This article describes how a theoretical analysis and empirical findings regarding number sense led to the development of an educational intervention that produces large and rapid increases in low-income children's mathematical knowledge. Roughly an hour of playing a simple numerical board game based on the mental number line construct led to substantial gains in their knowledge of numerical magnitudes, counting, numeral identification, number line estimation, and arithmetic. The gains remained present two months after the last game-playing session. Both physical features of the game board and the way in which children interact with it proved important in the size of the gains. Reasons why such a brief intervention produces such substantial learning were discussed.
One important cause of very low attainment in arithmetic (dyscalculia) seems to be a core deficit in an inherited foundational capacity for numbers. According to one set of hypotheses, arithmetic ability is built on an inherited system responsible for representing approximate numerosity. One account holds that this is supported by a system for representing exactly a small number (less than or equal to four4) of individual objects. In these approaches, the core deficit in dyscalculia lies in either of these systems. An alternative proposal holds that the deficit lies in an inherited system for sets of objects and operations on them (numerosity coding) on which arithmetic is built. I argue that a deficit in numerosity coding, not in the approximate number system or the small number system, is responsible for dyscalculia. Nevertheless, critical tests should involve both longitudinal studies and intervention, and these have yet to be carried out.
Geometry, etymologically the "science of measuring the Earth", is a mathematical formalization of space. Just as formal concepts of number may be rooted in an evolutionary ancient system for perceiving numerical quantity, the fathers of geometry may have been inspired by their perception of space. Is the spatial content of formal Euclidean geometry universally present in the way humans perceive space, or is Euclidean geometry a mental construction, specific to those who have received appropriate instruction? The spatial content of the formal theories of geometry may depart from spatial perception for two reasons: first, because in geometry, only some of the features of spatial figures are theoretically relevant; and second, because some geometric concepts go beyond any possible perceptual experience. Focusing in turn on these two aspects of geometry, we will present several lines of research on US adults and children from the age of three years, and participants from an Amazonian culture, the Mundurucu. Almost all the aspects of geometry tested proved to be shared between these two cultures. Nevertheless, some aspects involve a process of mental construction where explicit instruction seem to play a role in the US, but that can still take place in the absence of instruction in geometry.
Attaching meaning to arbitrary symbols (i.e. words) is a complex and lengthy process. In the case of numbers, it was previously suggested that this process is grounded on two early pre-verbal systems for numerical quantification: the approximate number system (ANS or 'analogue magnitude'), and the object tracking system (OTS or 'parallel individuation'), which children are equipped with before symbolic learning. Each system is based on dedicated neural circuits, characterized by specific computational limits, and each undergoes a separate developmental trajectory. Here, I review the available cognitive and neuroscientific data and argue that the available evidence is more consistent with a crucial role for the ANS, rather than for the OTS, in the acquisition of abstract numerical concepts that are uniquely human.
This chapter reviews the behavioral evidence for numerical capacities in animals. We show that animal number representations are ratio dependent, subject to the same numerical illusions as humans, illict semantic congruity effects, map across sensory modalities, enter into arithmetic computations, and support a precursor to the zero concept. The review illustrates that nonhuman animals share with humans a basic capacity to quantify the world around them that likely serves as a foundation for the rich and uniquely human mathematical mind.
It has been suggested that space, time and number are represented on a common subjective scale. Saccadic eye movements provide a fascinating test. Saccades compress the perceived magnitude of spatial separations and temporal intervals to approximately half of their true value. The question arises as to whether saccades also compress number. They do, and compression follows a very similar time course for all three attributes: it is maximal at saccadic onset and decreases to veridicality within a window of approximately 50 ms. These results reinforce the suggestion of a common perceptual metric, which is probably mediated by the intraparietal cortex; they further suggest that before each saccade the common metric for all three is reset, possibly to pave the way for a fresh analysis of the post-saccadic situation.
Among the most fundamental of mental capacities is the ability to represent magnitude information such as physical size, numerosity, and duration. Accumulating evidence suggests that such cues are processed as part of a general magnitude system with shared more vs less representational structure. Here we review recent research with young children and preverbal infants suggesting that this system is operational from early in human life and may be far more general than currently believed. We present data suggesting that from early in development, the representation of magnitude extends across sensory modalities (e.g., vision and audition) and beyond the "big three" dimensions of spatial extent, number, and time. We also speculate about particular properties of the general magnitude system, including the potentially special role of space in grounding magnitude information.
