The agent-based approach emphasizes the importance of learning through organism-environment interaction. This approach is part of a recent trend in computational models of learning and development toward studying autonomous organisms that are embedded in virtual or real environments. In this paper we introduce the concepts of online and offline sampling and highlight the role of online sampling in agent-based models. After comparing the strengths of each approach for modeling particular developmental phenomena and research questions, we describe a recent agent-based model of infant causal perception. We conclude by discussing some of the present limitations of agent-based models and suggesting how these challenges may be addressed. © 2001 Academic Press Computational models of learning and development are playing an increasingly critical role in child development research (Cassidy, 1990;

A new choice task was used to explore infants' spontaneous representations of more and less. Ten- and 12-month-old infants saw crackers placed sequentially into two containers, then were allowed to crawl and obtain the crackers from the container they chose. Infants chose the larger quantity with comparisons of 1 versus 2 and 2 versus 3, but failed with comparisons of 3 versus 4, 2 versus 4, and 3 versus 6. Success with visible arrays ruled out a motivational explanation for failure in the occluded 3-versus-6 condition. Control tasks ruled out the possibility that presentation duration guided choice, and showed that presentation complexity was not responsible for the failure with larger numbers. When crackers were different sizes, total surface area or volume determined choice. The infants' pattern of success and failure supports the hypothesis that they relied on object-file representations, comparing mental models via total volume or surface area rather than via one-to-one correspondence between object files.

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Are abstract representations of number -- representations that are independent of the particular type of entities that are enumerated -- a product of human language or culture, or do they trace back to human infancy? To investigate these questions, four experiments investigated whether human infants discriminate between sequences of actions (jumps of a puppet) on the basis of numerosity. At 6 months, infants successfully discriminated 4- vs. 8-jump sequences, when the continuous variables of sequence duration, jump duration, jump rate, jump interval and duration and extent of motion were controlled and rhythm was eliminated. In contrast, infants failed to discriminate 2- vs. 4-jump sequences, suggesting that infants fail to form cardinal number representations of small numbers of actions. Infants also failed to discriminate between sequences of 4 vs. 6 jumps at 6 months, and succeeded at 9 months, suggesting that infants' number representations are imprecise and increase in precision with age. All of these findings agree with those of studies using visual-spatial arrays and auditory sequences, providing evidence that a single, abstract system of number representation is present and functional in infancy. Infants' Enumeration of Actions: Numerical Discrimination and its Signature Limits Recent research provides evidence that human infants discriminate between large sets of elements on the basis of numerosity, when a variety of continuous quantitative variables are controlled. For example, 6-month-old infants discriminate visual arrays of 8 vs. 16 dots when array size and density, dot size, summed area and brightness, and summed contour length are equated either during habituation or during test (e.g., Brannon, 2002;Brannon, Abbott, & Lutz, in press; Xu & Spelke, 2000; Xu...

In two experiments, a manual search task explored 12- to 14-month-old infants' representations of small sets of objects. In this paradigm, patterns of searching revealed the number of objects infants represented as hidden in an opaque box. In Experiment 1, we obtained the set-size signature of object-file representations: infants succeeded at representing precisely 1, precisely 2, and precisely 3 objects in the box, but failed at representing 4 (or even that 4 is greater than 2). In Experiment 2, we showed that infants' expectations about the contents of the box were based on number of individual objects, and not on a continuous property such as total object volume. These findings support the hypothesis that infants maintained representations of individuals, that object-files were the underlying means of representing these individuals, and that object-file models can be compared via one-to-one correspondence to establish numerical equivalence.

Four experiments used a preferential looking method to investigate six-month-old infants' capacity to represent numerosity in visual-spatial displays. Building on previous findings that such infants discriminate between arrays of 8 vs. 16 discs, but not 8 vs. 12 discs (Xu & Spelke, 2000), Experiments 1 and 2 investigated whether infants' numerosity discrimination depends on the ratio of the two set sizes with even larger numerosities. Infants successfully discriminated between arrays of 16 vs. 32 discs, but not 16 vs. 24 discs, providing evidence that their discrimination shows the set-size ratio signature of numerosity discrimination in human adults, children, and many non-human animals. Experiments 3 and 4 addressed a controversy concerning infants' ability to discriminate large numerosities (observed under conditions that control for total filled area, array size and density, item size, and correlated properties such as brightness: Brannon, 2002; Xu, 2003; Xu & Spelke, 2000) vs. small numerosities (not observed under conditions that control for total contour length: Clearfield & Mix, 1999). To investigate the sources of these differing findings, Experiment 3 tested infants' large-number discrimination with controls for contour length, and Experiment 4 tested small-number discrimination with controls for total filled area. Infants successfully discriminated the large-number displays but showed no evidence of discriminating the small-number displays. These findings provide evidence that infants have robust abilities to represent large numerosities. In contrast, infants may fail to represent small numerosities in visual-spatial arrays with continuous quantity controls, consistent with the thesis that separate systems serve to represent large vs. small numerosities. A we...

