Development of calculation abilities in young children.

This study investigates the development of skills for solving verbally and nonverbally presented calculation problems in children between 4 and 6 years of age. Identical addition and subtraction calculations were presented in three problem-type formats: nonverbal problems, story problems, and number-fact problems. The nonverbal problems involved presenting sets of physical referents that were then transformed either by adding or removing elements. The child saw the initial set and the number of elements that were added or removed, but not the final set. The task was to construct an array that contained the number of elements in the final set. The story problems and number-fact problems were presented orally, without props. Results indicate that children as young as 4 years of age have some success on the nonverbal problems, showing that they can transform sets by adding or subtracting elements. In contrast, children do not achieve comparable levels of success on the story problems or number-fact problems until 5 1/2 to 6 1/2 years of age. Moreover, throughout the age range tested, children performed better on nonverbal problems than on either story problems or number-fact problems. These results suggest that children's earliest ability to add and subtract is based on experiences combining and separating sets of objects in the world and that this ability precedes the development of conventional verbal methods of calculating.     

Heterogeneity impairs numerical matching but not numerical ordering in preschool children

Do preschool children appreciate numerical value as an abstract property of a set of objects? We tested the influence of stimulus features such as size, shape, and color on preschool children's developing nonverbal numerical abilities. Children between 3 and 5 years of age were tested on their ability to estimate number when the sizes, shapes, and colors of the elements in an array were varied (heterogeneous condition) versus when they did not vary (homogeneous condition). One group of children was tested on an ordinal task in which the goal was to select the smaller of two arrays while another group of children was tested on a match‐to‐sample task in which the goal was to choose one of two visual arrays that matched the sample in number. Children performed above chance on both homogeneous and heterogeneous stimuli in both tasks. However, while children showed no impairment on heterogeneous relative to homogeneous arrays in the ordering task, performance was impaired by heterogeneity in the matching task. We suggest that nonverbal numerical abstraction occurs early in development, but specific task objectives may prevent children from engaging in numerical abstraction.     

Blocking of incidental information in young children

Methods of increasing learning efficiency through a blocking technique were examined in young children. An experimental group of children was trained on an initial task with one relevant dimension, followed by redundancy training in which an incidental dimension also became relevant. A control group given only redundancy training was compared to this group to test how much the experimental subjects blocked or screened out the incidental dimension. In order to increase blocking, three methods were used: a preset transfer problem, an overtraining on the initial task, and a delay following training. The most important finding was increased blocking through the Preset Condition. Overtraining and Delay Conditions had no effect. In addition, more slow learners on the initial task learned about the incidental dimension than fast or intermediate learners. Also, fewer subjects given form on the initial task learned about the incidental dimension than those given color. The Combined Blocking, Delay, and Overtraining group showed partial blocking, while the Preset group demonstrated complete blocking. No developmental differences occurred. The results were interpreted using a modification of Trabasso and Bower's attention model (Attention in learning: Theory and research. New York: Wiley, 1968).     

Perceptions of proportionality in young children: matching spatial ratios

Three experiments examined the ability of young children (4- and 5-year-olds) and adults to identify correspondences in spatial ratios. In all three experiments, children as young as 4 and 5 years of age made accurate spatial proportionality judgments. Moreover, their judgments were no less accurate when both response alternatives differed in spatial configuration from the sample than when all the configurations were alike, indicating that those judgments were based on relational information and not on the exact form of the stimuli. The findings, which suggest that the concept of ratio may have important foundations in the ways young children perceive relations within and between objects, have implications for theories of mathematical development and for methods of mathematics instruction.     

The development of ordinal numerical competence in young children

Two experiments assessed ordinal numerical knowledge in 2- and 3-year-old children and investigated the relationship between ordinal and verbal numerical knowledge. Children were trained on a 1 vs 2 comparison and then tested with novel numerosities. Stimuli consisted of two trays, each containing a different number of boxes. In Experiment 1, box size was held constant. In Experiment 2, box size was varied such that cumulative surface area was unrelated to number. Results show children as young as 2 years of age make purely numerical discriminations and represent ordinal relations between numerosities as large as 6. Children who lacked any verbal numerical knowledge could not make ordinal judgments. However, once children possessed minimal verbal numerical competence, further knowledge was entirely unrelated to ordinal competence. Number may become a salient dimension as children begin to learn to count. An analog magnitude representation of number may underlie success on the ordinal task.     

Varieties of numerical abilities.

This paper provides a tutorial introduction to numerical cognition, with a review of essential findings and current points of debate. A tacit hypothesis in cognitive arithmetic is that numerical abilities derive from human linguistic competence. One aim of this special issue is to confront this hypothesis with current knowledge of number representations in animals, infants, normal and gifted adults, and brain-lesioned patients. First, the historical evolution of number notations is presented, together with the mental processes for calculating and transcoding from one notation to another. While these domains are well described by formal symbol-processing models, this paper argues that such is not the case for two other domains of numerical competence: quantification and approximation. The evidence for counting, subitizing and numerosity estimation in infants, children, adults and animals is critically examined. Data are also presented which suggest a specialization for processing approximate numerical quantities in animals and humans. A synthesis of these findings is proposed in the form of a triple-code model, which assumes that numbers are mentally manipulated in an arabic, verbal or analogical magnitude code depending on the requested mental operation. Only the analogical magnitude representation seems available to animals and preverbal infants.     

