Log or linear? Distinct intuitions of the number scale in Western and Amazonian indigene cultures

Number sense-arithmetic link in Grade 1 and Grade 2: A case of fluency

BACKGROUND

Recent research suggested fluent processing as an explanation on why number sense contributes to simple arithmetic tasks-'Fluency hypothesis'.

AIMS

The current study investigates whether number sense contributes to such arithmetic tasks when other cognitive factors are controlled for (including those that mediate the link); and whether this contribution varies as a function of participants' individual maths fluency levels.

SAMPLE

Four hundred and thirty-seven Chinese schoolchildren (186 females; Mage = 83.49 months) completed a range of cognitive measures in Grade 1 (no previous classroom training) and in Grade 2 (a year later).

METHODS

Number sense, arithmetic (addition and subtraction), spatial ability, visuo-spatial working memory, perception, reaction time, character reading and general intelligence were measured.

RESULTS

Our data showed that the link between number sense and arithmetic was weaker in Grade 1 (Beta = .15 for addition and .06 (ns) for subtraction) compared to Grade 2 (.23-.28), but still persisted in children with no previous maths training. Further, math's performance in Grade 1 did not affect the link between number sense and maths performance in Grade 2.

CONCLUSION

Our data extended previous findings by showing that number sense is linked with simple maths task performance even after controlling for multiple cognitive factors. Our results brought some evidence that number sense-arithmetic link is somewhat sensitive to previous formal maths education. Further research is needed, as the differences in effects between grades were quite small, and arithmetic in Grade 1 did not moderate the link at question in Grade 2.

Young children intuitively organize numbers on straight, horizontal lines from left to right before the onset of formal instruction

The number line estimation task is frequently used to measure children's numerical magnitude understanding. It is unclear whether the resulting straight, horizontal, left-to-right-oriented estimate patterns indicate task constraints or children's intuitive number-space mapping. Three- to six-year-old children (N = 72, Mage = 4.89, 56% girls, 94% German citizenship) were asked to explain the meaning of numbers to a teddy by laying out a rope and attaching cards showing non-symbolic numerosities (dots) to it. Most children intuitively created straight, horizontal, and left-to-right-oriented representations. Characteristics of the line correlated with age, mathematical competencies, and home numeracy. This demonstrates the usefulness of the number line estimation task for assessing how children intuitively map numbers onto space.

Magnitude representation of preschool children with autism spectrum condition

LAY ABSTRACT

The mathematical abilities of children with autism spectrum condition have been understudied. Magnitude representation (e.g. presenting the number of a collection of objects) is a fundamental numerical ability presented since early infancy and is correlated with children's later learning of formal mathematics. It remains unclear about whether children with autism spectrum condition differ from their peers without autism spectrum condition in precision of magnitude representations. This study compared preschool children with and without autism spectrum condition in their precision of magnitude representation with an approximate number comparison task, in which children compared two sets of dots without counting and chose the set with more dots. Children with autism spectrum condition exhibited the lower numerical comparison accuracy (i.e. the weaker magnitude representation) than their peers without autism spectrum condition. This difference existed even when multiple general cognitive abilities (working memory, inhibitory control, and nonverbal intelligence) and language abilities were statistically controlled. Moreover, the individual difference of the numerical comparison accuracy was larger in children with autism spectrum condition than without autism spectrum condition. These findings suggest that children with autism spectrum condition are at risk of weaker magnitude representation from an early age, emphasizing the need for specialized mathematics education or interventions to support their learning. In addition, the large variance in the precision of their magnitude representation suggests that individualized mathematics interventions are needed for children with autism spectrum condition.

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.

Perceptual addition of continuous magnitudes in an 'artificial algebra'

Although there is substantial evidence for an innate 'number sense' that scaffolds learning about mathematics, whether the underlying representations are based on discrete or continuous perceptual magnitudes has been controversial. Yet the nature of the computations supported by these representations has been neglected in this debate. While basic computation of discrete non-symbolic quantities has been reliably demonstrated in adults, infants, and non-humans, far less consideration has been given to the capacity for computation of continuous perceptual magnitudes. Here we used a novel experimental task to ask if humans can learn to add non-symbolic, continuous magnitudes in accord with the properties of an algebraic group, by feedback and without explicit instruction. Three pairs of experiments tested perceptual addition under the group properties of commutativity (Experiments 1a-b), identity and inverses (Experiments 2a-b) and associativity (Experiments 3a-b), with both line length and brightness modalities. Transfer designs were used in which participants responded on trials with feedback based on sums of magnitudes and later were tested with novel stimulus configurations. In all experiments, correlations of average responses with magnitude sums were high on trials with feedback. Responding on transfer trials was accurate and provided strong support for addition under all of the group axioms with line length, and for all except associativity with brightness. Our results confirm that adult human subjects can implicitly add continuous quantities in a manner consistent with symbolic addition over the integers, and that an 'artificial algebra' task can be used to study implicit computation.

