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Background: Several theorists maintain that exact number abilities rely on language-relevant processes whereas approximate number calls on visuo-spatial skills. We chose two genetic disorders, Williams syndrome and Down's syndrome, which differ in their relative abilities in verbal versus spatial skills, to examine this hypothesis. Five experiments assessed number skills in these two genetic syndromes and in their mental age (MA) and chronological age (CA) matched controls. Methods: Experiment 1 used a preferential looking paradigm with infants and toddlers to measure sensitivity to changes in numerosity. Experiment 2 measured reaction times in older children and adults in a numerosity comparison task with dots in a random pattern. Experiment 3 comprised a number battery that measured various forms of counting and simple arithmetic. Results: The WS infants displayed a level of performance equal to that of their CA-controls, whereas the DS infants failed to reach even the level of their MA-controls. By contrast, the older DS children and adults outstripped the older WS group in their numerosity abilities, with different patterns of errors in the two clinical groups. Conclusions: Differences in the infant and adult number phenotypes between these two genetic disorders are discussed with reference to the processing styles used by each group and how these might impact on their developmental trajectories. Theoretically, we highlight our contention that one cannot infer the infant starting state from the adult end state. Rather, the development process itself must be taken into account.

While associations between number and space, in the form of a spatially oriented numerical representation, have been extensively reported in human adults, the origins of this phenomenon are still poorly understood. The commonly accepted view is that this number-space association is a product of human invention, with accounts proposing that culture, symbolic knowledge, and mathematics education are at the roots of this phenomenon. Here we show that preverbal infants aged 7 months, who lack symbolic knowledge and mathematics education, show a preference for increasing magnitude displayed in a left-to-right spatial orientation. Infants habituated to left-to-right oriented increasing or decreasing numerical sequences showed an overall higher looking time to new left-to-right oriented increasing numerical sequences at test (Experiment 1). This pattern did not hold when infants were presented with the same ordinal numerical information displayed from right to left (Experiment 2). The different pattern of results was congruent with the presence of a malleable, context-dependent baseline preference for increasing, left-to-right oriented, numerosities (Experiment 3). These findings are suggestive of an early predisposition in humans to link numerical order with a left-to-right spatial orientation, which precedes the acquisition of symbolic abilities, mathematics education, and the acquisition of reading and writing skills.

The SNARC effect, consisting of a systematic association between numbers and lateralized response, reflects the mental representation of magnitude along a left-to-right mental number line (Dehaene et al. in J Exp Psychol 122:371-396, 1993). Critically, this effect has been reported in the classification of overlearned non-numerical sequences such as letters, days and months (Gevers et al. in Cognition 87:B87-B95, 2003 and Cortex 40:171-172, 2004) suggesting that ordinal, rather than magnitude information, is critical for spatial coding. This study tests the hypothesis of an oriented spatial representation as the privileged way of mentally organizing serial information, by looking for stimulus-response compatibility effects in the processing of a newly acquired arbitrary sequence. Here we report an association between ordinal position of the items and spatial response preference for both order-relevant and order-irrelevant tasks. These results suggest that any ordered information, even when order is not intrinsically relevant to it, is spontaneously mapped in the representational space. This spatial representation is likely to acquire a left-to-right orientation, at least in western cultures.

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It is proposed that arithmetical facts are organized in memory in terms of a principle that is unique to numbers-the cardinal magnitudes of the addends. This implies that sums such as 4 + 2 and 2 + 4 are represented, and searched for, in terms of the maximum and minimum addends. This in turn implies that a critical stage in solving an addition problem is deciding which addend is the larger. The COMP model of addition fact retrieval incorporates a comparison stage, as well as a retrieval stage and a pronunciation stage. Three tasks, using the same subjects, were designed to assess the contribution of these three stages to retrieving the answers to single-digit addition problems. Task 3 was the addition task, which examined whether reaction times (RTs) were explained by the model; Task 1 was a number naming task to assess the contribution of the pronunciation stage; Task 2 was a magnitude comparison task to assess the contribution, if any, of the comparison stage. A regression equation that included just expressions of these three stages was found to account for 71% of the variance. It is argued that the COMP model fits not only the adult RT data better than do alternatives, but also the evidence from development of additional skills.The basic phenomena involved in single-digit addition performance are robust, widely replicated and well known, yet there has been much controversy as to the psychological processes involved. It is generally agreed that competent adults use some mixture of memory retrieval and procedures, but there is little agreement as to how the addition facts are represented and Requests for reprints should be sent to Brian Butterworth,

Despite the high incidence of numerical deficits in neurological patients, little attention has been paid to the development of diagnostic tools. In fact, most of the published reports on acquired numerical disorders, whether single case or group studies, do not refer to standardised measures of performance providing little, if any, control data specifically collected for the examination. In this study we present data of 282 healthy controls of different age groups and educational levels in a new battery of Number Processing and Calculation (NPC). The NPC battery includes a total of 35 tasks, assessing different counting abilities, various aspects of number comprehension (such as parity and magnitude judgements), numerical transcoding, calculation, arithmetic reasoning and conceptual knowledge. Special attention is paid to the assessment of different calculation abilities, including simple fact retrieval, rule based processing, mental calculation and written calculation in all four operations. Moreover, text problem solving is assessed as well as the understanding of arithmetic principles. Thus, the NPC battery differs from the EC 301 battery proposed by Deloche et al., 1994 (Dellatolas, Deloche, Basso, & Claros-Salinas, 2001) and allows a more fine grained diagnosis which is relevant for planning targeted interventions. The battery is easy to administer and does not require special materials or equipment.