Individual differences in numerical abilities in preschoolers

This study investigated individual differences in different aspects of early number concepts in preschoolers. Eighty 4-year-olds from Oxford nursery classes took part. They were tested on accuracy of counting sets of objects; the cardinal word principle; the order irrelevance principle; and predicting the results of repeated addition and subtraction by 1 from a set of objects. There were marked individual differences for most tasks. Most children were reasonably proficient at counting and 70% understood the cardinal word principle. Based on the results of a repeated addition and subtraction by 1 task, the children were divided into three approximately equal groups: those who were already able to use an internalized counting sequence for the simplest forms of addition and subtraction; those who relied on a repeated 'counting-all' procedure for such tasks; and those who were as yet unable to cope with such tasks. In each group, significant relationships between some, but not all, of the numerical tasks were found. However, for almost any two tasks, it was possible to find individuals who could carry out either one of the tasks but not the other. Thus, even before formal instruction, arithmetical cognition is not unitary but is made up of many components.     

Cognitive and social-communication abilities among young children conceived by assisted reproductive technologies

Assisted reproductive technologies (ART) lack biological filters that are part of the natural fertilization process and thus might enable the presence of abnormal genetic materials. Whereas the findings regarding neonatal and neurological risks among ART-conceived children are rather consistent, data regarding cognitive and social-emotional developmental outcomes are inconsistent. The aim of this study was to examine the association between ART and cognitive and social-communication outcomes among pre-school children. The results indicated that the cognitive and social-communicative abilities of the ART-conceived children were similar to those of the spontaneously conceived children; however, according to parental reports, children in the ART group had higher communicative skills and better motor abilities than spontaneously conceived children. These results should be interpreted with caution as we used measures that assess global cognitive abilities that may not be sensitive to more subtle differences of higher cognitive and social-communication abilities in infancy that may become more prominent later in life. Although infertile couples and professionals in the field of ART can be reassures by the current findings, further research is needed as well as follow-up evaluation of this population during school age.     

Multisensory integration in children with Developmental Coordination Disorder

This study examines how multisensory stimuli affect the performance of children with Developmental Coordination Disorder (DCD) on a choice reaction time (CRT) task. Ten children with DCD, identified using the Movement Assessment Battery for Children-2, aged 7–10 years (4F, M = 8 y 3 m, SD = 17 m) and 10 typically developing peers (TDC) (5F, M = 8 y 4 m, SD = 17 m) reached to unimodal (auditory (AO), visual (VO)) and bimodal (audiovisual (AV)) stimuli at one of three target locations. A multisensory (AV) stimulus reduced RTs for both groups (p < 0.001, η² = 0.36). While the children with DCD had a longer RT in all conditions, the AV stimulus produced RTs in children with DCD (494 ms) that were equivalent to those produced by the TDC to the VO stimulus (493 ms). Movement Time (DCD = 486 ms; TDC = 434 ms) and Path Length (DCD = 25.6 cm; TDC = 24.2 cm) were longer in children with DCD compared to TDC as expected (p < 0.05). Only the TDC benefited from the AV information for movement control, as deceleration time of the dominant hand was seen to decrease when moving to an AV stimulus (p < 0.05). Overall, data shows children with DCD do benefit from a bimodal stimulus to plan their movement, but do not for movement control. Further research is required to understand if this is a result of impaired multisensory integration.     

Parallel processing of position and form information in young children

Young children and adults were compared in their processing of position and form information when only one type of information was processed and when both types of information were processed. For adults, position information in both the single and combined tasks showed a linear increase in report accuracy with increasing duration. For children, this linear increase was found only in the single task, at longer durations, resulting in a loss of parity across tasks for position information among children. No loss of parity was found for form information for either age group. The children’s loss of parity in processing position information when combined with form information replicates their previously demonstrated inability to process multiple-form arrays using a parallel independent strategy.     

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.     

Processing of occurrence-rate and item information by children of different ages and abilities

Two experiments compare the performance of children who vary in both age and learning ability. In the first experiment, learning disabled and nondisabled children from three age groups were tested for their incidental memory for frequency of occurrence information. In the second experiment, learning disabled and nondisabled children from two age groups were tested for their intentional memory of item information on a free recall test. In agreement with the notion that frequency is automatically encoded, all groups extracted frequency information and neither age nor learning ability influenced performance. However, both age and ability influenced performance on the recall test.     

