Span and rate of apprehension in children and adults

Over-specification of small, borderline cardinalities and color in referential communication: the role of visual context, modifier position, and consistency

This paper reports on two flash-mode experiments that test redundant descriptions of small (2-4) cardinalities, borderline (5-8) cardinalities, and color in referential communication. It provides further support for the idea that small cardinalities are more salient (due to subitizing), less sensitive to visual context, and therefore give rise to higher over-specification rates than color. Because of greater salience, Russian speakers more often use prenominal positions for numerals than for color adjectives. The paper also investigates borderline cardinalities and argues for the order factor that affects their salience, since ordered items can be perceived in small subitized parts. The ordered mode of presentation of the borderline cardinalities leads to higher over-specification rates and to higher percentages of prenominal positions than the unordered one. The paper provides further evidence for the consistency of small, borderline cardinalities, and color in people's choices to minimally specify or over-specify given objects in referential communication.

The componential nature of arithmetical cognition: some important questions

Research on typically developing children and adults and people with developmental and acquired dyscalculia converges in indicating that arithmetical ability is not unitary but is made up of many different components. Categories of components include non-symbolic quantity representation and processing; symbolic quantity representation and processing; counting procedures and principles; arithmetic operations; arithmetical knowledge and understanding; multiple forms and applications of conceptual knowledge of arithmetic; and domain-general abilities such as attention, executive functions and working memory. There is much evidence that different components can and often do show considerable functional independence, not only in developmental and acquired dyscalculia, but in typically achieving children and adults. At the same time, it is possible to find complex interactions and bidirectional relationships between the different components, including between domain-specific and apparently domain-general abilities. There is a great deal that still needs to be discovered. In particular, we need to learn more about the origins in infancy of subitizing and approximate magnitude comparison, the extent to which these interact, the extent to which they may be further divisible, and the extent and ways in which they themselves may develop with age and the extent to which they may influence later-developing components. There also needs to be a lot more research on exactly how domain-general and domain-specific abilities contribute to mathematical development, and how they interact with one another.

Numerical Cognition after Brain Injury: Is There a Relationship between Subitizing and Arithmetical Abilities?

Subitizing is the ability to enumerate small quantities efficiently and automatically. Counting is a strategy adopted for larger numerosities resulting in a near linear increase in response time with each increase in quantity. Some developmental studies suggest that being able to subitize efficiently may be a predictor of later arithmetical ability. Being able to enumerate small quantities efficiently may be necessary for at least some aspects of arithmetical skill and understanding to develop. According to this view, arithmetic ability ultimately depends upon subitizing. If this were the case, when acquired brain injury results in impaired performance on subitizing tasks, mathematical performance may also be impaired. The following study tested eleven healthy control participants and nine chronic patients with acquired brain injury on tasks focused on visual enumeration, addition and multiplication to explore a potential relationship between subitizing ability and calculation performance. No overall correlations were found between subitizing and addition or multiplication speed. However, a very clear subitizing impairment was found in two patients who then demonstrated very different levels of preserved addition skills. The dissociations found and the large inter-individual variability supports a more componential view of arithmetical ability.

Manipulation of Attention Affects Subitizing Performance: A Systematic Review and Meta-analysis

Subitizing is the fast and accurate enumeration of small sets. Whether attention is necessary for subitizing remains controversial considering (1) subitizing is claimed to be "pre-attentive", and (2) existing experimental methods and results are inconsistent. To determine whether manipulations to attention demonstratively affect subitizing, the current study comprises a systematic review and meta-analysis. Results from fourteen studies (22 experiments, 35 comparisons) suggest that changes to attentional demands interferes with enumeration of small sets; leading to slower response times, lower accuracy, and poorer Weber acuity (p <.010; p <.001; p <.001; respectively)-notwithstanding a potential publication bias. A unifying framework is proposed to explain the role of attention in visual enumeration, with progressively greater attentional involvement from estimation to subitizing to counting. Our findings suggest attention is integral for subitizing and highlights the need to emphasise attentional mechanisms into neurocognitive models of numerosity processing. We also discuss the possible role of attention in numerical processing difficulties (e.g., dyscalculia).

