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Young children's estimates of numerical magnitude increase approximately logarithmically with actual magnitude. The conventional interpretation of this finding is that children's estimates reflect an innate logarithmic encoding of number. A recent set of findings, however, suggests that logarithmic number-line estimates emerge via a dynamic encoding mechanism that is sensitive to previously encountered stimuli. Here we examine trial-to-trial changes in logarithmicity of numerosity estimates to test an alternative dynamic model (D-MLLM) with both a strong logarithmic component and a weak response to previous stimuli. In support of D-MLLM, first-trial numerosity estimates in both adults (Study 1, 2, 3, and 4) and children (Study 4) were strongly logarithmic, despite zero previous stimuli. Additionally, although numerosity of a previous trial affected adults’ estimates, the influence of previous numbers always accompanied the logarithmic-to-linear shift predicted by D-MLLM. We conclude that a dynamic encoding mechanism is not necessary for compressive mapping, but sequential effects on response scaling are a possible source of linearity in adults’ numerosity estimation.

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... Studies conducted with preschool children are even more scarce and their results are mixed. Some authors report a linear fit within the 0-10 range and log-to-linear shift within the 0-100 range (Sasanguie et al., 2012a(Sasanguie et al., ,b, 2016, some logarithmic mapping function (for 0-30 range, Kim and Opfer, 2018; for 0-100 range, Praet and Desoete, 2014;Lee et al., 2022 for even larger ranges), and others two-segment linear model fit (Ebersbach et al., 2008). Interpretation of these studies is additionally hampered by the fact that some of them used a dual format of numerical stimuli presented as visual sets accompanied by number-words or digits. ...

... Additionally, Experiment 2 (but not Experiment 1) provided strong evidence for the scalar variance of estimates, which is expected in an uncompressed linear model. It is worth noting that although the logarithmic mapping of the sets' numerosities to a line was found in the study by Dehaene et al. (2008), and as an initial model in studies by Siegler and Opfer (2003) or Kim and Opfer (2018), the few other studies that used non-symbolic materials, especially those involving young children, showed mixed results, with the advantage of the linear model (either unitary or multi-segment; e.g., Ebersbach et al., 2008;Sasanguie et al., 2012b), at least in some numerical ranges, which is in line with our results. The question arises as to why non-symbolic NLE in children does not decisively fit the log-tolinear shift found in the symbolic number-line task. ...

... Already in the study by Opfer and Siegler (2007), the fit of the mapping changed after one-time feedback correcting the linear position of the number. Dynamic nature of mapping strategy shift was also shown by Kim and Opfer (2018) in children with non-symbolic task. In our Experiments 1 and 2, the children received training in which the position of the set was corrected according to the linear model, although the justification for shifting the set referred only to ordinal (not scaling) relations. ...

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.

... Recently, however, Kim and Opfer (2018) seriously challenged this interpretation by demonstrating that even the very first trials of number-to-space mappingwhich clearly cannot be subject to serial dependenceshow strong compressive nonlinearities. Indeed, mapping of the first trial had a greater logarithmic component than mapping of later trials. ...

... On the other hand, a static logarithmic nonlinearity should not affect mapping precision. We therefore repeated Kim and Opfer's (2018) experiment showing that mapping precision steadily improves over trials and that the improvement quantitatively predicts the results. We also asked observers to map color onto space, manipulating stimuli to simulate the presumed noise gradient limiting numerosity judgments by adding external noise. ...

... Sample size was determined following Kim and Opfer (2018), who employed 40 participants per study. We adapted this to our stimulus setup (nine numbers), which thus prescribed testing a minimum of 45 participants. ...

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.

... estimate of a current quantity, such as time, size, length, weight, and numerosity of a set (Gilden, Thornton, & Mallon, 1995;Helson, Michels, & Sturgeon, 1954;Parducci, 1963Parducci, , 1965Petrov & Anderson, 2005 & Stephan, 2015). Recently, researchers began noticing dynamic effects in number-line estimation tasks (Cicchini, Anobile, & Burr, 2014;D. Kim & Opfer, 2018), which are used in such diverse fields as numerical cognition and education. ...

... In one model, the internal scaling of numbers is linear, and compression results from dynamic updates (Cicchini et al., 2014). In another, the internal scaling of numbers is logarithmic, and decompression results from memory for previous trials (D. Kim & Opfer, 2018). Here, we tested a novel and unexpected prediction of the latter model. ...

... When estimates are perfectly logarithmic, meaning that there is no contribution of linear representations, λ equals 1. When estimates are perfectly linear, which indicates no influence of log representations in estimates, λ equals 0. The logarithmicity component (λ) adequately captures the log-to-linear changes over the course of development (D. Kim & Opfer, 2018Opfer et al., 2016Opfer et al., , 2019. Estimates of numbers in young children exhibit high values of logarithmicity components, which suggests that young children rely heavily on log representations. ...

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.

... Once students understand the mathematical properties of the number line, linear placements occur during a dynamic process as people adjust their placements during the act of positioning numerals on the visuospatial line (Dotan & Dehaene, 2016;Kim & Opfer, 2018). Such a dynamic process allows for strategic or contextual influences on number line performance, above and beyond an influence of the ANS (Barth & Paladino, 2011;Cicchini et al., 2014;Cohen & Blanc-Goldhammer, 2011;Rouder & Geary, 2014;Slusser et al., 2013). ...

... The use of such strategies need not preclude the existence of the ANS, as these representations could still manifest when students are in the early stages of learning the number line or under conditions (e.g., dual task) that disrupt the ability to use top-down strategies (Dotan & Dehaene, 2016;Kim & Opfer, 2018). In any case, a bias to represent quantities along a continuum provides a more direct link to spatial abilities than does the precision of ANS representations per se. ...

... Whatever the contributing factors, accurate placements on the number line require precise angular orientation as students move their hands to position the numeral on the line (Dotan & Dehaene, 2016;Kim & Opfer, 2018). Boys' heightened ability to attentionally focus on subtle deviation in the angular orientation of environmental cues appears to provide them with an advantage in precisely situating numerals on visuospatial representations of the mathematical number line. ...

