The left digit effect in an unbounded number line task.

The left digit effect in number line estimation refers to the phenomenon where numerals with similar magnitudes but different leftmost digits (e.g., 19 and 22) are estimated to be farther apart on a number line than is warranted.

Does instructional intervention reduce the left digit effect in number line estimation?

A robust left digit effect arises in number line estimation, whereby the leftmost digits of numerals have an undue influence on placements such that, for example, numbers like 298 are placed far to the left of numbers like 302. Past efforts to motivate more accurate performance using trial-by-trial and summary feedback have not led to a reduction in the left digit effect. In two experiments, we asked whether it is possible to reduce or eliminate the left digit effect in number line estimation through an instructional intervention in which one is explicitly taught about the left digit effect.

Modeling the left digit effect in adult number line estimation.

Number line estimation tasks are frequently used to study numerical cognition skills. In a typical version, the bounded number line task, target numerals must be placed on a bounded line labeled only at its endpoints (e.g.

Summary accuracy feedback and the left digit effect in number line estimation.

A robust left digit effect arises in number line estimation such that adults' estimates for numerals with different hundreds place digits but nearly identical magnitudes are systematically different from one another (e.g., 299 is placed too far to the left of 302).

Numerical cognition needs more and better distinctions, not fewer.

We agree that the approximate number system (ANS) truly represents number. We endorse the authors' conclusions on the arguments from confounds, congruency, and imprecision, although we disagree with many claims along the way. Here, we discuss some complications with the meanings that undergird theories in numerical cognition, and with the language we use to communicate those theories.

Do demand characteristics contribute to minimal ingroup preferences?

"Minimal group" paradigms investigate social preferences arising from mere group membership. We asked whether demand characteristics contribute to children's apparent minimal group bias in a preregistered experiment (N = 160). In a group condition, we attempted to replicate findings of bias following assignment to minimal groups.

Number line estimation and standardized test performance: The left digit effect does not predict SAT math score.

Introduction: Recent work reveals a new source of error in number line estimation (NLE), the left digit effect (Lai, Zax, et al., 2018), whereby numerals with different leftmost digits but similar magnitudes (e.g.

Partition dependence in financial aid distribution to income categories.

When allocating resources, people often diversify across categories even when those categories are arbitrary, such that allocations differ when identical sets of options are partitioned differently ("partition dependence"). The first goal of the present work (Experiment 1) was to replicate an experiment by Fox and colleagues in which graduate students exhibited partition dependence when asked how university financial aid should be allocated across arbitrarily partitioned income brackets. Our sample consisted of community members at a liberal arts college where financial aid practices have been recent topics of debate.

Developmental change in partition dependent resource allocation behavior.

Partition dependence, the tendency to distribute choices differently based on the way options are grouped, has important implications for decision making. This phenomenon, observed in adults across a variety of contexts such as allocating resources or making selections from a menu of items, can bias decision makers toward some choices and away from others. Only one study to date (Reichelson, Zax, Patalano, & Barth, Quarterly Journal of Experimental Psychology, 72, 1029-1036, 2019) has investigated the developmental trajectory of this phenomenon.

Intuitive symbolic magnitude judgments and decision making under risk in adults.

Performance on an intuitive symbolic number skills task-namely the number line estimation task-has previously been found to predict value function curvature in decision making under risk, using a cumulative prospect theory (CPT) model. However there has been no evidence of a similar relationship with the probability weighting function. This is surprising given that both number line estimation and probability weighting can be construed as involving proportion judgment, that is, involving estimating a number on a bounded scale based on its proportional relationship to the whole.

Spontaneous partitioning and proportion estimation in children's numerical judgments.

What do numerical estimates tell us about developing an understanding of number? One theory is that bounded number line estimation (NLE) tasks reveal a "representational shift" from logarithmically to linearly organized mental representations of number over development. According to a different theoretical framework, developmental change in estimation reflects changes in children's numerical knowledge and their ability to make appropriate relative judgments. Empirical support for this "proportion estimation" framework includes the fact that quantitative models of proportion estimation describe signature patterns of estimation bias.

Probability range and probability distortion in a gambling task.

In decision making under risk, adults tend to overestimate small and underestimate large probabilities (Tversky & Kahneman, 1992). This inverse S-shaped distortion pattern is similar to that observed in a wide variety of proportion judgment tasks (see Hollands & Dyre, 2000, for review). In proportion judgment tasks, distortion patterns tend not to be fixed but rather to depend on the reference points to which the targets are compared.

Suboptimality in perceptual decision making and beyond.

