The Approximate Number System Acuity Redefined: A Diffusion Model Approach

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.

Incentive motivation improves numerosity discrimination in children and adolescents

We recently showed that incentive motivation improves the precision of the Approximate Number System (ANS) in young adults. To shed light on the development of incentive motivation, the present study investigated whether this effect and its underlying mechanisms may also be observed in younger samples. Specifically, seven-year-old children (n = 23; 12 girls) and 14-year-old adolescents (n = 30; 15 girls) performed a dot comparison task with monetary reward incentives. Both age groups showed higher accuracy in a reward compared to a neutral condition and, similarly, higher processing efficiency as revealed by the drift rate parameter of the EZ-diffusion model. Furthermore, in line with the Incentive Salience Hypothesis, phasic pupil dilations—indicating the activation of the brain’s salience network—were greater in incentivized trials in both age groups. Together these finding suggest that incentive modulation improves numerosity discrimination in children and adolescents by enhancing the perceptual saliency of numerosity information. However, the observed reward anticipation effects were less pronounced in children relative to adolescents. Furthermore, unlike previous findings regarding young adults, the decision thresholds of children and adolescents were not raised by the monetary reward, which may indicate a more protracted development of incentive regulation of response caution than perceptual evidence accumulation.

Dogs (canis familiaris) underestimate the quantity of connected items: first demonstration of susceptibility to the connectedness illusion in non-human animals

In humans, numerical estimation is affected by perceptual biases, such as those originating from the spatial arrangement of elements. Different animal species can also make relative quantity judgements. This includes dogs, who have been proposed as a good model for comparative neuroscience. However, dogs do not show the same perceptual biases observed in humans. Thus, the exact perceptual/cognitive mechanisms underlying quantity estimations in dogs and their degree of similarity with humans are still a matter of debate. Here we explored whether dogs are susceptible to the connectedness illusion, an illusion based on the tendency to underestimate the quantity of interconnected items. Dogs were first trained to choose the larger of two food arrays. Then, they were presented with two arrays containing the same quantity of food, of which one had items interconnected by lines. Dogs significantly selected the array with unconnected items, suggesting that, like in humans, connectedness determines underestimation biases, possibly disrupting the perceptual system's ability to segment the display into discrete objects. The similarity in dogs' and humans' susceptibility to the connectedness, but not to other numerical illusions, suggests that different mechanisms are involved in the estimation of quantity of stimuli with different characteristics.

Numeral order and the operationalization of the numerical system

Recent years have witnessed an increase in research on how numeral ordering skills relate to children's and adults' mathematics achievement both cross-sectionally and longitudinally. Nonetheless, it remains unknown which core competency numeral ordering tasks measure, which cognitive mechanisms underlie performance on these tasks, and why numeral ordering skills relate to arithmetic and math achievement. In the current study, we focused on the processes underlying decision-making in the numeral order judgement task with triplets to investigate these questions. A drift-diffusion model for two-choice decisions was fit to data from 97 undergraduates. Findings aligned with the hypothesis that numeral ordering skills reflected the operationalization of the numerical system, where small numbers provide more evidence of an ordered response than large numbers. Furthermore, the pattern of findings suggested that arithmetic achievement was associated with the accuracy of the ordinal representations of numbers.

Developmental Changes in ANS Precision Across Grades 1-9: Different Patterns of Accuracy and Reaction Time

The main aim of this study was to analyze the patterns of changes in Approximate Number Sense (ANS) precision from grade 1 (mean age: 7.84 years) to grade 9 (mean age: 15.82 years) in a sample of Russian schoolchildren. To fulfill this aim, the data from a longitudinal study of two cohorts of children were used. The first cohort was assessed at grades 1-5 (elementary school education plus the first year of secondary education), and the second cohort was assessed at grades 5-9 (secondary school education). ANS precision was assessed by accuracy and reaction time (RT) in a non-symbolic comparison test ("blue-yellow dots" test). The patterns of change were estimated via mixed-effect growth models. The results revealed that in the first cohort, the average accuracy increased from grade 1 to grade 5 following a non-linear pattern and that the rate of growth slowed after grade 3 (7-9 years old). The non-linear pattern of changes in the second cohort indicated that accuracy started to increase from grade 7 to grade 9 (13-15 years old), while there were no changes from grade 5 to grade 7. However, the RT in the non-symbolic comparison test decreased evenly from grade 1 to grade 7 (7-13 years old), and the rate of processing non-symbolic information tended to stabilize from grade 7 to grade 9. Moreover, the changes in the rate of processing non-symbolic information were not explained by the changes in general processing speed. The results also demonstrated that accuracy and RT were positively correlated across all grades. These results indicate that accuracy and the rate of non-symbolic processing reflect two different processes, namely, the maturation and development of a non-symbolic representation system.