The study tested the hypotheses that boys will have an advantage learning the fractions number line and this advantage will be mediated by spatial abilities. Fractions number line and, as a contrast, fractions arithmetic performance were assessed for 342 adolescents, as was their intelligence, working memory, and various spatial abilities. Boys showed smaller placement errors on the fractions number line (d = −0.22) and correctly solved more fractions arithmetic problems (d = 0.23) than girls. Working memory and intelligence predicted performance on both fractions measures, and a measure of visuospatial attention uniquely predicted number line performance and fully mediated the sex difference. Visuospatial working memory uniquely predicted fractions arithmetic performance and fully mediated the sex difference. The results help to clarify the nuanced relations between spatial abilities and formal mathematics learning and the sex differences that often emerge in mathematical domains that have a visuospatial component. Boys are more accurate (lower error) in the placement of fractions on the mathematical number line. The sex difference in number line accuracy is fully mediated by boys' advantage in visuospatial attention

... The mental number line provides an intuitive but inexact understanding of smaller to larger, whereas children's understanding of the mathematical number line requires an explicit understanding of its properties and how Arabic numerals and rational numbers are situated on the line (Siegler, Thompson, & Schneider, 2011;Sullivan & Barner, 2014). Moreover, the actual placement of numerals or fractions on the physical number line often involves the explicit topdown use of strategies to decompose the line into segments (e.g., using the midpoint; Barth & Paladino, 2011;Rouder & Geary, 2014) and engagement of inherent visuospatial attentional systems that are sensitive to positional orientation (Dotan & Dehaene, 2016;Kim & Opfer, 2018;Longo & Lourenco, 2007). In other words, students' dynamic configuration of spatial and attentional systems during the act of placing numerals on the number line appears to be guided by their conceptual understanding of numerical magnitude as related to the mathematical number line, as opposed to awareness of mathematical knowledge implicit in the organization of primary spatial and magnitude systems. ...

... The finding for visuospatial attention provides more detail on the spatial competencies that contribute to some aspects of mathematics than is common in this literature. In this case, we see that sensitivity to angular deviations aids number line performance, potentially because accurate number line placements require precise angular orientation as individuals move their hands to position the numeral on the line (Dotan & Dehaene, 2016;Kim & Opfer, 2018). In other words, a heightened ability to focus on subtle deviation in angular orientation could provide boys with an advantage during the act of placing numerals on the line rather than boys' having a better conceptual understanding of the mathematical number line. ...

The articles in this special issue provide state-of-the-art reviews of the brain and cognitive systems that are engaged during some aspects of mathematical learning, as well as the self-beliefs, anxiety, and social factors that influence engagement with mathematics, along with discussion of any associated sex differences. These issues are integrated into an evolutionary perspective that includes discussion of how evolved brain and cognitive systems might be co-opted for learning in the evolutionarily novel domain of mathematics. Attitudes and beliefs about mathematics are considered in the context of the evolution of self-awareness that in turn explains why many students do not value mathematics, despites its importance in the modern world, as highly as many other personal traits, such as their physical appearance. The overall argument is that reflecting on academic learning and attitudes from an evolutionary perspective provides insights into student learning and self-beliefs about learning that might otherwise elude explanation.

... We investigated these relations in the domain of children's numerical predictions because young children often have less accurate number representations than adults (Kim and Opfer, 2018), which provides an opportunity to measure the degree to which their accuracy influences numerical predictions. The accuracy of number representations improves over development as symbolic and approximate number representations become aligned (Lourenco, 2016). ...

Predictions begin with an extrapolation of the properties of their underlying representations to forecast a future state not presently in evidence. For numerical predictions, sets of numbers are summarized and the result forms the basis of and constrains numerical predictions. One open question is how the accuracy of underlying representations influences predictions, particularly numerical predictions. It is possible that inaccuracies in individual number representations are randomly distributed and averaged over during summarization (e.g., wisdom of crowds). It is also possible that inaccuracies are not random and lead to errors in predictions. We investigated this question by measuring the accuracy of individual number representations of 279 children ages 8–12 years, using a 0–1,000 number line, and numerical predictions, measured using a home run derby task. Consistent with prior research, our results from mixed random effects models evaluating percent absolute error (PAE; prediction error) demonstrated that third graders’ representations of individual numbers were less accurate, characterized by overestimation errors, and were associated with overpredictions (i.e., predictions above the set mean). Older children had more accurate individual number representations and a slight tendency to underpredict (i.e., predictions below the set mean). The results suggest that large, systematic inaccuracies appear to skew predictions while small, random errors appear to be averaged over during summarization. These findings add to our understanding of summarization and its role in numerical predictions.

... Non-symbolic number-line estimation. This task paralleled the number-line estimation task with Arabic numerals used in Kim and Opfer [33] (Fig 2). Children were presented a number-line with a box of 30 dots on the right end and an empty box of dots on the left end. ...

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.

... We suggest that the overestimation in the similar condition is due to comparisons between similar stimuli, yielding the opposite of redundancy masked percept observed in the neutral and dissimilar conditions. An efficient way to overcome such contextual effects is to use a single trial paradigm [45][46][47] (see also 48 ). Although single trials cannot exclude all contextual factors that might influence observers' judgments, they often provide accurate estimates of the probed feature, and may yield less biased responses than typical psychophysical paradigms that present several exemplars of similar stimuli 45,46 (see also 9 , for a detailed discussion on single trials). ...

The perception of a target depends on other stimuli surrounding it in time and space. This contextual modulation is ubiquitous in visual perception, and is usually quantified by measuring performance on sets of highly similar stimuli. Implicit or explicit comparisons among the stimuli may, however, inadvertently bias responses and conceal strong variability of target appearance. Here, we investigated the influence of contextual stimuli on the perception of a repeating pattern (a line triplet), presented in the visual periphery. In the neutral condition, the triplet was presented a single time to capture its minimally biased perception. In the similar and dissimilar conditions, it was presented within stimulus sets composed of lines similar to the triplet, and distinct shapes, respectively. The majority of observers reported perceiving a line pair in the neutral and dissimilar conditions, revealing ‘redundancy masking’, the reduction of the perceived number of repeating items. In the similar condition, by contrast, the number of lines was overestimated. Our results show that the similar context did not reveal redundancy masking which was only observed in the neutral and dissimilar context. We suggest that the influence of contextual stimuli has inadvertently concealed this crucial aspect of peripheral appearance.

... However, a general association between quantity and space is not sufficient for correct positioning of quantities on a number line. Previous research has shown that children use various ordering strategies on a non-symbolic number line estimation task (Kim & Opfer, 2018;Van 't Noordende et al., 2018). A crucial aspect of number line estimation strategies is understanding the order of quantities and the distance between quantities. ...

The development of (early) numerical cognition builds on children’s ability to understand and manipulate quantities and numbers. However, previous research did not find conclusive evidence on the role of symbolic and non-symbolic skills in the development of (early) numerical cognition. The aim of the current study was to clarify the relation between different types of non-symbolic quantity skills, symbolic numerical skills and early numerical cognition. A sample of 43 children was tested at the age of 3.5 years and at the age of 5 years. At 3.5 years, non-symbolic number line estimation, non-symbolic quantity comparison and symbolic enumerating skills were measured. At 5 years, early numerical cognition, defined as symbolic number line estimation and counting, were measured It was found that non-symbolic number line estimation at 3.5 years could predict both symbolic number line estimation and counting at 5 years. Enumerating at 3.5 years could only predict counting at 5 years. This suggests that both non-symbolic and symbolic skills play a role in the development of early numerical cognition, although enumerating skills do not transfer to all types of early numerical cognition. Furthermore, not all non-symbolic skills seem to play an important role in the development of early numerical cognition. The results suggest that non-symbolic quantity comparison does not contribute much to the development of early numerical cognition. Associations between non-symbolic quantities and space, operationalized here as non-symbolic number line estimation, seem central to the development of early math from preschool to kindergarten age.

