficit characterized by severe visuospatial impairments, and normally developing children (4-6.5 years old). In Experiment 1, participants first explicitly counted displays of 1 to 8 squares that appeared for 5 s and reported "how many". They then reported "how many" for the same displays shown for 250 ms, a duration too brief to allow explicit counting, but sufficient for subitizing. All groups were highly accurate up to 8 objects when they explicitly counted. With the brief duration, people with WS showed almost perfect accuracy up to a limit of 3 objects, comparable to 4 year-olds but fewer than either 5 or 6.5 year-old children. In Experiment 2, participants were asked to report "how many" for displays that were presented for an unlimited duration, as rapidly as they could while remaining accurate. Individuals with WS responded as rapidly as 6.5 year-olds, and more rapidly than 4 year-olds. However, their accuracy was as in Experiment 1, comparable to 4 year-olds, and lower than older children. These results are consistent with previous results indicating that people with WS can simultaneously represent multiple objects, but that they have a smaller capacity than older children, on par with 4 year-olds. This pattern is discussed in the context of normal and abnormal development of visuospatial skills, in particular those linked to the representation of numerosity.     

Nonverbal estimation during numerosity judgements by adult humans

On an automated task, humans selected the larger of two sets of items, each created through the one-by-one addition of items. Participants repeated the alphabet out loud during trials so that they could not count the items. This manipulation disrupted counting without producing major effects on other cognitive capacities such as memory or attention, and performance of this experimental group was poorer than that of participants who counted the items. In Experiment 2, the size of individual items was varied, and performance remained stable when the larger numerical set contained a smaller total amount than the smaller numerical set (i.e., participants used numerical rather than nonnumerical quantity cues in making judgements). In Experiment 3, reports of the number of items in a single set showed scalar variability as accuracy decreased, and variability in responses increased with increases in true set size. These data indicate a mechanism for the approximate representation of numerosity in adult humans that might be shared with nonhuman animals.     

Is there a Relationship between Task Demand and Storage Space in Tests of Working Memory Capacity?

This paper considers working memory capacity, critically examining the hypothesis that counting span (the ability to count arrays of objects and store count totals) reflects a trade-off in resources available for processing and short-term storage. Previous evidence interpreted as favouring this hypothesis has confounded task difficulty with counting time. Experiment 1 validated a manipulation of the attentional demands of counting in which target objects were differentiated from non-targets by either a single feature (colour) or a feature conjunction (a combination of line orientations). The results confirmed that the two presentations involved qualitatively different attentional loads. Experiment 2 used these displays to compare counting span for children aged 6 to 11, both with and without an adjustment of target numerosity to control for differences in processing time. At all ages, span was lower when counting took longer, but there was no difference between feature and conjunction arrays once counting time was accounted for. These results argue against a resource trade-off interpretation of counting span. Rather, they support a hypothesis of resource-switching among children, implying that counting span acts as a measure of time-based forgetting.     

Nonverbal estimation during numerosity judgements by adult humans

On an automated task, humans selected the larger of two sets of items, each created through the one-by-one addition of items. Participants repeated the alphabet out loud during trials so that they could not count the items. This manipulation disrupted counting without producing major effects on other cognitive capacities such as memory or attention, and performance of this experimental group was poorer than that of participants who counted the items. In Experiment 2, the size of individual items was varied, and performance remained stable when the larger numerical set contained a smaller total amount than the smaller numerical set (i.e., participants used numerical rather than nonnumerical quantity cues in making judgements). In Experiment 3, reports of the number of items in a single set showed scalar variability as accuracy decreased, and variability in responses increased with increases in true set size. These data indicate a mechanism for the approximate representation of numerosity in adult humans that might be shared with nonhuman animals.     

