The repetition effect in time judgments is not a function of the intensity of the stimulus defining the interval: implications for an arousal theory

College students (N = 17 per group) estimated the durations of time intervals ranging from 8 to 20 sec. Intervals were defined by tones of 35, 55, and 75 db for each of three groups. The estimates of all groups declined across trials, but the degree of decline did not differ as a function of stimulus intensity. These data call into question the view that the repetition effect is attributable to a decrement in general arousal.     

Children's numerical skill and judgments of confidence in estimation

This study investigated how children's fluent use of a basic numerical skill is related to their perceptions of confidence on a task of quantitative estimation. One hundred four 10th graders (age 15.7 years) were categorized according to level of numerical fluency as assessed by timed tasks of counting in various nonunity increments. They then made estimates of large numbers of dots and rated how confident they were on each trial. The estimation task varied according to two within-subject factors of task difficulty. It was found that counting skill was related not only to accuracy in estimating, but also to the appropriateness of the children's confidence in their responses. Skillful children made judgments that corresponded with actual task difficulty. A derived calibration score corroborated ANOVA findings of the interaction of skill level and task difficulty on confidence ratings. It also revealed that while girls were less confident than boys, they were actually more realistic in their judgments. Findings are discussed in terms of (a) the effect of stimulus knowledge on awareness of task difficulty and (b) sex differences in achievement-related expectancies.     

Perceptual selectivity is task dependent: The pop-out effect poops out

An attempt was made to determine why evidence for perceptual selectivity based on conceptual category (e.g., digits vs. letters) has been found in some experiments but not in others. Experiments using the partial-report paradigm find no partial-report superiority when the report is cued by category, whereas, in recent visual search studies, evidence for perceptual selectivity has been obtained for arrays containing a single item that was categorically different from the other items (e.g., a digit among letters). Using a search task, Experiment 1 investigated the possibility that the number of categorically different items in the arrays could be a determinant of selectivity. One, two, or three digits and a variable number of letters were presented on each trial, and subjects determined if a particular digit was present. No evidence of selectivity was obtained, even for the one-digit condition. Experiment 2 verified this result, and Experiment 3 extended the failure of selectivity to a search task in which the possible targets differed in color from the distractor items. In Experiment 4, subjects counted the number of digits or red letters in arrays in which black letters were the distractor items. The counting task was used to eliminate the requirement in our previous tasks that the subjects search forspecific items. Evidence was obtained in the counting task for selectivity based on the color difference but not on the categorical difference. The color stimuli used in the counting task were essentially the same as those that did not yield any evidence of selectivity in the search task. The results suggest that task demands are an important determinant of whether or not perceptual selectivity will occur.     

Operational efficiency and the growth of short-term memory span

Four studies are reported which link developmental increases in memory span with developmental increases in operational speed or efficiency. In the first, a linear relationship is demonstrated between increases in word span and increases in speed of word repetition. In the second, adults and 6-year-olds are equated on speed of word repetition, by manipulating word familiarity. It is shown that their word spans are no longer different under these conditions. Similar findings are then reported for a test of M space called “Counting Span.” First, a linear relationship is demonstrated between increases in Counting Span and increases in speed of counting. Next, adults and 6-year-olds are equated on speed of counting, by forcing adults to count in an unfamiliar language. It is shown that their counting spans are no longer different under these conditions. The conclusion is that developmental increases in memory span do not result from increases in total processing space. Rather, with development, basic operations become faster and more efficient. This means that they require less processing space, and that more space becomes available for storage as a result.     

Number development in young children

This paper presents new data on children's acquisition of counting skills. Three aspects of counting were studied: the formation of the cardinality rule that the last number named during counting denotes the number of objects in an array, the mastery of the counting procedure or the coordination of ordered number names and objects counted, and the growth of the knowledge that x + 1 is greater than x. A model was outlined which posits the hierarchic integration of six number skills to account for the growth of the knowledge that x + 1 is greater than x and the development of number conservation. The six skills are: the cardinality rule, the counting procedure, acquisition of more x's, judgments of relative numerosity, pattern recognition of small numbers, and one-to-one correspondences.     