Over-Specification of Small Cardinalities in Referential Communication

The paper presents experimental evidence for the over-specification of small cardinalities in referential communication. The first experiment shows that when presented with a small set (2, 3, or 4) of unique objects, the speaker includes a numeral denoting a small cardinality into the description of given objects, although it is over-informative for the hearer (e.g., “three stars”). On the contrary, when presented with a large set of unique objects, the speaker does not include a numeral denoting a large cardinality into their description, so she produces a bare plural (e.g., “stars”). The effect of small cardinalities resembles the effect of over-specifying color in referential communication, which has been extensively studied in recent years (cf. Tarenskeen et al., 2015; Rubio-Fernández, 2016, among many others). This suggests that, like color, small cardinalities are absolute and salient. The second experiment demonstrates that when presented with an identical small set of monochrome objects, the speaker over-specifies a small cardinality to a much greater extent than a color. This suggests that small cardinalities are even more salient than color. The third experiment reveals that when slides are flashed on the screen one by one, highlighted objects of small cardinalities are still over-specified. We argue that a plausible explanation for the salience of small cardinalities is a subitizing effect, which is the human capacity to instantaneously grasp small cardinalities.

Common and distinct predictors of non-symbolic and symbolic ordinal number processing across the early primary school years

What are the cognitive mechanisms supporting non-symbolic and symbolic order processing? Preliminary evidence suggests that non-symbolic and symbolic order processing are partly distinct constructs. The precise mechanisms supporting these skills, however, are still unclear. Moreover, predictive patterns may undergo dynamic developmental changes during the first years of formal schooling. This study investigates the contribution of theoretically relevant constructs (non-symbolic and symbolic magnitude comparison, counting and storage and manipulation components of verbal and visuo-spatial working memory) to performance and developmental change in non-symbolic and symbolic numerical order processing. We followed 157 children longitudinally from Grade 1 to 3. In the order judgement tasks, children decided whether or not triplets of dots or digits were arranged in numerically ascending order. Non-symbolic magnitude comparison and visuo-spatial manipulation were significant predictors of initial performance in both non-symbolic and symbolic ordering. In line with our expectations, counting skills contributed additional variance to the prediction of symbolic, but not of non-symbolic ordering. Developmental change in ordering performance from Grade 1 to 2 was predicted by symbolic comparison skills and visuo-spatial manipulation. None of the predictors explained variance in developmental change from Grade 2 to 3. Taken together, the present results provide robust evidence for a general involvement of pair-wise magnitude comparison and visuo-spatial manipulation in numerical ordering, irrespective of the number format. Importantly, counting-based mechanisms appear to be a unique predictor of symbolic ordering. We thus conclude that there is only a partial overlap of the cognitive mechanisms underlying non-symbolic and symbolic order processing.

Canonical representations of fingers and dots trigger an automatic activation of number semantics: an EEG study on 10-year-old children

Over the course of development, children must learn to map a non-symbolic representation of magnitude to a more precise symbolic system. There is solid evidence that finger and dot representations can facilitate or even predict the acquisition of this mapping skill. While several behavioral studies demonstrated that canonical representations of fingers and dots automatically activate number semantics, no study so far has investigated their cerebral basis. To examine these questions, 10-year-old children were presented a behavioral naming task and a Fast Periodic Visual Stimulation EEG paradigm. In the behavioral task, children had to name as fast and as accurately as possible the numbers of dots and fingers presented in canonical and non-canonical configurations. In the EEG experiment, one category of stimuli (e.g., canonical representation of fingers or dots) was periodically inserted (1/5) in streams of another category (e.g., non-canonical representation of fingers or dots) presented at a fast rate (4 Hz). Results demonstrated an automatic access to number semantics and bilateral categorical responses at 4 Hz/5 for canonical representations of fingers and dots. Some differences between finger and dot configuration's processing were nevertheless observed and are discussed in light of an effortful-automatic continuum hypothesis.