We concur with the authors' overall approach and suggest that their analysis should be taken even further. First, the same points apply to areas beyond perceptual decision making. Second, the same points apply beyond issues of optimality versus suboptimality.

Partition dependence in development: Are children's decisions shaped by the arbitrary grouping of options?

The grouping of options into arbitrary categories influences adults' decisions about allocating choices or resources among those options; this is called "partition dependence." Partition dependence has been demonstrated in a wide range of contexts in adults and is often presented as a technique for designing choice architectures that nudge people towards better decisions. Whether children also make partition dependent decisions is unknown, as are potential patterns of developmental change.

Digit identity influences numerical estimation in children and adults.

Learning the meanings of Arabic numerals involves mapping the number symbols to mental representations of their corresponding, approximate numerical quantities. It is often assumed that performance on numerical tasks, such as number line estimation (NLE), is primarily driven by translating from a presented numeral to a mental representation of its overall magnitude. Part of this assumption is that the overall numerical magnitude of the presented numeral, not the specific digits that comprise it, is what matters for task performance.

Intuitive proportion judgment in number-line estimation: Converging evidence from multiple tasks.

How children's understanding of numerical magnitudes changes over the course of development remains a key question in the study of numerical cognition. In an ongoing debate about the source of developmental change, some argue that children maintain and access different mental representations of number, with evidence coming largely from common number-line estimation tasks. In contrast, others argue that a theoretical framework based on psychophysical models of proportion estimation accounts for typical performance on these tasks.

Partition dependence in consumer choice: Perceptual groupings do not reliably shape decisions.

The partitioning of options into arbitrary categories has been shown to influence decisions about allocating choices or resources among those options; this phenomenon is called partition dependence. While we do not call into question the validity of the partition dependence phenomenon in the present work, we do examine the robustness of one of the experimental paradigms reported by Fox, Ratner, and Lieb (Journal of Experimental Psychology: General, 134, 538-551, 2005, Study 4). In three experiments (N = 300) conducted here, participants chose from a menu of perceptually partitioned options (varieties of candy distributed across bowls).

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.

How feedback improves children's numerical estimation.

Developmental change in children's number-line estimation has been thought to reveal a categorical logarithmic-to-linear shift in mental representations of number. Some have claimed that the broad and rapid change in estimation patterns that occurs with corrective feedback provides strong evidence for this shift. However, quantitative models of proportion judgment may provide a better account of children's estimation patterns while also predicting broad and rapid change following feedback.

The role of numeracy and approximate number system acuity in predicting value and probability distortion.

It is well documented that individuals distort outcome values and probabilities when making choices from descriptions, and there is evidence of systematic individual differences in distortion. In the present study, we investigated the relationship between individual differences in such distortions and two measures of numerical competence, numeracy and approximate number system (ANS) acuity. Participants indicated certainty equivalents for a series of simple monetary gambles, and data were used to estimate individual-level value and probability distortion, using a cumulative prospect theory framework.

Spatial estimation: a non-Bayesian alternative.

A large collection of estimation phenomena (e.g. biases arising when adults or children estimate remembered locations of objects in bounded spaces; Huttenlocher, Newcombe & Sandberg, 1994) are commonly explained in terms of complex Bayesian models.

Preschoolers trust novel members of accurate speakers' groups and judge them favourably.

By age 3, children track a speaker's record of past accuracy and use it as a cue to current reliability. Two experiments (N=95 children) explored whether preschoolers' judgements about, and trust in, the accuracy of a previously reliable informant extend to other members of the informant's group. In Experiment 1, both 3- and 4-year-olds consistently judged an animated character who was associated with a previously accurate speaker more likely to be correct than a character associated with a previously inaccurate speaker, despite possessing no information about these characters' individual records of reliability.

Minimal-group membership influences children's responses to novel experience with group members.

Children, like adults, tend to prefer ingroup over outgroup individuals, but how this group bias affects children's processing of information about social groups is not well understood. In this study, 5- and 6-year-old children were assigned to artificial groups. They observed instances of ingroup and outgroup members behaving in either a positive (egalitarian) or a negative (stingy) manner.

Developmental change in numerical estimation.

Mental representations of numerical magnitude are commonly thought to undergo discontinuous change over development in the form of a "representational shift." This idea stems from an apparent categorical shift from logarithmic to linear patterns of numerical estimation on tasks that involve translating between numerical magnitudes and spatial positions (such as number-line estimation). However, the observed patterns of performance are broadly consistent with a fundamentally different view, based on psychophysical modeling of proportion estimation, that explains the data without appealing to discontinuous change in mental representations of numerical magnitude.