Variability signatures distinguish verbal from nonverbal counting for both large and small numbers

Arithmetic and the brain

Recent studies in human neuroimaging, primate neurophysiology, and developmental neuropsychology indicate that the human ability for arithmetic has a tangible cerebral substrate. The human intraparietal sulcus is systematically activated in all number tasks and could host a central amodal representation of quantity. Areas of the precentral and inferior prefrontal cortex also activate when subjects engage in mental calculation. A monkey analogue of these parieto-frontal regions has recently been identified, and a neuronal population code for number has been characterized. Finally, pathologies of this system, leading to acalculia in adults or to developmental dyscalculia in children, are beginning to be understood, thus paving the way for brain-oriented intervention studies.

“One-thousandone … one-thousandtwo …”: Chronometric counting violates the scalar property in interval timing

Weber's law applied to interval timing is called the scalar property. A hallmark of timing in the seconds-to-minutes range, the scalar property is characterized by proportionality between the standard deviation of a response distribution and the duration being timed. In this temporal reproduction study, we assessed whether the scalar property was upheld when participants chronometrically counted three visually presented durations (8, 16, and 24 sec) as compared with explicitly timing durations without counting. Accuracy for timing and accuracy for counting were similar. However, whereas timing variability showed the scalar property, counting variability did not. Counting variability across intervals was accurately modeled by summing a random variable representing an individual count. A second experiment replicated the first and demonstrated that task differences were not due to presentation order or practice effects. The distinct psychophysical properties of counting and timing behaviors argue for greater attention to participant strategies in timing studies.

Chimpanzees (Pan troglodytes) Respond to Nonvisible Sets After One-by-One Addition and Removal of Items

Two chimpanzees (Pan troglodytes) made numerousness judgments of nonvisible sets of items. In Experiment 1, 1-10 items were dropped 1 at a time into an opaque cup, and then an additional 1-10 items were dropped 1 at a time into another opaque cup. The chimpanzees' performance levels were high and were more dependent on factors indicative of an analogue-magnitude mechanism for representation of set size than on an object file mechanism. In Experiment 2, a 3rd visible set was made available after the sequential presentation of the first 2 sets. The chimpanzees again performed at high levels in selecting the largest of the 3 sets. In Experiment 3, 1 of the 2 initially presented sets was reduced in number by the sequential removal of 1, 2, or 3 items. Both chimpanzees performed above chance levels for the removal of 1, but not more than 1, item.

Origins of number sense. Large-number discrimination in human infants

Four experiments investigated infants' sensitivity to large, approximate numerosities in auditory sequences. Prior studies provided evidence that 6-month-old infants discriminate large numerosities that differ by a ratio of 2.0, but not 1.5, when presented with arrays of visual forms in which many continuous variables are controlled. The present studies used a head-turn preference procedure to test for infants' numerosity discrimination with auditory sequences designed to control for element duration, sequence duration, interelement interval, and amount of acoustic energy. Six-month-old infants discriminated 16 from 8 sounds but failed to discriminate 12 from 8 sounds, providing evidence that the same 2.0 ratio limits numerosity discrimination in auditory-temporal sequences and visual-spatial arrays. Nine-month-old infants, in contrast, successfully discriminated 12 from 8 sounds, but not 10 from 8 sounds, providing evidence that numerosity discrimination increases in precision over development, prior to the emergence of language or symbolic counting.

Human nonverbal counting estimated by response production and verbal report

In three experiments, people were shown sequential displays and were prevented from verbal counting by being required to perform other cognitive tasks. In Experiment 1, the subjects were shown three target (target = 8, 16, or 32) sequences of colored geometric shapes. On occasional question trials, the subjects were asked to estimate the target number after the final item in the sequence. On other test trials, items continued to appear beyond the target, and the subjects estimated the target manually by tapping a space bar. In Experiments 2 and 3, a matching-to-sample procedure required the subjects to estimate the same sequence of items (target = 8, 11, 14, 17, or 20) both verbally and manually. The results indicated that (1) manual and verbal estimates closely approximated target size in Experiments 1 and 2, (2) coefficients of variation were constant across target size, and (3) correlations between manual and verbal estimates were positive in Experiments 2 and 3. Requiring the subjects to perform a counting task during presentation of items led to underestimation of number in Experiment 3.

Numerosity discrimination in infants: evidence for two systems of representations

Two experiments investigated numerosity discrimination in 6-month-old infants, comparing their performance on both large numbers (4 vs. 8 elements) and small numbers (2 vs. 4 elements) with both total filled area and total contour length controlled for. These studies provide the first direct comparison between discrimination of small and large numbers in infants with the same methodology, the same type of stimuli, and the same continuous variable controls. Results showed that infants succeeded in discriminating 4 from 8 elements but failed to discriminate 2 from 4 elements, providing evidence for the existence of two systems of number representations in infancy.

Coding of cognitive magnitude: compressed scaling of numerical information in the primate prefrontal cortex

Whether cognitive representations are better conceived as language-based, symbolic representations or perceptually related, analog representations is a subject of debate. If cognitive processes parallel perceptual processes, then fundamental psychophysical laws should hold for each. To test this, we analyzed both behavioral and neuronal representations of numerosity in the prefrontal cortex of rhesus monkeys. The data were best described by a nonlinearly compressed scaling of numerical information, as postulated by the Weber-Fechner law or Stevens' law for psychophysical/sensory magnitudes. This nonlinear compression was observed on the neural level during the acquisition phase of the task and maintained through the memory phase with no further compression. These results suggest that certain cognitive and perceptual/sensory representations share the same fundamental mechanisms and neural coding schemes.

Approximate quantities and exact number words: dissociable systems

Numerical abilities are thought to rest on the integration of two distinct systems, a verbal system of number words and a non-symbolic representation of approximate quantities. This view has lead to the classification of acalculias into two broad categories depending on whether the deficit affects the verbal or the quantity system. Here, we test the association of deficits predicted by this theory, and particularly the presence or absence of impairments in non-symbolic quantity processing. We describe two acalculic patients, one with a focal lesion of the left parietal lobe and Gerstmann's syndrome and another with semantic dementia with predominantly left temporal hypometabolism. As predicted by a quantity deficit, the first patient was more impaired in subtraction than in multiplication, showed a severe slowness in approximation, and exhibited associated impairments in subitizing and numerical comparison tasks, both with Arabic digits and with arrays of dots. As predicted by a verbal deficit, the second patient was more impaired in multiplication than in subtraction, had intact approximation abilities, and showed preserved processing of non-symbolic numerosities.