Neurobiology of numerical learning

Numerical abilities are complex cognitive skills essential for dealing with requirements of the modern world. Although the brain structures and functions underlying numerical cognition in different species have long been appreciated, genetic and molecular techniques have more recently expanded the knowledge about the mechanisms underlying numerical learning. In this review, we discuss the status of the research related to the neurobiological bases of numerical abilities. We consider how genetic factors have been associated with mathematical capacities and how these link to the current knowledge of brain regions underlying these capacities in human and non-human animals. We further discuss the extent to which significant variations in the levels of specific neurotransmitters may be used as potential markers of individual performance and learning difficulties and take into consideration the therapeutic potential of brain stimulation methods to modulate learning and improve interventional outcomes. The implications of this research for formulating a more comprehensive view of the neural basis of mathematical learning are discussed.

Observers Efficiently Extract the Minimal and Maximal Element in Perceptual Magnitude Sets: Evidence for a Bipartite Format

The mind represents abstract magnitude information, including time, space, and number, but in what format is this information stored? We show support for the bipartite format of perceptual magnitudes, in which the measured value on a dimension is scaled to the dynamic range of the input, leading to a privileged status for values at the lowest and highest end of the range. In six experiments with college undergraduates, we show that observers are faster and more accurate to find the endpoints (i.e., the minimum and maximum) than any of the inner values, even as the number of items increases beyond visual short-term memory limits. Our results show that length, size, and number are represented in a dynamic format that allows for comparison-free sorting, with endpoints represented with an immediately accessible status, consistent with the bipartite model of perceptual magnitudes. We discuss the implications for theories of visual search and ensemble perception.

A single tDCS session can enhance numerical competence

While numerical skills are increasingly important in modern life, few interventions have been developed to support those with numeracy skills difficulties. Previous studies have demonstrated that applying transcranial Direct Current Stimulation (tDCS) can improve numerical skills. However, tDCS interventions designed to induce lasting changes typically involve reapplying brain-stimulation over several days. Repeated tDCS application can increase the risks associated with the procedure, as well as restricts the transferability of the method to a wider population, particularly those who may experience mobility issues, such as stroke survivors with acalculia. The current study investigated whether a single session of tDCS (anodal to right parietal lobe and cathodal to left parietal lobe), followed by four self-practice sessions without tDCS, could result in enhancement of numerical skills. Nineteen healthy adults (n = 10 tDCS, n = 9 sham control) implicitly learnt the magnitude association of nine arbitrary symbols, previously used by Cohen Kadosh et al. (2010). Numerical proficiency was assessed using number-to-space task, while automaticity was assessed with numerical Stroop. Results revealed that single-session tDCS had a significant effect on participants' accuracy on the number-to-space tasks, but not on the numerical Stroop task's congruity effect, implying automaticity may require longer practice. We conclude that a single session of tDCS should be considered as an avenue for interventions.

Understanding others' preferences: A comparison across primate species and human societies

We investigated children's and non-human great apes' ability to anticipate others' choices from their evident food preferences-regardless of whether these preferences deviate or align with one's own. We assessed children from three culturally-diverse societies (Namibia, Germany, and Samoa; N = 71; age range = 5-11) and four non-human great ape species (chimpanzees (Pan troglodytes), bonobos (Pan paniscus), gorillas (Gorilla gorilla), and orangutans (Pongo abelii); N = 25; age range = 7-29) regarding their choices in a dyadic food-retrieval task. Across conditions, participants' preferences were either aligned (same preference condition) or opposed (opposite preference condition) to those of their competitors. Children across societies altered their choices based on their competitor's preferences, indicating a cross-culturally recurrent capacity to anticipate others' choices relying on preferences-based inferences. In contrast to human children, all non-human great apes chose according to their own preferences but independent of those of their competitors. In sum, these results suggest that the tendency to anticipate others' choices based on their food preferences is cross-culturally robust and, among the great apes, most likely specific to humans.

The Future of Decisions From Experience: Connecting Real-World Decision Problems to Cognitive Processes

In many important real-world decision domains, such as finance, the environment, and health, behavior is strongly influenced by experience. Renewed interest in studying this influence led to important advancements in the understanding of these decisions from experience (DfE) in the last 20 years. Building on this literature, we suggest ways the standard experimental design should be extended to better approach important real-world DfE. These extensions include, for example, introducing more complex choice situations, delaying feedback, and including social interactions. When acting upon experiences in these richer and more complicated environments, extensive cognitive processes go into making a decision. Therefore, we argue for integrating cognitive processes more explicitly into experimental research in DfE. These cognitive processes include attention to and perception of numeric and nonnumeric experiences, the influence of episodic and semantic memory, and the mental models involved in learning processes. Understanding these basic cognitive processes can advance the modeling, understanding and prediction of DfE in the laboratory and in the real world. We highlight the potential of experimental research in DfE for theory integration across the behavioral, decision, and cognitive sciences. Furthermore, this research could lead to new methodology that better informs decision-making and policy interventions.

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.