Estimating and operating on discrete quantities in orangutans (Pongo pygmaeus)

This study investigated the ability of 3 male orangutans (Pongo pygmaeus; 1 subadult, 2 adults) to estimate, compare, and operate on 2 sets of small quantities (1-6 cereal bits). Experiment 1 investigated the orangutans' ability to choose the larger of 2 quantities when they were presented successively as opposed to simultaneously, thus being perceptually unavailable at the time of choice. Experiment 2 investigated the orangutans' ability to select the larger quantity after the original quantities were augmented or reduced. Orangutans were capable of selecting the larger of 2 quantities in Experiment 1. There was also some evidence from Experiment 2, albeit weaker, that orangutans may mentally combine quantities (but not dissociate) to obtain the larger of 2 quantities. This study suggests that orangutans use a representational mechanism (especially when comparing quantities) to select the larger of 2 sets of items.     

Young Natural-Number Arithmeticians

ABSTRACT— When preschoolers count to check their arithmetic predictions, their counts are better than when they simply count a set of items on count-only tasks. This is so even for 2 1/2- and 3-year-olds dealing with small values. Such results lend support to the view that learning about verbal counting benefits from a nonverbal count-arithmetic system and challenge theories that place understanding of verbal counting at 4 1/2 or 5 years. That preschoolers readily engage in predicting-and-checking number tasks has implications for educational programs.     

Toddler subtraction with large sets: further evidence for an analog-magnitude representation of number

Two experiments were conducted to test the hypothesis that toddlers have access to an analog-magnitude number representation that supports numerical reasoning about relatively large numbers. Three-year-olds were presented with subtraction problems in which initial set size and proportions subtracted were systematically varied. Two sets of cookies were presented and then covered. The experimenter visibly subtracted cookies from the hidden sets, and the children were asked to choose which of the resulting sets had more. In Experiment 1, performance was above chance when high proportions of objects (3 versus 6) were subtracted from large sets (of 9) and for the subset of older participants (older than 3 years, 5 months; n = 15), performance was also above chance when high proportions (10 versus 20) were subtracted from the very large sets (of 30). In Experiment 2, which was conducted exclusively with older 3-year-olds and incorporated an important methodological control, the pattern of results for the subtraction tasks was replicated. In both experiments, success on the tasks was not related to counting ability. The results of these experiments support the hypothesis that young children have access to an analog-magnitude system for representing large approximate quantities, as performance on these subtraction tasks showed a Weber's Law signature, and was independent of conventional number knowledge.     

Rhesus monkeys (Macaca mulatta) enumerate large and small sequentially presented sets of items using analog numerical representations

Two rhesus monkeys selected the larger of two sequentially presented sets of items on a computer monitor. In Experiment 1, performance was related to the ratio of set sizes, and the monkeys discriminated between sets with up to 10 items. Performance was not disrupted when 1 set had fewer than 4 items and 1 set had more than 4 items, a critical trial type for differentiating object file and analog models of numerical representation. Experiment 2 controlled the interitem rate of presentation. Experiment 3 included some trials on which number and amount (visual surface area) offered conflicting cues. Experiment 4 varied the total duration of set presentation and the duration of item visibility. In all of the experiments, performance remained high, although total set presentation duration also acted as a partial cue for the monkeys. Overall, the data indicated that rhesus monkeys estimate the approximate number of items in sequentially presented sets and that they are not relying solely on nonnumerical cues such as rate, duration, or cumulative amount.     

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.     

Summation and numerousness judgments of sequentially presented sets of items by chimpanzees (Pan troglodytes).

Summation and numerousness judgments by 2 chimpanzees (Pan troglodytes) were investigated when 2 quantities of M&Ms were presented sequentially, and the quantities were never viewed in their totality. Each M&M was visible only before placement in 1 of 2 cups. In Experiment 1, sets of 1 to 9 M&Ms were presented. In Experiment 2, the quantities in each cup were presented as 2 smaller sets (e.g., 2 + 2 vs. 4 + 1). In Experiment 3, the quantities were presented as 3 smaller sets (e.g., 2 + 2 + 3 vs. 3 + 4 + 1). In Experiment 4, an M&M was removed from 1 set before the chimpanzees' selection. In Experiments 1 and 2, the chimpanzees selected the larger quantity on significantly more trials than would be predicted by chance. In Experiments 3 and 4, 1 chimpanzee performed at a level significantly better than chance. Therefore, chimpanzees mentally represent quantity and successfully combine and compare nonvisible, sequentially presented sets of items.     

