Decade breaks in the mental number line? Putting the tens and units back in different bins

(Dis)similarities between non-symbolic and symbolic number representations: Insights from vector space models

Empirical evidence in support of a shared system for non-symbolic and symbolic number processing has been inconclusive. The current study aims to address this question in a novel way, specifically by testing whether the efficient coding principle based on co-occurrence of number symbols in natural language holds for both non-symbolic and symbolic number processing. The efficient coding principle postulates that perception is optimized when stimuli frequently co-occur in a natural environment. We hypothesized that both numerical ratios and co-occurrence frequencies of symbolic numbers would significantly influence participants' performance on a non-symbolic and symbolic number comparison task. To test this hypothesis, we employed latent semantic analysis on a TASA corpus to quantify number co-occurrence in natural language and calculate language similarity estimates. We engaged 73 native English speakers (mean age = 19.36, standard deviation = 1.83) with normal or corrected vision and no learning disorders in a number comparison task involving non-symbolic (dot arrays) and symbolic stimuli (Arabic numerals and English number words). Results showed that numerical ratios significantly predicted participants' performances across all number formats (ps < 0.001). Language similarity estimates derived from everyday language also predicted performance on the non-symbolic task and the symbolic task involving number words (ps < 0.007). Our results highlight the complex nature of numerical processing, pointing to the co-occurrence of number symbols in natural language as an auxiliary factor in understanding the shared characteristics between non-symbolic and symbolic number representations. Given that our study focused on a limited number range (5 to 16) and a specific task type, future studies should explore a wider range of tasks and numbers to further test the role of the efficient coding principle in number processing.

Age-related effects in magnitude and place-value processing

While general cognitive skills decline during aging, numerical skills seem to be mainly preserved. Such skills are essential for an independent life up to old age, e.g., when dealing with money or time. Operating with numbers usually requires number magnitude and place-value processing. The question is whether these processes are negatively affected by aging due to the general cognitive decline or positively affected due to lifelong experience with numbers. Therefore, we investigated age-related changes in the distance and compatibility effects in single-digit, two-digit, and four-digit number comparison. On the one hand, older adults took longer for number processing and showed a smaller distance effect, indicating altered number magnitude representations. On the other hand, older adults were better in place-value processing as indicated by a smaller compatibility effect than in younger adults. We conclude that aging differentially affects basic numerical skills.

Children's comparison of different-length numbers: Managing different attributes in multidigit number processing

In everyday life the comparison of numbers usually occurs between numbers with different numbers of digits. However, experimental research here is scarce. Recent research has shown that adults respond faster to congruent pairs (the initial digit in the number with more digits is larger, e.g., 2384 vs. 107) than to incongruent pairs (the initial digit is larger in the number with fewer digits, e.g., 2675 vs. 398). This has been interpreted as support for the processing of multiple attributes in parallel and against serial accounts. The current research asked whether there is a change in the relevance of these attributes as school grades increase. School-age children from the second to sixth grades (N = 206) were presented with pairs of numbers that had either the same number of digits (3 vs. 3 or 4 vs. 4) or a different number of digits (3 vs. 4). In this latter condition, the stimuli, matched by distance, could be either length/digit congruent (e.g., 2384 vs. 107) or length/digit incongruent (e.g., 2675 vs. 398). Linear mixed models showed a length/digit congruity effect from second graders. Interestingly, in the response time measure, congruity interacted with school grade and the side in which the larger number of the pair was presented. Whereas these results support a model that considers number comparison as a process that weighs different attributes in parallel, it is also argued that developmental changes are associated with differences in the level of automatization of the componential skills involved in the comparison.

The left digit effect in an unbounded number line task

The left digit effect in number line estimation refers to the phenomenon where numerals with similar magnitudes but different leftmost digits (e.g., 19 and 22) are estimated to be farther apart on a number line than is warranted. The effect has been studied using a bounded number line task, a task in which a line is bounded by two endpoints (e.g., 0 and 100), and where one must indicate the correct location of a target numeral on the line. The goal of the present work is to investigate the left digit effect in an unbounded number line task, a task that involves using the size of one unit to determine a target numeral's location, and that elicits strategies different from those used in the bounded number line task. In a preregistered study, participants (N = 58 college students) completed four blocks of 38 trials each of an unbounded number line task, with target numerals ranging between 0 and 100. We found a medium and statistically reliable left digit effect (d = 0.70). The study offers further evidence that the effect is not driven by response strategies specific to the bounded number line task. We discuss other possible sources of the effect including conversion of symbols to magnitudes in these and other contexts.