Competing numerical magnitude codes in decimal comparison: Whole number and rational number distance both impact performance

A critical difference between decimal and whole numbers is that among whole numbers the number of digits provides reliable information about the size of the number, e.g., double-digit numbers are larger than single-digit numbers. However, for decimals, fewer digits can sometimes denote a larger number (i.e., 0.8 > 0.27). Accordingly, children and adults perform worse when comparing such Inconsistent decimal pairs relative to Consistent pairs, where the larger number also has more digits (i.e., 0.87 > 0.2). Two explanations have been posited for this effect. The string length congruity account proposes that participants compare each position in the place value system, and they additionally compare the number of digits. The semantic interference account suggests that participants additionally activate the whole number referents of numbers - the numbers unadorned with decimal points (e.g., 8 < 27) - and compare these. The semantic interference account uniquely predicts that for Inconsistent problems with the same actual rational distance, those with larger whole number distances should be harder, e.g., 0.9 vs. 0.81 should be harder than 0.3 vs. 0.21 because 9 < < 81 whereas 3 < 21. Here we test this prediction in two experiments with college students (Study 1: n = 58 participants, Study 2: n = 78). Across both, we find a main effect of consistency, demonstrating string length effects, and also that whole number distance interferes with processing conflicting decimals, demonstrating semantic interference effects. Evidence for both effects supports the semantic interference account, highlighting that decimal comparison difficulties arise from multiple competing numerical codes. Finally, for accuracy we found no relationship between whole number distance sensitivity and math achievement, indicating that whole number magnitude interference affects participants similarly across the spectrum of math achievement.

Numerical Representation for Action in Crows Obeys the Weber-Fechner Law

The psychophysical laws governing the judgment of perceived numbers of objects or events, called the number sense, have been studied in detail. However, the behavioral principles of equally important numerical representations for action are largely unexplored in both humans and animals. We trained two male carrion crows (Corvus corone) to judge numerical values of instruction stimuli from one to five and to flexibly perform a matching number of pecks. Our quantitative analysis of the crows' number production performance shows the same behavioral regularities that have previously been demonstrated for the judgment of sensory numerosity, such as the numerical distance effect, the numerical magnitude effect, and the logarithmical compression of the number line. The presence of these psychophysical phenomena in crows producing number of pecks suggests a unified sensorimotor number representation system underlying the judgment of the number of external stimuli and internally generated actions.

Trend judgment as a perceptual building block of graphicacy and mathematics, across age, education, and culture

Data plots are widely used in science, journalism and politics, since they efficiently allow to depict a large amount of information. Graphicacy, the ability to understand graphs, has thus become a fundamental cultural skill comparable to literacy or numeracy. Here, we introduce a measure of intuitive graphicacy that assesses the perceptual ability to detect a trend in noisy scatterplots ("does this graph go up or down?"). In 3943 educated participants, responses vary as a sigmoid function of the t-value that a statistician would compute to detect a significant trend. We find a minimum level of core intuitive graphicacy even in unschooled participants living in remote Namibian villages (N = 87) and 6-year-old 1st-graders who never read a graph (N = 27). The sigmoid slope that we propose as a proxy of intuitive graphicacy increases with education and tightly correlates with statistical and mathematical knowledge, showing that experience contributes to refining graphical intuitions. Our tool, publicly available online, allows to quickly evaluate and formally quantify a perceptual building block of graphicacy.

Numeracy, gist, literal thinking and the value of nothing in decision making

The onus on the average person is greater than ever before to make sense of large amounts of readily accessible quantitative information, but the ability and confidence to do so are frequently lacking. Many people lack practical mathematical skills that are essential for evaluating risks, probabilities and numerical outcomes such as survival rates for medical treatments, income from retirement savings plans or monetary damages in civil trials. In this Review, we integrate research on objective and subjective numeracy, focusing on cognitive and metacognitive factors that distort human perceptions and foment systematic biases in judgement and decision making. Paradoxically, an important implication of this research is that a literal focus on objective numbers and mechanical number crunching is misguided. Numbers can be a matter of life and death but a person who uses rote strategies (verbatim representations) cannot take advantage of the information contained in the numbers because 'rote' strategies are, by definition, processing without meaning. Verbatim representations (verbatim is only surface form, not meaning) treat numbers as data as opposed to information. We highlight a contrasting approach of gist extraction: organizing numbers meaningfully, interpreting them qualitatively and making meaningful inferences about them. Efforts to improve numerical cognition and its practical applications can benefit from emphasizing the qualitative meaning of numbers in context - the gist - building on the strengths of humans as intuitive mathematicians. Thus, we conclude by reviewing evidence that gist training facilitates transfer to new contexts and, because it is more durable, longer-lasting improvements in decision making.

Information Theory Opens New Dimensions in Experimental Studies of Animal Behaviour and Communication

Over the last 40–50 years, ethology has become increasingly quantitative and computational. However, when analysing animal behavioural sequences, researchers often need help finding an adequate model to assess certain characteristics of these sequences while using a relatively small number of parameters. In this review, I demonstrate that the information theory approaches based on Shannon entropy and Kolmogorov complexity can furnish effective tools to analyse and compare animal natural behaviours. In addition to a comparative analysis of stereotypic behavioural sequences, information theory can provide ideas for particular experiments on sophisticated animal communications. In particular, it has made it possible to discover the existence of a developed symbolic “language” in leader-scouting ant species based on the ability of these ants to transfer abstract information about remote events.