How special are objects? Children's reasoning about objects, parts, and holes

Discrete physical objects have a special status in cognitive and linguistic development. Infants track and enumerate objects, young children are biased to construe novel words as referring to objects, and, when asked to count an array of items, preschool children tend to count the discrete objects, even if explicitly asked to do otherwise. We address here the question of whether discrete physical objects are the only entities that have this special status, or whether other individuals are salient as well. In two experiments, we found that 3-year-olds are just as good at identifying, tracking, and counting certain nonobject entities (holes in Experiment 1; holes and parts in Experiment 2) as they are with objects. These results are discussed in light of different theories of the nature and development of children's object bias.     

The nature of gestures' beneficial role in spatial problem solving

Co-thought gestures are hand movements produced in silent, noncommunicative, problem-solving situations. In the study, we investigated whether and how such gestures enhance performance in spatial visualization tasks such as a mental rotation task and a paper folding task. We found that participants gestured more often when they had difficulties solving mental rotation problems (Experiment 1). The gesture-encouraged group solved more mental rotation problems correctly than did the gesture-allowed and gesture-prohibited groups (Experiment 2). Gestures produced by the gesture-encouraged group enhanced performance in the very trials in which they were produced (Experiments 2 & 3). Furthermore, gesture frequency decreased as the participants in the gesture-encouraged group solved more problems (Experiments 2 & 3). In addition, the advantage of the gesture-encouraged group persisted into subsequent spatial visualization problems in which gesturing was prohibited: another mental rotation block (Experiment 2) and a newly introduced paper folding task (Experiment 3). The results indicate that when people have difficulty in solving spatial visualization problems, they spontaneously produce gestures to help them, and gestures can indeed improve performance. As they solve more problems, the spatial computation supported by gestures becomes internalized, and the gesture frequency decreases. The benefit of gestures persists even in subsequent spatial visualization problems in which gesture is prohibited. Moreover, the beneficial effect of gesturing can be generalized to a different spatial visualization task when two tasks require similar spatial transformation processes. We concluded that gestures enhance performance on spatial visualization tasks by improving the internal computation of spatial transformations. (PsycINFO Database Record (c) 2010 APA, all rights reserved).     

计数与匹配:5~6岁儿童量比较策略的发展特点

选取56名5~6岁儿童,采用计算机呈现刺激的方式,实现了刺激的动态呈现来模拟策略的使用过程,并通过对于被试正确率和反应时的考察确定了每一名被试使用计数和匹配策略进行量比较的特点。结果表明:(1)与计数策略相比,儿童对于匹配策略的掌握更加成熟,5岁和5岁半组被试使用匹配策略解决量比较问题的正确率更高;(2)在已经掌握计数和匹配两种策略的情况下,仍有一部分儿童更倾向于使用匹配策略来解决量比较问题。     

Monkeys (Macaca mulatta and Cebus apella) track, enumerate, and compare multiple sets of moving items

Despite many demonstrations of numerical competence in nonhuman animals, little is known about how well animals enumerate moving stimuli. In this series of experiments, rhesus monkeys (Macaca mulatta) and capuchin monkeys (Cebus apella) performed computerized tasks in which they had to enumerate sets of stimuli. In Experiment 1, rhesus monkeys compared two sets of moving stimuli. Experiment 2 required comparisons of a moving set and a static set. Experiment 3 included human participants and capuchin monkeys to assess all 3 species' performance and to determine whether responding was to the numerical properties of the stimulus sets rather than to some other stimulus property such as cumulative area. Experiment 4 required both monkey species to enumerate subsets of each moving array. In all experiments, monkeys performed above chance levels, and their responses were controlled by the number of items in the arrays as opposed to nonnumerical stimulus dimensions. Rhesus monkeys performed comparably to adult humans when directly compared although capuchin performance was lower.     