Place-value and physical size converge in automatic processing of multi-digit numbers

Previous research has shown that multi-digit number processing is modulated by both place-value and physical size of the digits. By pitting place-value against physical size, the present study examined whether one of the attributes had a greater impact on the automatic processing of multi-digit numbers. In three experiments, participants were presented with two-digit number pairs that appeared in frames. They were instructed to select the larger frame while ignoring the numbers within the frames. Importantly, we manipulated the physical size of the digits (i.e., both decade/unit digits were physically larger) within the frames, the unit-decade compatibility (i.e., the relationship between the numerical values of both decade and unit digits was consistent or inconsistent), and the congruity between the numerical values of the decade digits and the frames' physical size (i.e., decade-value-frame-size congruity). In Experiment 1, where all pairs were unit-decade compatible, a decade-value-frame-size congruity effect emerged for pairs with physically larger decade, but not unit, digits. However, when adding unit-decade incompatible pairs (Experiments 2-3), in unit-decade compatible pairs, there was a decade-value-frame-size congruity effect regardless of the digits' physical size. In contrast, in unit-decade incompatible pairs, there was no decade-value-frame-size congruity effect, even when the physically larger digit (i.e., unit) contradicted the place-value information, presumably due to the cancellation of the opposing influences of the digits' physical sizes their place-values. Overall, these findings suggest that place-value and physical size are intertwined in the Hindu-Arabic numerical system and are processed as one.

Why fractions are difficult? Modeling optimal and sub-optimal integration strategies of numerators and denominators by educated adults

Many children and educated adults experience difficulties in understanding and manipulating fractions. In this study, we argue that a major cause of this challenge is rooted in the need to integrate information from two separate informational sources (i.e., denominator and numerator) according to a normative arithmetic rule (i.e., division). We contend that in some tasks, the correct arithmetic rule is replaced by an inadequate (sub-optimal) operation (e.g., multiplication), which leads to inaccurate representation of fractions. We tested this conjecture by applying two rigorous models of information integration : (a) functional measurement (Experiments 1-3) and (b) conjoint measurement (Experiment 4-5) to data from number-to-line and comparative judgment tasks. These allowed us to compare participants' integration strategies with that of an ideal-observer model. Functional measurement analyses on data from the number-to-line task, revealed that participants could represent the global magnitude of proper and improper fractions quite accurately and combine the fractions' components according to an ideal-observer model. However, conjoint measurement analyses on data from the comparative judgment task, showed that most participants combined these fractions' components according to a sub-optimal (saturated) observer model, that is inconsistent with an ideal-observer (additive) model. These results support the view that educated adults are capable of extracting multiple types of representations of fractions depending on the task at-hand. These representations can be either accurate and conform with normative arithmetic or approximated and inconsistent with normative arithmetic. The latter may lead to the observed difficulties people experience with fractions.

tDCS effects in basic symbolic number magnitude processing are not significantly lateralized

Functional lateralization was previously established for various cognitive domains-but not for number processing. Although numbers are considered to be bilaterally represented in the intraparietal sulcus (IPS), there are some indications of different functional roles of the left vs. right IPS in processing number pairs with small vs. large distance, respectively. This raises the question whether number size plays a distinct role in the lateralization within the IPS. In our preregistered study, we applied anodal transcranial direct current stimulation (tDCS) over the left vs. right IPS to investigate the effect of stimulation as compared to sham on small vs. large distance, in both single-digit and two-digit number comparison. We expected that anodal tDCS over the left IPS facilitates number comparison with small distance, while anodal tDCS over the right IPS facilitates number comparison with large distance. Results indicated no effect of stimulation; however, exploratory analyses revealed that tDCS over the right IPS slowed down single-digit number processing after controlling for the training effect. In conclusion, number magnitude processing might be bilaterally represented in the IPS, however, our exploratory analyses emphasise the need for further investigation on functional lateralization of number processing.