A probabilistic approach for quantifying children's subitizing span

The development of enumeration skills over childhood is thought to reflect improvements in both subitizing (for small sets) and serial counting (for larger sets). However, investigations into the contribution of subitizing to advancing mathematics ability are limited by challenges in measuring subitizing capacity across developmental populations. Subitizing capacity in adults is traditionally assessed by calculating the bilinear inflection point for reaction times or accuracy across set sizes, but in children greater variability and dramatic improvements in counting ability introduce problems with this approach. This study demonstrates this limitation in a sample of elementary school children and proposes a novel probabilistic approach to measuring subitizing capacity. This metric captures well-established trends in the development of children's subitizing. Furthermore, the proposed metric predicts unique variance in symbolic arithmetic ability, corroborating previous research that suggests a foundational role for subitizing in the development of numerical cognition. Findings demonstrate the advantages of a probabilistic approach to determining subitizing capacity in young children and suggest that it may be practically and theoretically well-suited for investigating subitizing and its role in mathematics development.

Enumeration strategy differences revealed by saccade-terminated eye tracking

Brain regions involved in saccadic eye movements partially overlap with a frontoparietal network implicated in encoding numerosities. Eye movement patterns may plausibly reflect strategic scanning behaviours to resolve the open-ended task of efficiently enumerating visual arrays. If so, these patterns may help explain individual differences in enumeration acuity in terms of well-understood visual attention mechanisms. Most enumeration eye-tracking paradigms, however, do not allow for direct manipulation of eye movement behaviours to test these claims. In the current study we terminated trials after a specified number of saccades to systematically probe the time course of enumeration strategies. Fifteen adults (11 naïve, 4 informed) enumerated random dot arrays under three conditions: (1) a novel saccade-terminated design where arrays were visible until one, two or four saccades had occurred; (2) a duration-terminated design where arrays were shown for 250, 500 or 1000 ms; and (3) a response-terminated design where arrays were visible until a response. Participants gave more accurate responses when enumerating saccade-terminated trials despite taking a similar time as in the duration-terminated trials. When participants were informed how trials would terminate, their saccade onset latencies shifted to match task demands. Rotating saccade vectors to align with salient image locations accounted for variability in the orientation of saccade trajectories. These findings (1) show a combination of stimulus-derived visual processing and task-based strategic demands account for enumeration eye movements patterns, (2) validate a novel saccade-contingent trial termination procedure for studying sequences of enumeration eye movements, and (3) highlight the need to include analyses of spatial and temporal eye movement patterns into models of visual enumeration strategies.

Possible Associations between Subitizing, Estimation and Visuospatial Working Memory (VSWM) in Children

Researchers have focused on identifying the mechanisms involved in subitizing and its differences with estimation. Some suggest that subitizing relies on a visual indexing system in charge of the simultaneous individuation of objects that is also used by visuospatial working memory (VSWM). In adults, studies found associations between subitizing and VSWM, in the absence of correlation between VSWM and estimation. The present study analyzed the performance of 120 4 and 6-year-old children in three tasks: dot enumeration to measure subitizing capacity, quantity discrimination for estimation, and Corsi Block-tapping task for VSWM. In the enumeration task RTs (F(9, 1062)=720.59, MSE=734394, p<.001, η2=.86) and errors (F(9, 1062)=42.15, MSE=.194, p<.001, η2=.26.) increased with the array, but this growth was statistically significant only from 4 dots onward. Each subject's subitizing range was estimated by fitting RTs with a sigmoid function of number of dots and obtaining the bend point of the curve. Data fit (age 4: R 2 = .88; SD = .08; age 6: R 2 = .91, SD = .08) showed a mean subitizing range of 2.79 (SD = .66) for 4 year-olds and of 3.11 (SD = .64) for 6 year-olds. Subitizing ranges and average RTs showed low association with storage (r = .274; p < .05; r = -.398; p < .001) and average RTs with concurrent processing (r = -.412; p < .001) in VSWM. Subitizing range and speed showed no association with estimation speed and a poor association with accuracy (r = .234, p < .01; r = -.398, p < .001), which suggests independent systems for small and large quantities. Subitizing and estimation measures correlated with VSWM (p < .01), which suggests that both processes may require VSWM resources.