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.

Kim and Opfer (2017) found that number-line estimates increased approximately logarithmically with number when an upper bound (e.g., 100 or 1000) was explicitly marked (bounded condition) and when no upper bound was marked (unbounded condition). Using procedural suggestions from Cohen and Ray (2020), we examined whether this logarithmicity might come from restrictions on the response space provided. Consistent with our previous findings, logarithmicity was evident whether tasks were bounded or unbounded, with the degree of logarithmicity tied to the numerical value of the estimates rather than the response space per se. We also found a clear log-to-linear shift in numerical estimates. Results from Bayesian modeling supported the idea that unbounded tasks are qualitatively similar to bounded ones, but unbounded ones lead to greater logarithmicity. Our findings support the original findings of Kim and Opfer (2017) and extend their generality to more age groups and more varieties of number-line estimation. (PsycINFO Database Record (c) 2020 APA, all rights reserved).

Dyscalculia is often associated with poor numerosity sensitivity. However, it is not known whether the perceptual systems of dyscalculics have implicit access to the sensory noise of numerosity judgements, and whether their perceptual systems take the noise levels into account in optimizing their perception. We tackled this question by measuring central tendency and serial dependence with a numberline task on dyscalculics and math-typical preadolescents. Numerosity thresholds were also measured with a separate 2AFC discrimination task. Our data confirmed that dyscalculics had poorer numerosity sensitivity and less accurate numberline mapping. Importantly, numberline responses, as well as central tendency and serial dependence strengths, were well predicted by sensory thresholds and could be modelled by a performance-optimizing Bayesian model based on sensory thresholds, suggesting that the functional architecture of systems encoding numerosity in dyscalculia is preserved. We speculate that the numerosity system of dyscalculics has retained those perceptual strategies that are useful to cope with and compensate for low sensory resolution.

There is an ongoing debate over the psychophysical functions that best fit human data from numerical estimation tasks. To test whether one psychophysical function could account for data across diverse tasks, we examined 40 kindergartners, 38 first graders, 40 second graders and 40 adults’ estimates using two fully crossed 2 × 2 designs, crossing symbol (symbolic, non-symbolic) and boundedness (bounded, unbounded) on free number-line tasks (Experiment 1) and crossing the same factors on anchored tasks (Experiment 2). Across all 8 tasks, 88.84% of participants provided estimates best fit by a mixed log-linear model, and the weight of the logarithmic component (λ) decreased with age. After controlling for age, the λ significantly predicted arithmetic skills, whereas parameters of other models failed to do so. Results suggest that the logarithmic-to-linear shift theory provides a unified account of numerical estimation and provides uniquely accurate predictions for mathematical proficiency.

BACKGROUND: The objective of this study was to scrutinize number line estimation behaviors displayed by children in mathematics classrooms during the first three years of schooling. We extend existing research by not only mapping potential logarithmic-linear shifts but also provide a new perspective by studying in detail the estimation strategies of individual target digits within a number range familiar to children. METHODS: Typically developing children (n = 67) from Years 1-3 completed a number-to-position numerical estimation task (0-20 number line). Estimation behaviors were first analyzed via logarithmic and linear regression modeling. Subsequently, using an analysis of variance we compared the estimation accuracy of each digit, thus identifying target digits that were estimated with the assistance of arithmetic strategy. RESULTS: Our results further confirm a developmental logarithmic-linear shift when utilizing regression modeling; however, uniquely we have identified that children employ variable strategies when completing numerical estimation, with levels of strategy advancing with development. CONCLUSION: In terms of the existing cognitive research, this strategy factor highlights the limitations of any regression modeling approach, or alternatively, it could underpin the developmental time course of the logarithmic-linear shift. Future studies need to systematically investigate this relationship and also consider the implications for educational practice.

Children's number-line estimation has produced a lively debate about representational change, supported by apparently incompatible data regarding descriptive adequacy of logarithmic (Opfer, Siegler, & Young, 2011) and cyclic power models (Slusser, Santiago, & Barth, 2013). To test whether methodological differences might explain discrepant findings, we created a fully crossed 2. ×. 2 design and assigned 96 children to one of four cells. In the design, we crossed anchoring (free, anchored) and sampling (over-, even-), which were candidate factors to explain discrepant findings. In three conditions (free/over-sampling, free/even-sampling, and anchored/over-sampling), the majority of children provided estimates better fit by the logarithmic than cyclic power function. In the last condition (anchored/even-sampling), the reverse was found. Results suggest that logarithmically-compressed numerical estimates do not depend on sampling, that the fit of cyclic power functions to children's estimates is likely an effect of anchors, and that a mixed log/linear model provides a useful model for both free and anchored numerical estimation.

Links between spatial and numerical thinking are well established in studies of adult cognition. Here, we review recent research on the origins and development of these links, with an emphasis on the formative role of experience in typical development and on the theoretical insights to be gained from infant cognition. This research points to three important influences on the development of spatial-numerical associations: innate mechanisms linking space and nonsymbolic number, gross and fine motor activity that couples spatial location to both symbolic and nonsymbolic number, and culturally bound activities (e.g., reading, writing, and counting) that shape the relationship between spatial direction and symbolic number.

Significance
The ability to map numbers onto space is fundamental to measurement and mathematics. The mental “numberline” is an important predictor of math ability, thought to reflect an internal, native logarithmic representation of number, later becoming linearized by education. Here we demonstrate that the nonlinearity results not from a static logarithmic transformation but from dynamic processes that incorporate past history into numerosity judgments. We show strong and significant correlations between the response to the current trial and the magnitude of the previous stimuli and that subjects respond with a weighted average of current and recent stimuli, explaining completely the logarithmic-like nonlinearity. We suggest that this behavior reflects a general strategy akin to predictive coding to cope adaptively with environmental statistics.

In the last couple of decades, there has been a growing number of reports on space-based representation of numbers and serial order in humans. In the present study, to explore evolutionary origins of such representations, we examined whether our closest evolutionary relatives, chimpanzees, map an acquired sequence onto space in a similar way to humans. The subjects had been trained to perform a number sequence task in which they touched a sequence of "small" to "large" Arabic numerals presented in random locations on the monitor. This task was presented in sessions that also included test trials consisting of only two numerals (1 and 9) horizontally arranged. On half of the trials 1 was located to the left of 9, whereas on the other half 1 was to the right to 9. The Chimpanzees' performance was systematically influenced by the spatial arrangement of the stimuli; specifically, they responded quicker when 1 was on the left and 9 on the right compared to the other way around. This result suggests that chimpanzees, like humans, spontaneously map a learned sequence onto space.