Directionality in the interrelations between approximate number, verbal number, and mathematics in preschool-aged children

A fundamental question in numerical development concerns the directional relation between an early-emerging non-verbal approximate number system (ANS) and culturally acquired verbal number and mathematics knowledge. Using path models on longitudinal data collected in preschool children (M_age  = 3.86 years; N = 216; 99 males; 80.8% White; 10.8% Multiracial, 3.8% Latino; 1.9% Black; collected 2013-2017) over 1 year, this study showed that earlier verbal number knowledge was associated with later ANS precision (average β = .32), even after controlling for baseline differences in numerical, general cognitive, and language abilities. In contrast, earlier ANS precision was not associated with later verbal number knowledge (β = -.07) or mathematics abilities (average β = .10). These results suggest that learning about verbal numbers is associated with a sharpening of pre-existing non-verbal numerical abilities.

A Perception Study for Unit Charts in the Context of Large-Magnitude Data Representation

Unit charts are a common type of chart for visualizing scientific data. A unit chart is a chart used to communicate quantities of things by making the number of symbols on the chart proportional to the number of items represented. An accurate perception of the order of magnitude is essential to evaluating whether a unit chart can effectively convey information. Previous studies have primarily focused on perceptual properties at small order-of-magnitude scales or the efficacy of pictographs in unit charts. However, few researchers have explored the perceptual effectiveness of unit charts when representing large orders of magnitude. In this study, we performed a series of sampling measurements to investigate the visual–perceptual characteristics of unit charts when representing asymmetric interactions such as large-scale numbers. The results showed that under the restriction of the current conventional display medium, unit charts still offer a significant advantage over bar charts in a single-scale visual overview. However, this comes at the cost of a longer response time. Although this study constitutes basic research, accumulating evidence about how people reason about magnitudes beyond human perception is critical to the field of information science. This study may contribute to understanding how viewers perceive unit charts and the factors that influence graphical perception. This article provides some specific guidelines for designing unit charts that may be useful to visualization designers.

Is my visualization better than yours? Analyzing factors modulating exponential growth bias in graphs

Humans tend to systematically underestimate exponential growth and perceive it in linear terms, which can have severe consequences in a variety of fields. Recent studies attempted to examine the origins of this bias and to mitigate it by using the logarithmic vs. the linear scale in graphical representations. However, they yielded conflicting results as to which scale induces more perceptual errors. In the current study, in an experiment with a short educational intervention, we further examine the factors modulating the exponential bias in graphs and suggest a theoretical explanation for our findings. Specifically, we test the hypothesis that each of the scales can induce misperceptions in a particular context. In addition to this, we explore the effect of mathematical education by testing two groups of participants (with a background in humanities vs. formal sciences). The results of this study confirm that when used in an inadequate context, these scales can have a dramatic effect on the interpretation of visualizations representing exponential growth. In particular, while the log scale leads to more errors in graph description tasks, the linear scale misleads people when they have to make predictions on the future trajectory of exponential growth. The second part of the study revealed that the difficulties with both scales can be reduced by means of a short educational intervention. Importantly, while no difference between participants groups was observed prior to the intervention, participants with a better mathematical education showed a stronger learning effect at posttest. The findings of this study are discussed in light of a dual-process model.

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.

The number line estimation task is a valid tool for assessing mathematical achievement: A population-level study with 6484 Luxembourgish ninth-graders

The number line estimation task is an often-used measure of numerical magnitude understanding. The task also correlates substantially with broader measures of mathematical achievement. This raises the question of whether the task would be a useful component of mathematical achievement tests and instruments to diagnose dyscalculia or mathematical giftedness and whether a stand-alone version of the task can serve as a short screener for mathematical achievement. Previous studies on the relation between number line estimation accuracy and broader mathematical achievement were limited in that they used relatively small nonrepresentative samples and usually did not account for potentially confounding variables. To close this research gap, we report findings from a population-level study with nearly all Luxembourgish ninth-graders (N = 6484). We used multilevel regressions to test how a standardized mathematical achievement test relates to the accuracy in number line estimation on bounded number lines with whole numbers and fractions. We also investigated how these relations were moderated by classroom characteristics, person characteristics, and trial characteristics. Mathematical achievement and number line estimation accuracy were associated even after controlling for potentially confounding variables. Subpopulations of students showed meaningful differences in estimation accuracy, which can serve as benchmarks in future studies. Compared with the number line estimation task with whole numbers, the number line estimation task with fractions was more strongly related to mathematical achievement in students across the entire mathematical achievement spectrum. These results show that the number line estimation task is a valid and useful tool for diagnosing and monitoring mathematical achievement.

Does auditory numerosity and non-numerical magnitude affect visual non-symbolic numerical representation?

In the present study, we investigated the effects of auditory numerosity and magnitude (loudness) on visual numerosity processing. Participants compared numerosities of two sequential dot arrays. The second dot array was paired with a tone array that was independent of visual comparison. The numerosity (One-tone vs. Multiple-tone) and the non-numerical magnitude of tones (loudness) were manipulated in Experiments 1 and 2, respectively. In Experiment 1, participants' inverse efficiency score (IES), that is, the quotient between response time and accuracy, was significantly smaller in the One-tone and Multiple-tone conditions than that in the No-tone condition, and linear trend analyses showed that the IES decreased with the number of tones. In Experiment 2, the IES in the Loud-tone condition was significantly smaller than that in the No-tone condition, and the IES decreased as the loudness of the tones increased. In Experiment 3, both auditory numerosity and magnitude were manipulated. For soft tones, the IES was smaller in the Multiple-tone condition than in the One-tone condition, whereas no significant difference was found between two conditions in loud tones. In sum, these findings suggest that the visual numerical representation can be spontaneously affected by the numerosity and non-numerical magnitude of stimuli from another modality.

