Dynamics Versus Development in Numerosity Estimation: A Computational Model Accurately Predicts a Developmental Reversal

Perceptual judgments result from a dynamic process, but little is known about the dynamics of number-line estimation. A recent study proposed a computational model that combined a model of trial-to-trial changes with a model for the internal scaling of discrete numbers. Here, we tested a surprising prediction of the model—a situation in which children's estimates of numerosity would be better than those of adults. Consistent with the model simulations, task contexts led to a clear developmental reversal: children made more adult-like, linear estimates when to-be-estimated numbers were descending over trials (i.e., backward condition), whereas adults became more like children with logarithmic estimates when numbers were ascending (i.e., forward condition). In addition, adults’ estimates were subject to inter-trial differences regardless of stimulus order. In contrast, children were not able to use the trial-to-trial dynamics unless stimuli varied systematically, indicating the limited cognitive capacity for dynamic updates. Together, the model adequately predicts both developmental and trial-to-trial changes in number-line tasks.     

Baby arithmetic: one object plus one tone

Recent studies using a violation-of-expectation task suggest that preverbal infants are capable of recognizing basic arithmetical operations involving visual objects. There is still debate, however, over whether their performance is based on any expectation of the arithmetical operations, or on a general perceptual tendency to prefer visually familiar and complex displays. Here we provide new evidence that 5-month-old infants recognize basic arithmetic operations across sensory modalities. Using a violation-of-expectation task that eliminated the possibility of the familiarity and complexity preference, 5-month-old infants were presented alternatively with two types of arithmetical events: the expected, correct outcomes of operations (1 object+1 tone=2 objects and 1 object+2 tones=3 objects) and the unexpected, incorrect ones (1 object+2 tones=2 objects and 1 object+1 tone=3 objects). Results showed that subjects looked significantly longer at the unexpected events than at the expected events, suggesting that infants are able to recognize basic arithmetic operations across sensory modalities.     

When Intertemporal Rewards Are Hedonic,Larger Units of Wait Time Boost Patience

In intertemporal choices between smaller‐sooner (SS) and larger‐later (LL) rewards, five studies reveal that patience for the LL option is influenced by an interactive effect of the measurement units used to express wait time (large vs. small) and the type of the reward (hedonic vs. utilitarian). Specifically, larger time units boost patience, but more so when rewards are hedonic rather than utilitarian. In line with the numerosity heuristic, the effect of time units on patience is mediated by larger time units shrinking wait time perception. This effect arises because hedonic (vs. utilitarian) rewards promote a reliance on heuristics rather than systematic calculations. Therefore, a more calculative mindset diminishes the effect of units even for hedonic rewards and eliminates the hedonic‐utilitarian asymmetry. These results contribute to research on numerosity, intertemporal choice, and hedonic‐utilitarian differences, and offer a simple tool for practitioners to influence patience.     

How do people apprehend large numerosities?

People discriminate remarkably well among large numerosities. These discriminations, however, need not entail numerical representation of the quantities being compared. This research evaluated the role of both non-numerical and numerical information in adult judgments of relative numerosity for large-numerosity spatial arrays. Results of Experiment 1 indicated that judgments of relative numerosity were affected by the amount of open space in the arrays being compared. Further, the accuracy of verbal estimates of the numerosities of the arrays made upon completion of the comparison task bore little relation to performance on that task. Experiment 2, however, showed that numerical estimates for individually presented arrays were affected in much the same way by open space within or around the edges of the array as were the comparative judgments examined in Experiment 1. The findings suggest that adults heuristically utilize non-numerical cues as well as numerical information in apprehending large numerosities.     

On the limits of infants' quantification of small object arrays

Recent work suggests that infants rely on mechanisms of object-based attention and short-term memory to represent small numbers of objects. Such work shows that infants discriminate arrays containing 1, 2, or 3 objects, but fail with arrays greater than 3 [Feigenson, L., & Carey, S. (2003). Tracking individuals via object-files: Evidence from infants' manual search. Developmental Science, 6, 568-584; Feigenson, L., Carey, S., & Hauser, M. (2002). The representations underlying infants' choice of more: Object files versus analog magnitudes. Psychological Science, 13(2), 150-156]. However, little is known about how infants represent arrays exceeding the 3-item limit of parallel representation. We explored possible formats by which infants might represent a 4-object array. Experiment 1 used a manual search paradigm to show that infants successfully discriminated between arrays of 1 vs. 2, 2 vs. 3, and 1 vs. 3 objects. However, infants failed to discriminate 1 vs. 4 despite the highly discriminable ratio, providing the strongest evidence to date for object-file representations underlying performance in this task. Experiment 2 replicated this dramatic failure to discriminate 1 from 4 in a second paradigm, a cracker choice task. We then showed that infants in the choice task succeeded at choosing the larger quantity with 0 vs. 4 crackers and with 1 small vs. 4 large crackers. These results suggest that while infants failed to represent 4 as “exactly 4”, “approximately 4”, “3”, or as even as “a plurality”, they did represent information about the array, including the existence of a cracker or cracker-material and the size of the individual objects in the array.     

