Finger counting, finger number gesturing, and basic numerical skills: A cross-sectional study in 3- to 5-year-olds

Recent evidence suggests that using finger-based strategies is beneficial for the acquisition of basic numerical skills. There are basically two finger-based strategies to be distinguished: (a) finger counting (i.e., extending single fingers successively) and (b) finger number gesturing (i.e., extending fingers simultaneously to represent magnitudes). In this study, we investigated both spontaneous and prompted finger counting and finger number gesturing as well as their contribution to basic numerical skills in 3- to 5-year-olds (N = 156). Results revealed that only 6% of children spontaneously used their fingers for counting when asked to name a specific number of animals, whereas 59% applied finger number gesturing to show their age. This indicates that the spontaneous use of finger-based strategies depends heavily on the specific context. Moreover, children performed significantly better in prompted finger counting than in finger number gesturing, suggesting that both strategies build on each other. Finally, both prompted finger counting and finger number gesturing significantly and individually predicted counting, cardinal number knowledge, and basic arithmetic. These results indicate that finger counting and finger number gesturing follow and positively relate to numerical development.

Influences of cognitive control on number processing: New evidence from switching between two numerical tasks

A growing body of research suggests that basic numerical abilities such as number magnitude and number parity processing are influenced by cognitive control. So far, however, evidence for number processing being influenced by cognitive control came primarily from observed adaptations to stimulus set characteristics (e.g., ratio or order of specific stimulus types) and switches between a numerical and non-numerical task. Complementing this previous research, the present study employed a task switching paradigm exclusively involving numerical tasks (i.e., magnitude comparisons and parity judgements) to examine how cognitive control processes influence number processing. Participants were presented with a single-digit number and had to either judge its parity or compare its magnitude with a standard of 5, depending on a preceding cue. Based on previous results, we expected the numerical distance effect and the spatial-numerical association of response codes (SNARC) effect to be modulated in switch trials requiring the exertion of cognitive control. Partly in line with our expectations, the numerical distance effect was reduced in switch trials. However, no modulation of the SNARC effect was observed. The results pattern suggests that number processing is influenced by cognitive control, depending on task requirements and the type of numerical information (i.e., numerical magnitude vs spatial association of numbers) that is processed. To reconcile the present and previous results, we propose an information prioritisation account, suggesting that cognitive control primarily influences the processing of the information type that requires the most explicit processing.

Numeracy and COVID-19: examining interrelationships between numeracy, health numeracy and behaviour

During the COVID-19 pandemic, people across the globe have been exposed to large amounts of statistical data. Previous studies have shown that individuals' mathematical understanding of health-related information affects their attitudes and behaviours. Here, we investigate the relation between (i) basic numeracy, (ii) COVID-19 health numeracy, and (iii) COVID-19 health-related attitudes and behaviours. An online survey measuring these three variables was distributed in Canada, the United States (US) and the United Kingdom (UK) (n = 2032). In line with predictions, basic numeracy was positively related to COVID-19 health numeracy. However, predictions, neither basic numeracy nor COVID-19 health numeracy was related to COVID-19 health-related attitudes and behaviours (e.g. follow experts' recommendations on social distancing, wearing masks etc.). Multi-group analysis was used to investigate mean differences and differences in the strength of the correlation across countries. Results indicate there were no between-country differences in the correlations between the main constructs but there were between-country differences in latent means. Overall, results suggest that while basic numeracy is related to one's understanding of data about COVID-19, better numeracy alone is not enough to influence a population's health-related attitudes about disease severity and to increase the likelihood of following public health advice.

Cognitive control in number processing: new evidence from number compatibility effects in task-switching

A growing body of research suggests that basic numerical abilities such as number magnitude processing are influenced by cognitive control processes. So far, evidence for number processing being affected by cognitive control processes stems primarily from observed adaptations of numerical effects to stimulus set characteristics (e.g. order or ratio of specific stimulus types). Complementing previous research on adaptation to stimulus set characteristics as an index of influences of cognitive control, the present study employed a task-switching paradigm to examine how cognitive control processes influence number processing. Participants were presented with a two-digit number and had to either judge its parity or compare its magnitude to a standard depending on a preceding cue. We expected numerical congruency effects (i.e. the unit-decade compatibility effect for magnitude comparisons and the parity congruity effect for parity judgements) to be larger in switch trials, as persisting activation of the task set of the preceding trial should increase interference. In contrast to our expectations, both numerical congruity effects were reduced following task switches as compared to repetitions. This interaction of task-switching with numerical congruency effects suggests an influence of cognitive control on basic number processing in form of persisting inhibition of previously abandoned task sets, so that these exert less influence on current number processing demands.