Evidence for spontaneous mappings between the dimensions of number and length, time and length, and number and time, has been recently described in preverbal infants. It is unclear, however, whether these abilities reflect the existence of privileged mappings between certain quantitative dimensions, like number, space and time, or instead the existence of a magnitude system underlying the representation of any quantitative dimension, and allowing mappings across those dimensions. Four experiments, using the same methods from previous research that revealed a number-length mapping in eight-month-old infants, investigated whether infants of the same age establish mappings between number and a different, non-spatial continuous dimension: level of brightness. We show that infants are able to learn and productively use mappings between brightness and number when they are positively related, i.e., larger numbers paired with brighter or higher contrast levels, and fail when they are inversely related, i.e., smaller numbers paired with brighter or higher contrast levels, suggesting that they are able to learn this mapping in a specific direction. However, infants not only do not show any baseline preference for any direction of the number-brightness mapping, but fail at transferring the discrimination from one dimension (number) to the other (brightness). Although infants can map multiple dimensions to one another, the number-length mapping may be privileged early in development, as it is for adults.

Abstract Despite their lack of language, human infants and several animal species possess some elementary abilities for numerical processing. These include the ability to recognize that a given numerosity is being presented visually or auditorily, and, at a later stage of development, the ability to compare two nume-rosities and to decide which is larger. We propose a model for the development of these abilities in a formal neuronal network. Initially, the model is equipped only with unordered numerosity detectors. It can therefore detect the numerosity of an input set and can be conditioned to react accordingly. In a later stage, the addition of a short-term memory network is shown to be sufficient for number comparison abilities to develop. Our computer simulations account for several phenomena in the numerical domain, including the distance effect and Fechner's law for numbers. They also demonstrate that infants' numerosity detection abilities may be explained without assuming that infants can count. The neurobiological bases of the critical components of the model are discussed.

In this study, we investigated in school-age children the relationship among mathematical performance, the perception of numerosity (discrimination and mapping to number line), and sustained visual attention. The results (on 68 children between 8 and 11years of age) show that attention and numerosity perception predict math scores but not reading performance. Even after controlling for several variables, including age, gender, nonverbal IQ, and reading accuracy, attention remained correlated with math skills and numerosity discrimination. These findings support previous reports showing the interrelationship between visual attention and both numerosity perception and math performance. It also suggests that attentional deficits may be implicated in disturbances such as developmental dyscalculia.

Experience engenders learning, but not all learning involves representational change. In this paper, we provide a dramatic case study of the distinction between learning and representational change. Specifically, we examined long- and short-term changes in representations of numeric magnitudes by asking individuals with Williams syndrome (WS) and typically developing (TD) children to estimate the position of numbers on a number line. As with TD children, accuracy of WS children's numerical estimates improved with age (Experiment 1) and feedback (Experiment 2). Both long- and short-term changes in estimates of WS individuals, however, followed an atypical developmental trajectory: as TD children gained in age and experience, increases in accuracy were accompanied by a logarithmic-to-linear shift in estimates of numerical magnitudes, whereas in WS individuals, accuracy increased but logarithmic estimation patterns persisted well into adulthood and after extensive training. These findings suggest that development of numerical estimation in WS is both arrested and atypical.

2 studies with 3 undergraduates each investigated the effects of previous stimuli on responses in an absolute judgment of loudnesses situation when feedback was and was not provided. Whether or not information feedback was provided, responses were assimilated to the value of the immediately previous stimulus in the series. The effects of stimuli more than 1 trial back in the sequence depend on the presence or absence of feedback. When the entire stimulus scale was shifted up or down to 5 db. from the level on the previous day, a substantial shift occurred in the constant error of judgment in the direction of the scale shift, providing evidence that a relatively long-term (24-hr) memory process was being used in the judgment situation. None of the currently available models is adequate to account for both these results and those of earlier studies. The form of the sequential dependencies observed may depend at least partially on the presence or absence of an identification function from stimuli to responses. (18 ref.) (PsycINFO Database Record (c) 2012 APA, all rights reserved)

Reviews the status of similarity as an explanatory construct with a focus on similarity judgments. For similarity to be a useful construct, one must be able to specify the ways or respects in which 2 things are similar. One solution to this problem is to restrict the notion of similarity to hard-wired perceptual processes. It is argued that this view is too narrow and limiting. Instead, it is proposed that an important source of constraints derives from the similarity comparison process itself. Both new experiments and other evidence are described that support the idea that respects are determined by processes internal to comparisons. (PsycINFO Database Record (c) 2012 APA, all rights reserved)

Similarity comparisons are highly sensitive to judgment context. Three experiments explore context effects that occur within
a single comparison rather than across several trials. Experiment 1 shows reliable intransitivities in which a target is judged
to be more similar to stimulus A than to stimulus B, more similar to B than to stimulus C, and more similar to C than to A.
Experiment 2 explores the locus of Tversky’s (1977) diagnosticity effect in which the relative similarity of two alternatives
to a target is influenced by a third alternative. Experiment 3 demonstrates a new violation of choice independence which is
explained by object dimensions’ becoming foregrounded or backgrounded, depending upon the set of displayed objects. The observed
violations of common assumptions to many models of similarity and choice can be accommodated in terms of a dynamic property-weighting
process based on the variability and diagnosticity of dimensions.

Studies have reported high correlations in accuracy across estimation contexts, robust transfer of estimation training to novel numerical contexts, and adults drawing mistaken analogies between numerical and fractional values. We hypothesized that these disparate findings may reflect the benefits and costs of learning linear representations of numerical magnitude. Specifically, children learn that their default logarithmic representations are inappropriate for many numerical tasks, leading them to adopt more appropriate linear representations despite linear representations being inappropriate for estimating fractional magnitude. In Experiment 1, this hypothesis accurately predicted a developmental shift from logarithmic to linear estimates of numerical magnitude and a negative correlation between accuracy of numerical and fractional magnitude estimates (r = −.80). In Experiment 2, training that improved numerical estimates also led to poorer fractional magnitude estimates. Finally, both before and after training that eliminated age differences in estimation accuracy, complementary sex differences were observed across the two estimation contexts.

Data on numerical processing by verbal (human) and non-verbal (animal and human) subjects are integrated by the hypothesis that a non-verbal counting process represents discrete (countable) quantities by means of magnitudes with scalar variability. These appear to be identical to the magnitudes that represent continuous (uncountable) quantities such as duration. The magnitudes representing countable quantity are generated by a discrete incrementing process, which defines next magnitudes and yields a discrete ordering. In the case of continuous quantities, the continuous accumulation process does not define next magnitudes, so the ordering is also continuous (‘dense’). The magnitudes representing both countable and uncountable quantity are arithmetically combined in, for example, the computation of the income to be expected from a foraging patch. Thus, on the hypothesis presented here, the primitive machinery for arithmetic processing works with real numbers (magnitudes).

