Darko Odic

Intuitive sense of number correlates with math scores on college-entrance examination

Many educated adults possess exact mathematical abilities in addition to an approximate, intuitive sense of number, often referred to as the Approximate Number System (ANS). Here we investigate the link between ANS precision and mathematics performance in adults by testing participants on an ANS-precision test and collecting their scores on the Scholastic Aptitude Test (SAT), a standardized college-entrance exam in the USA. In two correlational studies, we found that ANS precision correlated with SAT-Quantitative (i.e., mathematics) scores. This relationship remained robust even when controlling for SAT-Verbal scores, suggesting a small but specific relationship between our primitive sense for number and formal mathematical abilities.

Developmental change in the acuity of approximate number and area representations.

From very early in life, humans can approximate the number and surface area of objects in a scene. The ability to discriminate between 2 approximate quantities, whether number or area, critically depends on the ratio between the quantities, with the most difficult ratio that a participant can reliably discriminate known as the Weber fraction. While developmental improvements in the Weber fraction have been demonstrated for number, the developmental trajectory of improvement in area discrimination remains unknown. Here we investigated whether the development of area discrimination parallels that of number discrimination. We tested forty 3- to 6-year-old children and adults in both a number and an area discrimination task in which participants selected the greater of 2 quantities across a range of ratios. We used formal psychophysical models to derive, for each participant and each age group, the Weber fraction for both number and area discrimination. We found that, like number acuity, area acuity steadily improves during childhood. However, we also found area acuity to be consistently higher than number acuity, suggesting a potential difference in the underlying mechanisms that encode and/or represent approximate area and approximate number. We discuss these findings in the context of quantity processing and its development.

Children’s mappings between number words and the approximate number system

Humans can represent number either exactly--using their knowledge of exact numbers as supported by language, or approximately--using their approximate number system (ANS). Adults can map between these two systems--they can both translate from an approximate sense of the number of items in a brief visual display to a discrete number word estimate (i.e., ANS-to-Word), and can generate an approximation, for example by rapidly tapping, when provided with an exact verbal number (i.e., Word-to-ANS). Here we ask how these mappings are initially formed and whether one mapping direction may become functional before the other during development. In two experiments, we gave 2-5 year old children both an ANS-to-Word task, where they had to give a verbal number response to an approximate presentation (i.e., after seeing rapidly flashed dots, or watching rapid hand taps), and a Word-to-ANS task, where they had to generate an approximate response to a verbal number request (i.e., rapidly tapping after hearing a number word). Replicating previous results, children did not successfully generate numerically appropriate verbal responses in the ANS-to-Word task until after 4 years of age--well after they had acquired the Cardinality Principle of verbal counting. In contrast, children successfully generated numerically appropriate tapping sequences in the Word-to-ANS task before 4 years of age--well before many understood the Cardinality Principle. We further found that the accuracy of the mapping between the ANS and number words, as captured by error rates, continues to develop after this initial formation of the interface. These results suggest that the mapping between the ANS and verbal number representations is not functionally bidirectional in early development, and that the mapping direction from number representations to the ANS is established before the reverse.

Changing the precision of preschoolers' approximate number system representations changes their symbolic math performance.

From early in life, humans have access to an approximate number system (ANS) that supports an intuitive sense of numerical quantity. Previous work in both children and adults suggests that individual differences in the precision of ANS representations correlate with symbolic math performance. However, this work has been almost entirely correlational in nature. Here we tested for a causal link between ANS precision and symbolic math performance by asking whether a temporary modulation of ANS precision changes symbolic math performance. First, we replicated a recent finding that 5-year-old children make more precise ANS discriminations when starting with easier trials and gradually progressing to harder ones, compared with the reverse. Next, we show that this brief modulation of ANS precision influenced childrens performance on a subsequent symbolic math task but not a vocabulary task. In a supplemental experiment, we present evidence that children who performed ANS discriminations in a random trial order showed intermediate performance on both the ANS task and the symbolic math task, compared with children who made ordered discriminations. Thus, our results point to a specific causal link from the ANS to symbolic math performance.

Acquisition of the Cardinal Principle Coincides with Improvement in Approximate Number System Acuity in Preschoolers.

Human mathematical abilities comprise both learned, symbolic representations of number and unlearned, non-symbolic evolutionarily primitive cognitive systems for representing quantities. However, the mechanisms by which our symbolic (verbal) number system becomes integrated with the non-symbolic (non-verbal) representations of approximate magnitude (supported by the Approximate Number System, or ANS) are not well understood. To explore this connection, forty-six children participated in a 6-month longitudinal study assessing verbal number knowledge and non-verbal numerical acuity. Cross-sectional analyses revealed a strong relationship between verbal number knowledge and ANS acuity. Longitudinal analyses suggested that increases in ANS acuity were most strongly related to the acquisition of the cardinal principle, but not to other milestones of verbal number acquisition. These findings suggest that experience with culture and language is intimately linked to changes in the properties of a core cognitive system.

The precision of mapping between number words and the approximate number system predicts children's formal math abilities.