Predictors of early numeracy: Is there a place for mistakes when learning about number?

It is one thing to be able to count and share items proficiently, but it is another thing to know how counting and sharing establish and identify quantity. The aim of the study was to identify which measures of numerical knowledge predict children's success on simple number problems, where counting and set equivalence are at issue. Seventy‐two 5‐year‐olds were given a battery of nine tasks on each of three sessions (at 3‐monthly intervals). Tasks measured procedural proficiency, conceptual understanding (using an error‐detection paradigm) and the ability to compare sets using number knowledge. Procedural skills remained fairly stable over the 6‐month period, and preceded children's ability to detect another's violations to those procedures. Regression analysis revealed that children who are sensitive to procedural errors in another's counting and sharing are more likely to recognize the significance of cardinal numbers for set comparisons. We suggest that although children's conceptual understanding of well‐rehearsed routines is often limited, conceptual insight might be achieved by setting tasks that require reflection rather than practice.     

The need to explain

How do reasoners deal with inconsistencies? James (1907) believed that the rational solution is to revise your beliefs and to do so in a minimal way. We propose an alternative: You explain the origins of an inconsistency, which has the side effect of a revision to your beliefs. This hypothesis predicts that individuals should spontaneously create explanations of inconsistencies rather than refute one of the assertions and that they should rate explanations as more probable than refutations. A pilot study showed that participants spontaneously explain inconsistencies when they are asked what follows from inconsistent premises. In three subsequent experiments, participants were asked to compare explanations of inconsistencies against minimal refutations of the inconsistent premises. In Experiment 1, participants chose which conclusion was most probable; in Experiment 2 they rank ordered the conclusions based on their probability; and in Experiment 3 they estimated the mean probability of the conclusions' occurrence. In all three studies, participants rated explanations as more probable than refutations. The results imply that individuals create explanations to resolve an inconsistency and that these explanations lead to changes in belief. Changes in belief are therefore of secondary importance to the primary goal of explanation.     

Negative facial expression captures attention and disrupts performance

In two experiments, participants counted features of schematic faces with positive, negative, or neutral emotional expressions. In Experiment 1 it was found that counting features took longer when they were embedded in negative as opposed to positive faces. Experiment 2 replicated the results of Experiment 1 and also demonstrated that more time was required to count features of negative relative to neutral faces. However, in both experiments, when the faces were inverted to reduce holistic face perception, no differences between neutral, positive, and negative faces were observed, even though the feature information in the inverted faces was the same as in the upright faces. We suggest that, relative to neutral and positive faces, negative faces are particularly effective at capturing attention to the global face level and thereby make it difficult to count the local features of faces.     

The roles of the central executive and visuospatial storage in mental arithmetic: A comparison across strategies

Previous research has demonstrated that working memory plays an important role in arithmetic. Different arithmetical strategies rely on working memory to different extents—for example, verbal working memory has been found to be more important for procedural strategies, such as counting and decomposition, than for retrieval strategies. Surprisingly, given the close connection between spatial and mathematical skills, the role of visuospatial working memory has received less attention and is poorly understood. This study used a dual-task methodology to investigate the impact of a dynamic spatial n-back task (Experiment 1) and tasks loading the visuospatial sketchpad and central executive (Experiment 2) on adults' use of counting, decomposition, and direct retrieval strategies for addition. While Experiment 1 suggested that visuospatial working memory plays an important role in arithmetic, especially when counting, the results of Experiment 2 suggested this was primarily due to the domain-general executive demands of the n-back task. Taken together, these results suggest that maintaining visuospatial information in mind is required when adults solve addition arithmetic problems by any strategy but the role of domain-general executive resources is much greater than that of the visuospatial sketchpad.