Young Children Intuitively Divide Before They Recognize the Division Symbol.

Children bring intuitive arithmetic knowledge to the classroom before formal instruction in mathematics begins. For example, children can use their number sense to add, subtract, compare ratios, and even perform scaling operations that increase or decrease a set of dots by a factor of 2 or 4. However, it is currently unknown whether children can engage in a true division operation before formal mathematical instruction. Here we examined the ability of 6- to 9-year-old children and college students to perform symbolic and non-symbolic approximate division. Subjects were presented with non-symbolic (dot array) or symbolic (Arabic numeral) dividends ranging from 32 to 185, and non-symbolic divisors ranging from 2 to 8. Subjects compared their imagined quotient to a visible target quantity. Both children (Experiment 1 N = 89, Experiment 2 N = 42) and adults (Experiment 3 N = 87) were successful at the approximate division tasks in both dots and numeral formats. This was true even among the subset of children that could not recognize the division symbol or solve simple division equations, suggesting intuitive division ability precedes formal division instruction. For both children and adults, the ability to divide non-symbolically mediated the relation between Approximate Number System (ANS) acuity and symbolic math performance, suggesting that the ability to calculate non-symbolically may be a mechanism of the relation between ANS acuity and symbolic math. Our findings highlight the intuitive arithmetic abilities children possess before formal math instruction.

Increasing entropy reduces perceived numerosity throughout the lifespan.

Numerical illusions may provide a powerful window into the mechanisms that give rise to our visual number sense. Recent research has shown that similarly oriented elements appear more numerous than randomly oriented elements in an array. Here we examine whether the orientation coherence illusion is a more general byproduct of the effect of entropy on numerical information-processing. Participants engaged in an ordinal numerical comparison task where the color entropy of arrays was manipulated. We found that arrays with low color entropy were perceived as more numerous than arrays with high color entropy (Experiments 1 and 2), suggesting that the coherence illusion on numerosity perception is not specific to a particular visual property (e.g., orientation) but instead that the entropy of visual arrays more generally affects numerical processing. In Experiment 3, we explored the developmental trajectory of the color entropy effect in children aged 5 to 17 and found that the strength of the coherence illusion increases into adulthood, raising intriguing questions as to how perceptual experiences influence the progression of this numerosity illusion. We consider a recently proposed resource-rational model as a framework for understanding the entropy effect on numerosity perception under an information-theoretic perspective.

Number sense biases children's area judgments.

Humans are thought to use the approximate number system (ANS) to make quick approximations based on quantity even before learning to count. However, there has long been controversy regarding the salience of number versus other stimulus dimensions throughout development, including a recent proposal that number sense is derived from a sense of general magnitude. Here, we used a regression approach to disentangle numerical acuity from sensitivity to total surface area in both 5-year-old children and adults. We found that both children and adults displayed higher acuity when making numerosity judgments than total surface area judgments. Adults were largely able to ignore irrelevant stimulus features when making numerosity or total area judgments. Children were more biased by numerosity when making total area judgments than by total area when making numerosity judgments. These results provide evidence that number is more salient than total surface area even before the start of formal education and are inconsistent with the Sense of Magnitude proposal.

Similarly oriented objects appear more numerous.

Several non-numerical factors influence the numerical estimation of visual arrays, including the spacing of items and whether they are arranged randomly or symmetrically. Here we report a novel numerosity illusion we term the coherence illusion. When items in an array have a coherent orientation (all pointing in the same direction) they seem to be more numerous than when items are oriented randomly. Participants show parametric effects of orientation coherence in three distinct numerical judgment tasks. These findings are not predicted by any current model of numerical estimation. We discuss array entropy as a possible framework for explaining both the coherence illusion and the previously reported regular-random illusion.

Numerical encoding in early visual cortex.

The ability to estimate numerosity in a visual array arose early in evolution, develops early in human development, and is correlated with mathematical ability. Previous work with visually presented arrays indicates that the intraparietal sulcus (IPS) represents number. However, it is not clear if the number signal originates in IPS or is propagated from earlier visual areas. Previous work from our group has demonstrated a rapidly instantiated representation of number in low-level regions of visual cortex using the high temporal resolution of event-related electro-encephalography (EEG). Here, we use a rapid event-related functional magnetic resonance imaging (fMRI) paradigm and find convergent evidence for a number signal in low-level visual cortex (areas V1, V2, and V3). Employing a stringent set of stimulus controls, we demonstrate that this signal cannot be explained by the total extent of the array, the density of the items in the array, the aggregate visual area of the items, the size of individual items, the proportion of the array covered by items, nor the overall scale of the array and items. Our findings thus provide strong support for the hypothesis that number is rapidly and directly encoded early in the visual processing stream.

