Assessing the Approximate Number System: no relation between numerical comparison and estimation tasks

The relationship between numerical magnitude processing and math anxiety, and their joint effect on adult math performance, varied by indicators of numerical tasks

According to the hypothesis of Maloney et al. (Cognition 114(2):293-297, 2010. https://doi.org/10.1016/j.cognition.2009.09.013), math anxiety is related to deficits in numerical magnitude processing, which in turn compromises the development of advanced math skills. Because previous studies on this topic are contradictory, which may be due to methodological differences in the measurement of numerical magnitude processing, we tested Maloney et al.'s hypothesis using different tasks and their indicators: numerical magnitude processing (symbolic and non-symbolic comparison tasks: accuracy, reaction time, numerical ratio, distance and size effects, and Weber fraction; number line estimation task: estimation error), math anxiety (combined scores of learning, testing, math problem solving, and general math anxiety), and math performance. The results of our study conducted on 119 young adults mostly support the hypothesis proposed by Maloney et al. that deficiency in symbolic magnitude processing is related to math anxiety, but the relationship between non-symbolic processes and math anxiety was opposite to the assumptions. Moreover, the results indicate that estimation processes (but not comparison processes) and math anxiety are related to math performance in adults. Finally, high math anxiety moderated the relationship between reaction time in the symbolic comparison task, reaction time in the non-symbolic comparison task, numerical ratio effect in the symbolic comparison task, and math performance. Because the results of the joint effect of numerical magnitude processing and math anxiety on math performance were inconsistent, this part of the hypothesis is called into question.

Anatomical connectivity in children with developmental dyscalculia: A graph theory study

Current theories postulate that numerical processing depends upon a brain circuit formed by regions and their connections; specialized in the representation and manipulation of the numerical properties of stimuli. It has been suggested that the damage of these network may cause Developmental Dyscalculia (DD): a persistent neurodevelopmental disorder that significantly interferes with academic performance and daily life activities that require mastery of mathematical notions and operations. However, most of the studies on the brain foundations of DD have focused on regions of interest associated with numerical processing, and have not addressed numerical cognition as a complex network phenomenon. The present study explored DD using a Graph Theory network approach. We studied the association between topological measures of integration and segregation of information processing in the brain proposed by Graph Theory; and individual variability in numerical performance in a group of 11 school-aged children with DD (5 of which presented with comorbidity with Developmental Dyslexia, the specific learning disorder for reading) and 17 typically developing peers. A statistically significant correlation was found between the Weber fraction (a measure of numerical representations' precision) and the Clustering Index (a measure of segregation of information processing) in the whole sample. The DD group showed significantly lower Characteristic Path Length (average shortest path length among all pairs of regions in the brain network) compared to controls. Also, differences in critical regions for the brain network performance (hubs) were found between groups. The presence of limbic hubs characterized the DD brain network while right Temporal and Frontal hubs found in controls were absent in the DD group. Our results suggest that the DD may be associated with alterations in anatomical brain connectivity that hinder the capacity to integrate and segregate numerical information.

Numerosity Comparison, Estimation and Proportion Estimation Abilities May Predict Numeracy and Cognitive Reflection in Adults

This study explores whether and how different tasks associated with approximate number system (ANS) ability are related to numeracy and cognitive reflection in adults. We conducted an online experiment using a sample of 300 Japanese adults aged 20–39. Participants were given three ANS tasks (numerosity comparison, numerosity estimation, and proportion estimation) as well as Rasch-based numeracy scale and cognitive reflection test, and we tested the correlation among the measures of these tasks. We explored the hypothesis that the typical measures used to gauge ANS ability, numerosity comparison and numerosity estimation may mediate different cognitive mechanisms in adults. We also introduced a task measuring proportion estimation, added because such estimation requires numerosity perception and the ability to map symbolic numerals. Our findings suggest that there is a weak, but significant correlation among the three ANS-related tasks. Moreover, there is a significant relationship between each of these measures and the numeracy and CRT score, suggesting that the ANS-related ability may be associated with higher cognitive abilities such as numeracy and cognitive reflection. In addition, we found that performances on the numerosity and proportion estimation are more clearly related to CRT score than the numerosity comparison task.

Video-game play and non-symbolic numerical comparison

Visual display was used by Stäb and Ilg to examine the abilities of video-game players and non-players to determine simple mathematical abilities.