Both animals and humans represent and compare numerical quantities, but only humans have evolved multi-digit place-value number systems. This article develops a Spatial Number Network, or SpaN, model to explain how these shared numerical capabilities are computed using a spatial representation of number quantities in the Where cortical processing stream, notably the inferior parietal cortex. Multi-digit numerical representations that obey a place-value principle are proposed to arise through learned interactions between categorical language representations in the What cortical processing stream and the Where spatial representation. Learned semantic categories that symbolize separate digits, as well as place markers like `ty,' `hundred,' and `thousand,' are associated through learning with the corresponding spatial locations of the Where representation. Such What-to-Where auditory-to-visual learning generates place-value numbers as an emergent property, and may be compared with other examples of multi-modal cross-modality learning, including synesthesia. The model quantitatively simulates error rates in quantification and numerical comparison tasks, and reaction times for number priming and numerical assessment and comparison tasks. In the Where cortical process, transient responses to inputs are integrated before they activate an ordered spatial map that selectively responds to the number of events in a sequence and exhibits Weber law properties. Numerical comparison arises from activity pattern changes across the spatial map that define a `directional comparison wave.' Variants of these model mechanisms have elsewhere been used to explain data about other Where stream phenomena, such as motion perception, spatial attention, and target tracking. The model is compared wi...

Developmental research suggests that some of the mechanisms that underlie numerical cognition are present and functional in human infancy. To investigate these mechanisms and their developmental course, psychologists have turned to behavioral and electrophysiological methods using briefly presented displays. These methods, however, depend on the assumption that young infants can extract numerical information rapidly. Here we test this assumption and begin to investigate the speed of numerical processing in 5-month-old infants. Infants successfully discriminated between arrays of 4 vs. 8 dots on the basis of number when a new array appeared every 2 seconds, but not when a new array appeared every 1.0 or 1.5 seconds. These results suggest alternative interpretations of past findings, provide constraints on the design of future experiments, and introduce a new method for probing infants' enumeration process. Further experiments using this method provide initial evidence that infants' enumeration mechanism operates in parallel and yields increasingly accurate numerical representations over time, as does the enumeration mechanism used by adults in symbolic and nonsymbolic tasks. Over the past two decades, a wealth of research has focused on the nature and origins of numerical knowledge. Although reports that infants represent small numbers of objects have been interpreted in multiple ways (e.g., Carey, 2001; Clearfield & Mix, 1999; Feigenson, Carey & Spelke, 2002; Simon, 1997; Starkey & Cooper, 1980; Treiber & Wilcox, 1984; Wynn, 1992; Wynn, Bloom & Chiang, 2002), recent research provides clear evidence that infants as young as 6 months represent the approximate cardinal values of large sets of entities. In studies using a looking time method, for example, 6-month-old infant...

Mathematicians frequently evoke their “intuition” when they are able to quickly and automatically solve a problem, with little introspection into their insight. Cognitive neuroscience research shows that mathematical intuition is a valid concept that can be studied in the laboratory in reduced paradigms, and that relates to the availability of “core knowledge” associated with evolutionarily ancient and specialized cerebral subsystems. As an illustration, I discuss the case of elementary arithmetic. Intuitions of numbers and their elementary transformations by addition and subtraction are present in all human cultures. They relate to a brain system, located in the intraparietal sulcus of both hemispheres, which extracts numerosity of sets and, in educated adults, maps back and forth between numerical symbols and the corresponding quantities. This system is available to animal species and to preverbal human infants. Its neuronal organization is increasingly being uncovered, leading to a precise mathematical theory of how we perform tasks of number comparison or number naming. The next challenge will be to understand how education changes our core intuitions of number.