Spatial skill is highly related to success in math and science (e.g., Casey, Nuttall, Pezaris, & Benbow, 1995). However, little work has investigated the cognitive pathways by which the relation between spatial skill and math achievement emerges. We hypothesized that spatial skill plays a crucial role in the development of numerical reasoning by helping children to create a spatially meaningful, powerful numerical representation-the linear number line. In turn, a strong linear number representation improves other aspects of numerical knowledge such as arithmetic estimation. We tested this hypothesis using 2 longitudinal data sets. First, we found that children's spatial skill (i.e., mental transformation ability) at the beginning of 1st and 2nd grades predicted improvement in linear number line knowledge over the course of the school year. Second, we found that children's spatial skill at age 5 years predicted their performance on an approximate symbolic calculation task at age 8 and that this relation was mediated by children's linear number line knowledge at age 6. The results are consistent with the hypothesis that spatial skill can improve children's development of numerical knowledge by helping them to acquire a linear spatial representation of numbers.

Children (n = 130; M(age) = 8.51-15.68 years) and college-aged adults (n = 72; M(age) = 20.50 years) completed numerosity discrimination and lexical decision tasks. Children produced longer response times (RTs) than adults. R. Ratcliff's (1978) diffusion model, which divides processing into components (e.g., quality of evidence, decision criteria settings, nondecision time), was fit to the accuracy and RT distribution data. Differences in all components were responsible for slowing in children in these tasks. Children extract lower quality evidence than college-aged adults, unlike older adults who extract a similar quality of evidence as college-aged adults. Thus, processing components responsible for changes in RTs at the beginning of the life span are somewhat different from those responsible for changes occurring with healthy aging.

Barth and Paladino (2011) argue that changes in numerical representations are better modeled by a power function whose exponent gradually rises to 1 than as a shift from a logarithmic to a linear representation of numerical magnitude. However, the fit of the power function to number line estimation data may simply stem from fitting noise generated by averaging over changing proportions of logarithmic and linear estimation patterns. To evaluate this possibility, we used conventional model fitting techniques with individual as well as group average data; simulations that varied the proportion of data generated by different functions; comparisons of alternative models' prediction of new data; and microgenetic analyses of rates of change in experiments on children's learning. Both new data and individual participants' data were predicted less accurately by power functions than by logarithmic and linear functions. In microgenetic studies, changes in the best fitting power function's exponent occurred abruptly, a finding inconsistent with Barth and Paladino's interpretation that development of numerical representations reflects a gradual shift in the shape of the power function. Overall, the data support the view that change in this area entails transitions from logarithmic to linear representations of numerical magnitude.

Previous investigations on the subjective scale of numerical representations assumed that the scale type can be inferred directly from stimulus-response mapping. This is not a valid assumption, as mapping from the subjective scale into behavior may be nonlinear and/or distorted by response bias. Here we present a method for differentiating between logarithmic and linear hypotheses robust to the effect of distorting processes. The method exploits the idea that a scale is defined by transformational rules and that combinatorial operations with stimulus magnitudes should be closed under admissible transformations on the subjective scale. The method was implemented with novel variants of the number line task. In the line-marking task, participants marked the position of an Arabic numeral within an interval defined by various starting numbers and lengths. In the line construction task, participants constructed an interval given its part. Two alternative approaches to the data analysis, numerical and analytical, were used to evaluate the linear and log components. Our results are consistent with the linear hypothesis about the subjective scale with responses affected by a bias to overestimate small magnitudes and underestimate large magnitudes. We also observed that in the line-marking task, participants tended to overestimate as the interval start increased, and in the line construction task, they tended to overconstruct as the interval length increased. This finding suggests that magnitudes were encoded differently in the 2 tasks: in terms of their absolute magnitudes in the line-marking task and in terms of numerical differences in the line construction task.

How does understanding the decimal system change with age and experience? Second, third, sixth graders, and adults (Experiment 1: N = 96, mean ages = 7.9, 9.23, 12.06, and 19.96 years, respectively) made number line estimates across 3 scales (0-1,000, 0-10,000, and 0-100,000). Generation of linear estimates increased with age but decreased with numerical scale. Therefore, the authors hypothesized highlighting commonalities between small and large scales (15:100::1500:10000) might prompt children to generalize their linear representations to ever-larger scales. Experiment 2 assigned second graders (N = 46, mean age = 7.78 years) to experimental groups differing in how commonalities of small and large numerical scales were highlighted. Only children experiencing progressive alignment of small and large scales successfully produced linear estimates on increasingly larger scales, suggesting analogies between numeric scales elicit broad generalization of linear representations.

We investigated the relation between children's numerical-magnitude representations and their memory for numbers. Results of three experiments indicated that the more linear children's magnitude representations were, the more closely their memory of the numbers approximated the numbers presented. This relation was present for preschoolers and second graders, for children from low-income and middle-income backgrounds, for the ranges 0 through 20 and 0 through 1,000, and for four different tasks (categorization and number-line, measurement, and numerosity estimation) measuring numerical-magnitude representations. Other types of numerical knowledge-numeral identification and counting-were unrelated to recall of the same numerical information. The results also indicated that children's representations vary from trial to trial with the numbers they need to represent and remember and that general strategy-choice mechanisms may operate in selection of numerical representations, as in other domains.

Humans represent numbers along a mental number line (MNL), where smaller values are
located on the left and larger on the right. The origin of the MNL and its connections
with cultural experience are unclear: Pre-verbal infants and nonhuman species master a
variety of numerical abilities, supporting the existence of evolutionary ancient precursor
systems. In our experiments, 3-day-old domestic chicks, once familiarized with a target
number (5), spontaneously associated a smaller number (2) with the left space and a
larger number (8) with the right space. The same number (8), though, was associated with
the left space when the target number was 20. Similarly to humans, chicks associate
smaller numbers with the left space and larger numbers with the right space.

