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 possible 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 findings, 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.     

Stochastic λ-calculi: An extended abstract

It is shown how the operators in the “graph model” for λ-calculus (which can function as a programming language for Recursive Function Theory) can be expanded to allow for “random combinators”. The result then is a model for a new language for random algorithms.     

The role of vision in the development of finger-number interactions: Finger-counting and finger-montring in blind children

Previous research has suggested that the use of the fingers may play a functional role in the development of a mature counting system. However, the role of developmental vision in the elaboration of a finger numeral representation remains unexplored. In the current study, 14 congenitally blind children and 14 matched sighted controls undertook three different test batteries that examined (a) general cognitive abilities, (b) the spontaneous use of finger-counting and finger-montring strategies (where “finger-montring” is a term used to characterize the way people raise their fingers to show numerosities to other people), and (c) the canonicity level of the finger-counting and finger-montring habits. Compared with sighted controls, blind children used their fingers less spontaneously to count and in a less canonical way to count and show quantities. These results demonstrate that the absence of vision precludes the development of a typical finger numeral representation and suggest that the use of canonical finger-counting and finger-montring strategies relies on the visual recognition of particular hand shapes.     

The role of attention in subitizing

The process of rapidly and accurately enumerating small numbers of items without counting, i.e. subitizing, is often believed to rest on parallel preattentive processes. However, the possibility that enumeration of small numbers of items would also require attentional processes has remained an open question. The present study is the first that directly contrasts the preattentive and attentive models of subitizing. We used an inattentional blindness paradigm to manipulate the availability of attentional resources during enumeration. In the inattention condition, the items to be enumerated were presented unexpectedly while participants focused on a line length comparison task. Divided- and full-attention conditions were also included. The results showed that only numbers one and two could be enumerated when the effects of attention were minimized. Freeing attentional resources increased the enumeration accuracies considerably, including for number two. The results suggest that even for enumerating small numbers, the attentional demands increase as the number of objects increases.     

Parallel non-verbal enumeration is constrained by a set-based limit

Adults can represent approximate numbers of items independently of language. This approximate number system can discriminate and compare entities as varied as dots, sounds, or actions. But can multiple different types of entities be enumerated in parallel and stored as independent numerosities? Subjects who were prevented from verbally counting watched an experimenter hide sequences of objects in two locations. The number of object types, which contrasted in category membership, color, shape, and texture, varied from 1 to 5, and object types were completely temporally intermixed. Subjects were then asked how many objects of each type were in each location. In three experiments, subjects successfully enumerated the objects of each type in each location when 1-3 types were presented, but failed with 4 or 5 types, regardless of the total number of objects seen. Thus, adults can perform simultaneous enumeration of multiple sets that unfold in temporally intermixed fashion, but are limited to 3 such sets at a time. Furthermore, they perform these parallel enumerations in the absence of training or instruction, and can do so for sets of objects that are hidden in distinct locations. The convergence of this 3-set capacity limit with the 3-item capacity limit widely observed in studies of working memory suggests that each enumeration requires a single slot in memory, and that storage in memory is required before enumeration can occur.     

Enumeration versus multiple object tracking: the case of action video game players

Here, we demonstrate that action video game play enhances subjects' ability in two tasks thought to indicate the number of items that can be apprehended. Using an enumeration task, in which participants have to determine the number of quickly flashed squares, accuracy measures showed a near ceiling performance for low numerosities and a sharp drop in performance once a critical number of squares was reached. Importantly, this critical number was higher by about two items in video game players (VGPs) than in non-video game players (NVGPs). A following control study indicated that this improvement was not due to an enhanced ability to instantly apprehend the numerosity of the display, a process known as subitizing, but rather due to an enhancement in the slower more serial process of counting. To confirm that video game play facilitates the processing of multiple objects at once, we compared VGPs and NVGPs on the multiple object tracking task (MOT), which requires the allocation of attention to several items over time. VGPs were able to successfully track approximately two more items than NVGPs. Furthermore, NVGPs trained on an action video game established the causal effect of game playing in the enhanced performance on the two tasks. Together, these studies confirm the view that playing action video games enhances the number of objects that can be apprehended and suggest that this enhancement is mediated by changes in visual short-term memory skills.     

A context-free language for binary multinomial processing tree models

This paper provides a new formalization for the class of binary multinomial processing tree (BMPT) models, and theorems for the class are developed using the formalism. MPT models are a popular class of information processing models for categorical data in specific cognitive paradigms. They have a recursive structure that is productively described with the tools of formal language and computation theory. We provide an axiomatization that characterizes BMPT models as strings in a context-free language, and then we add model-theoretic axioms and definitions to interpret the strings as parameterized probabilistic models for categorical data. The language for BMPT models is related to the Full Binary Tree language, a well-studied context-free language. Once BMPT models are viewed from the perspective of the Full Binary Tree language, a number of theoretical and computational results can be developed. In particular, we have a number of results concerning the enumerations of BMPT models as well as the identifiability of subclasses of these models.     

Infants’ auditory enumeration: Evidence for analog magnitudes in the small number range

Vigorous debate surrounds the issue of whether infants use different representational mechanisms to discriminate small and large numbers. We report evidence for ratio-dependent performance in infants’ discrimination of small numbers of auditory events, suggesting that infants can use analog magnitudes to represent small values, at least in the auditory domain. Seven-month-old infants in the present study reliably discriminated two from four tones (a 1:2 ratio) in Experiment 1, when melodic and continuous temporal properties of the sequences were controlled, but failed to discriminate two from three tones (a 2:3 ratio) under the same conditions in Experiment 2. A third experiment ruled out the possibility that infants in Experiment 1 were responding to greater melodic variety in the four-tone sequences. The discrimination function obtained here is the same as that found for infants’ discrimination of large numbers of visual and auditory items at a similar age, as well as for that obtained for similar-aged infants’ duration discriminations, and thus adds to a growing body of evidence suggesting that human infants may share with adults and nonhuman animals a mechanism for representing quantities as “noisy” mental magnitudes.     

Enumeration of collective entities by 5-month-old infants

Recent findings suggest that infants are capable of distinguishing between different numbers of objects, and of performing simple arithmetical operations. But there is debate over whether these abilities result from capacities dedicated to numerical cognition, or whether infants succeed in such experiments through more general, non-numerical capacities, such as sensitivity to perceptual features or mechanisms of object tracking. We report here a study showing that 5-month-olds can determine the number of collective entities -- moving groups of items -- when non-numerical perceptual factors such as contour length, area, density, and others are strictly controlled. This suggests both that infants can represent number per se, and that their grasp of number is not limited to the domain of objects.