Conservation Accidents Revisited

McGarrigle and Donaldson's (1975) claim that the Genevan conservation test procedure underestimates the child's cognitive competence is examined. Experiment 1 reports a successful replication of their original work. We then consider several objections to McGarrigle and Donaldson's claim that the tasks they used did in fact test the child's ability to conserve number. Firstly, as only four counters were used in their study the child's judgment of numerosity could have been perceptually rather than conceptually based (Winer 1974). Secondly, teddy may have acted as a distracting agent leading the child to ignore the post-transformational state of the arrays and to answer the experimenter's post-transformational question (correctly) by mere repetition of his earlier response. In experiment 2 a procedure which is free from these objections, but otherwise similar to McGarrigle and Donaldson's procedure, is employed, together with appropriate comparison conditions. This procedure results in a decrease in the overall incidence of conservation responses, but fails to eradicate the difference between accidental and intentional transformation of the arrays (p < 0.001). In the final part of the paper we consider the implications of these and McGarrigle and Donaldson's results for our understanding of the social psychology of the conservation task.

The Young Child's Processing of Dot Patterns: A Chronometric and Eye Movement Analysis

The present study reports the measurement of response latencies and the recording of eye movement in a task where children of about 5.5 years had to count arrangements of 1-8 dots in different configurations. Consistent with earlier findings, response latencies for numbers up to 5 suggested subitizing rather than counting strategies. Data from concomittant eye movement recordings clearly showed that even the processing of the small numbers required at least four fixations per response. Records of eye movements under the conditions of numbers of dots larger than n = 5 were found to reflect mixed strategies and not elementary one-by-one counting procedures. It is hypothesized that large processing times in comparison with adults were mainly due to interim verifications of results already established: children were, much more than adults, mentally loaded by the double task of storing partial results and processing new information at the same time.

Infants' Individuation and Enumeration of Actions

Two experiments examined 6-month-old infants' ability to individuate and enumerate physical actions—the sequential jumps of a puppet In both experiments, which employed a habituation paradigm, infants successfully discriminated two-jump from three-jump sequences The sequences of activity in the two experiments provided for an initial exploration of the cues infants use to individuate actions Results show that (a) infants can individuate and enumerate actions in a sequence, indicating that their enumeration mechanism is quite general in the kinds of entities over which it will operate, (b) actions whose temporal boundaries are characterized by a contrast between motion and absence of motion are especially easy to individuate and enumerate, but nonetheless (c) infants can individuate and enumerate actions embedded in a sequence of continuous motion, indicating that infants possess procedures for parsing an ongoing motionful scene into distinct portions of activity

Children's mathematical performance: five cognitive tasks across five grades

Children in elementary school, along with college adults, were tested on a battery of basic mathematical tasks, including digit naming, number comparison, dot enumeration, and simple addition or subtraction. Beyond cataloguing performance to these standard tasks in Grades 1 to 5, we also examined relationships among the tasks, including previously reported results on a number line estimation task. Accuracy and latency improved across grades for all tasks, and classic interaction patterns were found, for example, a speed-up of subitizing and counting, increasingly shallow slopes in number comparison, and progressive speeding of responses especially to larger addition and subtraction problems. Surprisingly, digit naming was faster than subitizing at all ages, arguing against a pre-attentive processing explanation for subitizing. Estimation accuracy and speed were strong predictors of children's addition and subtraction performance. Children who gave exponential responses on the number line estimation task were slower at counting in the dot enumeration task and had longer latencies on addition and subtraction problems. The results provided further support for the importance of estimation as an indicator of children's current and future mathematical expertise.