Related consistent lures increase the judgment of multiplication facts: Evidence using event-related potential technique

Simple multiplication errors are primarily shown in whether the lures are related to the operands (relatedness, such as 3 × 4 = 15 vs. 17) or whether the same decades are shared with the correct answers (consistency, such as 3 × 4 = 16 vs. 21). This study used a delayed verification paradigm and event-related potential technique to investigate the effects of relatedness and consistency in simple multiplication mental arithmetic for 30 college students in an experiment of presenting probes in auditory channels. We found that, compared to the related inconsistent lures, the related consistent lures showed significantly faster reaction time and induced significantly large amplitudes of N400 and late positive component. The findings suggest that related consistent lures are less affected by the activation diffusion of the arithmetic problem, and the credibility of being perceived as the correct answer is less; the lures related to operands and sharing the same decades with the accurate results can promote the judgment of multiplication mental arithmetic, and the results support the Interacting Neighbors Model.

A language compatibility effect in fraction processing

A language compatibility effect occurs when there is a match between what a language provides and what a mathematical task demands. Here, we investigated whether such an effect exists for fraction processing in English, which names the numerator first, versus Korean, which names the denominator first. We developed two new tasks: a fraction span task where participants view and then recall four fractions and a fraction identification task where they view one fraction and then another and judge whether the two fractions are the same or not. We generally found that English speakers were advantaged when the numerator drove task performance and Korean speakers were advantaged when the denominator was critical. These findings, particularly from the fraction identification task, were inconsistent with the attentional focus hypothesis, which proposes that the serialisation bias of a language guides which fraction component is attended to first. Rather, they were better explained by the verbal encoding hypothesis, which states that a necessary condition for observing language compatibility effects may be that the fraction components must be encoded in verbal working memory and rehearsed there.

Modeling the left digit effect in adult number line estimation

Number line estimation tasks are frequently used to study numerical cognition skills. In a typical version, the bounded number line task, target numerals must be placed on a bounded line labeled only at its endpoints (e.g., with 0 and 100). Placements by adults, while highly accurate, reveal a cyclical pattern of over- and underestimation of target numerals. The pattern suggests use of proportion judgment strategies and is well-captured by cyclical power models. Another systematic number line bias that has recently been observed, but has not yet been considered in modeling efforts, is the left digit effect. Numerals with different leftmost digits (e.g., 39 and 41) are placed farther apart on a line than is warranted. In the current study (N = 60), adult estimates were obtained for all numerals on a 0-100 number line estimation task, and fit of the standard cyclical power model was compared with two modified versions of the model. One modified version included a parameter that underweights the rightward digit's place value (e.g., the ones digit here), and the other used the same parameter to underweight all digits' place values. We found that both modifications provided a considerably better fit for individual and median data than the standard model, and we discuss their relative merits and cognitive interpretations. The data and models suggest how a left digit bias might impact estimates across the number line.

Length is not all that matters: testing the role of number identity and the ratio of fillers in comparisons of multi-digits with different digit length

Research in multi-digit number comparison usually considers stimuli with the same number of digits (e.g., 3452 vs. 7831). Surprisingly, there is almost no research on the comparison of numbers that differ in length (e.g., 995 vs. 1000), which demands a focus on the number of digits in each multi-digit, despite the fact that the role of number length has been explicitly acknowledged in componential models of multi-digit processing. Our study explores whether the comparison of pairs of natural numbers that differ in length is affected by the identity of the leftmost digit of each multi-digit, and asks what is the effect of having variable proportions of trials with pairs of numbers of the same-length in the task. Across three studies participants compared numbers in blocks with different proportions of same-length multi-digit pairs (Experiment 1 and 2: 25% vs. 50% vs. 75%; Experiment 3: 0% vs. 50%). Stimuli in the different-length condition were length-digit congruent (the number with more digits starting with a larger digit: 2384 vs. 107) or length-digit incongruent (the number with more digits starting with a smaller number: 2675 vs. 398). Response times were shorter in length-digit congruent pairs than in the incongruent pairs. Unexpectedly, this effect was only slightly modulated by the proportion of same-/different-length multi-digit pairs in the experimental set. Despite its perceptual saliency, length is not the only information considered when comparing different-length numbers. The leftmost-digit is also taken into account, with variable relevance here, depending on the characteristics of the stimuli set.