Representations of numerical value have been assessed by using bounded (e.g., 0-1,000) and unbounded (e.g., 0-?) number-line tasks, with considerable debate regarding whether 1 or both tasks elicit unique cognitive strategies (e.g., addition or subtraction) and require unique cognitive models. To test this, we examined how well a mixed log-linear model accounted for 86 5- to 9-year-olds' estimates on bounded and unbounded number-line tasks and how well it predicted mathematical performance. Compared with mixtures of 4 alternative models, the mixed log-linear model better predicted 76% of individual children's estimates on bounded number lines and 100% of children's estimates on unbounded number lines. Furthermore, the distribution of estimates was fit better by a Bayesian log-linear model than by a Bayesian distributional model that depicted estimates as being anchored to varying number of reference points. Finally, estimates were generally more logarithmic on unbounded than bounded number lines, but logarithmicity scores on both tasks predicted addition and subtraction skills, whereas model parameters of alternative models failed to do so. Results suggest that the logarithmic-to-linear shift theory provides a simple, unified framework for numerical estimation with high descriptive adequacy and yields uniquely accurate predictions for children's early math proficiency. (PsycINFO Database Record

Cognitive and neural research over the past few decades has produced sophisticated models of the representations and algorithms underlying numerical reasoning in humans and other animals. These models make precise predictions for how humans and other animals should behave when faced with quantitative decisions, yet primarily have been tested only in laboratory tasks. We used data from wild baboons’ troop movements recently reported by Strandburg-Peshkin, Farine, Couzin, and Crofoot (2015) to compare a variety of models of quantitative decision making. We found that the decisions made by these naturally behaving wild animals rely specifically on numerical representations that have key homologies with the psychophysics of human number representations. These findings provide important new data on the types of problems human numerical cognition was designed to solve and constitute the first robust evidence of true numerical reasoning in wild animals.

The number-to-position task, in which children and adults are asked to place numbers on a spatial number line, has become a classic measure of number comprehension. We present a detailed experimental and theoretical dissection of the processing stages that underlie this task. We used a continuous finger-tracking technique, which provides detailed information about the time course of processing stages. When adults map the position of 2-digit numbers onto a line, their final mapping is essentially linear, but intermediate finger location show a transient logarithmic mapping. We identify the origins of this log effect: Small numbers are processed faster than large numbers, so the finger deviates toward the target position earlier for small numbers than for large numbers. When the trajectories are aligned on the finger deviation onset, the log effect disappears. The small-number advantage and the log effect are enhanced in dual-task setting and are further enhanced when the delay between the 2 tasks is shortened, suggesting that these effects originate from a central stage of quantification and decision making. We also report cases of logarithmic mapping—by children and by a brain-injured individual—which cannot be explained by faster responding to small numbers. We show that these findings are captured by an ideal-observer model of the number-to-position mapping task, comprising 3 distinct stages: a quantification stage, whose duration is influenced by both exact and approximate representations of numerical quantity; a Bayesian accumulation-of-evidence stage, leading to a decision about the target location; and a pointing stage.

How do speed and accuracy trade off, and what components of information processing develop as children and adults make simple numeric comparisons? Data from symbolic and non-symbolic number tasks were collected from 19 first graders (Mage = 7.12 years), 26 second/third graders (Mage = 8.20 years), 27 fourth/fifth graders (Mage = 10.46 years), and 19 seventh/eighth graders (Mage = 13.22 years). The non-symbolic task asked children to decide whether an array of asterisks had a larger or smaller number than 50, and the symbolic task asked whether a two-digit number was greater than or less than 50. We used a diffusion model analysis to estimate components of processing in tasks from accuracy, correct and error response times, and response time (RT) distributions. Participants who were accurate on one task were accurate on the other task, and participants who made fast decisions on one task made fast decisions on the other task. Older participants extracted a higher quality of information from the stimulus arrays, were more willing to make a decision, and were faster at encoding, transforming the stimulus representation, and executing their responses. Individual participants’ accuracy and RTs were uncorrelated. Drift rate and boundary settings were significantly related across tasks, but they were unrelated to each other. Accuracy was mainly determined by drift rate, and RT was mainly determined by boundary separation. We concluded that RT and accuracy operate largely independently.

It is well-known in numerical cognition that higher numbers are represented with less absolute fidelity than lower numbers, often formalized as a logarithmic mapping. Previous derivations of this psychological law have worked by assuming that relative change in physical magnitude is the key psychologically-relevant measure (Fechner, 1860; Sun et al., 2012; Portugal & Svaiter, Minds and Machines, 21(1), 73-81, 2011). Ideally, however, this property of psychological scales would be derived from more general, independent principles. This paper shows that a logarithmic number line is the one which minimizes the error between input and representation relative to the probability that subjects would need to represent each number. This need probability is measured here through natural language and matches the form of need probabilities found in other literatures. The derivation does not presuppose anything like Weber's law and makes minimal assumptions about both the nature of internal representations and the form of the mapping. More generally, the results prove in a general setting that the optimal psychological scale will change with the square root of the probability of each input. For stimuli that follow a power-law need distribution this approach recovers either a logarithmic or power-law psychophysical mapping (Stevens, 1957, 1961, 1975).

This article studies the joint roles of similarity and frequency in determining graded category structure. Perceptual classification learning experiments were conducted in which presentation frequencies of individual exemplars were manipulated. The exemplars had varying degrees of similarity to members of the target and contrast categories. Classification accuracy and typicality ratings increased for exemplars presented with high frequency and for members of the target category that were similar to the high-frequency exemplars. Typicality decreased for members of the contrast category that were similar to the high-frequency exemplars. A frequency-sensitive similarity-to-exemplars model provided a good quantitative account of the classification learning and typicality data. The interactive relations among similarity, frequency, and categorization are considered in the General Discussion.

In the number-to-position task, with increasing age and numerical expertise, children's pattern of estimates shifts from a biased (nonlinear) to a formal (linear) mapping. This widely replicated finding concerns symbolic numbers, whereas less is known about other types of quantity estimation. In Experiment 1, Preschool, Grade 1, and Grade 3 children were asked to map continuous quantities, discrete nonsymbolic quantities (numerosities), and symbolic (Arabic) numbers onto a visual line. Numerical quantity was matched for the symbolic and discrete nonsymbolic conditions, whereas cumulative surface area was matched for the continuous and discrete quantity conditions. Crucially, in the discrete condition children's estimation could rely either on the cumulative area or numerosity. All children showed a linear mapping for continuous quantities, whereas a developmental shift from a logarithmic to a linear mapping was observed for both nonsymbolic and symbolic numerical quantities. Analyses on individual estimates suggested the presence of two distinct strategies in estimating discrete nonsymbolic quantities: one based on numerosity and the other based on spatial extent. In Experiment 2, a non-spatial continuous quantity (shades of gray) and new discrete nonsymbolic conditions were added to the set used in Experiment 1. Results confirmed the linear patterns for the continuous tasks, as well as the presence of a subset of children relying on numerosity for the discrete nonsymbolic numerosity conditions despite the availability of continuous visual cues. Overall, our findings demonstrate that estimation of numerical and non-numerical quantities is based on different processing strategies and follow different developmental trajectories. (PsycINFO Database Record
(c) 2015 APA, all rights reserved).