Further evidence for a spatial-numerical association in children before formal schooling

Given the robust finding that number and space are associated systematically at least in school children and adults, it has been concluded that this association might be based on the frequent practice of reading or writing skills, which are usually consolidated by formal schooling. However, first studies contradict this assumption demonstrating that associations of "small" magnitudes with left space and of "large" magnitudes with right space exist already in preschoolers. The present study used a non-symbolic magnitude comparison task to examine whether kindergartners who have not yet been formally instructed in reading and writing show a SNARC effect, that is, whether they would respond more rapidly with the right hand to larger numbers and with the left hand to smaller numbers. This assumption was confirmed by the data. In view of further evidence for an association between number and space that evolves before children are proficient in reading and writing, the role of potential alternative culture-specific, individual, and universal foundations of this association is emphasized and discussed.

Preschoolers' dot enumeration abilities are markers of their arithmetic competence

The abilities to enumerate small sets of items (e.g., dots) and to compare magnitudes are claimed to be indexes of core numerical competences that scaffold early math development. Insofar as this is correct, these abilities may be diagnostic markers of math competence in preschoolers. However, unlike magnitude comparison abilities, little research has examined preschoolers' ability to enumerate small sets, or its significance for emerging math abilities; which is surprising since dot enumeration is a marker of school-aged children's math competence. It is nevertheless possible that general cognitive functions (working memory, response inhibition in particular) are associated with preschoolers' math abilities and underlie nascent dot enumeration abilities. We investigated whether preschoolers' dot enumeration abilities predict their non-verbal arithmetic ability, over and above the influence of working memory and response inhibition. Two measures of dot enumeration ability were examined-inverse efficiency and paradigm specific (response time profiles) measures-to determine which has the better diagnostic utility as a marker of math competence. Seventy-eight 42-to-57 month-olds completed dot enumeration, working memory, response inhibition, and non-verbal addition and subtraction tasks. Dot enumeration efficiency predicted arithmetic ability over and above the influence of general cognitive functions. While dot enumeration efficiency was a better predictor of arithmetic ability than paradigm specific response time profiles; the response time profile displaying the smallest subitizing range and steepest subitizing slope, also displayed poor addition abilities, suggesting a weak subitizing profile may have diagnostic significance in preschoolers. Overall, the findings support the claim that dot enumeration abilities and general cognitive functions are markers of preschoolers' math ability.

The role of pattern recognition in children's exact enumeration of small numbers

Enumeration can be accomplished by subitizing, counting, estimation, and combinations of these processes. We investigated whether the dissociation between subitizing and counting can be observed in 4- to 6-year-olds and studied whether the maximum number of elements that can be subitized changes with age. To detect a dissociation between subitizing and counting, it is tested whether task manipulations have different effects in the subitizing than in the counting range. Task manipulations concerned duration of presentation of elements (limited, unlimited) and configuration of elements (random, line, dice). In Study 1, forty-nine 4- and 5-year-olds were tested with a computerized enumeration task. Study 2 concerned data from 4-, 5-, and 6-year-olds, collected with Math Garden, a computer-adaptive application to practice math. Both task manipulations affected performance in the counting, but not the subitizing range, supporting the conclusion that children use two distinct enumeration processes in the two ranges. In all age groups, the maximum number of elements that could be subitized was three. The strong effect of configuration of elements suggests that subitizing might be based on a general ability of pattern recognition.

A cross-sectional analysis of orienting of visuospatial attention in child and adult carriers of the fragile X premutation

Background

Fragile X premutation carriers (fXPCs) have an expansion of 55–200 CGG repeats in the FMR1 gene. Male fXPCs are at risk for developing a neurodegenerative motor disorder (fragile X-associated tremor/ataxia syndrome (FXTAS)) often accompanied by cognitive decline. Several broad domains are implicated as core systems of dysfunction in fXPCs, including perceptual processing of spatial information, orienting of attention to space, and inhibiting attention to irrelevant distractors. We tested whether orienting of spatial attention is impaired in fXPCs.

Methods

Participants were fXPCs or healthy controls (HCs) asymptomatic for FXTAS. In experiment 1, they were male and female children and adults (aged 7–45 years). They oriented attention in response to volitional (endogenous) and reflexive (exogenous) cues. In experiment 2, the participants were men (aged 18–48 years). They oriented attention in an endogenous cueing task that manipulated the amount of information in the cue.