The cultural origins of symbolic number

It is popular in psychology to hypothesize that representations of exact number are innately determined-in particular, that biology has endowed humans with a system for manipulating quantities which forms the primary representational substrate for our numerical and mathematical concepts. While this perspective has been important for advancing empirical work in animal and child cognition, here we examine six natural predictions of strong numerical nativism from a multidisciplinary perspective, and find each to be at odds with evidence from anthropology and developmental science. In particular, the history of number reveals characteristics that are inconsistent with biological determinism of numerical concepts, including a lack of number systems across some human groups and remarkable variability in the form of numerical systems that do emerge. Instead, this literature highlights the importance of economic and social factors in constructing fundamentally new cognitive systems to achieve culturally specific goals. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

An illustrated tutorial for logarithmic scales and decibels in acoustics

Acoustics is generally defined as the science that deals with the production, transmission, and reception of sound and the understanding and control of its effects. In fact, the fields of acoustics cover an especially broad range of subjects and domains, and comprehensive acoustics textbooks are usually quite thick as a consequence. While they are valuable resources for researchers, these books might appear a little daunting for a young audience or for people who are new to acoustics. This paper is an example of how educational comics can be designed and used to introduce one of the most commonly discussed topics when the basics of acoustics are taught: decibel level. Seven drawn pages constitute a visual support to explain the origin and history of the decibel, together with examples from acoustics and other domains on the use of logarithmic scales and classical decibel calculations. Several comments and comprehensive bibliographical references are also provided for each drawn page to enlarge the range of subjects or exercises that can be discussed in courses and foster further readings.

More linear than log? Non-symbolic number-line estimation in 3- to 5-year-old children

The number-line estimation task has become one of the most important methods in numerical cognition research. Originally applied as a direct measure of spatial number representation, it became also informative regarding various other aspects of number processing and associated strategies. However, most of this work and associated conclusions concerns processing numbers in a symbolic format, by school children and older subjects. Symbolic number system is formally taught and trained at school, and its basic mathematical properties (e.g., equidistance, ordinality) can easily be transferred into a spatial format of an oriented number line. This triggers the question on basic characteristics of number line estimation before children get fully familiar with the symbolic number system, i.e., when they mostly rely on approximate system for non-symbolic quantities. In our three studies, we examine therefore how preschool children (3–5-years old) estimate position of non-symbolic quantities on a line, and how this estimation is related to the developing symbolic number knowledge and cultural (left-to-right) directionality. The children were tested with the Give-a-number task, then they performed a computerized number-line task. In Experiment 1, lines bounded with sets of 1 and 20 elements going left-to-right or right-to-left were used. Even in the least numerically competent group, the linear model better fit the estimates than the logarithmic or cyclic power models. The line direction was irrelevant. In Experiment 2, a 1–9 left-to-right oriented line was used. Advantage of linear model was found at group level, and variance of estimates correlated with tested numerosities. In Experiment 3, a position-to-number procedure again revealed the advantage of the linear model, although the strategy of selecting an option more similar to the closer end of the line was prevalent. The precision of estimation increased with the mastery of counting principles in all three experiments. These results contradict the hypothesis of the log-to-linear shift in development of basic numerical representation, rather supporting the linear model with scalar variance. However, the important question remains whether the number-line task captures the nature of the basic numerical representation, or rather the strategies of mapping that representation to an external space.

The distribution of initial estimates moderates the effect of social influence on the wisdom of the crowd

Whether, and under what conditions, groups exhibit “crowd wisdom” has been a major focus of research across the social and computational sciences. Much of this work has focused on the role of social influence in promoting the wisdom of the crowd versus leading the crowd astray and has resulted in conflicting conclusions about how social network structure determines the impact of social influence. Here, we demonstrate that it is not enough to consider the network structure in isolation. Using theoretical analysis, numerical simulation, and reanalysis of four experimental datasets (totaling 2885 human subjects), we find that the wisdom of crowds critically depends on the interaction between (i) the centralization of the social influence network and (ii) the distribution of the initial individual estimates. By adopting a framework that integrates both the structure of the social influence and the distribution of the initial estimates, we bring previously conflicting results under one theoretical framework and clarify the effects of social influence on the wisdom of crowds.

Reduced choice-confidence in negative numerals

Negative numbers are central in math. However, they are abstract, hard to learn, and manipulated slower than positive numbers regardless of math ability. It suggests that confidence, namely the post-decision estimate of being correct, should be lower than positives. We asked participants to pick the larger single-digit numeral in a pair and collected their implicit confidence with button pressure (button pressure was validated with three empirical signatures of confidence). We also modeled their choices with a drift-diffusion decision model to compute the post-decision estimate of being correct. We found that participants had relatively low confidence with negative numerals. Given that participants compared with high accuracy the basic base-10 symbols (0–9), reduced confidence may be a general feature of manipulating abstract negative numerals as they produce more uncertainty than positive numerals per unit of time.