Strategies used by children when solving simple subtractions

Ten children, 9–11 years old, solved all subtraction problems in the form of M?N=…, where 0 ? M ? 13, 0 ? N ? 13 and M ? N. The solution times were analysed and used for the formulation of a process model for subtraction. The model involves memory processes on two different levels, called reproductive and reconstructive respectively. When M=N, N=1, and M=2N the answers were quickly retrieved in reproductive memory processes. The reconstructive processes were found to be analogous to one of two counting procedures, viz. counting up and counting down. In general, the counting process starts either on N (when M < 2N) and counts up to reach the answer, or on M (when M > 2N) and counts down to reach the answer. This may reflect an effort to minimize the number of steps to be counted. However, when M > 10 and N < 10 a problem is always solved in a decrementing counting process. When M=10 many subtractions are solved in a reproductive memory process and the number 10 is also important as a point of reference for solving subtractions when M > 10 and N < 10.     

Computer-assisted intervention for children with low numeracy skills

We present results of a computer-assisted intervention (CAI) study on number skills in kindergarten children. Children with low numeracy skill (n = 30) were randomly allocated to two treatment groups. The first group played a computer game (The Number Race) which emphasized numerical comparison and was designed to train number sense, while the other group played a game (Graphogame-Math) which emphasized small sets of exact numerosities by training matching of verbal labels to visual patterns and number symbols. Both groups participated in a daily intervention session for three weeks. Children's performance in verbal counting, number comparison, object counting, arithmetic, and a control task (rapid serial naming) were measured before and after the intervention. Both interventions improved children's skills in number comparison, compared to a group of typically performing children (n = 30), but not in other areas of number skills. These findings, together with a review of earlier computer-assisted intervention studies, provide guidance for future work on CAI aiming to boost numeracy development of low performing children.     

Preschoolers' counting: principles before skill

Three- to 5-year-old children participated in one of 4 counting experiments. On the assumption that performance demands can mask the young child's implicit knowledge of the counting principles, 3 separate experiments assessed a child's ability to detect errors in a puppet's application of the one-one, stable-order and cardinal count principles. In a fourth experiment children counted in different conditions designed to vary performance demands. Since children in the errror-detection experiments did not have to do the counting, we predicted excellent performance even on set sizes beyond the range a young child counts accurately. That they did well on these experiments supports the view that errors in counting—at least for set sizes up to 20—reflect performance demands and not the absence of implicit knowledge of the counting principles. In the final experiment, where children did the counting themselves, set size did affect their success. So did some variations in conditions, the most difficult of which was the one where children had to count 3-dimensional objects which were under a plexiglass cover. We expected that this condition would interfere with the child's tendency to point and touch objects in order to keep separate items which have been counted from those which have not been counted.     

The production and verification tasks in mental addition: An empirical comparison

We respond to A. Baroody's comment (1984, Developmental Review, 4, 148–156) with an empirical comparison of the production and verification tasks. With the exception of performance at the first grade level, the two tasks yield essentially identical conclusions. The results of an adjunct task, in which the rate of mental counting was assessed, suggest that children as young as second grade are relying on memory retrieval to a significant degree. In contrast to Baroody's speculation, there appear to be no widespread difficulties associated with results from the verification task. Furthermore, the task permits a more analytic examination of performance and underlying mental process than is afforded by the production task. We conclude by reiterating the empirical support for a model of fact retrieval, and suggesting that accessibility of the arithmetic facts is the basic factor which underlies both fact retrieval and procedural knowledge performance.     