The quandary of diagnosing mathematical difficulties in a generally low performing population

Brazilian students' mathematical achievement was repeatedly observed to fall below average levels of mathematical attainment in international studies such as PISA.

Objective

In this article, we argue that this general low level of mathematical attainment may interfere with the diagnosis of developmental dyscalculia when a psychometric criterion is used establishing an arbitrary cut-off (e.g., performance<percentile 10) may result in misleading diagnoses.

Methods

Therefore, the present study evaluated the performance of 706 Brazilian school children from 3^rd to 5^th grades on basic arithmetic operations addition, subtraction, and multiplication.

Results

In line with PISA results, children presented difficulties in all arithmetic operations investigated. Even after five years of formal schooling, less than half of 5^th graders performed perfectly on simple addition, subtraction, or multiplication problems.

Conclusions

As such, these data substantiate the argument that the sole use of a psychometric criterion might not be sensible to diagnose dyscalculia in the context of a generally low performing population, such as Brazilian children of our sample. When the majority of children perform poorly on the task at hand, it is hard to distinguish atypical from typical numerical development. As such, other diagnostic approaches, such as Response to Intervention, might be more suitable in such a context.

Neurofunctional plasticity in fraction learning: An fMRI training study

BACKGROUND

Fractions are known to be difficult for children and adults. Behavioral studies suggest that magnitude processing of fractions can be improved via number line estimation (NLE) trainings, but little is known about the neural correlates of fraction learning.

METHOD

To examine the neuro-cognitive foundations of fraction learning, behavioral performance and neural correlates were measured before and after a five-day NLE training.

RESULTS

In all evaluation tasks behavioral performance increased after training. We observed a fronto-parietal network associated with number magnitude processing to be recruited in all tasks as indicated by a numerical distance effect. For symbolic fractions, the distance effect on intraparietal activation was only observed after training.

CONCLUSION

The absence of a distance effect of symbolic fractions before the training could indicate an initially less automatic access to their overall magnitude. NLE training facilitates processing of overall fraction magnitude as indicated by the distance effect in neural activation.

Cognitive control in number processing: new evidence from task switching

Recently, it was demonstrated that even basic numerical cognition such as the processing of number magnitude is under cognitive control. However, evidence so far primarily came from adaptation effects to stimulus characteristics (e.g., relative frequency of specific stimulus categories). Expanding this approach, we evaluated a possible influence of more active exertion of cognitive control on basic number processing in task switching. Participants had to perform a magnitude comparison task while we manipulated the order of compatible and incompatible input–output modalities (i.e., auditory/vocal input–visual/manual output vs. auditory/visual input–manual/vocal output, respectively) on the trial level, differentiating repeat vs. switch trials. Results indicated that the numerical distance effect but not the problem size effect was increased after a switch in input–output modality compatibility. In sum, these findings substantiate that basic number processing is under cognitive control by providing first evidence that it is influenced by the active exertion of cognitive control as required in task switching.

Inversion effects on mental arithmetic in English- and Polish-speaking adults

In some languages the order of tens and units in number words is inverted compared with the symbolic digital notation (e.g., German 23 → "dreiundzwanzig," literally: "three-and-twenty"). In other languages only teen-numbers are inverted (e.g., English 17 → "seventeen"; Polish 17 → "siedemnaście" literally "seventeen"). Previous studies have focused on between group comparisons of inverted and non-inverted languages and showed that number word inversion impairs performance on basic numerical tasks and arithmetic. In two independent experiments, we investigated whether number word inversion affects addition performance within otherwise non-inverted languages (Exp. 1: English, Exp. 2: Polish). In particular, we focused on the influence of inverted (I; English: teen-numbers ⩾ 13, Polish: numbers 11-19) and non-inverted (N) summands with sums between 13 and 39. Accordingly, three categories of addition problems were created: N + N, N + I, and I + I with problem size matched across categories. Across both language groups, we observed that problems with results in the 20 and 30 number range were responded to faster when only non-inverted summands were part of the problems as opposed to problems with one or two inverted summands. In line with this, the cost of a carry procedure was the largest for two inverted summands. The results support the notion that both language-specific and language-invariant aspects contribute to addition problem-solving. In particular though, regarding language-specific aspects, the results indicate that inverted number word formation of teens influences place-value processing of Arabic digits even in otherwise non-inverted languages.