The dopamine circuit as a reward-taxis navigation system

Studying the brain circuits that control behavior is challenging, since in addition to their structural complexity there are continuous feedback interactions between actions and sensed inputs from the environment. It is therefore important to identify mathematical principles that can be used to develop testable hypotheses. In this study, we use ideas and concepts from systems biology to study the dopamine system, which controls learning, motivation, and movement. Using data from neuronal recordings in behavioral experiments, we developed a mathematical model for dopamine responses and the effect of dopamine on movement. We show that the dopamine system shares core functional analogies with bacterial chemotaxis. Just as chemotaxis robustly climbs chemical attractant gradients, the dopamine circuit performs ‘reward-taxis’ where the attractant is the expected value of reward. The reward-taxis mechanism provides a simple explanation for scale-invariant dopaminergic responses and for matching in free operant settings, and makes testable quantitative predictions. We propose that reward-taxis is a simple and robust navigation strategy that complements other, more goal-directed navigation mechanisms.

Large numbers cause magnitude neglect: The case of government expenditures

Four studies demonstrate that the public's understanding of government budgetary expenditures is hampered by difficulty in representing large numerical magnitudes. Despite orders of magnitude difference between millions and billions, study participants struggle with the budgetary magnitudes of government programs. When numerical values are rescaled as smaller magnitudes (in the thousands or lower), lay understanding improves, as indicated by greater sensitivity to numerical ratios and more accurate rank ordering of expenses. A robust benefit of numerical rescaling is demonstrated across a variety of experimental designs, including policy relevant choices and incentive-compatible accuracy measures. This improved sensitivity ultimately impacts funding choices and public perception of respective budgets, indicating the importance of numerical cognition for good citizenship.

A number-line task with a Bayesian active learning algorithm provides insights into the development of non-symbolic number estimation

To characterize numerical representations, the number-line task asks participants to estimate the location of a given number on a line flanked with zero and an upper-bound number. An open question is whether estimates for symbolic numbers (e.g., Arabic numerals) and non-symbolic numbers (e.g., number of dots) rely on common processes with a common developmental pathway. To address this question, we explored whether well-established findings in symbolic number-line estimation generalize to non-symbolic number-line estimation. For exhaustive investigations without sacrificing data quality, we applied a novel Bayesian active learning algorithm, dubbed Gaussian process active learning (GPAL), that adaptively optimizes experimental designs. The results showed that the non-symbolic number estimation in participants of diverse ages (5-73 years old, n = 238) exhibited three characteristic features of symbolic number estimation.

Numerosity perception is tuned to salient environmental features

Numerosity perception is a key ability to guide behavior. However, current models propose that number units encode an abstract representation of numerosity regardless of the non-numerical attributes of the stimuli, suggesting rather coarse environmental tuning. Here we investigated whether numerosity systems spontaneously adapt to all visible items, or to subsets segregated by salient attributes such as color or pitch. We measured perceived numerosity after participants adapted to highly numerous stimuli with color either matched to or different from the test. Matched colors caused a 25% underestimation of numerosity, while different colors had virtually no effect. This was true both for physically different colors, and for the same colors perceived as different, via a color-assimilation illusion. A similar result occurred in the acoustic domain, where adaptation magnitude was halved when the adaptor and test differed in pitch. Taken together, our results support the idea that numerosity perception is selectively tuned to salient environmental attributes.

Understanding the Effects of Transcranial Electrical Stimulation in Numerical Cognition: A Systematic Review for Clinical Translation

Atypical development of numerical cognition (dyscalculia) may increase the onset of neuropsychiatric symptoms, especially when untreated, and it may have long-term detrimental social consequences. However, evidence-based treatments are still lacking. Despite plenty of studies investigating the effects of transcranial electrical stimulation (tES) on numerical cognition, a systematized synthesis of results is still lacking. In the present systematic review (PROSPERO ID: CRD42021271139), we found that the majority of reports (20 out of 26) showed the effectiveness of tES in improving both number (80%) and arithmetic (76%) processing. In particular, anodal tDCS (regardless of lateralization) over parietal regions, bilateral tDCS (regardless of polarity/lateralization) over frontal regions, and tRNS (regardless of brain regions) strongly enhance number processing. While bilateral tDCS and tRNS over parietal and frontal regions and left anodal tDCS over frontal regions consistently improve arithmetic skills. In addition, tACS seems to be more effective than tDCS at ameliorating arithmetic learning. Despite the variability of methods and paucity of clinical studies, tES seems to be a promising brain-based treatment to enhance numerical cognition. Recommendations for clinical translation, future directions, and limitations are outlined.

Analyzing the misperception of exponential growth in graphs

Exponential growth is frequently underestimated, an error that can have a heavy social cost in the context of epidemics. To clarify its origins, we measured the human capacity (N = 521) to extrapolate linear and exponential trends in scatterplots. Four factors were manipulated: the function underlying the data (linear or exponential), the response modality (pointing or venturing a number), the scale on the y axis (linear or logarithmic), and the amount of noise in the data. While linear extrapolation was precise and largely unbiased, we observed a consistent underestimation of noisy exponential growth, present for both pointing and numerical responses. A biased ideal-observer model could explain these data as an occasional misperception of noisy exponential graphs as quadratic curves. Importantly, this underestimation bias was mitigated by participants' math knowledge, by using a logarithmic scale, and by presenting a noiseless exponential curve rather than a noisy data plot, thus suggesting concrete avenues for interventions.