The development of many-to-one counting in 4-year-old children

This study investigated the development of children's knowledge of many-to-one counting. Four-year-olds (n = 51) either participated in a training or in a control group. The training taught children how to allot Y physically not present items to each of X objects and how to count these items in the process. It was found that most children could be taught how to perform a many-to-one count. Counting performance was related to insight into the concept of multiplication. More specifically, learning of the many-to-one strategy was related to a better insight into aggregation over sets of items.     

Finger counting habit and spatial–numerical association in children and adults

Sensory-motor experiences are known to build up concrete and abstract concepts during the lifespan. The present study aimed to test how finger counting habits (right-hand vs. left-hand starters) could influence the spatial–numerical representation in number-to-position (explicit) and digit-string bisection (implicit) tasks. The subjects were Italian primary school children (N = 184, from the first to the fifth year) and adults (N = 42). No general preference for right- or left-starting in the finger counting was found. In the explicit task, right- or left-starting did not affect performance. In the implicit task, the right-hand starters shifted from the left to the right space when bisecting small and large numbers respectively, while the left-hand starters shifted from the right to the left space with higher leftward bias for large numbers. The finger configuration in Italian children and adults influences the spatial–numerical representation, but only when implicit number processing is required by the task.     

On the problems of being more or less the same

Sixty children aged 2–3 to 5-2 were given four quantity tasks which tested their understanding of more and same. Two tasks involved addition, two involved judgment of static quantities. One of each type of task required a manipulative response, and one of each required a yes/no judgment. The tasks involved judgments of equality and inequality. Tasks involving a manipulative response were significantly easier than those involving a yes/no judgment, indicating that the nature of the response required of the child is crucial. All other differences were negligible, indicating that the other task variables investigated do not affect the child's ability to respond correctly. The ability to make accurate responses was not strongly associated with counting ability. There was a significant difference in the children's counting responses depending on the method used to elicit it.     

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。     

To infinity and beyond: Children generalize the successor function to all possible numbers years after learning to count

Recent accounts of number word learning posit that when children learn to accurately count sets (i.e., become “cardinal principle” or “CP” knowers), they have a conceptual insight about how the count list implements the successor function – i.e., that every natural number n has a successor defined as n + 1 (Carey, 2004, 2009; Sarnecka & Carey, 2008). However, recent studies suggest that knowledge of the successor function emerges sometime after children learn to accurately count, though it remains unknown when this occurs, and what causes this developmental transition. We tested knowledge of the successor function in 100 children aged 4 through 7 and asked how age and counting ability are related to: (1) children’s ability to infer the successors of all numbers in their count list and (2) knowledge that all numbers have a successor. We found that children do not acquire these two facets of the successor function until they are about 5½ or 6 years of age – roughly 2 years after they learn to accurately count sets and become CP-knowers. These findings show that acquisition of the successor function is highly protracted, providing the strongest evidence yet that it cannot drive the cardinal principle induction. We suggest that counting experience, as well as knowledge of recursive counting structures, may instead drive the learning of the successor function.     

Counting on fine motor skills: links between preschool finger dexterity and numerical skills

Finger counting is widely considered an important step in children's early mathematical development. Presumably, children's ability to move their fingers during early counting experiences to aid number representation depends in part on their early fine motor skills (FMS). Specifically, FMS should link to children's procedural counting skills through consistent repetition of finger‐counting procedures. Accordingly, we hypothesized that (a) FMS are linked to early counting skills, and (b) greater FMS relate to conceptual counting knowledge (e.g., cardinality, abstraction, order irrelevance) via procedural counting skills (i.e., one–one correspondence and correctness of verbal counting). Preschool children (N = 177) were administered measures of procedural counting skills, conceptual counting knowledge, FMS, and general cognitive skills along with parent questionnaires on home mathematics and fine motor environment. FMS correlated with procedural counting skills and conceptual counting knowledge after controlling for cognitive skills, chronological age, home mathematics and FMS environments. Moreover, the relationship between FMS and conceptual counting knowledge was mediated by procedural counting skills. Findings suggest that FMS play a role in early counting and therewith conceptual counting knowledge.     

