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What are the brain and cognitive systems that allow humans to play baseball, compute square roots, cook soufflés, or navigate the Tokyo subways? It may seem that studies of human infants and of non-human animals will tell us little about these abilities, because only educated, enculturated human adults engage in organized games, formal mathematics, gourmet cooking, or map-reading. In this chapter, we argue against this seemingly sensible conclusion. When human adults exhibit complex, uniquely human, culture-specific skills, they draw on a set of psychological and neural mechanisms with two distinctive properties: they evolved before humanity and thus are shared with other animals, and they emerge early in human development and thus are common to infants, children, and adults. These core knowledge systems form the building blocks for uniquely human skills. Without them we wouldn’t be able to learn about different kinds of games, mathematics, cooking, or maps. To understand what is special about human intelligence, therefore, we must study both the core knowledge systems on which it rests and the mechanisms by which these systems are orchestrated to permit new kinds of concepts and cognitive processes. What is core knowledge? A wealth of research on non-human primates and on human

Human adults are thought to possess two dissociable systems to represent numbers: an approximate quantity system akin to a mental number line, and a verbal system capable of representing numbers exactly. Here, we study the interface between these two systems using an estimation task. Observers were asked to estimate the approximate numerosity of dot arrays. We show that, in the absence of calibration, estimates are largely inaccurate: responses increase monotonically with numerosity, but underestimate the actual numerosity. However, insertion of a few inducer trials, in which participants are explicitly (and sometimes misleadingly) told that a given display contains 30 dots, is sufficient to calibrate their estimates on the whole range of stimuli. Based on these empirical results, we develop a model of the mapping between the numerical symbols and the representations of numerosity on the number line.

Can human adults perform arithmetic operations with large approximate numbers, and what effect, if any, does an internal spatial–numerical representation of numerical magnitude have on their responses? We conducted a psychophysical study in which subjects viewed several hundred short videos of sets of objects being added or subtracted from one another and judged whether the final numerosity was correct or incorrect. Over a wide range of possible outcomes, the subjects ’ responses peaked at the approximate location of the true numerical outcome and gradually tapered off as a function of the ratio of the true and proposed outcomes (Weber’s law). Furthermore, an operational momentum effect was observed, whereby addition problems were overestimated and subtraction problems were underestimated. The results show that approximate arithmetic operates according to precise quantitative rules, perhaps analogous to those characterizing movement on an internal continuum. Human adults possess an ability to estimate and manipulate approximate numerical magnitudes, which has been termed number sense (Dehaene, 1997). This ability appears to be largely independent of language and other symbol systems, since it is present in both infants (Xu & Spelke, 2000) and other animal species (Brannon & Roitman, 2003;

This article explores the effect of external representations on numeric tasks. Through several minor modifications on the previously reported two-digit number comparison task, we obtained different results. Rather than a holistic comparison, we found parallel comparison. We argue that this difference was a reflection of different representational forms: the comparison was based on internal representations in previous studies but on external representations in our present study. This representational effect was discussed under a framework of distributed number representations. We propose that in numerical tasks involving external representations, numbers should be considered as distributed representations and the behavior in these tasks should be considered as the interactive processing of internal and external information through the interplay of perceptual and cognitive processes. We suggest that theories of number representations and process models of numerical tasks should consider external representations as an essential component.

SUMMARY—Amid ongoing public speculation about the reasons for sex differences in careers in science and mathematics, we present a consensus statement that is based on the best available scientific evidence. Sex differences in science and math achievement and ability are smaller for the mid-range of the abilities distribution than they are for those with the highest levels of achievement and ability. Males are more variable on most measures of quantitative and visuospatial ability, which necessarily results in more males at both high- and low-ability extremes; the reasons why males are often more variable remain elusive. Successful careers in math and science require many types of cognitive abilities. Females tend to excel in verbal abilities, with large differences between females and males found when assessments include writing

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Specific types of multi-net neural computing systems can give improved generalisation performance over single network solutions. In single-net systems learning is one way in which good generalisation can be achieved, where a number of neurons are combined through a process of collaboration. In this thesis we examine collaboration in multi-net systems through in-situ learning. Here we explore how generalisation can be improved through learning in the components and their combination at the same time. To achieve this we present a formal way in which multi-net systems can be described in an attempt to provide a method with which the general properties of multi-net systems can be explored. We then explore two novel learning algorithms for multi-net systems that exploit in-situ learning, evaluating them in comparison with multi-net and single-net solutions. Last, we simulate two cognitive processes with in-situ learning to examine the interaction between different numerical abilities in multi-net systems. Using single-net simulations of subitization and counting we build a multi-net simulation of quantification. Similarly, we combine single-net simulations of the fact retrieval and ‘count all ’ addition strategies into a multi-net simulation of addition. Our results are encouraging, with improved generalisation performance obtained on benchmark problems, and the interaction of strategies with in-situ learning used to describe well known numerical ability phenomena. This learning through interaction in connectionist simulations we call integrated learning.

Summary. In this article we review scientific work on the perception of time, that is, the feeling of time as perceived by individuals. The phenomenon of time being felt passing faster with growing age is well-known, and there are numerous interesting studies to shed light on the question why this is so. A considerable number of these are based on studies in psychology and social sciences. Others range from symptoms of the ageing process and related symptoms of decreasing memory capacities over symptoms of diminished activity. Again other explanations, quite different in nature from the preceding ones, involve event intensities in the life of individuals. The relative decrease of interesting new events as one grows older, is seen as an important factor contributing to the feeling that time is thinned out. The last type of possible explanations can be made more explicit in a mathematical model. Quantitative conclusions about the rate of decrease of the feeling of time can be drawn, and, interestingly, without restrictive assumptions. It is shown that, under this model, the feeling of time is thinned out at least logarithmically. Numerical constants will depend on specific hypotheses but

Running title: Weber’s law and neuronal spike train