Results

In women, fXPCs exhibited slower reaction times than HCs in both the endogenous and exogenous conditions. In men, fXPCs exhibited slower reaction times than HCs in the exogenous condition and in the challenging endogenous cueing task with probabilistic cues. In children, fXPCs did not differ from HCs.

Conclusions

Because adult fXPCs were slower even when controlling for psychomotor speed, results support the interpretation that a core dysfunction in fXPCs is the allocation of spatial attention, while perceptual processing and attention orienting are intact. These findings indicate the importance of considering age and sex when interpreting and generalizing studies of fXPCs.

Young adult male carriers of the fragile X premutation exhibit genetically modulated impairments in visuospatial tasks controlled for psychomotor speed

Background

A previous study reported enhanced psychomotor speed, and subtle but significant cognitive impairments, modulated by age and by mutations in the fragile X mental retardation 1 (FMR1) gene in adult female fragile X premutation carriers (fXPCs). Because male carriers, unlike females, do not have a second, unaffected FMR1 allele, male fXPCs should exhibit similar, if not worse, impairments. Understanding male fXPCs is of particular significance because of their increased risk of developing fragile X-associated tremor/ataxia syndrome (FXTAS).

Methods

Male fXPCs (n = 18) and healthy control (HC) adults (n = 26) aged less than 45 years performed two psychomotor speed tasks (manual and oral) and two visuospatial tasks (magnitude comparison and enumeration). In the magnitude comparison task, participants were asked to compare and judge which of two bars was larger. In the enumeration task, participants were shown between one and eight green bars in the center of the screen, and asked to state the total number displayed. Enumeration typically proceeds in one of two modes: subitizing, a fast and accurate process that works only with a small set of items, and counting, which requires accurate serial-object detection and individuation during visual search. We examined the associations between the performance on all tasks and the age, full-scale intelligent quotient, and CGG repeat length of participants.

Results

We found that in the magnitude comparison and enumeration tasks, male fXPCs exhibited slower reaction times relative to HCs, even after controlling for simple reaction time.

Conclusions

Our results indicate that male fXPCs as a group show impairments (slower reaction times) in numerical visuospatial tasks, which are consistent with previous findings. This adds to a growing body of literature characterizing the phenotype in fXPCs who are asymptomatic for FXTAS. Future longitudinal studies are needed to determine how these impairments relate to risk of developing FXTAS.

The development of individuation in autism

Evidence suggests that people with autism rely less on holistic visual information than typical adults. The current studies examine this by investigating core visual processes that contribute to holistic processing--namely, individuation and element grouping--and how they develop in participants with autism and typically developing (TD) participants matched for age, IQ, and gender. Individuation refers to the ability to "see" approximately four elements simultaneously; grouping elements can modify how many elements can be individuated. We examined these processes using two well-established paradigms, rapid enumeration and multiple object tracking (MOT). In both tasks, a performance limit of four elements in typical adults is thought to reflect individuation capacity. Participants with autism displayed a smaller individuation capacity than TD controls, regardless of whether they were enumerating static elements or tracking moving ones. To manipulate the holistic information available via element grouping, elements were arranged into a design in rapid enumeration, or moved together in MOT. Performance in participants with autism was affected to a similar degree as TD participants by element grouping, whether the manipulation helped or hurt performance, consistent with evidence that some types of gestalt/grouping information are processed typically in autism. There was substantial development from childhood to adolescence in the speed of individuation in those with autism, but not from adolescence to adulthood, a pattern distinct from TD participants. These results reveal how core visual processes function in autism, and provide insight into the architecture of vision (i.e., individuation appears distinct from visual strengths in autism, such as visual search).