The ability to integrate information over time is a fundamental operation of the brain, but the neuronal mechanisms underlying it are poorly understood. Here we describe how rhythmic variation in the activity of neurons-a common observation in neural recordings-could provide one such mechanism. In particular, we review a model that uses interference between neuronal oscillations to account for the spatio-temporal firing patterns of place cells in the hippocampus and grid cells in entorhinal cortex. The mechanism integrates movement information by varying the frequencies of the oscillations according to running velocity, thus mapping temporal oscillations into phase codes for distance traveled and, in combination with environmental information, spatial location. This provides a specific model of path integration and spatial orientation, but also provides an example of how, more generally, dynamic neural oscillations could be used to integrate and encode information. In this vein, we suggest how representations of sequential (or "rank") order and numerosity could also be generated using such a system.
An understanding of sensory and motor processing will require elucidation of the mechanisms by which the brain tells time. Open questions relate to whether timing relies on dedicated or intrinsic mechanisms and whether distinct mechanisms underlie timing across scales and modalities. Although experimental and theoretical studies support the notion that neural circuits are intrinsically capable of sensory timing on short scales, few general models of motor timing have been proposed. For one class of models, population clocks, it is proposed that time is encoded in the time-varying patterns of activity of a population of neurons. We argue that population clocks emerge from the internal dynamics of recurrently connected networks, are biologically realistic and account for many aspects of motor timing.
How does the human brain support abstract concepts such as seven or square? Studies of non-human animals, of human infants, and of children and adults in diverse cultures suggest these concepts arise from a set of cognitive systems that are phylogenetically ancient, innate, and universal across humans: systems of core knowledge. Two of these systems-for tracking small numbers of objects and for assessing, comparing and combining the approximate cardinal values of sets-capture the primary information in the system of positive integers. Two other systems-for representing the shapes of small-scale forms and the distances and directions of surfaces in the large-scale navigable layout-capture the primary information in the system of Euclidean plane geometry. As children learn language and other symbol systems, they begin to combine their core numerical and geometrical representations productively, in uniquely human ways. These combinations may give rise to the first truly abstract concepts at the foundations of mathematics.
Observers represent only a tiny fraction of the total amount of information available at any given moment. This small amount of information has been quantified: throughout the lifespan we typically maintain only three or four visual items in working memory at a time. Yet we are also capable of impressive quantificational feats: we can count the objects in arrays containing hundreds, or estimate that a scene contains "about 100" people. Given the strict limits on working memory, how do observers accomplish this? Here I propose that although working memory is limited in the number of items it can store, it is also flexible in what counts as an item. At least three types of representations can serve as an item in working memory: an individual object, a set, and an ensemble. Shifting between these types of representations allows us to bypass some of the strict constraints imposed by WM, thereby empowering quantification.
Since the seminal observation of the SNARC effect by Dehaene, Bossini, and Giraux [(1993) Journal of Experimental Psychology: General, 122(3), 371-396] several studies have indicated the existence of an intrinsic-automatic spatial coding of number magnitudes. In the first part of this chapter we summarize recent work with healthy participants and expand on this original claim. Some of our evidence can be used to support a theory of spatial mapping of mental numbers onto mental space where smaller numbers are associated with the left and larger numbers with the right side of space. In the second part of the chapter we review investigations of spatial neglect and relate them to "small number neglect", which initially seemed to provide crucial support for a tight link between mechanisms of spatial attentional orienting and the mental manipulation of number magnitudes [Zorzi et al. (2002). Nature 417, 138-139]. We will see that although left unilateral neglect after right-brain damage can occur in both visual and number space, recent behavioral dissociations, controlled studies and neuroanatomical correlations have consistently confirmed the functional dissociation of these two deficits and the absence of a causal effect of lateralized spatial-attentional impairments on numerical cognition. Finally, based on recent data gathered from experiments specifically designed to generate a mismatch in the "default" association of small numbers with the left side of space and large numbers with the right side of space, we argue that the pathological deviation toward larger numbers shown by right-brain-damaged patients in the bisection of number intervals may not arise from a basic spatial-attentional impairment. Taken together, these findings suggest that to assume a close phenomenological, functional and anatomical equivalence between orienting in visual space and orienting in representational number space could be partially misleading. It is concluded that careful reassessment of empirical evidence and consideration of the combined contributions of sensorimotor, conceptual, and working memory factors to mathematical cognition may provide a more coherent understanding of the adaptive interaction between spatial and mathematical thought.
Numerous studies have claimed that eye movements are a critical or even obligatory part of explicit counting whereas here we will show that counting relies on a set of attention pointers that individuate targets of interest and specify their locations independently of eye movements. We demonstrate that explicit counting can proceed to very high numbers without error in afterimages where eye movements are not possible. Previous studies with afterimages had used displays too dense to allow individuation of items by attention: the displays suffered from crowding. We also show that explicit counting is defeated for displays of more than about six items in motion because there is no mechanism available to mark already-counted items and keep that marking linked to the items as they move. In this case, only the approximate number system can operate and, interestingly, this system shows fairly accurate estimates, rather than the underestimation typically seen for denser displays.