Psychophysics is a lively account by one of experimental psychology's seminal figures of his lifelong scientific quest for general laws governing human behavior. It is a landmark work that captures the fundamental themes of Steven's experimental research and his vision of what psychophysics and psychology are and can be. The context of this modern classic is detailed by Lawrence Mark's pungent and highly revealing introduction. The search for a general psychophysical law-a mathematical equation relating sensation to stimulus-pervades this work, first published in 1975. Stevens covers methods of measuring human psychophysical behavior; magnitude estimation, magnitude production, and cross-modality matching are used to examine sensory mechanisms, perceptual processes, and social consensus. The wisdom in this volume lies in its exposition of an approach that can apply generally to the study of human behavior.

Learning of the mathematical number line has been hypothesized to be dependent on an inherent sense of approximate quantity. Children's number line placements are predicted to conform to the underlying properties of this system; specifically, placements are exaggerated for small numerals and compressed for larger ones. Alternative hypotheses are based on proportional reasoning; specifically, numerals are placed relative to set anchors such as end points on the line. Traditional testing of these alternatives involves fitting group medians to corresponding regression models which assumes homogenous residuals and thus does not capture useful information from between- and within-child variation in placements across the number line. To more fully assess differential predictions, we developed a novel set of hierarchical statistical models that enable the simultaneous estimation of mean levels of and variation in performance, as well as developmental transitions. Using these techniques we fitted the number line placements of 224 children longitudinally assessed from first to fifth grade, inclusive. The compression pattern was evident in mean performance in first grade, but was the best fit for only 20% of first graders when the full range of variation in the data are modeled. Most first graders' placements suggested use of end points, consistent with proportional reasoning. Developmental transition involved incorporation of a mid-point anchor, consistent with a modified proportional reasoning strategy. The methodology introduced here enables a more nuanced assessment of children's number line representation and learning than any previous approaches and indicates that developmental improvement largely results from midpoint segmentation of the line.

When placing numbers along a number line with endpoints 0 and 1000, children generally space numbers logarithmically until around the age of 7, when they shift to a predominantly linear pattern of responding. This developmental shift of responding on the number placement task has been argued to be indicative of a shift in the format of the underlying representation of number (Siegler & Opfer, ). In the current study, we provide evidence from both child and adult participants to suggest that performance on the number placement task may not reflect the structure of the mental number line, but instead is a function of the fluency (i.e. ease) with which the individual can work with the values in the sequence. In Experiment 1, adult participants respond logarithmically when placing numbers on a line with less familiar anchors (1639 to 2897), despite linear responding on control tasks with standard anchors involving a similar range (0 to 1287) and a similar numerical magnitude (2000 to 3000). In Experiment 2, we show a similar developmental shift in childhood from logarithmic to linear responding for a non-numerical sequence with no inherent magnitude (the alphabet). In conclusion, we argue that the developmental trend towards linear behavior on the number line task is a product of successful strategy use and mental fluency with the values of the sequence, resulting from familiarity with endpoints and increased knowledge about general ordering principles of the sequence.A video abstract of this article can be viewed at:http://www.youtube.com/watch?v=zg5Q2LIFk3M

Diminishing marginal utility (DMU) is a basic tenet of economic and psychological models of judgment and choice, but its determinants are little understood. In the research reported here, we tested whether insensitivities in valuations of dollar amounts (e.g., $40, $100) may be due to inexact mappings of symbolic numbers (i.e., "40," "100") onto mental magnitudes. In three studies, we demonstrated that inexact mappings appear to guide valuation and mediate numeracy's relations with riskless valuations (Studies 1 and 1a) and risky choices (Study 2). The results highlight the fundamental notion that individuals' valuations of $100 depend critically on how individuals perceive and map the symbolic quantity "100." This notion has implications for conceptualizations of value, risk aversion, intertemporal choice, and dual-process theories of decision making. Normative implications are also briefly discussed.

This study investigated the relative importance of language and education to the development of numerical knowledge. Consistent with previous research suggesting that counting systems that transparently reflect the base-10 system facilitate an understanding of numerical concepts, Chinese and Chinese American kindergartners' and second graders' number line estimation (0-100 and 0-1000) was 1 to 2years more advanced than that of American children tested in previous studies. However, Chinese children performed better than their Chinese American peers, who were fluent in Chinese but had been educated in America, at kindergarten on 0-100 number lines, at second grade on 0-1000 number lines, and at both time points on complex addition problems. Overall, the pattern of findings suggests that educational approach may have a greater influence on numerical development than the linguistic structure of the counting system. The findings also demonstrate that, despite generating accurate estimates of numerical magnitude on 0-100 number lines earlier, it still takes Chinese children approximately 2years to demonstrate accurate estimates on 0-1000 number lines, which raises questions about how to promote the mapping of knowledge across numerical scales.

How do we understand two-digit numbers such as 42? Models of multi-digit number comprehension differ widely. Some postulate that the decades and units digits are processed separately and possibly serially. Others hypothesize a holistic process which maps the entire 2-digit string onto a magnitude, represented as a position on a number line. In educated adults, the number line is thought to be linear, but the "number sense" hypothesis proposes that a logarithmic scale underlies our intuitions of number size, and that this compressive representation may still be dormant in the adult brain. We investigated these issues by asking adults to point to the location of two-digit numbers on a number line while their finger location was continuously monitored. Finger trajectories revealed a linear scale, yet with a transient logarithmic effect suggesting the activation of a compressive and holistic quantity representation. Units and decades digits were processed in parallel, without any difference in left-to-right vs. right-to-left readers. The late part of the trajectory was influenced by spatial reference points placed at the left end, middle, and right end of the line. Altogether, finger trajectory analysis provides a precise cognitive decomposition of the sequence of stages used in converting a number to a quantity and then a position.

In this chapter, I put together the first elements of a mathematical theory relating neuro- biological observations to psychological laws in the domain of numerical cognition. The starting point is the postulate of a neuronal code whereby numerosity—the cardinal of a set of objects—is represented approximately by the firing of a population of numerosity detectors. Each of these neurons fires to a certain preferred numerosity, with a tuning curve which is a Gaussian function of the logarithm of numerosity. From this log- Gaussian code, decisions are taken using Bayesian mechanisms of log-likelihood compu- tation and accumulation. The resulting equations for response times and errors in classical tasks of number comparison and same-different judgments are shown to tightly fit behavioral and neural data. Two more speculative issues are discussed. First, new chronometric evidence is presented supporting the hypothesis that the acquisition of number symbols changes the mental number line, both by increasing its precision and by changing its coding scheme from logarithmic to linear. Second, I examine how symbolic and nonsymbolic representations of numbers affect performance in arithmetic compu- tations such as addition and subtraction.