Anticipation of difficult tasks: neural correlates of negative emotions and emotion regulation

BACKGROUND

Difficult cognitive tasks are often associated with negative feelings. This can be already the case for the mere anticipation of having to do a difficult task. For the case of difficult math tasks, it was recently suggested that such a negative emotional response may be exclusive to highly math-anxious individuals. However, it is also conceivable that negative emotional responses simply reflect that math is perceived as difficult. Here we investigated whether non-math-anxious individuals also experience negative emotional responses when anticipating to do difficult math tasks.

METHODS

We compared brain activation following the presentation of a numerical cue indicating either difficult or easy upcoming proportion magnitude comparison tasks.

RESULTS

Comparable to previous results for highly math-anxious individuals we observed a network associated with negative emotions to be activated in non-math-anxious individuals when facing cues indicating a difficult upcoming task. Importantly, however, math anxiety scores did not predict the neural response. Furthermore, we observed activation in areas associated with processes of cognitive control areas such as anterior cingulate cortex, which were suggested to play a key role in emotion regulation.

CONCLUSION

Activation in the emotion processing network was observed when anticipating an upcoming difficult (math) task. However, this activation was not predicted by individual' degree of math anxiety. Therefore, we suggest that negative emotional responses to difficult math tasks might be a rather common reaction not specific to math-anxious individuals. Whether or not this initial negative response impairs math performance, however, might depend on the ability to regulate those emotions effectively.

Processing symbolic and non-symbolic proportions: Domain-specific numerical and domain-general processes in intraparietal cortex

Previous studies on the processing of fractions and proportions focused mainly on the processing of their overall magnitude information in the intraparietal sulcus (IPS). However, the IPS is also associated with domain-general cognitive functions beyond processing overall magnitude, which may nevertheless be involved in operating on magnitude information of proportions. To pursue this issue, the present study aimed at investigating whether there is a shared neural correlate for proportion processing in the intraparietal cortex beyond overall magnitude processing and how part-whole relations are processed on the neural level. Across four presentation formats (i.e., fractions, decimals, dot patterns, and pie charts) we observed a shared neural substrate in bilateral inferior parietal cortex, slightly anterior and inferior to IPS areas recently found for overall magnitude proportion processing. Nevertheless, when evaluating the neural correlates of part-whole processing (i.e., contrasting fractions, dot patterns, and pie charts vs. decimals), we found wide-spread activation in fronto-parietal brain areas. These results indicate involvement of domain-general cognitive processes in part-whole processing beyond processing the overall magnitude of proportions. The dissociation between proportions involving part-whole relations and decimals was further substantiated by a representational similarity analysis, which revealed common neural processing for fractions, pie charts, and dot patterns, possibly representing their bipartite part-whole structure. In contrast, decimals seemed to be processed differently on the neural level, possibly reflecting missing processes of actual proportion calculation in decimals.

More than simple facts: cross-linguistic differences in place-value processing in arithmetic fact retrieval

Linguistic specificities such as the inversion property of number words (e.g., in German 43 is spoken dreiundvierzig, literally three and  forty) moderate Arabic number processing. So far, cross-linguistic studies have mostly focused on inversion-related effects on simple (e.g., number comparison) and calculation-based (e.g., multi-digit addition) magnitude processing of numerical information. Despite the assumption that multiplication facts are represented in verbal format, not much attention has been paid to inversion-related influences on multiplication fact retrieval. Accordingly, the current study evaluated inversion-related effects on the processing of place-value information in multiplication. In a verification paradigm, the decade consistency effect (i.e., more errors when the decade of a solution probe shares the decade digit with the correct solution) was larger for English- than German-speaking participants for table-related probes. Processing of decade digits might be prioritised in English-speaking participants because the decade digit is named first in English number words, whereas in German number words the unit digit is named first. Our results indicate that (1) the influence of specificities of a verbal number word formation on place-value processing generalise to arithmetic fact retrieval and (2) inversion of number words might even be advantageous in specific cases.

Direct evidence for linguistic influences in two-digit number processing

Language-specific differences in number words influence number processing even in nonverbal numerical tasks. For instance, the unit-decade compatibility effect in two-digit number magnitude comparison (compatible number pairs [42_57: 4 < 5 and 2 < 7] are responded to faster than incompatible pairs [47_62: 4 < 6 but 7 > 2]) was shown to be influenced by the inversion of number words (e.g., in German the number word for 42 is zweiundvierzig [literally: two-and-forty]). In two studies, we used articulatory suppression to investigate whether previously observed cross-linguistic differences in two-digit number processing are indeed driven by differences in number word formation. In a two-digit number comparison task, German- and English-speaking participants had to identify the larger of two numbers presented in Arabic digits. In Study 1, participants performed the same task twice, with and without articulatory suppression. In Study 2, the percentage of within-decade filler items (36_39) was manipulated additionally. As expected, in both studies between-groups differences in the compatibility effect disappeared under articulatory suppression irrespective of the percentage of fillers included. Furthermore, paralleling results of previous studies including 33% or less filler items, we found that the compatibility effect was larger in German compared with English speakers in the 20% filler condition. However, this pattern was reversed in the 50% filler condition in both studies. Thus, results provide first direct evidence for influences of verbal number word formation on symbolic number processing. Moreover, these new findings suggest that linguistic influences and those of cognitive control processes associated with characteristics of the stimulus set interact in symbolic number processing. (PsycINFO Database Record (c) 2019 APA, all rights reserved).