Children can represent number in at least two ways: by using their non-verbal, intuitive approximate number system (ANS) and by using words and symbols to count and represent numbers exactly. Furthermore, by the time they are 5years old, children can map between the ANS and number words, as evidenced by their ability to verbally estimate numbers of items without counting. How does the quality of the mapping between approximate and exact numbers relate to childrens math abilities? The role of the ANS-number word mapping in math competence remains controversial for at least two reasons. First, previous work has not examined the relation between verbal estimation and distinct subtypes of math abilities. Second, previous work has not addressed how distinct components of verbal estimation-mapping accuracy and variability-might each relate to math performance. Here, we addressed these gaps by measuring individual differences in ANS precision, verbal number estimation, and formal and informal math abilities in 5- to 7-year-old children. We found that verbal estimation variability, but not estimation accuracy, predicted formal math abilities, even when controlling for age, expressive vocabulary, and ANS precision, and that it mediated the link between ANS precision and overall math ability. These findings suggest that variability in the ANS-number word mapping may be especially important for formal math abilities.

Solving the correspondence problem within the Ternus display: The differential-activation theory

The Ternus display produces a bistable illusion of motion: at very short interstimulus intervals (ISIs; < 30 ms) observers perceive element motion while at longer ISIs (> 30 ms) observers perceive group motion. In experiment 1, however, we find that, when the Ternus displays ISI contains an occluding box, group motion is mostly eliminated. These results do not fit the predictions made by the short-range/long-range two-process theory [Braddick and Adlard, 1978, in Visual Psychophysics and Psychology (New York: Academic Press)]. We propose that the differential-activation theory (Gilroy et al, 2001 Perception & Psychophysics 63 847–861) accounts for our results. We then extend the differential-activation theory as an explanatory mechanism for the Ternus display in experiment 2 by selectively placing an occluder over the first, second, or third Ternus display element. As predicted by the differential-activation theory, the occlusion of the far-left element produced a normal distribution of group motion increasing with ISI, while the occlusion of the other two elements produced an illusion of occluded elements remaining stationary throughout the display. Furthermore, as predicted by the differential-activation theory, each moving element was assigned to its nearest neighbour, producing, in the case of second and third element occlusion, a novel Ternus display motion illusion where only two out of three elements are perceived as moving.

Chapter 12 - The Precision and Internal Confidence of Our Approximate Number Thoughts

The approximate number system (ANS) is a portion of your cognition that is active across your entire life, and is in the business of giving you a rapid and intuitive sense for numbers and their relations (e.g., how many blue versus yellow dots are on a screen or how many voices you hear speaking at the dinner table). In this chapter, we discuss approximate number representations, Weber fractions, and individual differences in these representations (e.g., individuals who may struggle with ANS number perceptions that are less precise than is typical). A key organizing theme throughout the chapter is “internal confidence.” We suggest that internal confidence (a technical notion related to signal fidelity) is the major causal factor for the observed links between our sense of approximate number and our performance in school mathematics.

A Developmental Vocabulary Assessment for Parents (DVAP): Validating Parental Report of Vocabulary Size in 2- to 7-Year-Old Children

Measuring individual differences in childrens emerging language abilities is important to researchers and clinicians alike. The 2 most widely used methods for assessing childrens vocabulary both have limitations: Experimenter-administered tests are time-consuming and expensive, and parent questionnaires have only been designed for children up to 37 months of age. Here, we test the validity of a new assessment to fill this gap: the Developmental Vocabulary Assessment for Parents (DVAP). In 4 experiments, we assess the reliability of this measure and its concurrent and predictive validity in samples of 2- to 7-year-old children. We found that the DVAP provides a rapid, cost-effective alternative to experimenter-administered vocabulary tests for children.

Eye movements reveal distinct encoding patterns for number and cumulative surface area in random dot arrays

Humans can quickly and intuitively represent the number of objects in a scene using visual evidence through the Approximate Number System (ANS). But the computations that support the encoding of visual number-the transformation from the retinal input into ANS representations-remain controversial. Two types of number encoding theories have been proposed: those arguing that number is encoded through a dedicated, enumeration computation, and those arguing that visual number is inferred from nonnumber specific visual features, such as surface area, density, convex hull, etc. Here, we attempt to adjudicate between these two theories by testing participants on both a number and a cumulative area task while also tracking their eye-movements. We hypothesize that if approximate number and surface area depend on distinct encoding computations, saccadic signatures should be distinct for the two tasks, even if the visual stimuli are identical. Consistent with this hypothesis, we find that discriminating number versus cumulative area modulates both where participants look (i.e., participants spend more time looking at the more numerous set in the number task and the larger set in the cumulative area task), and how participants look (i.e., cumulative area encoding shows fewer, longer saccades, while number encoding shows many short saccades and many switches between targets). We further identify several saccadic signatures that are associated with task difficulty and correct versus incorrect trials for both dimensions. These results suggest distinct encoding algorithms for number and cumulative area extraction, and thereby distinct representations of these dimensions.