Comparing Eight Parameter Estimation Methods for the Ratcliff Diffusion Model Using Free Software

The Ratcliff Diffusion Model has become an important and widely used tool for the evaluation of psychological experiments. Concurrently, numerous programs and routines have appeared to estimate the model's parameters. The present study aims at comparing some of the most widely used tools with special focus on freely available routines (i.e., open source). Our simulations show that (1) starting point and non-decision time were recovered better than drift rate, (2) the Bayesian approach outperformed all other approaches when the number of trials was low, (3) the Kolmogorov-Smirnov and χ² approaches revealed more bias than Bayesian or Maximum Likelihood based routines, and (4) EZ produced substantially biased estimates of threshold separation, non-decision time and drift rate when starting point z ≠ a/2. We discuss the implications for the choice of parameter estimation approaches for real data and suggest that if biased starting point cannot be excluded, EZ will produce deviant estimates and should be used with great care.

Quantifying the benefits of using decision models with response time and accuracy data

Response time and accuracy are fundamental measures of behavioral science, but discerning participants' underlying abilities can be masked by speed-accuracy trade-offs (SATOs). SATOs are often inadequately addressed in experiment analyses which focus on a single variable or which involve a suboptimal analytic correction. Models of decision-making, such as the drift diffusion model (DDM), provide a principled account of the decision-making process, allowing the recovery of SATO-unconfounded decision parameters from observed behavioral variables. For plausible parameters of a typical between-groups experiment, we simulate experimental data, for both real and null group differences in participants' ability to discriminate stimuli (represented by differences in the drift rate parameter of the DDM used to generate the simulated data), for both systematic and null SATOs. We then use the DDM to fit the generated data. This allows the direct comparison of the specificity and sensitivity for testing of group differences of different measures (accuracy, reaction time, and the drift rate from the model fitting). Our purpose here is not to make a theoretical innovation in decision modeling, but to use established decision models to demonstrate and quantify the benefits of decision modeling for experimentalists. We show, in terms of reduction of required sample size, how decision modeling can allow dramatically more efficient data collection for set statistical power; we confirm and depict the non-linear speed-accuracy relation; and we show how accuracy can be a more sensitive measure than response time given decision parameters which reasonably reflect a typical experiment.

Effects of Visual Training of Approximate Number Sense on Auditory Number Sense and School Math Ability

Research with children and adults suggests that people's math performance is predicted by individual differences in an evolutionarily ancient ability to estimate and compare numerical quantities without counting (the approximate number system or ANS). However, previous work has almost exclusively used visual stimuli to measure ANS precision, leaving open the possibility that the observed link might be driven by aspects of visuospatial competence, rather than the amodal ANS. We addressed this possibility in an ANS training study. Sixty-eight 6-year-old children participated in a 5-week study that either trained their visual ANS ability or their phonological awareness (an active control group). Immediately before and after training, we assessed children's visual and auditory ANS precision, as well as their symbolic math ability and phonological awareness. We found that, prior to training, children's precision in a visual ANS task related to their math performance - replicating recent studies. Importantly, precision in an auditory ANS task also related to math performance. Furthermore, we found that children who completed visual ANS training showed greater improvements in auditory ANS precision than children who completed phonological awareness training. Finally, children in the ANS training group showed significant improvements in math ability but not phonological awareness. These results suggest that the link between ANS precision and school math ability goes beyond visuospatial abilities and that the modality-independent ANS is causally linked to math ability in early childhood.

Incentive motivation improves numerosity discrimination: Insights from pupillometry combined with drift-diffusion modelling

Recent studies show that training the approximate number system (ANS) holds promise for improving symbolic math abilities. Extending this line of research, the present study aims to shed light on incentive motivation of numerosity discrimination and the underlying mechanisms. Thirty-two young adults performed a novel incentivized dot comparison task, that we developed, to discern the larger of two numerosities. An EZ-diffusion model was applied to decompose motivational effects on component processes of perceptual decision-making. Furthermore, phasic pupil dilation served as an indicator of the involvement of the salience network. The results of improved accuracy and a higher information accumulation rate under the reward condition suggest that incentive motivation boosts the precision of the ANS. These novel findings extend earlier evidence on reward-related enhancements of perceptual discrimination to the domain of numerosity perception. In light of the Adaptive Gain Theory, we interpret the results in terms of two processes of gain modulation driven by the locus coeruleus-norepinephrine system. Specifically, the reward-induced increase in pupil dilation may reflect incentive modulation of (i) salience attention during reward anticipation towards incentivized stimuli to upregulate stimulus processing that results in a larger drift rate; and (ii) response caution that leads to an increased decision threshold.