Many entities, no identity

The aim of this paper is to argue that some objections raised by Jantzen (Synthese, 2010) against the separation of the concepts of ‘counting’ and ‘identity’ are misled. We present a definition of counting in the context of quasi-set theory requiring neither the labeling nor the identity and individuality of the counted entities. We argue that, contrary to what Jantzen poses, there are no problems with the technical development of this kind of definition. As a result of being able to keep counting and identity apart for those entities, we briefly suggest that one venerable tradition concerning the nature of quantum particles may be consistently held. According to that tradition, known as the Received View on particles non-individuality, quantum particles may be seen as entities having both features: (i) identity and individuality do not apply to them, (ii) they may be gathered in collections comprising a plurality of them.     

Find the picture of eight turtles: a link between children's counting and their knowledge of number word semantics

An essential part of understanding number words (e.g., eight) is understanding that all number words refer to the dimension of experience we call numerosity. Knowledge of this general principle may be separable from knowledge of individual number word meanings. That is, children may learn the meanings of at least a few individual number words before realizing that all number words refer to numerosity. Alternatively, knowledge of this general principle may form relatively early and proceed to guide and constrain the acquisition of individual number word meanings. The current article describes two experiments in which 116 children (2½- to 4-year-olds) were given a Word Extension task as well as a standard Give-N task. Results show that only children who understood the cardinality principle of counting successfully extended number words from one set to another based on numerosity—with evidence that a developing understanding of this concept emerges as children approach the cardinality principle induction. These findings support the view that children do not use a broad understanding of number words to initially connect number words to numerosity but rather make this connection around the time that they figure out the cardinality principle of counting.     

Recognition memory and intelligence

The assumption that individual differences in recognition memory are associated with individual differences in intelligence was explored by administering intelligence tests and tests of immediate visual recognition memory to a sample of 52 5-year-old children expected to vary widely from one another in intelligence. Each child was given the Peabody Picture Vocabulary Test (Form L) and two tests of immediate recognition memory: one test for 27 abstract patterns and one test for 27 unfamiliar cartoon faces. The mean PPVT-IQ for the sample was in the average range at 98.1. Interindividual variability in IQ proved to be high as reflected in the group SD of 22.6, with scores ranging from 40 to 136. The recognition tasks proved to be of moderate difficulty. Individual differences in memory for patterns were highly related to memory for faces (r = .76), indicating that the overall recognition test was reliable. The most important result of the present study was the strong association between recognition memory performance and PPVT-IQ of .70. The relation between recognition memory and IQ could not be accounted for by the inclusion of a few very low IQ children, since the association remained high at .61 when children with IQs below 75 were omitted from analysis. In short, the present results indicate that immediate recognition memory is highly associated with intelligence.     

Nonlinear analysis of the movement variability structure can detect aging-related differences among cognitively healthy individuals

Studying the dynamics of nonlinear systems can provide additional information about the variability structure of the system. Within the current study, we examined the application of regularity and local stability measures to capture motor function alterations due to dual-tasking using a previously validated upper-extremity function (UEF). We targeted young (ages 18 and 30 years) and older adults (65 years or older) with normal cognition based on clinical screening. UEF involved repetitive elbow flexion without counting (ST) and while counting backward by one (DT1) or three (DT3). We measured the regularity (measured by sample entropy (SE)), local stability (measured by the largest Lyapunov exponent (LyE)), as well as conventional peak-dependent variability measures (coefficient of variation of kinematics parameters) to capture motor dynamic alterations due to dual-tasking. Within both groups, only SE showed significant differences between all pairs of UEF condition comparisons, even ST vs DT1 (p = 0.007, effect size = 0.507), for which no peak-dependent parameter showed significant difference. Among all measures, the only parameter that showed a significant difference between young and older adults was LyE (p < 0.001, effect size = 0.453). Current findings highlight the potential of nonlinear analysis to detect aging-related alterations among cognitively healthy participants.