A Taxonomy Proposal for Types of Interactions of Language and Place-Value Processing in Multi-Digit Numbers

Research on associations between language and number processing has seen growing interest in the last years – in particular with respect to place-value processing in multi-digit numbers. Recently, Dowker and Nuerk (2016) proposed a taxonomy of linguistic influences on number processing. However, this taxonomy does not address the generality or specificity of linguistic influences across different levels of number processing. In contrast, Nuerk et al. (2015) proposed different levels of place-value processing in multi-digit numbers. However, the authors did not specify if and how linguistic factors influence these levels of place-value processing. The present perspective aims at addressing this conceptual gap by suggesting an integrated taxonomy representing how different linguistic factors may influence different levels of place-value processing. We show that some effects of different linguistic levels have already been observed on different levels of place-value processing. Moreover, while some linguistic influences (e.g., lexical influences) have been studied for all levels of place-value processing, other influences have been studied for only one level or even none. Beyond categorizing existing research, we argue that the explicit consideration of research gaps may inspire new research paradigms complementing the picture of language influences on place-value processing. We conclude by outlining the importance of a differential approach for levels of both linguistic and number processing to evaluate linguistic obstacles and facilitators of different languages and their relevance for numerical development.

Magnitude processing of symbolic and non-symbolic proportions: an fMRI study

Background

Recent research indicates that processing proportion magnitude is associated with activation in the intraparietal sulcus. Thus, brain areas associated with the processing of numbers (i.e., absolute magnitude) were activated during processing symbolic fractions as well as non-symbolic proportions. Here, we investigated systematically the cognitive processing of symbolic (e.g., fractions and decimals) and non-symbolic proportions (e.g., dot patterns and pie charts) in a two-stage procedure. First, we investigated relative magnitude-related activations of proportion processing. Second, we evaluated whether symbolic and non-symbolic proportions share common neural substrates.

Methods

We conducted an fMRI study using magnitude comparison tasks with symbolic and non-symbolic proportions, respectively. As an indicator for magnitude-related processing of proportions, the distance effect was evaluated.

Results

A conjunction analysis indicated joint activation of specific occipito-parietal areas including right intraparietal sulcus (IPS) during proportion magnitude processing. More specifically, results indicate that the IPS, which is commonly associated with absolute magnitude processing, is involved in processing relative magnitude information as well, irrespective of symbolic or non-symbolic presentation format. However, we also found distinct activation patterns for the magnitude processing of the different presentation formats.

Conclusion

Our findings suggest that processing for the separate presentation formats is not only associated with magnitude manipulations in the IPS, but also increasing demands on executive functions and strategy use associated with frontal brain regions as well as visual attention and encoding in occipital regions. Thus, the magnitude processing of proportions may not exclusively reflect processing of number magnitude information but also rather domain-general processes.

Spatial Arrangement and Set Size Influence the Coding of Non-symbolic Quantities in the Intraparietal Sulcus

Performance in visual quantification tasks shows two characteristic patterns as a function of set size. A precise subitizing process for small sets (up to four) was contrasted with an approximate estimation process for larger sets. The spatial arrangement of elements in a set also influences visual quantification performance, with frequently perceived arrangements (e.g., dice patterns) being faster enumerated than random arrangements. Neuropsychological and imaging studies identified the intraparietal sulcus (IPS), as key brain area for quantification, both within and above the subitizing range. However, it is not yet clear if and how set size and spatial arrangement of elements in a set modulate IPS activity during quantification. In an fMRI study, participants enumerated briefly presented dot patterns with random, canonical or dice arrangement within and above the subitizing range. We evaluated how activity amplitude and pattern in the IPS were influenced by size and spatial arrangement of a set. We found a discontinuity in the amplitude of IPS response between subitizing and estimation range, with steep activity increase for sets exceeding four elements. In the estimation range, random dot arrangements elicited stronger IPS response than canonical arrangements which in turn elicited stronger response than dice arrangements. Furthermore, IPS activity patterns differed systematically between arrangements. We found a signature in the IPS response for a transition between subitizing and estimation processes during quantification. Differences in amplitude and pattern of IPS activity for different spatial arrangements indicated a more precise representation of non-symbolic numerical magnitude for dice and canonical than for random arrangements. These findings challenge the idea of an abstract coding of numerosity in the IPS even within a single notation.