Analog number representations in mongoose lemurs (Eulemur mongoz): evidence from a search task

A wealth of data demonstrating that monkeys and apes represent number have been interpreted as suggesting that sensitivity to number emerged early in primate evolution, if not before. Here we examine the numerical capacities of the mongoose lemur (Eulemur mongoz), a member of the prosimian suborder of primates that split from the common ancestor of monkeys, apes and humans approximately 47–54 million years ago. Subjects observed as an experimenter sequentially placed grapes into an opaque bucket. On half of the trials the experimenter placed a subset of the grapes into a false bottom such that they were inaccessible to the lemur. The critical question was whether lemurs would spend more time searching the bucket when food should have remained in the bucket, compared to when they had retrieved all of the food. We found that the amount of time lemurs spent searching was indicative of whether grapes should have remained in the bucket, and furthermore that lemur search time reliably differentiated numerosities that differed by a 1:2 ratio, but not those that differed by a 2:3 or 3:4 ratio. Finally, two control conditions determined that lemurs represented the number of food items, and neither the odor of the grapes, nor the amount of grape (e.g., area) in the bucket. These results suggest that mongoose lemurs have numerical representations that are modulated by Webers Law.     

The role of feedback and dot presentation format in younger and older adults’ number estimation

Numerosity estimation, the rapid assessment of the number of items in a visual scene, is historically inaccurate. We assessed whether providing feedback regarding the correct numerosity on either 0%, 50%, or 100% of the trials would affect younger and older adults’ estimation accuracy for randomized, clustered (i.e., groups of 3 or 7 dots), and stacked (i.e., column) dot formats. Participants provided estimates and confidence ratings in six blocks, each containing 48 trials (16 numerosities shown in each format). Feedback frequency was manipulated between participants during blocks 1–4; no feedback was provided during blocks 5 and 6, which contained old and new numerosities and previously estimated presentations rotated 90°. Estimation accuracy was age equivalent across blocks despite younger adults initially being more accurate than older adults. Feedback improved both age groups’ accuracy. Stacked presentations were most accurately estimated but were more likely to be over-estimated than clustered and randomized presentations. Older adults gave lower confidence ratings than younger adults despite both age groups showing increased confidence across blocks, for more structured presentation formats, and as feedback frequency increased. These results expand our understanding of the role of presentation format and feedback in producing age equivalence or age-related differences in numerosity estimation.     

Incidental encoding of numerosity in visual long-term memory

The visual system can readily extract numerosity information from brief experiences. This numerical perception is characterized by diminishing accuracy as numerosity increases, and impaired discrimination for similar quantities and large magnitudes. Here we assess whether these properties apply more broadly to numerosity in visual long-term memory. In surprise memory tests, we observed: Remarkable accuracy in estimating the number of repetitions of an exemplar image (Experiment 1a), that this accuracy decreased but remained high when estimating over categories (Experiments 1b and 1c), that numerical discrimination from memory exhibited psychophysical distance and size effects (Experiment 2), that these effects may derive from stored representations rather than post hoc approximation (Experiment 3a), and that they can reflect total elapsed experience in addition to discrete counts (Experiment 3b). Similar to how numerosity is readily extracted during visual perception, our results suggest that numerosity is encoded incidentally in visual long-term memory.     

Excessive confidence in visually-based estimates

People exhibit excessive confidence in visually-based estimates, which in turn biases decision making. Three experiments support this assertion. Experiment 1 shows a strong impact of presentation format on estimation of proportions. Experiment 2 shows that people rely on these erroneous estimates to make incentive-compatible decisions even when objective information can be easily obtained. Experiment 3 demonstrates that the biased decisions disappear when confidence in visually-based estimates is called into question by the perceived complexity of the stimulus.     

Its own reward: lessons to be drawn from the reversed-reward contingency paradigm

This is a review of the reversed-reward contingency (RRC) paradigm in animals and the cognitive functions on which it is founded. I shall present the RRC basic paradigm and the ensuing modifications it underwent, the animals tested, the results obtained and the analyses offered within the literature. Then I would the claim that RRC is a case of a compound cognitive behavior, one that is the result of interactions between three other cognitive functions: crude numerical assessment and economic choice (uniting value assignment and behavioral inhibition). I will present data concerning these three fields and will demonstrate how they are both affecting and affected by the findings of the RRC scheme. RRC is treated here as a test case for a broader type of analysis, one which, hopefully, will show that in order to fully understand composite and complex behavior we need to meticulously explore its building blocks and their dynamic interplay.