Transforming Social Perspectives with Cognitive Maps

Growing evidence suggests that cognitive maps represent relations between social knowledge similar to how spatial locations are represented in an environment. Notably, the extant human medial temporal lobe literature assumes associations between social stimuli follow a linear associative mapping from an egocentric viewpoint to a cognitive map. Yet, this form of associative social memory doesn't account for a core phenomenon of social interactions in which social knowledge learned via comparisons to the self, other individuals, or social networks are assimilated within a single frame of reference. We argue that hippocampal-entorhinal coordinate transformations, known to integrate egocentric and allocentric spatial cues, inform social perspective switching between the self and others. We present evidence that the hippocampal formation helps inform social interactions by relating self versus other social attribute comparisons to society in general, which can afford rapid and flexible assimilation of knowledge about the relationship between the self and social networks of varying proximities. We conclude by discussing the ramifications of cognitive maps in aiding this social perspective transformation process in states of health and disease.

Prone versus supine free-breathing for right-sided whole breast radiotherapy

Prone setup has been advocated to improve organ sparing in whole breast radiotherapy without impairing breast coverage. We evaluate the dosimetric advantage of prone setup for the right breast and look for predictors of the gain. Right breast cancer patients treated in 2010-2013 who had a dual supine and prone planning were retrospectively identified. A penalty score was computed from the mean absolute dose deviation to heart, lungs, breasts, and tumor bed for each patient's supine and prone plan. Dosimetric advantage of prone was assessed by the reduction of penalty score from supine to prone. The effect of patients' characteristics on the reduction of penalty was analyzed using robust linear regression. A total of 146 patients with right breast dual plans were identified. Prone compared to supine reduced the penalty score in 119 patients (81.5%). Lung doses were reduced by 70.8%, from 4.8 Gy supine to 1.4 Gy prone. Among patient's characteristics, the only significant predictors were the breast volumes, but no cutoff could identify when prone would be less advantageous than supine. Prone was associated with a dosimetric advantage in most patients. It sets a benchmark of achievable lung dose reduction.Trial registration: ClinicalTrials.gov NCT02237469, HUGProne, September 11, 2014, retrospectively registered.

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.

Uncertainty and Prior Assumptions, Rather Than Innate Logarithmic Encoding, Explain Nonlinear Number-to-Space Mapping

Mapping number to space is natural and spontaneous but often nonveridical, showing a clear compressive nonlinearity that is thought to reflect intrinsic logarithmic encoding of numerical values. We asked 78 adult participants to map dot arrays onto a number line across nine trials. Combining participant data, we confirmed that on the first trial, mapping was heavily compressed along the number line, but it became more linear across trials. Responses were well described by logarithmic compression but also by a parameter-free Bayesian model of central tendency, which quantitatively predicted the relationship between nonlinearity and number acuity. To experimentally test the Bayesian hypothesis, we asked 90 new participants to complete a color-line task in which they mapped noise-perturbed color patches to a "color line." When there was more noise at the high end of the color line, the mapping was logarithmic, but it became exponential with noise at the low end. We conclude that the nonlinearity of both number and color mapping reflects contextual Bayesian inference processes rather than intrinsic logarithmic encoding.

Constructing rationals through conjoint measurement of numerator and denominator as approximate integer magnitudes in tradeoff relations

To investigate mechanisms of rational representation, I consider (1) construction of an ordered continuum of psychophysical scale of magnitude of sensation; (2) counting mechanism leading to an approximate numerosity scale for integers; and (3) conjoint measurement structure pitting the denominator against the numerator in tradeoff positions. Number sense of resulting rationals is neither intuitive nor expedient in their manipulation.

Crowd control: Reducing individual estimation bias by sharing biased social information

Cognitive biases are widespread in humans and animals alike, and can sometimes be reinforced by social interactions. One prime bias in judgment and decision-making is the human tendency to underestimate large quantities. Previous research on social influence in estimation tasks has generally focused on the impact of single estimates on individual and collective accuracy, showing that randomly sharing estimates does not reduce the underestimation bias. Here, we test a method of social information sharing that exploits the known relationship between the true value and the level of underestimation, and study if it can counteract the underestimation bias. We performed estimation experiments in which participants had to estimate a series of quantities twice, before and after receiving estimates from one or several group members. Our purpose was threefold: to study (i) whether restructuring the sharing of social information can reduce the underestimation bias, (ii) how the number of estimates received affects the sensitivity to social influence and estimation accuracy, and (iii) the mechanisms underlying the integration of multiple estimates. Our restructuring of social interactions successfully countered the underestimation bias. Moreover, we find that sharing more than one estimate also reduces the underestimation bias. Underlying our results are a human tendency to herd, to trust larger estimates than one's own more than smaller estimates, and to follow disparate social information less. Using a computational modeling approach, we demonstrate that these effects are indeed key to explain the experimental results. Overall, our results show that existing knowledge on biases can be used to dampen their negative effects and boost judgment accuracy, paving the way for combating other cognitive biases threatening collective systems.