Fact learning in complex arithmetic-the role of the angular gyrus revisited

In recent theoretical considerations as well as in neuroimaging findings the left angular gyrus (AG) has been associated with the retrieval of arithmetic facts. This interpretation was corroborated by higher AG activity when processing trained as compared with untrained multiplication problems. However, so far neural correlates of processing trained versus untrained problems were only compared after training. We employed an established learning paradigm (i.e., extensive training of multiplication problems) but measured brain activation before and afte training to evaluate neural correlates of arithmetic fact acquisition more specifically. When comparing activation patterns for trained and untrained problems of the post-training session, higher AG activation for trained problems was replicated. However, when activation for trained problems was compared to activation for the same problems in the pre-training session, no signal change in the AG was observed. Instead, our results point toward a central role of hippocampal, para-hippocampal, and retrosplenial structures in arithmetic fact retrieval. We suggest that the AG might not be associated with the actual retrieval of arithmetic facts, and outline an attentional account of the role of the AG in arithmetic fact retrieval that is compatible with recent attention to memory hypotheses. Hum Brain Mapp 37:3061-3079, 2016. © 2016 Wiley Periodicals, Inc.

On the limits of language influences on numerical cognition - no inversion effects in three-digit number magnitude processing in adults

The inversion of number words influences numerical cognition even in seemingly non-verbal tasks, such as Arabic number comparison. However, it is an open question whether inversion of decades and units also influences number processing beyond the two-digit number range. The current study addresses this question by investigating compatibility effects in both German- (a language with inverted) and English-speaking (a language with non-inverted number words) university students (mean age 22 years) in a three-digit number comparison task. We observed reliable hundred-decade as well as hundred-unit compatibility effects for three-digit number comparison. This indicates that, comparable two-digit numbers, three-digit numbers are processed in a parallel decomposed fashion. However, in contrast to previous results on two-digit numbers as well as on children’s processing of three-digit numbers, no reliable modulation of these compatibility effects through language was observed in adults. The present data indicate that inversion-related differences in multi-digit number processing are limited. They seem to be restricted to the number range involving those digits being inverted (i.e., tens and units in two-digit numbers) but do not generalize to neighboring digits. Possible reasons for this lack of generalization are discussed.

An integration of competing accounts on children's number line estimation

Children’s estimation patterns in bounded number line estimation (NLE) reveal marked developmental changes. Three different theoretical accounts were proposed to explain these changes: a log-to-linear shift account, a proportion-judgment account and a two-linear account considering familiarity with numbers or the understanding of the place-value structure of the Arabic number system. However, only the first two accounts are considered prominently in the ongoing scientific debate. Therefore, we first present a reanalysis of NLE data of Austrian first-graders contrasting all three accounts. Results indicate that the two-linear account is a reliable alternative to the log-to-linear shift as well as the proportion-judgment account. However, we do not claim the two-liner account to provide an exhaustive explanation for the observed developmental changes. We rather introduce the idea that aspects of all three accounts may complement – instead of exclude – each other. Jointly considering conceptual (i.e., familiarity, place-value) and procedural (i.e., proportion-judgments) aspects will allow for a more comprehensive understanding of children’s development in NLE.

Language influences on numerical development-Inversion effects on multi-digit number processing

In early numerical development, children have to become familiar with the Arabic number system and its place-value structure. The present review summarizes and discusses evidence for language influences on the acquisition of the highly transparent structuring principles of digital-Arabic digits by means of its moderation through the transparency of the respective language's number word system. In particular, the so-called inversion property (i.e., 24 named as “four and twenty” instead of “twenty four”) was found to influence number processing in children not only in verbal but also in non-verbal numerical tasks. Additionally, there is first evidence suggesting that inversion-related difficulties may influence numerical processing longitudinally. Generally, language-specific influences in children's numerical development are most pronounced for multi-digit numbers. Yet, there is currently only one study on three-digit number processing for German-speaking children. A direct comparison of additional new data from Italian-speaking children further corroborates the assumption that language impacts on cognitive (number) processing as inversion-related interference was found most pronounced for German-speaking children. In sum, we conclude that numerical development may not be language-specific but seems to be moderated by language.