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. The effect has been studied using a bounded number line task, a task in which a line is bounded by two endpoints (e.g., 0 and 100), and where one must indicate the correct location of a target numeral on the line. The goal of the present work is to investigate the left digit effect in an unbounded number line task, a task that involves using the size of one unit to determine a target numeral's location, and that elicits strategies different from those used in the bounded number line task. In a preregistered study, participants (N = 58 college students) completed four blocks of 38 trials each of an unbounded number line task, with target numerals ranging between 0 and 100. We found a medium and statistically reliable left digit effect (d = 0.70). The study offers further evidence that the effect is not driven by response strategies specific to the bounded number line task. We discuss other possible sources of the effect including conversion of symbols to magnitudes in these and other contexts.

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. In Experiment 1 (N = 134), participants completed two blocks (60 trials per block) of a self-paced 0-1,000 number line estimation task and were randomly assigned to either an instruction or a control condition. In Experiment 2 (N = 143), the procedure was enhanced with a learning check, and with additional measures to assess changes in behaviour as a result of instruction. In both experiments, a left digit effect was found in each block of each condition. Although there was evidence that instruction changed behaviour, these changes did not result in any reduction in the left digit effect relative to the control condition. These findings demonstrate that the left digit effect cannot be easily reduced by making people aware of it.

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., with 0 and 100). Placements by adults, while highly accurate, reveal a cyclical pattern of over- and underestimation of target numerals. The pattern suggests use of proportion judgment strategies and is well-captured by cyclical power models. Another systematic number line bias that has recently been observed, but has not yet been considered in modeling efforts, is the left digit effect. Numerals with different leftmost digits (e.g., 39 and 41) are placed farther apart on a line than is warranted. In the current study (N = 60), adult estimates were obtained for all numerals on a 0-100 number line estimation task, and fit of the standard cyclical power model was compared with two modified versions of the model. One modified version included a parameter that underweights the rightward digit's place value (e.g., the ones digit here), and the other used the same parameter to underweight all digits' place values. We found that both modifications provided a considerably better fit for individual and median data than the standard model, and we discuss their relative merits and cognitive interpretations. The data and models suggest how a left digit bias might impact estimates across the number line.

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). In two experiments, we investigate whether brief feedback interventions designed to increase task effort can reduce or eliminate the left digit effect in a self-paced 0-1,000 number line estimation task. Participants were assigned to complete three blocks of 120 trials each where the middle block contained feedback or no feedback. Feedback was in the form of summary accuracy scores (Experiment 1; N = 153) or competitive (summary) accuracy scores (Experiment 2; N = 145). In both experiments, planned analyses revealed large left digit effects in all blocks regardless of feedback condition. Feedback did not lead to a reduction in the left digit effect in either experiment, but improvements in overall accuracy were observed. We conclude that there are no changes in the left digit effect resulting from either summary accuracy feedback or competitive accuracy feedback. Also reported are exploratory analyses of trial characteristics (e.g., whether 299 is presented before or after 302) and the left digit effect.

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. A second closely matched no-group condition retained potential demand characteristics while removing group assignment. Parallel bias in the no-group condition would suggest that demand characteristics contribute to findings of apparent ingroup bias. Three main findings emerged. First, in the group condition, ingroup preference emerged in one of three bias measures only. Second, this preference emerged even though participants evaluated ingroup/outgroup photos varying in race/ethnicity between trials. Third, the measure that yielded ingroup preferences in the group condition produced no parallel bias in the no-group condition, consistent with the view that mere membership in a group, not experimental demand, leads to minimal ingroup preferences.

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., 399, 401) are placed farther apart on a number line (e.g., 0 to 1,000) than is warranted. The goals of the present study were to: (1) replicate the left digit effect, and (2) assess whether it is related to mathematical achievement. METHOD: Participants were all individuals (adult college students) who completed the NLE task in the laboratory between 2014 and 2019 for whom SAT scores were available (n = 227). RESULTS: We replicated the left digit effect but found its size was not correlated with SAT math score, although it was negatively correlated with SAT verbal score for one NLE task version. CONCLUSIONS: These findings provide further evidence that individual digits strongly influence estimation performance and suggest that this effect may have different cognitive contributors, and predict different complex skills, than overall NLE accuracy.

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. Because stronger intrinsic preferences can reduce partition dependence, these participants might display little partition dependence with financial aid allocations. Alternatively, a demonstration of strong partition dependence in this population would emphasize the robustness of the effect. The second goal was to extend a "high transparency" modification to the present task context (Experiment 2) in which participants were shown both possible income partitions and randomly assigned themselves to one, to determine whether partition dependence in this paradigm would be reduced by revealing the study design (and the arbitrariness of income categories). Participants demonstrated clear partition dependence in both experiments. Results demonstrate the robustness of partition dependence in this context.

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. In the current study we investigate children's and adults' susceptibility to partitioning effects in a child-friendly resource allocation task. In Experiment 1 (N = 80), adults distributed 12 food tokens to animals at the zoo. Based on previous findings that older children show weaker partition dependence in this task, we predicted that adults might exhibit reduced partition dependent behavior: they showed none. In Experiment 2 (N = 272), we used a less transparent task with only five food tokens, predicting that both adults and children (ages 3-10 years) would show partition dependence. Children, but not adults, made partition dependent resource allocations, with younger children exhibiting greater effects than older children. These experiments provide further evidence that children's decisions, like adults' (in other tasks), are influenced by the arbitrary partitioning of the available options. This work supports previous findings that younger children may be more susceptible to these effects, and maps developmental change in partition dependent behavior from early childhood to adulthood on this child-friendly partition dependence task.