Pharmacological inactivation does not support a unique causal role for intraparietal sulcus in the discrimination of visual number.

The "number sense" describes the intuitive ability to quantify without counting. Single neuron recordings in non-human primates and functional imaging in humans suggest the intraparietal sulcus is an important neuroanatomical locus of numerical estimation. Other lines of inquiry implicate the IPS in numerous other functions, including attention and decision making. Here we provide a direct test of whether IPS has functional specificity for numerosity judgments. We used muscimol to reversibly and independently inactivate the ventral and lateral intraparietal areas in two monkeys performing a numerical discrimination task and a color discrimination task, roughly equilibrated for difficulty. Inactivation of either area caused parallel impairments in both tasks and no evidence of a selective deficit in numerical processing. These findings do not support a causal role for the IPS in numerical discrimination, except insofar as it also has a role in the discrimination of color. We discuss our findings in light of several alternative hypotheses of IPS function, including a role in orienting responses, a general cognitive role in attention and decision making processes and a more specific role in ordinal comparison that encompasses both number and color judgments.

The contributions of numerical acuity and non-numerical stimulus features to the development of the number sense and symbolic math achievement.

Numerical acuity, frequently measured by a Weber fraction derived from nonsymbolic numerical comparison judgments, has been shown to be predictive of mathematical ability. However, recent findings suggest that stimulus controls in these tasks are often insufficiently implemented, and the proposal has been made that alternative visual features or inhibitory control capacities may actually explain this relation. Here, we use a novel mathematical algorithm to parse the relative influence of numerosity from other visual features in nonsymbolic numerical discrimination and to examine the strength of the relations between each of these variables, including inhibitory control, and mathematical ability. We examined these questions developmentally by testing 4-year-old children, 6-year-old children, and adults with a nonsymbolic numerical comparison task, a symbolic math assessment, and a test of inhibitory control. We found that the influence of non-numerical features decreased significantly over development but that numerosity was a primary determinate of decision making at all ages. In addition, numerical acuity was a stronger predictor of math achievement than either non-numerical bias or inhibitory control in children. These results suggest that the ability to selectively attend to number contributes to the maturation of the number sense and that numerical acuity, independent of inhibitory control, contributes to math achievement in early childhood.

Significant Inter-Test Reliability across Approximate Number System Assessments.

The approximate number system (ANS) is the hypothesized cognitive mechanism that allows adults, infants, and animals to enumerate large sets of items approximately. Researchers usually assess the ANS by having subjects compare two sets and indicate which is larger. Accuracy or Weber fraction is taken as an index of the acuity of the system. However, as Clayton et al. (2015) have highlighted, the stimulus parameters used when assessing the ANS vary widely. In particular, the numerical ratio between the pairs, and the way in which non-numerical features are varied often differ radically between studies. Recently, Clayton et al. (2015) found that accuracy measures derived from two commonly used stimulus sets are not significantly correlated. They argue that a lack of inter-test reliability threatens the validity of the ANS construct. Here we apply a recently developed modeling technique to the same data set. The model, by explicitly accounting for the effect of numerical ratio and non-numerical features, produces dependent measures that are less perturbed by stimulus protocol. Contrary to their conclusion we find a significant correlation in Weber fraction across the two stimulus sets. Nevertheless, in agreement with Clayton et al. (2015) we find that different protocols do indeed induce differences in numerical acuity and the degree of influence of non-numerical stimulus features. These findings highlight the need for a systematic investigation of how protocol idiosyncrasies affect ANS assessments.

Rapid and Direct Encoding of Numerosity in the Visual Stream.

Humans are endowed with an intuitive number sense that allows us to perceive and estimate numerosity without relying on language. It is controversial, however, as to whether there is a neural mechanism for direct perception of numerosity or whether numerosity is perceived indirectly via other perceptual properties. In this study, we used a novel regression-based analytic method, which allowed an assessment of the unique contributions of visual properties, including numerosity, to explain visual evoked potentials of participants passively viewing dot arrays. We found that the human brain is uniquely sensitive to numerosity and more sensitive to changes in numerosity than to changes in other visual properties, starting extremely early in the visual stream: 75 ms over a medial occipital site and 180 ms over bilateral occipitoparietal sites. These findings provide strong evidence for the existence of a neural mechanism for rapidly and directly extracting numerosity information in the human visual pathway.

Using cognitive training studies to unravel the mechanisms by which the approximate number system supports symbolic math ability.