Comparing Numerical Comparison Tasks: A Meta-Analysis of the Variability of the Weber Fraction Relative to the Generation Algorithm

Since more than 15 years, researchers have been expressing their interest in evaluating the Approximate Number System (ANS) and its potential influence on cognitive skills involving number processing, such as arithmetic. Although many studies reported significant and predictive relations between ANS and arithmetic abilities, there has recently been an increasing amount of published data that failed to replicate such relationship. Inconsistencies lead many researchers to question the validity of the assessment of the ANS itself. In the current meta-analysis of over 68 experimental studies published between 2004 and 2017, we show that the mean value of the Weber fraction (w), the minimal amount of change in magnitude to detect a difference, is very heterogeneous across the literature. Within young adults, w might range from < 10 to more than 60, which is critical for its validity for research and diagnostic purposes. We illustrate here the concern that different methods controlling for non-numerical dimensions lead to substantially variable performance. Nevertheless, studies that referred to the exact same method (e.g., Panamath) showed high consistency among them, which is reassuring. We are thus encouraging researchers only to compare what is comparable and to avoid considering the Weber fraction as an abstract parameter independent from the context. Eventually, we observed that all reported correlation coefficients between the value of w and general accuracy were very high. Such result calls into question the relevance of computing and reporting at all the Weber fraction. We are thus in disfavor of the systematic use of the Weber fraction, to discourage any temptation to compare given data to some values of w reported from different tasks and generation algorithms.

Modeling numerosity representation with an integrated diffusion model

Models of the representation of numerosity information used in discrimination tasks are integrated with a diffusion decision model. The representation models assume distributions of numerosity either with means and SD that increase linearly with numerosity or with means that increase logarithmically with constant SD. The models produce coefficients that are applied to differences between two numerosities to produce drift rates and these drive the decision process. The linear and log models make differential predictions about how response time (RT) distributions and accuracy change with numerosity and which model is successful depends on the task. When the task is to decide which of two side-by-side arrays of dots has more dots, the log model fits decreasing accuracy and increasing RT as numerosity increases. When the task is to decide, for dots of two colors mixed in a single array, which color has more dots, the linear model fits decreasing accuracy and decreasing RT as numerosity increases. For both tasks, variables such as the areas covered by the dots affect performance, but if the task is changed to one in which the subject has to decide whether the number of dots in a single array is more or less than a standard, the variables have little effect on performance. Model parameters correlate across tasks suggesting commonalities in the abilities to perform them. Overall, results show that the representation used depends on the task and no single representation can account for the data from all the paradigms. (PsycINFO Database Record

A Diffusion Model Analysis of Magnitude Comparison in Children with and without Dyscalculia: Care of Response and Ability Are Related to Both Mathematical Achievement and Stimuli

The respective roles of the approximate number system (ANS) and an access deficit (AD) in developmental dyscalculia (DD) are not well-known. Most studies rely on response times (RTs) or accuracy (error rates) separately. We analyzed the results of two samples of elementary school children in symbolic magnitude comparison (MC) and non-symbolic MC using a diffusion model. This approach uses the joint distribution of both RTs and accuracy in order to synthesize measures closer to ability and response caution or response conservatism. The latter can be understood in the context of the speed-accuracy tradeoff: It expresses how much a subject trades in speed for improved accuracy. We found significant effects of DD on both ability (negative) and response caution (positive) in MC tasks and a negative interaction of DD with symbolic task material on ability. These results support that DD subjects suffer from both an impaired ANS and an AD and in particular support that slower RTs of children with DD are indeed related to impaired processing of numerical information. An interaction effect of symbolic task material and DD (low mathematical ability) on response caution could not be refuted. However, in a sample more representative of the general population we found a negative association of mathematical ability and response caution in symbolic but not in non-symbolic task material. The observed differences in response behavior highlight the importance of accounting for response caution in the analysis of MC tasks. The results as a whole present a good example of the benefits of a diffusion model analysis.

A Systematic Investigation of Accuracy and Response Time Based Measures Used to Index ANS Acuity

The approximate number system (ANS) was proposed to be a building block for later mathematical abilities. Several measures have been used interchangeably to assess ANS acuity. Some of these measures were based on accuracy data, whereas others relied on response time (RT) data or combined accuracy and RT data. Previous studies challenged the view that all these measures can be used interchangeably, because low correlations between some of the measures had been observed. These low correlations might be due to poor reliability of some of the measures, since the majority of these measures are mathematically related. Here we systematically investigated the relationship between common ANS measures while avoiding the potential confound of poor reliability. Our first experiment revealed high correlations between all accuracy based measures supporting the assumption that all of them can be used interchangeably. In contrast, not all RT based measures were highly correlated. Additionally, our results revealed a speed-accuracy trade-off. Thus, accuracy and RT based measures provided conflicting conclusions regarding ANS acuity. Therefore, we investigated in two further experiments which type of measure (accuracy or RT) is more informative about the underlying ANS acuity, depending on participants’ preferences for accuracy or speed. To this end, we manipulated participants’ preferences for accuracy or speed both explicitly using different task instructions and implicitly varying presentation duration. Accuracy based measures were more informative about the underlying ANS acuity than RT based measures. Moreover, the influence of the underlying representations on accuracy data was more pronounced when participants preferred accuracy over speed after the accuracy instruction as well as for long or unlimited presentation durations. Implications regarding the diffusion model as a theoretical framework of dot comparison as well as regarding the relationship between ANS acuity and math performance are discussed.