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. In the present work, we re-evaluated the relationship between number line estimation and probability weighting through the lens of proportion judgment. Using a CPT model with a two-parameter probability weighting function, we found a double dissociation: number line estimation bias predicted probability weighting curvature while performance on a different number skills task, number comparison, predicted probability weighting elevation. Interestingly, while degree of bias was correlated across tasks, the direction of bias was not. The findings provide support for proportion judgment as a plausible account of the shape of the probability weighting function, and suggest directions for future work.

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. A recent study argued against this latter theory by suggesting that patterns of curvature in number line placements are simply artifacts of a task procedure in which participants receive explicit information about the location of the numerical midpoint. We tested this claim in two experiments with children aged 6-8 years (Experiment 1: N = 47; Experiment 2: N = 104). Results demonstrated that the proportion estimation framework provides a good explanation of children's number line placement in the absence of explicit midpoint cues, that explicit cues to the midpoint are associated with more frequent use of middle reference points in young children, and that children can use a middle reference point spontaneously in the absence of explicit cues (with this tendency increasing with age). These findings provide novel support for the idea that psychophysical models of proportion estimation successfully account for numerical estimates across development regardless of whether spatial and numerical midpoint cues are provided as part of the NLE task.

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. Here, we tested the novel hypothesis that probability distortion in decision making under risk might also be influenced by reference points-in this case, references implied by the probability range. Adult participants were assigned to either a full-range (probabilities from 0-100%), upper-range (50-100%), or lower-range (0-50%) condition, where they indicated certainty equivalents for 176 hypothetical monetary gambles (e.g., "a 50% chance of $100, otherwise $0"). Using a modified cumulative prospect theory model, we found only minimal differences in probability distortion as a function of condition, suggesting no differences in use of reference points by condition, and broadly demonstrating the robustness of distortion pattern across contexts. However, we also observed deviations from the curve across all conditions that warrant further research.

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. In this experiment ( N = 159), we examined whether children exhibit partition dependence using a novel resource allocation task. This novel task, distributing food tokens to zoo animals, did elicit partition dependence in our developmental sample. Both older children (ages 7-10 years) and younger children (ages 3-6 years) made partition dependent allocations, and younger children exhibited a larger partition dependence effect than did older children. This work provides the first evidence that children's decisions, like adults', are influenced by the arbitrary grouping of the options, and suggests that younger children may be more susceptible to this influence, at least in the context explored here.

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. Here we ask whether the magnitudes of the presented target numerals drive symbolic number line performance, or whether specific digits influence estimates. If the former is true, estimates of numerals with very similar magnitudes but different hundreds digits (such as 399 and 402) should be placed in similar locations. However, if the latter is true, these placements will differ significantly. In two studies (N = 262), children aged 7-11 and adults completed 0-1000 NLE tasks with target values drawn from a set of paired numerals that fell on either side of "Hundreds" boundaries (e.g., 698 and 701) and "Fifties" boundaries (e.g., 749 and 752). Study 1 used an atypical speeded NLE task, while Study 2 used a standard non-speeded NLE task. Under both speeded and non-speeded conditions, specific hundreds digits in the target numerals exerted a strong influence on estimates, with large effect sizes at all ages, showing that the magnitudes of target numerals are not the primary influence shaping children's or adults' placements. We discuss patterns of developmental change and individual difference revealed by planned and exploratory analyses.

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). We found no clear evidence of partition dependent choice in children (Experiment 1) and no evidence at all of partition dependence in adults' choices (Experiments 1-3). This was true even when methods were closely matched to those of Fox et al.'s Study 4 (Experiment 3). We conclude that the candy-bowl choice task does not reliably elicit partition dependence and propose possible explanations for the discrepancy between these findings and prior reports. Future work will explore the conditions under which partition dependence in consumer choice does reliably arise.

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.

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. The current study explored children's (n=71) and adults' (n=27) performance on two number-line tasks: the "number to position" (or NP) task and the inverse "position to number" (or PN) task. Estimates on both tasks are consistent with the predictions of the proportion estimation account and do not support the hypothesis that a fundamental shift in mental representations underlies developmental change in numerical estimation and, in turn, mathematical ability. Converging evidence across the tasks also calls into question the utility of bounded number-line tasks as an evaluation of mental representations of number.

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.

Early group bias in the Faroe Islands: Cultural variation in children's group-based reasoning.

Recent developmental research demonstrates that group bias emerges early in childhood. However, little is known about the extent to which bias in minimal (i.e., arbitrarily assigned) groups varies with children's environment and experience, and whether such bias is universal across cultures. In this study, the development of group bias was investigated using a minimal groups paradigm with 46 four- to six-year-olds from the Faroe Islands. Children observed in-group and out-group members exhibiting varying degrees of prosocial behaviour (egalitarian or stingy sharing). Children did not prefer their in-group in the pretest, but a pro-in-group and anti-out-group sentiment emerged in both conditions in the posttest. Faroese children's response patterns differ from those of American children [Schug, M. G., Shusterman, A., Barth, H., & Patalano, A. L. (2013). Minimal-group membership influences children's responses to novel experience with group members. Developmental Science, 16(1), 47-55], suggesting that intergroup bias shows cultural variation even in a minimal groups context.

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. Here we test the hypothesis that local corrective feedback provides children with additional reference points, rather than catalyzing a shift to a different mental representation of number. We tested 117 children from several second-grade classrooms in a number-line feedback study. Data indicate that the proportion-judgment framework accounts for individual differences in estimation patterns, and that the effects of feedback are consistent with the unique quantitative predictions of the framework. They do not provide evidence supporting the representational shift hypothesis or, more broadly, for the proposal that cognitive change can occur rapidly at the level of entire mental representations.