Language does arithmetic: linguistic differences in children's place-value processing

The representation and retrieval of multiplication facts is dependent on linguistic specificities such as number word inversion (i.e., 23 is spoken dreiundzwanzig in German which translates to three and twenty). Previous research has evaluated these language influences in adults. Now this study aims to follow-up on earlier findings and takes a closer look at inversion-related effects on place-value processing during multiplication in children. In a task of choice 46 children, either German- or Italian-speaking, had to pick the right answer out of two options for a given multiplication problem. Already established effects in adult participants such as decade-consistency and table-relatedness were also present in elementary school children, but different between the language groups. For decade-consistent items the effect of table-relatedness was larger for Italian-speaking students than for German-speaking. This indicates that the inversion property in the German language leads to those children putting less emphasis on the tens digit when solving multiplication problems, than Italian-speaking children.

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.

Patterns of symbolic numerical magnitude processing and working memory as predictors of early mathematics performance

Although the roles of symbolic numerical magnitude processing (SNMP) and working memory (WM) in mathematics performance are well acknowledged, studies examining their joint effects are few. Here, we investigated the profiles of SNMP (1- and 2-digit comparison) and WM (verbal, visual and central executive) among Norwegian first graders (N = 256), and how these predict performance in counting, arithmetic facts and word problem–solving. Using latent class cluster analysis, four groups were identified: (1) weak SNMP (33.6%), (2) strong SNMP (25.8%), (3) weak SNMP and WM (23.4%) and (4) strong WM (17.2%). Group differences in mathematics performance were significant with explained variance ranging from 7 to 16%, even after controlling for relevant demographics and domain-general cognitive skills. Our findings suggest that children may display relative strengths in SNMP and WM, and that they both have a unique, even compensatory role in mathematics performance.

Running Online Behavioral Experiments Using R: Implementation of a Response-Time Decision Making Task as an R-Shiny App

Online experiments allow for fast, massive, cost-efficient data collection. However, uncontrolled conditions in online experiments can be problematic, particularly when inferences hinge on response-times (RTs) in the millisecond range. To address this challenge, we developed a mobile-friendly open-source application using R-Shiny, a popular R package. In particular, we aimed to replicate the numerical distance effect, a well-established cognitive phenomenon. In the task, 169 participants (109 with a mobile device, 60 on a desktop computer) completed 116 trials displaying two-digit target numbers and decided whether they were larger or smaller than a fixed standard number. Sessions lasted ~7-minutes. Using generalized linear mixed models estimated with Bayesian inference methods, we observed a numerical distance effect: RTs decreased with the logarithm of the absolute difference between the target and the standard. Our results support the use of R-Shiny for RT-data collection. Furthermore, our method allowed us to measure systematic shifts in recorded RTs related to different OSs, web browsers, and devices, with mobile devices inducing longer shifts than desktop devices. Our work shows that precise RT measures can be reliably obtained online across mobile and desktop devices. It further paves the ground for the design of simple experimental tasks using R, a widely popular programming framework among cognitive scientists.

Non-symbolic and symbolic number and the approximate number system

The distinction between non-symbolic and symbolic number is poorly addressed by the authors despite being relevant in numerical cognition, and even more important in light of the proposal that the approximate number system (ANS) represents rational numbers. Although evidence on non-symbolic number and ratios fits with ANS representations, the case for symbolic number and rational numbers is still open.

Reactive and proactive cognitive control as underlying processes of number processing in children

Cognitive control is crucial to resolve conflict in tasks such as the flanker task. Reactive control is used when conflict is rare, whereas proactive control is more efficient in situations where conflict is frequent. Macizo and Herrera (Psychological Research, 2013, Vol. 77, pp. 651-658) found that these two control processes can also underlie two-digit number comparison in adults. Specifically, they observed that the unit-decade compatibility effect decreased in a block containing many conflict trials as compared with a block containing few conflict trials (i.e., a list-wide proportion congruency effect). In the current study, we assessed whether this finding also applies to children (7-, 9-, and 11-year-olds). Participants performed a flanker task and a two-digit number comparison task. In both tasks, the proportion of conflict was manipulated (80% vs. 20%). Results from the flanker task showed a typical list-wide proportion congruency effect in reaction times in all participating age groups. In the number comparison task, we observed list-wide proportion congruency effects in both reaction times and error rates, which did not interact with age. Our findings support the assumption that children as young as 7 years can effectively use proactive and reactive control strategies. We showed that this effect is not limited to standardized artificial laboratory tasks, such as the flanker task, but also underlies more daily life tasks, such as the processing of Arabic numbers.

