Numbers, not value, motivate cooperation in humans and orangutans

Cooperation among competitors-whether sharing the burden of wind resistance in the Tour de France, forming price-fixing cartels in economic markets, or adhering to arms-control agreements in international treaties-seldom spreads in proportion to the potential benefits. To gain insight into the minds of uncooperative agents, economists and social psychologists have used the prisoner's dilemma task to examine factors leading to cooperation among competitors. Two types of factors have emerged in these studies: the relative rewards of defecting versus cooperating and breakdowns in trust, forgiveness and communication. The generalizability of economic and social psychological factors, however, relies on the assumption that agents' comparisons of gains and losses (whether for themselves, others, or both) preserves ratio information over arbitrary units, such as dollars and cents, and real rewards, such as food. This assumption is inconsistent with psychophysical studies on how the brain represents quantitative information, which suggests that mental magnitudes increase logarithmically with actual value. Thus, discrimination of two numerical magnitudes improves as the numerical distance between them increases and decreases as the magnitudes increase. Here we show an important consequence of this representational system for economic decision making: in the prisoner's dilemma game, purely nominal increases in the numerical magnitude of payoffs (such as, converting dollar values to cents or whole grapes into grape-parts) has a large effect on cooperative behaviour. Moreover, a logarithmic scaling of the ratio of rewards for cooperation versus defection predicted 97% of variability in observed cooperation, whereas the objective ratio predicted 0% of variability. By linking the brain's system of representing the magnitude of rewards to motivations for cooperative behaviour, these findings suggest that the nature of numerical representations may also account for the subjective value function described by Bernoulli, in which the apparent value of monetary incentives increases logarithmically with actual value

Dynamics vs. Development in Numerosity Estimation: A Computational ModelAccurately Predicts a Developmental Reversal

Perceptual judgments result from a dynamic process, but little is known about the dynamics of numerosity estimation. Arecent study proposed a computational model (D-MLLM) that combined a model of trial-to-trial changes with a modelfor the internal scaling of discrete number. Here, we tested a surprising prediction of the model – a situation in whichchildren’s estimates of numerosity would be better than those of adults. Consistent with the model simulations, taskcontexts led to a clear developmental reversal: children made more adult-like, linear estimates when to-be-estimatednumbers were descending over trials (backward), whereas adults became more like children with log estimates whennumbers were ascending (forward). In addition, adults’ estimates were subject to inter-trial differences regardless ofstimulus order. In contrast, children were not able to use the trial-to-trial dynamics unless task contexts were salient(forward or backward order), indicating the limited memory capacity for dynamic updates. Together, the model adequatelypredicts both developmental and trial-to-trial changes in number-line tasks

From Integers to Fractions: Developing a Coherent Understanding ofProportional Magnitude

Children display an early sensitivity to implicit proportions (e.g., 1 of 5 apples vs 3 of 4 apples) but have considerabledifficulty in learning the explicit, symbolic proportions denoted by fractions (e.g., 1/5 vs 3/4). Theoretically, reducingthe gap between representations of implicit vs explicit proportions would improve understanding of fractions, but littleis known about how the representations develop and interact with one another. To address this, we asked 163 third tofifth graders to estimate the position of proportionally-equivalent integers and fractions on number lines (e.g., 3 on 0-8number line vs 3/8 on 0-1 number line). We found that, with increasing age, children were more accurate and linear inrepresenting both integers and fractions. More importantly, childrens estimates of implicit and explicit proportions becamemore coherent, such that a childs estimates of fractions on a 0-1 number-line was a linear function of the same childsestimates of equivalent integers. This representational coherence independently predicted childrens fraction proficiency inother tasks, suggest- ing that building a coherent understanding of proportions is an educationally-important goal

Linear Spatial–Numeric Associations Aid Memory for Single Numbers

Memory for numbers improves with age. One source of this improvement may be learning linear spatial–numeric associations, but previous evidence for this hypothesis likely confounded memory span with quality of numerical magnitude representations and failed to distinguish spatial–numeric mappings from other numeric abilities, such as counting or number word-cardinality mapping. To obviate the influence of memory span on numerical memory, we examined 39 3- to 5-year-olds’ ability to recall one spontaneously produced number (1–20) after a delay, and the relation between numeric recall (controlling for non-numeric recall) and quality of mapping between symbolic and non-symbolic quantities using number-line estimation, give-a-number estimation, and counting tasks. Consistent with previous reports, mapping of numerals to space, to discrete quantities, and to numbers in memory displayed a logarithmic-to-linear shift. Also, linearity of spatial–numeric mapping correlated strongly with multiple measures of numeric recall (percent correct and percent absolute error), even when controlling for age and non-numeric memory. Results suggest that linear spatial–numeric mappings may aid memory for number over and above children’s other numeric skills

Number Sense and Mathematics: Which, When and How?