A picture is emerging that preverbal nonsymbolic numerical representations derived from the approximate number system (ANS) play an important role in mathematical development and sustained mathematical thinking. Functional imaging studies are revealing developmental trends in how the brain represents number. We propose that combining behavioral and neuroimaging techniques with cognitive training approaches will help identify the fundamental relationship between the ANS and symbolic mathematics. Understanding this relationship should ultimately benefit educators by providing ways to harness the ANS and hopefully improve math readiness in young children.

Non-invasive primate head restraint using thermoplastic masks.

BACKGROUND: The success of many neuroscientific studies depends upon adequate head fixation of awake, behaving animals. Typically, this is achieved by surgically affixing a head-restraint prosthesis to the skull. NEW METHOD: Here we report the use of thermoplastic masks to non-invasively restrain monkeys' heads. Mesh thermoplastic sheets become pliable when heated and can then be molded to an individual monkey's head. After cooling, the custom mask retains this shape indefinitely for day-to-day use. RESULTS: We successfully trained rhesus macaques (Macaca mulatta) to perform cognitive tasks while wearing thermoplastic masks. Using these masks, we achieved a level of head stability sufficient for high-resolution eye-tracking and intracranial electrophysiology. COMPARISON WITH EXISTING METHOD: Compared with traditional head-posts, we find that thermoplastic masks perform at least as well during infrared eye-tracking and single-neuron recordings, allow for clearer magnetic resonance image acquisition, enable freer placement of a transcranial magnetic stimulation coil, and impose lower financial and time costs on the lab. CONCLUSIONS: We conclude that thermoplastic masks are a viable non-invasive form of primate head restraint that enable a wide range of neuroscientific experiments.

Modeling the approximate number system to quantify the contribution of visual stimulus features.

The approximate number system (ANS) subserves estimation of the number of items in a set. Typically, ANS function is assessed by requiring participants to compare the number of dots in two arrays. Accuracy is determined by the numerical ratio of the sets being compared, and each participant's Weber fraction (w) provides a quantitative index of ANS acuity. When making numerical comparisons, however, performance is also influenced by non-numerical features of the stimuli, such as the size and spacing of dots. Current models of numerosity comparison do not account for these effects and consequently lead to different estimates of w depending on the methods used to control for non-numerical features. Here we proffer a new model that teases apart the effects of ANS acuity from the effects of non-numerical stimulus features. The result is an estimate of w that is a more theoretically valid representation of numerical acuity and novel terms that denote the degree to which a participant's perception of number is affected by non-numerical features. We tested this model in a sample of 20 adults and found that, by correctly attributing errors due to non-numerical stimulus features, the w obtained was more reliable across different stimulus conditions. We found that although non-numerical features biased numerosity discriminations in all participants, number was the primary feature driving discriminations in most of them. Our findings support the idea that, while numerosity is a distinct visual quantity, the internal representation of number is tightly bound to the representation of other magnitudes. This tool for identifying the different effects of the numerical and non-numerical features of a stimulus has important implications not only for the behavioral investigation of the ANS, but also for the collection and analyses of neural data sets associated with ANS function.

Lemurs and macaques show similar numerical sensitivity.

We investigated the precision of the approximate number system (ANS) in three lemur species (Lemur catta, Eulemur mongoz, and Eulemur macaco flavifrons), one Old World monkey species (Macaca mulatta) and humans (Homo sapiens). In Experiment 1, four individuals of each nonhuman primate species were trained to select the numerically larger of two visual arrays on a touchscreen. We estimated numerical acuity by modeling Weber fractions (w) and found quantitatively equivalent performance among all four nonhuman primate species. In Experiment 2, we tested adult humans in a similar procedure, and they outperformed the four nonhuman species but showed qualitatively similar performance. These results indicate that the ANS is conserved over the primate order.

Malleability of the approximate number system: effects of feedback and training.

Prior research demonstrates that animals and humans share an approximate number system (ANS), characterized by ratio dependence and that the precision of this system increases substantially over human development. The goal of the present research was to investigate the malleability of the ANS (as measured by Weber fraction) in adult subjects in response to feedback and to explore the relationship between ANS acuity and acuity on another magnitude comparison task. We tested each of 20 subjects over six 1-h sessions. The main findings were that (a) Weber fractions rapidly decreased when trial-by-trial feedback was introduced in the second session and remained stable over continued training, (b) Weber fractions remained steady when trial-by-trial feedback was removed in session 6, (c)Weber fractions from the number comparison task were positively correlated with Weber fractions from a line length comparison task, (d) improvement in Weber fractions in response to feedback for the number task did not transfer to the line length task, (e) finally, the precision of the ANS was positively correlated with math, but not verbal, standardized aptitude scores. Potential neural correlates of the perceptual information and decision processes are considered, and predictions regarding the neural correlates of ANS malleability are discussed.