Adult Female Fragile X Premutation Carriers Exhibit Age- and CGG Repeat Length-Related Impairments on an Attentionally Based Enumeration Task

The high frequency of the fragile X premutation in the general population and its emerging neurocognitive implications highlight the need to investigate the effects of the premutation on lifespan cognitive development. Until recently, cognitive function in fragile X premutation carriers (fXPCs) was presumed to be unaffected by the mutation. Although as a group fXPCs did not differ from healthy controls (HCs), we show that young adult female fXPCs show subtle age- and significant fragile X mental retardation 1 (FMR1) gene mutation-modulated cognitive function as tested by a basic numerical enumeration task. These results indicate that older women with the premutation and fXPCs with greater CGG repeat lengths were at higher risk for difficulties in the deployment of volitional attention required to count 5–8 items, but spared performance when spatial shifts of attention were minimized to subitize a few (1–3). Results from the current study add to a growing body of evidence that suggests the premutation allele is associated with a subtle phenotype and implies that the cognitive demands necessary for counting are less effectively deployed in female fXPCs compared to HCs.

Précis ofUnified theories of cognition

The book presents the case that cognitive science should turn its attention to developing theories of human cognition that cover the full range of human perceptual, cognitive, and action phenomena. Cognitive science has now produced a massive number of high-quality regularities with many microtheories that reveal important mechanisms. The need for integration is pressing and will continue to increase. Equally important, cognitive science now has the theoretical concepts and tools to support serious attempts at unified theories. The argument is made entirely by presenting an exemplar unified theory of cognition both to show what a real unified theory would be like and to provide convincing evidence that such theories are feasible. The exemplar is SOAR, a cognitive architecture, which is realized as a software system. After a detailed discussion of the architecture and its properties, with its relation to the constraints on cognition in the real world and to existing ideas in cognitive science, SOAR is used as theory for a wide range of cognitive phenomena: immediate responses (stimulus-response compatibility and the Sternberg phenomena); discrete motor skills (transcription typing); memory and learning (episodic memory and the acquisition of skill through practice); problem solving (cryptarithmetic puzzles and syllogistic reasoning); language (sentence verification and taking instructions); and development (transitions in the balance beam task). The treatments vary in depth and adequacy, but they clearly reveal a single, highly specific, operational theory that works over the entire range of human cognition, SOAR is presented as an exemplar unified theory, not as the sole candidate. Cognitive science is not ready yet for a single theory – there must be multiple attempts. But cognitive science must begin to work toward such unified theories.

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.

22q11.2 microdeletions: linking DNA structural variation to brain dysfunction and schizophrenia

Recent studies are beginning to paint a clear and consistent picture of the impairments in psychological and cognitive competencies that are associated with microdeletions in chromosome 22q11.2. These studies have highlighted a strong link between this genetic lesion and schizophrenia. Parallel studies in humans and animal models are starting to uncover the complex genetic and neural substrates altered by the microdeletion. In addition to offering a deeper understanding of the effects of this genetic lesion, these findings may guide analysis of other copy-number variants associated with cognitive dysfunction and psychiatric disorders.

Numerical competence in animals: Definitional issues, current evidence, and a new research agenda

Numerical competence is one of the many aspects of animal cognition that have enjoyed a resurgence of interest during the past decade. Evidence for numerical abilities in animals has followed a tortuous path to respectability, however, from Clever Hans, the counting horse, to modern experimental studies. Recent surveys of the literaturereveal theoretical as well as definitional confusion arising from inconsistent terminology for numerical processes and procedures. The term “counting” has been applied to situations having little to do with its meaning in the human literature. We propose a consistent vocabulary and theoretical framework for evaluating numerical competence. Relative numerousness judgments, subitizing, counting, and estimation may be the essential processes by which animals perform numerical discriminations. Ordinality, cardinality, and transitivity also play an important role in these processes. Our schema is applied to a variety of recent experimental situations. Some evidence of transfer is essential in demonstrating higher-order ability such as counting or “sense of number.” Those instances of numerical competence in which all viable alternatives to counting (e.g., subitizing) have been precluded, but no evidence of transfer has been demonstrated might be described as “protocounting.” To show that animals are capable of “true” counting future research will have to demonstrate generality across situations.