In absolute judgment experiments with feedback, the events on a given trial,
n, exert a biphasic effect on succeeding responses: The response on trial
n + 1 is displaced toward the stimulus (or feedback) on trial
n (assimilation), and the response on each of several subsequent trials is displaced in the opposite direction (contrast). The possibility that the response on trial
n +
k can be explained as the weighted sum of events on that and preceding trials (linear model) was examined. It is concluded that (a) data from a typical absolute judgment experiment are not clearly consistent with the linear approach; (b) a 1st-order (1 trial back) linear model cannot account for the typical biphasic weighting sequence, but a 2nd-order model can do so; and (c) the possibility of real effects extending over several preceding trials cannot be excluded. (24 ref) (PsycINFO Database Record (c) 2012 APA, all rights reserved)

To date, a number of studies have demonstrated the existence of mismatches between children's implicit and explicit knowledge at certain points in development that become manifest by their gestures and gaze orientation in different problem solving contexts. Stimulated by this research, we used eye movement measurement to investigate the development of basic knowledge about numerical magnitude in primary school children. Sixty-six children from grades one to three (i.e. 6–9 years) were presented with two parallel versions of a number line estimation task of which one was restricted to behavioural measures, whereas the other included the recording of eye movement data. The results of the eye movement experiment indicate a quantitative increase as well as a qualitative change in children's implicit knowledge about numerical magnitudes in this age group that precedes the overt, that is, behavioural, demonstration of explicit numerical knowledge. The finding that children's eye movements reveal substantially more about the presence of implicit precursors of later explicit knowledge in the numerical domain than classical approaches suggests further exploration of eye movement measurement as a potential early assessment tool of individual achievement levels in numerical processing. Copyright © 2009 John Wiley & Sons, Ltd.

The relation between short-term and long-term change (also known as learning and development) has been of great interest throughout the history of developmental psychology. Werner and Vygotsky believed that the two involved basically similar progressions of qualitatively distinct knowledge states; behaviorists such as Kendler and Kendler believed that the two involved similar patterns of continuous growth; Piaget believed that the two were basically dissimilar, with only development involving qualitative reorganization of existing knowledge and acquisition of new cognitive structures. This article examines the viability of these three accounts in accounting for the development of numerical representations. A review of this literature indicated that Werner's and Vygotsky's position (and that of modern dynamic systems and information processing theorists) provided the most accurate account of the data. In particular, both changes over periods of years and changes within a single experimental session indicated that children progress from logarithmic to linear representations of numerical magnitudes, at times showing abrupt changes across a large range of numbers. The pattern occurs with representations of whole number magnitudes at different ages for different numerical ranges; thus, children progress from logarithmic to linear representations of the 0–100 range between kindergarten and second grade, whereas they make the same transition in the 0–1,000 range between second and fourth grade. Similar changes are seen on tasks involving fractions; these changes yield the paradoxical finding that young children at times estimate fractional magnitudes more accurately than adults do. Several different educational interventions based on this analysis of changes in numerical representations have yielded promising results.

This paper outlines a classificatory theory of cognitive similarity and compares it to feature approaches. The theory assumes that similarity is a function of the number of classes contained in the universe referred to in the judgments and the number of elements contained in the class defined by the two comparison stimuli. The theory is tested in a number of studies in which verbal stimuli are used. Most of the experiments concern context effects predicted by the theory. An empirical comparison is made between the classificatory theory and a feature theory. It is argued that the classificatory theory accounts for some of the data more easily than a feature approach does. Some implications for the use of similarity data in multidimensional scaling are also discussed.

A tendency for judgments of stimulus magnitude to be biased in the direction of the value of the immediately preceding stimulus
is found in magnitude estimations of loudness. This produces a bias in the empirical psychophysical function that results
in underestimation of the exponent of the unbiased function presumed to relate number and stimulus intensity, N = aSn. The biased judgment can be represented as a power product of focal and preceding stimulus intensity, Nij= aS
m Sj
b. A bias-free estimate of the correct exponent, n, can be obtained from the relation n = m + b.

Various measures have been used to investigate number processing in children, including a number comparison or a number line estimation task. The present study aimed to examine whether and to which extent these different measures of number representation are related to performance on a curriculum-based standardized mathematics achievement test in kindergarteners, first, second, and sixth graders. Children completed a number comparison task and a number line estimation task with a balanced set of symbolic (Arabic digits) and non-symbolic (dot patterns) stimuli. Associations with mathematics achievement were observed for the symbolic measures. Although the association with number line estimation was consistent over grades, the association with number comparison was much stronger in kindergarten compared to the other grades. The current data indicate that a good knowledge of the numerical meaning of Arabic digits is important for children's mathematical development and that particularly the access to the numerical meaning of symbolic digits rather than the representation of number per se is important.

To date, researchers investigating nonsymbolic number processes devoted little attention to the visual properties of their stimuli. This is unexpected, as nonsymbolic number is defined by its visual characteristics. When number changes, its visual properties change accordingly. In this study, we investigated the influence of different visual properties on nonsymbolic number processes and show that the current assumptions about the relation between number and its visual characteristics are incorrect. Similar to previous studies, we controlled the visual cues: Each visual cue was not predictive of number. Nevertheless, participants showed congruency effects induced by the visual properties of the stimuli. These congruency effects scaled with the number of visual cues manipulated, implicating that people do not extract number from a visual scene independent of its visual cues. Instead, number judgments are based on the integration of information from multiple visual cues. Consequently, current ways to control the visual cues of the number stimuli are insufficient, as they control only a single variable at the time. And, more important, the existence of an approximate number system that can extract number independent of the visual cues appears unlikely. We therefore propose that number judgment is the result of the weighing of several distinct visual cues. (PsycINFO Database Record (c) 2011 APA, all rights reserved).

We tested children in Grades 1 to 5, as well as college students, on a number line estimation task and examined latencies and errors to explore the cognitive processes involved in estimation. The developmental trends in estimation were more consistent with the hypothesized shift from logarithmic to linear representation than with an account based on a proportional judgment application of a power function model; increased linear responding across ages, as predicted by the log-to-lin shift position, yielded reasonable developmental patterns, whereas values derived from the cyclical power model were difficult to reconcile with expected developmental patterns. Neither theoretical position predicted the marked "M-shaped" pattern that was observed, beginning in third graders' errors and fourth graders' latencies. This pattern suggests that estimation comes to rely on a midpoint strategy based on children's growing number knowledge (i.e., knowledge that 50 is half of 100). As found elsewhere, strength of linear responding correlated significantly with children's performance on standardized math tests.

How do our mental representations of number change over development? The dominant view holds that children (and adults) possess multiple representations of number, and that age and experience lead to a shift from greater reliance upon logarithmically organized number representations to greater reliance upon more accurate, linear representations. Here we present a new theoretically motivated and empirically supported account of the development of numerical estimation, based on the idea that number-line estimation tasks entail judgments of proportion. We extend existing models of perceptual proportion judgment to the case of abstract numerical magnitude. Two experiments provide support for these models; three likely sources of developmental change in children's estimation performance are identified and discussed. This work demonstrates that proportion-judgment models provide a unified account of estimation patterns that have previously been explained in terms of a developmental shift from logarithmic to linear representations of number.