Extensive training extends numerical abilities of guppies

Recent studies on animal mathematical abilities suggest that all vertebrates show comparable abilities when they are given spontaneous preference tests, such as selecting the larger number of food items, but that mammals and birds generally achieve much better performance than fish when tested with training procedures. At least part of these differences might be due to the fact that fish are usually trained with only one or two dozen trials while extensive training, sometimes with thousands of trials, is normally performed in studies of mammals and birds. To test this hypothesis, female guppies were trained on four consecutive numerical discriminations of increasing difficulty (from 2 vs. 3 to 5 vs. 6 items), with up to 120 trials with each discrimination. Five out of eight subjects discriminated all contrasts up to 4 versus 5 objects at levels significantly better than chance, a much higher limit than the 2 versus 3 limit previously reported in studies that provided fish with only short training sequences. Our findings indicate that the difference in numerical cognition between teleosts and warm-blooded vertebrates might be smaller than previously supposed.     

Working memory and access to numerical information in children with disability in mathematics

The relationship among working memory, mathematic ability, and the cognitive impairment of children with difficulties in mathematics was examined. A group of children with difficulties in mathematics (MD) was compared with a group of children with a normal level of achievement matched for vocabulary, age, and gender (N = 49). The children were required to perform a variety of working memory and short-term memory tasks that had been administered 1 year previously. Moreover, the children were asked to perform tasks designed to provide information about speed of articulation. The results suggest a general working memory deficit in children with MD, specifically in the central executive component of Baddeley's model and primarily in the ability to inhibit irrelevant information. However, the MD children were not impaired in speech rate and counting speed tasks, which mainly involve the role of the articulatory loop.     

Multisensory processing of redundant information in go/no-go and choice responses

In multisensory research, faster responses are commonly observed when multimodal stimuli are presented, as compared to unimodal target presentations. This so-called redundant-signals effect can be explained by several frameworks, including separate-activation and coactivation models. The redundant-signals effect has been investigated in a large number of studies; however, most of those studies have been limited to the rejection of separate-activation models. Coactivation models have been analyzed in only a few studies, primarily using simple response tasks. Here, we investigated the mechanism of multisensory integration underlying go/no-go and choice responses to redundant auditory–visual stimuli. In the present study, the mean and variance of response times, as well as the accuracy rates of go/no-go and choice responses, were used to test a coactivation model based on the linear superposition of diffusion processes (Schwarz, 1994) within two absorbing barriers. The diffusion superposition model accurately describes the means and variances of response times as well as the proportions of correct responses observed in the two tasks. Linear superposition thus seems to be a general principle in the integration of redundant information provided by different sensory channels, and is not restricted to simple responses. The results connect existing theories of multisensory integration with theories on choice behavior.     

Role of biological-motion information in recognition of facial expressions by young children

Data on the development of the perception of facial biological motion during preschool years are disproportionately scarce. We investigated the ability of preschoolers to recognise happy, angry, and surprised expressions, and eye-closing facial movements on the basis of facial biological motion. Children aged 4 years (n = 18) and 5-6 years (n = 19), and adults (n = 17) participated in a matching task, in which they were required to match the point-light displays of facial expressions to prototypic schematic images of facial expressions and facial movement. The results revealed that the ability to recognise facial expressions from biological motion emerges as early as the age of 4 years. This ability was evident for happy expressions at the age of 4 years; 5-6-year-olds reliably recognised surprised as well as happy expressions. The theoretical significances of these findings are discussed.     

Association between basic numerical abilities and mathematics achievement

Various measures have been used to investigate number processing in children, including a number comparison or a number line estimation task. The present study aimed to examine whether and to which extent these different measures of number representation are related to performance on a curriculum‐based standardized mathematics achievement test in kindergarteners, first, second, and sixth graders. Children completed a number comparison task and a number line estimation task with a balanced set of symbolic (Arabic digits) and non‐symbolic (dot patterns) stimuli. Associations with mathematics achievement were observed for the symbolic measures. Although the association with number line estimation was consistent over grades, the association with number comparison was much stronger in kindergarten compared to the other grades. The current data indicate that a good knowledge of the numerical meaning of Arabic digits is important for children's mathematical development and that particularly the access to the numerical meaning of symbolic digits rather than the representation of number per se is important.     

The interplay among temperament,neuropsychological abilities,and global functioning in young hyperactive/inattentive children

Cognition and emotion have been shown to interact and influence psychological functioning. However, to date these interactions have only been examined cross‐sectionally among inattentive and/or hyperactive/impulsive children. This study investigated the moderating effects of neuropsychological functioning at age 3–4 years on the relation between negative emotionality at age 3–4 years and global functioning 1 year later, at age 4–5 years. Hyperactive/inattentive (H/I; n = 114) preschoolers entered the study (BL: baseline) and were seen again 1 year later (F1). Children's BL scores on a neuropsychological test (NEPSY) and their temperament as rated by parents (Child Behavior Questionnaire) and teachers (Temperament Assessment Battery for Children‐Revised) were obtained, as were clinicians’ ratings of their global functioning (Children's Global Assessment Scale) at F1. Hierarchical linear regression analyses revealed that BL temperament variables accounted for significant variance in F1 Global Functioning. Significant interactions indicated that higher Verbal Executive abilities were associated with better child functioning when parent‐rated Effortful Control was high, but not when Effortful Control was low. Additionally, high levels of Nonverbal Executive skills were associated with higher child global functioning when both parent‐ and teacher‐rated negative affect was low, but not when negative affect was high.