Boosting Numerical Cognition in Children and Adolescents with Mathematical Learning Disabilities by a Brain-Based Intervention: A Study Protocol for a Randomized, Sham-Controlled Clinical Trial

Numbers are everywhere, and supporting difficulties in numerical cognition (e.g., mathematical learning disability (MLD)) in a timely, effective manner is critical for their daily use. To date, only low-efficacy cognitive-based interventions are available. The extensive data on the neurobiology of MLD have increased interest in brain-directed approaches. The overarching goal of this study protocol is to provide the scientific foundation for devising brain-based and evidence-based treatments in children and adolescents with MLD. In this double-blind, between-subject, sham-controlled, randomized clinical trial, transcranial random noise stimulation (tRNS) plus cognitive training will be delivered to participants. Arithmetic, neuropsychological, psychological, and electrophysiological measures will be collected at baseline (T0), at the end of the interventions (T1), one week (T2) and three months later (T3). We expect that tRNS plus cognitive training will significantly improve arithmetic measures at T1 and at each follow-up (T2, T3) compared with placebo and that such improvements will correlate robustly and positively with changes in the neuropsychological, psychological, and electrophysiological measures. We firmly believe that this clinical trial will produce reliable and positive results to accelerate the validation of brain-based treatments for MLD that have the potential to impact quality of life.

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.

Psychology Within and Without the State

Psychological research in small-scale societies is crucial for what it stands to tell us about human psychological diversity. However, people in these communities, typically Indigenous communities in the global South, have been underrepresented and sometimes misrepresented in psychological research. Here I discuss the promises and pitfalls of psychological research in these communities, reviewing why they have been of interest to social scientists and how cross-cultural comparisons have been used to test psychological hypotheses. I consider factors that may be undertheorized in our research, such as political and economic marginalization, and how these might influence our data and conclusions. I argue that more just and accurate representation of people from small-scale communities around the world will provide us with a fuller picture of human psychological similarity and diversity, and it will help us to better understand how this diversity is shaped by historical and social processes. Expected final online publication date for the Annual Review of Psychology, Volume 73 is January 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

Response bias in numerosity perception at early judgments and systematic underestimation

Mental number representation relies on mapping numerosity based on nonsymbolic stimuli to symbolic magnitudes. It is known that mental number representation builds on a logarithmic scale, and thus numerosity decisions result in underestimation. In the current study, we investigated the temporal dynamics of numerosity perception in four experiments by employing the response-deadline SAT procedure. We presented random number of dots and required participants to make a numerosity judgment by comparing the perceived number of dots to 50. Using temporal dynamics in numerosity perception allowed us to observe a response bias at early decisions and a systematic underestimation at late decisions. In all three experiments, providing feedback diminished the magnitude of underestimation, whereas in Experiment 3 the absence of feedback resulted in greater underestimation errors. These results were in accordance with the findings that suggested feedback is necessary for the calibration of the mental number representation.

Cognitive mediators of US-China differences in early symbolic arithmetic

Chinese children routinely outperform American peers in standardized tests of mathematics knowledge. To examine mediators of this effect, 95 Chinese and US 5-year-olds completed a test of overall symbolic arithmetic, an IQ subtest, and three tests each of symbolic and non-symbolic numerical magnitude knowledge (magnitude comparison, approximate addition, and number-line estimation). Overall Chinese children performed better in symbolic arithmetic than US children, and all measures of IQ and number knowledge predicted overall symbolic arithmetic. Chinese children were more accurate than US peers in symbolic numerical magnitude comparison, symbolic approximate addition, and both symbolic and non-symbolic number-line estimation; Chinese and U.S. children did not differ in IQ and non-symbolic magnitude comparison and approximate addition. A substantial amount of the nationality difference in overall symbolic arithmetic was mediated by performance on the symbolic and number-line tests.

Operational momentum during children's approximate arithmetic relates to symbolic math skills and space-magnitude association

Operational momentum (OM) refers to the behavioral tendency to overestimate or underestimate the results of addition or subtraction, respectively. The cognitive mechanism of the OM effect and how it is related to the development of symbolic math abilities are not well understood. The current study examined whether individual differences in the OM effect are related to symbolic arithmetic abilities, number line estimation performance, and the space-magnitude association effect in young children. In this study, first-grade elementary school children manifested the OM effect during approximate addition and subtraction. Individual differences in the OM effect were not correlated with number line estimation error. Interestingly, children who showed a greater degree of the OM effect performed not worse, but better on the symbolic arithmetic task. In addition, the OM effect was correlated with the space-magnitude association (size congruity) effect measured with the Numerical Stroop task. More specifically, the OM bias was correlated with the ability to inhibit interference from competing information on the incongruent trials of the Numerical Stroop task. Our results suggest that the inaccuracy of numerical magnitude representations is not the source of the OM effect. Given that children with better math ability showed a greater OM bias, a stronger OM effect may reflect better intuition in arithmetic operations. Altogether, we carefully interpret these findings as suggesting that a greater OM effect reflects superior intuition or fundamental knowledge of arithmetic operations and a more adult-like maturation of the reorienting component of the attentional system.