Categorical Perception of p-Values

Traditional statistics instruction emphasizes a .05 significance level for hypothesis tests. Here, we investigate the consequences of this training for researchers' mental representations of probabilities - whether .05 becomes a boundary, that is, a discontinuity of the mental number line, and alters their reasoning about p-values. Graduate students with statistical training (n = 25) viewed pairs of p-values and judged whether they were "similar" or "different." After controlling for several covariates, participants were more likely and faster to judge p-values as "different" when they crossed the .05 boundary (e.g., .046 vs. .052) compared to when they did not (e.g., .026 vs. .032). This result suggests a categorical perception-like effect for the processing of p-values. It may be a consequence of traditional statistical instruction creating a psychologically real divide between so-called statistical "significance" and "nonsignificance." Such a distortion is undesirable given modern approaches to statistical reasoning that de-emphasize dichotomizing the p-value continuum.

Incongruity in fraction verification elicits N270 and P300 ERP effects

Understanding how the numerical magnitudes of fractions are accessed is a topic of major interest in numerical cognition and mathematics education. Only a few studies have investigated fraction processing using EEG methods. In the present study, 24 adult participants completed a fraction magnitude verification task while EEGs were recorded. Similar to other arithmetic verification tasks, behavioral results show increased response times to validate mismatching magnitudes compared to matching ones. ERP results show an early frontal N270 component to mismatching trials and a late parietal P300 component during matching trials. These ERP results highlight that participants treat matching fractions as targets and suggest that additional cognitive resources are needed to process mismatching targets. These results provide evidence that fractions processing shares a similar neurocognitive process as those observed during the processing of arithmetic operations and open the door to further explore fraction processing using ERP methods.

Deficits in or preservation of basic number processing in Parkinson's disease? A registered report

Neurodegenerative diseases such as Parkinson's disease (PD) have a huge impact on patients, caregivers, and the health-care system. To date, the diagnosis of mild cognitive impairments in PD has been established based on domain-general functions such as executive functions, attention, or working memory. However, specific numerical deficits observed in clinical practice have not yet been systematically investigated. PD-immanent deterioration of domain-general functions and domain-specific numerical areas suggests the mechanisms of both primary and secondary dyscalculia. The current study will systematically investigate basic number processing performance in PD patients for the first time, targeting domain-specific cognitive representations of numerosity and the influence of domain-general factors. The overall sample consists of patients with a diagnosis of PD, according to consensus guidelines, and healthy controls. PD patients will be stratified into patients with normal cognition or mild cognitive impairment (level I-PD-MCI based on cognitive screening). Basic number processing will be assessed using transcoding, number line estimation, and (non)symbolic number magnitude comparison tasks. Discriminant analysis will be employed to assess whether basic number processing tasks can differentiate between a healthy control group and both PD groups. All participants will be subjected to a comprehensive numerical and a neuropsychological test battery, as well as sociodemographic and clinical measures. Study results will give the first broad insight into the extent of basic numerical deficits in different PD patient groups and will help us to understand the underlying mechanisms of the numerical deficits faced by PD patients in daily life.

Linking inhibitory control to math achievement via comparison of conflicting decimal numbers

The relationship between executive functions (EF) and academic achievement is well-established, but leveraging this insight to improve educational outcomes remains elusive. Here, we propose a framework for relating the role of specific EF on specific precursor skills that support later academic learning. Starting from the premise that executive functions contribute to general math skills both directly - supporting the execution of problem solving strategies - and indirectly - supporting the acquisition of precursor mathematical content, we hypothesize that the contribution of domain-general EF capacities to precursor skills that support later learning can help explain relations between EF and overall math skills. We test this hypothesis by examining whether the contribution of inhibitory control on general math knowledge can be explained by inhibition's contribution to processing rational number pairs that conflict with individual's prior whole number knowledge. In 97 college students (79 female, age = 20.58 years), we collected three measures of EF: working memory (backwards spatial span), inhibition (color-word Stroop) and cognitive flexibility (task switching), and timed and untimed standardized measures of math achievement. Our target precursor skill was a decimals comparison task where correct responses were inconsistent with prior whole number knowledge (e.g., 0.27 vs. 0.9). Participants performed worse on these trials relative to the consistent decimals pairs (e.g., 0.2 vs. 0.87). Individual differences in the Stroop task predicted performance on inconsistent decimal comparisons, which in turn predicted general math achievement. With respect to relating inhibitory control to math achievement, Stroop performance was an independent predictor of achievement after accounting for age, working memory and cognitive flexibility, but decimal performance mediated this relationship. Finally, we found inconsistent decimals performance mediated the relationship of inhibition with rational number performance, but not other advanced mathematical concepts. These results pinpoint the specific contribution of inhibitory control to rational number understanding, and more broadly are consistent with the hypothesis that acquisition of foundational mathematical content can explain the relationships between executive functions and academic outcomes, making them promising targets for intervention.