Individual differences in number sense correlate with mathematical ability and performance, although the presence and strength of this relationship differs across studies. Inconsistencies in the literature may stem from heterogeneity of number sense and mathematical ability constructs. Sample characteristics may also play a role as changes in the relationship between number sense and mathematics may differ across development and cultural contexts. In this study, 4,984 16-year-old students were assessed on estimation ability, one aspect of number sense. Estimation was measured using two different tasks: number line and dot-comparison. Using cognitive and achievement data previously collected from these students at ages 7, 9, 10, 12, and 14 years of age, the study explored for which of the measures and when in development these links are observed; how strong these links are and how much these links are moderated by other cognitive abilities. The two number sensemeasures correlated modestly with each other (r = .22), but moderately with mathematics at age 16. Both measures were also associated with earlier mathematics; but this association was uneven across development and was moderated by other cognitive abilities

The nature of the association between number line and mathematical performance: An international twin study

Background: The number line task assesses the ability to estimatenumerical magnitudes. People vary greatly in this abilityand this variability has been previously associated with mathematical skills. However, the sources of individual differences in number line estimation and its association with mathematics are not fully understood. Aims: This large scale genetically sensitive studyuses a twin design to estimate the magnitude of the effects of genes and environments on: (1) individualvariation in number line estimation and (2) the co-variation of number line estimation with mathematics. Samples: We used over3,0008-16 years-old twins from US, Canada,UK, and Russia, and a sample of 1,456 8-18 years-old singleton Russian students. Methods: Twins were assessed on: (1)estimation of numerical magnitudes using a numberline task and (2) two mathematics components: fluency and problemsolving. Results: Results suggest that environments largelydrive individual differences in numberline estimation.Both genes and environments contribute to different extents to the number line estimationandmathematics correlation, depending on the sample and mathematics component. Conclusions: Taken together, the results suggest that in more heterogeneous school settings, environments may be more important in driving variation in number line estimation and its associationwith mathematics, whereas in more homogeneous school settings, genetic effects drive the covariation between number line estimationand mathematics. These results are discussed in light of development and educational settings

Developing a biological understanding of goal -directed action.

To reason competently about novel entities, people must discover whether the entity is alive and/or sentient. Exactly how people make this discovery is unknown, although past researchers have proposed that young children---unlike adults---rely chiefly on whether the object can move itself. Three studies examined the effect of self-versus goal-directed movement on children's and adults' attributions of biological and psychological capacities to novel entities. In Study 1, 16 adults were shown videos of unfamiliar blobs moving independently, and another 16 were shown videos of identical blobs moving identically but toward a goal. Adults were unlikely to attribute any biological or psychological properties to the blobs moving independently, but they were likely to attribute biological but not psychological properties to the blobs that moved toward goals. In Study 2, these same videos were presented to 4-, 5-, 7-, 10-year-olds, and adults (Ns = 32) to test whether goal-directedness affects biological and psychological inferences across documented periods of change in biological reasoning. Again, no age group was likely to attribute biological or psychological capacities to the self-moving blobs. However, for 5-year-olds through adults, goal-directed movement reliably elicited life-judgments, and it elicited more biological and psychological attributions overall. Adults differed from children in that goal-directed movement affected their attributions of biological properties more than their attributions of psychological properties. Because adults often identified these goal-directed blobs as (insentient) microorganisms, Study 3 presented adults (N = 48) with videos of triangles that moved with the same rotation, speed, and trajectory as the blobs in Studies 1 and 2. Again, adults were unwilling to attribute biological and psychological capacities to triangles that moved independently; however, they were likely to say that the goal-directed triangles were alive and could want, and goal-directedness affected their biological and psychological attributions equally. The results of these three studies suggest that both young children and adults consider the capacity for goal-directed movement to be a decisive factor in whether something unfamiliar is alive, though other factors may be important in deciding whether the thing is sentient.Ph.D.Cognitive psychologyDevelopmental psychologyExperimental psychologyPsychologyUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/132857/2/9990956.pd