Ratio abstraction by 6-month-old infants

Human infants appear to be capable of the rudimentary mathematical operations of addition, subtraction, and ordering. To determine whether infants are capable of extracting ratios, we presented 6-month-old infants with multiple examples of a single ratio. After repeated presentations of this ratio, the infants were presented with new examples of a new ratio, as well as new examples of the previously habituated ratio. Infants were able to successfully discriminate two ratios that differed by a factor of 2, but failed to detect the difference between two numerical ratios that differed by a factor of 1.5. We conclude that infants can extract a common ratio across test scenes and use this information while examining new displays. The results support an approximate magnitude-estimation system, which has also been found in animals and human adults.     

Core multiplication in childhood

A dedicated, non-symbolic, system yielding imprecise representations of large quantities (approximate number system, or ANS) has been shown to support arithmetic calculations of addition and subtraction. In the present study, 5–7-year-old children without formal schooling in multiplication and division were given a task requiring a scalar transformation of large approximate numerosities, presented as arrays of objects. In different conditions, the required calculation was doubling, quadrupling, or increasing by a fractional factor (2.5). In all conditions, participants were able to represent the outcome of the transformation at above-chance levels, even on the earliest training trials. Their performance could not be explained by processes of repeated addition, and it showed the critical ratio signature of the ANS. These findings provide evidence for an untrained, intuitive process of calculating multiplicative numerical relationships, providing a further foundation for formal arithmetic instruction.     

The relationship between non-symbolic multiplication and division in childhood

Children without formal education in addition and subtraction are able to perform multi-step operations over an approximate number of objects. Further, their performance improves when solving approximate (but not exact) addition and subtraction problems that allow for inversion as a shortcut (e.g., a?+?b???b?=?a). The current study examines children's ability to perform multi-step operations, and the potential for an inversion benefit, for the operations of approximate, non-symbolic multiplication and division. Children were trained to compute a multiplication and division scaling factor (*2 or /2, *4 or /4), and were then tested on problems that combined two of these factors in a way that either allowed for an inversion shortcut (e.g., 8*4/4) or did not (e.g., 8*4/2). Children's performance was significantly better than chance for all scaling factors during training, and they successfully computed the outcomes of the multi-step testing problems. They did not exhibit a performance benefit for problems with the a*b/b structure, suggesting that they did not draw upon inversion reasoning as a logical shortcut to help them solve the multi-step test problems.     

Large-number addition and subtraction by 9-month-old infants

Do genuinely numerical computational abilities exist in infancy? It has recently been argued that previous studies putatively illustrating infants' ability to add and subtract tapped into specialized object-tracking processes that apply only with small numbers. This argument contrasts with the original interpretation that successful performance was achieved via a numerical system for estimating and calculating magnitudes. Here, we report that when continuous variables (such as area and contour length) are controlled, 9-month-old infants successfully add and subtract over numbers of items that exceed object-tracking limits. These results support the theory that infants possess a magnitude-based estimation system for representing numerosities that also supports procedures for numerical computation.     

Moving along the number line: operational momentum in nonsymbolic arithmetic

Can human adults perform arithmetic operations with large approximate numbers, and what effect, if any, does an internal spatial-numerical representation of numerical magnitude have on their responses? We conducted a psychophysical study in which subjects viewed several hundred short videos of sets of objects being added or subtracted from one another and judged whether the final numerosity was correct or incorrect. Over a wide range of possible outcomes, the subjects' responses peaked at the approximate location of the true numerical outcome and gradually tapered off as a function of the ratio of the true and proposed outcomes (Weber's law). Furthermore, an operational momentum effect was observed, whereby addition problems were overestimated and subtraction problems were underestimated. The results show that approximate arithmetic operates according to precise quantitative rules, perhaps analogous to those characterizing movement on an internal continuum.     

Operational momentum in large-number addition and subtraction by 9-month-olds

Recent studies on nonsymbolic arithmetic have illustrated that under conditions that prevent exact calculation, adults display a systematic tendency to overestimate the answers to addition problems and underestimate the answers to subtraction problems. It has been suggested that this operational momentum results from exposure to a culture-specific practice of representing numbers spatially; alternatively, the mind may represent numbers in spatial terms from early in development. In the current study, we asked whether operational momentum is present during infancy, prior to exposure to culture-specific representations of numbers. Infants (9-month-olds) were shown videos of events involving the addition or subtraction of objects with three different types of outcomes: numerically correct, too large, and too small. Infants looked significantly longer only at those incorrect outcomes that violated the momentum of the arithmetic operation (i.e., at too-large outcomes in subtraction events and too-small outcomes in addition events). The presence of operational momentum during infancy indicates developmental continuity in the underlying mechanisms used when operating over numerical representations.     

Non‐symbolic halving in an Amazonian indigene group

Much research supports the existence of an Approximate Number System (ANS) that is recruited by infants, children, adults, and non‐human animals to generate coarse, non‐symbolic representations of number. This system supports simple arithmetic operations such as addition, subtraction, and ordering of amounts. The current study tests whether an intuition of a more complex calculation, division, exists in an indigene group in the Amazon, the Mundurucu, whose language includes no words for large numbers. Mundurucu children were presented with a video event depicting a division transformation of halving, in which pairs of objects turned into single objects, reducing the array's numerical magnitude. Then they were tested on their ability to calculate the outcome of this division transformation with other large‐number arrays. The Mundurucu children effected this transformation even when non‐numerical variables were controlled, performed above chance levels on the very first set of test trials, and exhibited performance similar to urban children who had access to precise number words and a surrounding symbolic culture. We conclude that a halving calculation is part of the suite of intuitive operations supported by the ANS.     

How capuchin monkeys (Cebus apella) quantify objects and substances

Humans and nonhuman animals appear to share a capacity for nonverbal quantity representations. But what are the limits of these abilities? Results of previous research with human infants suggest that the ontological status of an entity as an object or a substance affects infants' ability to quantify it. We ask whether the same is true for another primate species-the New World monkey Cebus apella. We tested capuchin monkeys' ability to select the greater of two quantities of either discrete objects or a nonsolid substance. Participants performed above chance with both objects (Experiment 1) and substances (Experiment 2); in both cases, the observed performance was ratio dependent. This finding suggests that capuchins quantify objects and substances similarly and do so via analog magnitude representations.     

Children's and adults' judgments of equitable resource distributions

This study explored the criteria that children and adults use when evaluating the niceness of a character who is distributing resources. Four- and five-year-olds played the 'Giving Game', in which two puppets with different amounts of chips each gave some portion of these chips to the children. Adults played an analogous task that mimicked the situations presented to children in the Giving Game. For all groups of participants, we manipulated the absolute amount and proportion of chips given away. We found that children and adults used different cues to establish which puppet was nicer: 4-year-olds focused exclusively on absolute amount, 5-year-olds showed some sensitivity to proportion, and adults focused exclusively on proportion. These results are discussed in light of their implications for equity theory and for theories of the development of social evaluation.