Intuitive mapping between nonsymbolic quantity and observed action across development

Adults' concurrent processing of numerical and action information yields bidirectional interference effects consistent with a cognitive link between these two systems of representation. This link is in place early in life: infants create expectations of congruency across numerical and action-related stimuli (i.e., a small [large] hand aperture associated with a smaller [larger] numerosity). Although these studies point to a developmental continuity of this mapping, little is known about the later development and thus how experience shapes such relationships. We explored how number-action intuitions develop across early and later childhood using the same methodology as in adults. We asked 3-, 6-, and 8-year-old children, as well as adults, to relate the magnitude of an observed action (a static hand shape, open vs. closed, in Experiment 1; a dynamic hand movement, opening vs. closing, in Experiment 2) to either a small or large nonsymbolic quantity (numerosity in Experiment 1 and numerosity and/or object size in Experiment 2). From 6 years of age, children started performing in a systematic congruent way in some conditions, but only 8-year-olds (added in Experiment 2) and adults performed reliably above chance in this task. We provide initial evidence that early intuitions guiding infants' mapping between magnitude across nonsymbolic number and observed action are used in an explicit way only from late childhood, with a mapping between action and size possibly being the most intuitive. An initial coarse mapping between number and action is likely modulated with extensive experience with grasping and related actions directed to both arrays and individual objects.

Children's understanding of the abstract logic of counting

When children learn to count, do they understand its logic independent of the number list that they learned to count with? Here we tested CP-knowers' (ages three to five) understanding of how counting reveals a set's cardinality, even when non-numerical lists are used to count. Participants watched an agent count unobservable objects in two boxes and were asked to identify the larger set. Participants successfully identified the box with more objects when the agent counted using their familiar number list (Experiment 1) and when the agent counted using a non-numeric ordered list, as long as the items in the list were not linguistically used as number words (Experiments 2-3). Additionally, children's performance was strongly influenced by visual cues that helped them link the list's order to representations of magnitude (Experiment 4). Our findings suggest that three- to six-year-olds who can count also understand how counting reveals a set's cardinality, but they require additional time to understand how symbols on any arbitrary ordered list can be used as numerals.

Absence perception and the philosophy of zero

Zero provides a challenge for philosophers of mathematics with realist inclinations. On the one hand it is a bona fide cardinal number, yet on the other it is linked to ideas of nothingness and non-being. This paper provides an analysis of the epistemology and metaphysics of zero. We develop several constraints and then argue that a satisfactory account of zero can be obtained by integrating (1) an account of numbers as properties of collections, (2) work on the philosophy of absences, and (3) recent work in numerical cognition and ontogenetic studies.

Is an intact hippocampus necessary for answering 3 × 3? - Evidence from Alzheimer's disease

Recent evidence has suggested that the hippocampus supports learning and retrieval of arithmetic facts during childhood and adolescence. Whether the hippocampus is also involved in retrieving overlearned arithmetic facts (such as 3 × 5 = 15) during adult age is open for investigation. In this study, we assessed whether patients with hippocampal atrophy due to Alzheimer's disease (AD) are still able to retrieve overlearned arithmetic facts from memory. Sixteen patients (n = 13 with AD, n = 3 with Mild Cognitive Impairment - MCI) were evaluated using standard radiological, neurological, and neuropsychological test procedures. We adopted a multiple single-case analysis in order to acknowledge possible dissociations between hippocampal degeneration and intact arithmetic fact retrieval. All patients performed a neuropsychological screening battery assessing episodic memory as well as arithmetic processing, and underwent a 3-Tesla MRI procedure. A morphometric analysis comprising estimation of both cortical thickness and hippocampal volume, which also included a subfield analysis, was conducted. All patients had marked hippocampal atrophy (bilateral n = 15, unilateral n = 1) in comparison to healthy matched controls and showed deficits in episodic memory (delayed recall). However, 13 out of 16 patients performed in the average range of standardised norms during retrieval of overlearned arithmetic facts (i.e. multiplication tables). Our results suggest that intact retrieval of consolidated arithmetic facts from memory does not depend on the integrity of the hippocampus. This is in line with the view that the hippocampus plays a dynamic and time-limited role in arithmetic processing. While the hippocampus seems to be necessary for learning and consolidating new arithmetic facts in memory, it might not be critically involved in retrieving arithmetic facts when these are well consolidated in memory.

Confidence judgments during ratio comparisons reveal a Bayesian bias

Rational numbers are essential in mathematics and decision-making but humans often and erroneously rely on the magnitude of the numerator or denominator to determine the relative size of a quotient. The source of this flawed whole number strategy is poorly understood. Here we test the Bayesian hypothesis that the human bias toward large values in the numerator or denominator of a ratio estimate is the result of higher confidence in large samples. Larger values are considered a better (more certain) instance of that ratio than the same ratio composed of smaller values. We collected confidence measures explicitly (Experiment 1) and implicitly (Experiment 2) during subjects' comparisons of non-symbolic proportions (images with arrays of orange and blue dots). We manipulated the discernibility of the fractions to control difficulty and varied the cardinality and congruency of the numerators, denominators, and ratio values (e.g. 8/20 vs. 5/10 and 16/40 vs. 10/20). The results revealed that subjects' confidence during ratio comparisons was modulated by the numerical magnitude of the fraction's components, consistent with a Bayesian perception of relative ratios. The results suggest that the large number bias could arise from greater confidence in large samples.