Response bias in numerosity perception at early judgments and systematic underestimation

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

Bias and noise in proportion estimation: A mixture psychophysical model

The importance of proportional reasoning has long been recognized by psychologists and educators, yet we still do not have a good understanding of how humans mentally represent proportions. In this paper we present a psychophysical model of proportion estimation, extending previous approaches. We assumed that proportion representations are formed by representing each magnitude of a proportion stimuli (the part and its complement) as Gaussian activations in the mind, which are then mentally combined in the form of a proportion. We next derived the internal representation of proportions, including bias and internal noise parameters -capturing respectively how our estimations depart from true values and how variable estimations are. Methodologically, we introduced a mixture of components to account for contaminating behaviors (guessing and reversal of responses) and framed the model in a hierarchical way. We found empirical support for the model by testing a group of 4th grade children in a spatial proportion estimation task. In particular, the internal density reproduced the asymmetries (skewedness) seen in this and in previous reports of estimation tasks, and the model accurately described wide variations between subjects in behavior. Bias estimates were in general smaller than by using previous approaches, due to the model's capacity to absorb contaminating behaviors. This property of the model can be of especial relevance for studies aimed at linking psychophysical measures with broader cognitive abilities. We also recovered higher levels of noise than those reported in discrimination of spatial magnitudes and discuss possible explanations for it. We conclude by illustrating a concrete application of our model to study the effects of scaling in proportional reasoning, highlighting the value of quantitative models in this field of research.

Ordinality: The importance of its trial list composition and examining its relation with adults' arithmetic and mathematical reasoning

Understanding whether a sequence is presented in an order or not (i.e., ordinality) is a robust predictor of adults’ arithmetic performance, but the mechanisms underlying this skill and its relationship with mathematics remain unclear. In this study, we examined (a) the cognitive strategies involved in ordinality inferred from behavioural effects observed in different types of sequences and (b) whether ordinality is also related to mathematical reasoning besides arithmetic. In Experiment 1, participants performed an arithmetic, a mathematical reasoning test, and an order task, which had balanced trials on the basis of order, direction, regularity, and distance. We observed standard distance effects (DEs) for ordered and non-ordered sequences, which suggest reliance on magnitude comparison strategies. This contradicts past studies that reported reversed distance effects (RDEs) for some types of sequences, which suggest reliance on retrieval strategies. Also, we found that ordinality predicted arithmetic but not mathematical reasoning when controlling for fluid intelligence. In Experiment 2, we investigated whether the aforementioned absence of RDEs was because of our trial list composition. Participants performed two order tasks: in both tasks, no RDE was found demonstrating the fragility of the RDE. In addition, results showed that the strategies used when processing ordinality were modulated by the trial list composition and presentation order of the tasks. Altogether, these findings reveal that ordinality is strongly related to arithmetic and that the strategies used when processing ordinality are highly dependent on the context in which the task is presented.

Probing the mechanisms underlying numerosity-to-numeral mappings and their relation to math competence

Numerosity estimation performance (e.g., how accurate, consistent, or proportionally spaced (linear) numerosity-numeral mappings are) has previously been associated with math competence. However, the specific mechanisms that underlie such a relation is unknown. One possible mechanism is the mapping process between numerical sets and symbolic numbers (e.g., Arabic numerals). The current study examined two hypothesized mechanisms of numerosity-numeral mappings (item-based "associative" and holistic "structural" mapping) and their roles in the estimation-and-math relation. Specifically, mappings for small numbers (e.g., 1-10) are thought to be associative and resistant to calibration (e.g., feedback on accuracy of estimates), whereas holistic "structural" mapping for larger numbers (e.g., beyond 10) may be supported by flexibly aligning a numeral "response grid" (akin to a ruler) to an analog "mental number line" upon calibration. In 57 adults, we used pre- and post-calibration estimates to measure the range of continuous associative mappings among small numbers (e.g., a base range of associative mappings from 1 to 10), and obtained measures of math competence and delayed multiple-choice strategy reports. Consistent with previous research, uncalibrated estimation performance correlated with calculation competence, controlling for reading fluency and working memory. However, having a higher base range of associative mappings was not related to estimation performance or any math competence measures. Critically, discontinuity in calibration effects was typical at the individual level, which calls into question the nature of "holistic structural mapping". A parsimonious explanation to integrate previous and current findings is that estimation performance is likely optimized by dynamically constructing numerosity-numeral mappings through the use of multiple strategies from trial to trial.