Experimental Evidence From Newborn Chicks Enriches Our Knowledge on Human Spatial-Numerical Associations

Núñez and Fias raised concerns on whether our results demonstrate a linear number-space mapping. Patro and Nuerk urge caution on the use of animal models to understand the origin (cultural vs. biological) of the orientation of spatial-numerical association. Here, we discuss why both objections are unfounded.

Mapping number to space in the two hemispheres of the avian brain

Pre-verbal infants and non-human animals associate small numbers with the left space and large numbers with the right space. Birds and primates, trained to identify a given position in a sagittal series of identical positions, whenever required to respond on a left/right oriented series, referred the given position starting from the left end. Here, we extended this evidence by selectively investigating the role of either cerebral hemisphere, using the temporary monocular occlusion technique. In birds, lacking the corpus callosum, visual input is fed mainly to the contralateral hemisphere. We trained 4-day-old chicks to identify the 4th element in a sagittal series of 10 identical elements. At test, the series was identical but left/right oriented. Test was conducted in right monocular, left monocular or binocular condition of vision. Right monocular chicks pecked at the 4th right element; left monocular and binocular chicks pecked at the 4th left element. Data on monocular chicks demonstrate that both hemispheres deal with an ordinal (sequential) task. Data on binocular chicks indicate that the left bias is linked to a right hemisphere dominance, that allocates the attention toward the left hemispace. This constitutes a first step towards understanding the neural basis of number space mapping.

A cross-sectional developmental examination of the SNARC effect in a visually-guided grasping task

The present study documents the influence of numerical processing on hand and space use during a reach-to-grasp task. Three questions regarding the SNARC (spatial-numerical association of response codes) effect were asked: (1) would the SNARC effect influence hand and/or space preference for grasping?; (2) would the SNARC effect be demonstrated during the processing of one-digit numbers, two-digit numbers, or both?; and (3) would developmental age influence the strength of the SNARC effect? A total of 84 participants in three age/school level groups (Primary, Secondary, and Post-secondary) took part in the study. Two identical sets of small wooden blocks numbered from 0 to 19 were used. Each set was presented to the right and to the left of each participant. A number was called and participants were asked to find and grasp a block with the corresponding number as fast and accurately as possible. Hand and space used (L/R) was recorded for each grasp. Number magnitude was shown to influence the selection of hand and hemi-space in accordance with the SNARC effect. In the small percentage of trials where the left hand was used, it was more commonly recruited to grasp blocks displaying low numbers than high numbers. Participants grasped blocks from left and right space with equal frequency, but respectively left/right space was accessed more often for blocks displaying low/high numbers. Regression analyses revealed that developmental age is a powerful predictor of the SNARC effect on hand and space selection for grasping. This study provides the first description of the SNARC effect on hand and space preference for the reach-to-grasp action. Results are discussed with relevant literature of numerical processing in the human brain.

Residual number processing in dyscalculia

Developmental dyscalculia – a congenital learning disability in understanding numerical concepts – is typically associated with parietal lobe abnormality. However, people with dyscalculia often retain some residual numerical abilities, reported in studies that otherwise focused on abnormalities in the dyscalculic brain. Here we took a different perspective by focusing on brain regions that support residual number processing in dyscalculia. All participants accurately performed semantic and categorical colour-decision tasks with numerical and non-numerical stimuli, with adults with dyscalculia performing slower than controls in the number semantic tasks only. Structural imaging showed less grey-matter volume in the right parietal cortex in people with dyscalculia relative to controls. Functional MRI showed that accurate number semantic judgements were maintained by parietal and inferior frontal activations that were common to adults with dyscalculia and controls, with higher activation for participants with dyscalculia than controls in the right superior frontal cortex and the left inferior frontal sulcus. Enhanced activation in these frontal areas was driven by people with dyscalculia who made faster rather than slower numerical decisions; however, activation could not be accounted for by response times per se, because it was greater for fast relative to slow dyscalculics but not greater for fast controls relative to slow dyscalculics. In conclusion, our results reveal two frontal brain regions that support efficient number processing in dyscalculia.

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Dyscalculics (DD) show congenital number impairment due to parietal abnormalities.

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However DD often show residual number skills which have not been studied before.

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We studied the brain networks supporting residual skills and individual differences

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DD: reduced parietal grey-matter, accurate but slower than controls in number tasks

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Faster DD responses over-activated two frontal areas