The Relationship Between Non-symbolic and Symbolic Numerosity Representations in Elementary School: The Role of Intelligence

This study aimed to estimate the extent to which the development of symbolic numerosity representations relies on pre-existing non-symbolic numerosity representations that refer to the Approximate Number System. To achieve this aim, we estimated the longitudinal relationships between accuracy in the Number Line (NL) test and “blue–yellow dots” test across elementary school children. Data from a four-wave longitudinal study involving schoolchildren in grades 1–4 in Russia and Kyrgyzstan (N = 490, mean age 7.65 years in grade 1) were analyzed. We applied structural equation modeling and tested several competing models. The results revealed that at the start of schooling, the accuracy in the NL test predicted subsequent accuracy in the “blue–yellow dots” test, whereas subsequently, non-symbolic representation in grades 2 and 3 predicted subsequent symbolic representation. These results indicate that the effect of non-symbolic representation on symbolic representation emerges after a child masters the basics of symbolic number knowledge, such as counting in the range of twenty and simple arithmetic. We also examined the extent to which the relationships between non-symbolic and symbolic representations might be explained by fluid intelligence, which was measured by Raven’s Standard Progressive Matrices test. The results revealed that the effect of symbolic representation on non-symbolic representation was explained by fluid intelligence, whereas at the end of elementary school, non-symbolic representation predicted subsequent symbolic representation independently of fluid intelligence.

A primarily serial, foveal accumulator underlies approximate numerical estimation

The approximate number system (ANS) has attracted broad interest due to its potential importance in early mathematical development and the fact that it is conserved across species. Models of the ANS and behavioral measures of ANS acuity both assume that quantity estimation is computed rapidly and in parallel across an entire view of the visual scene. We present evidence instead that ANS estimates are largely the product of a serial accumulation mechanism operating across visual fixations. We used an eye-tracker to collect data on participants' visual fixations while they performed quantity-estimation and -discrimination tasks. We were able to predict participants' numerical estimates using their visual fixation data: As the number of dots fixated increased, mean estimates also increased, and estimation error decreased. A detailed model-based analysis shows that fixated dots contribute twice as much as peripheral dots to estimated quantities; people do not "double count" multiply fixated dots; and they do not adjust for the proportion of area in the scene that they have fixated. The accumulation mechanism we propose explains reported effects of display time on estimation and earlier findings of a bias to underestimate quantities.

Malleability of mappings between Arabic numerals and approximate quantities: Factors underlying individual differences and the relation to math

Humans tend to be inaccurate and inconsistent when estimating a large number of objects. Furthermore, we modify our estimates when feedback or a reference array is provided, indicating that the mappings between perceived numerosity and their corresponding numerals are largely malleable in response to calibration. However, there is great variability in response to calibration across individuals. Using uncalibrated and calibrated numerosity estimation conditions, the current study explored the factors underlying individual differences in the extent and nature of the malleability of numerosity estimation performance as a result of calibration in a sample of 71 undergraduate students. We found that individual differences in performance were reliable across conditions, and participants' responses to calibration varied greatly. Participants who were less consistent or had more proportionally spaced (i.e., linear) estimates before calibration tended to shift the distributions of their estimates to a greater extent. Higher calculation competence also predicted an increase in how linear participants' estimates were after calibration. Moreover, the effect of calibration was not continuous across numerosities within participants. This suggests that the mechanisms underlying numeral-numerosity mappings may be less systematic than previously thought and likely depend on cognitive mechanisms beyond representation of numerosities. Taken together, the mappings between numerosities and numerical symbols may not be stable and direct, but transient and mediated by task-related (e.g., strategic) mechanisms. Rather than estimation skills being foundational for math competence, math competence may also influence estimation skills. Therefore, numerosity estimation tasks are not a pure measure of number representations.

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.

The measurement of approximate number system acuity across the lifespan is compromised by congruency effects

Recent studies have highlighted the influence of visual cues such as dot size and cumulative surface area on the measurement of the approximate number system (ANS). Previous studies assessing ANS acuity in ageing have all applied stimuli generated by the Panamath protocol, which does not control nor measure the influence of convex hull. Crucially, convex hull has recently been identified as an influential visual cue present in dot arrays, with its impact on older adults’ ANS acuity yet to be investigated. The current study therefore investigated the manipulation of convex hull by the Panamath protocol, and its effect on the measurement of ANS acuity in younger and older participants. First, analyses of the stimuli generated by Panamath revealed a confound between numerosity ratio and convex hull ratio. Second, although older adults were somewhat less accurate than younger adults on convex hull incongruent trials, ANS acuity was broadly similar between the groups. These findings have implications for the valid measurement of ANS acuity across all ages, and suggest that the Panamath protocol produces stimuli that do not adequately control for the influence of convex hull on numerosity discrimination.