Two processing stages of the SNARC effect

The spatial-numerical association of response codes (SNARC) effect showed that small/large numbers represented in the left/right space facilitated left/right responses, respectively. However, the processing stage (semantic representation or response selection) of the SNARC effect is still controversial. To investigate this issue, we adopted a modified magnitude comparison task in which the effects of SNARC, Stroop (semantic-representation stage), and Simon (response-selection stage) could be simultaneously induced. The processing stages of the SNARC effect were investigated by examining the interactions among these effects. According to the additive factor logic, if two effects are interactive, then they occur in the same stage; if two effects are additive, then they occur in different stages. Across two experiments, the SNARC effect interacted with the Stroop effect and with the Simon effect. These results suggested that the SNARC effect occurred in both the semantic-representation and response-selection stages and provided insight into that the SNARC effect might have two originating sources.

The processing of prices across numerical formats

We evaluated whether the format in which prices are presented determines the processing of their magnitude. A price comparison task was used in which two-digit prices with Arabic digits, written number words and auditory number words were presented in the euro currency. Prices were number-monetary category (NMC) compatible (49 euros - 36 cents) when the number and monetary category of one price were larger than those of the other (49 > 36, euros > cents); or NMC incompatible (49 cents - 36 euros) when the number of one price was larger but the monetary category smaller than those of the other (49 > 36, cents < euros). In addition, there were unit-decade (UD) compatible prices when the decade and unit of one price were larger than those of the other (49 euros - 36 cents, 4 > 3, 9 > 6); and UD incompatible prices when the decade of one price was larger but the unit smaller than those of the other (46 euros - 39 cents, 4 > 3, 6 < 9). The results showed NMC compatibility effects in all numerical formats. However, the UD compatibility effect was not found in any numerical format. The results are discussed within the hybrid model of multisymbolic magnitude processing.

Eye-movements during number comparison: Associations to sex and sex hormones

Multi-digit numbers are of a hierarchical nature with whole number magnitudes depending on digit magnitudes. Processing of multi-digit numbers can occur in a holistic or decomposed fashion. The unit-decade compatibility effect during number comparison is often used as a measure of decomposed processing. It refers to the fact that performance is reduced when the larger number contains the smaller unit digit (e.g. 73 vs. 26). It has been demonstrated that women show a larger compatibility effect than men, which is in accordance with their general tendency towards focusing on stimulus details during processing of visual hierarchical stimuli (local processing style). Such a local processing style has been related to higher progesterone and lower testosterone levels. One method to study individual processing styles is eye-tracking. The aim of the present study was to examine whether sex and sex hormones (estradiol, progesterone, testosterone) relate to eye movement behavior in the number comparison task. Unlike previous studies we found no evidence for sex differences in the behavioral compatibility effect. Nevertheless, women look more often and longer at individual digits and show a stronger compatibility effect in fixation durations compared to men, while men show more saccades between numbers than women. Estradiol and progesterone were related to fewer fixations and shorter fixation durations and more saccades between numbers in men, but not in women. Furthermore, the compatibility effect in the number of fixations and fixation durations was negatively related to testosterone in women. In summary, this is the first study to demonstrate sex differences and sex hormone influences on eye gaze behavior during number comparison.

Number line estimation and standardized test performance: The left digit effect does not predict SAT math score

INTRODUCTION

Recent work reveals a new source of error in number line estimation (NLE), the left digit effect (Lai, Zax, et al., 2018), whereby numerals with different leftmost digits but similar magnitudes (e.g., 399, 401) are placed farther apart on a number line (e.g., 0 to 1,000) than is warranted. The goals of the present study were to: (1) replicate the left digit effect, and (2) assess whether it is related to mathematical achievement.

METHOD

Participants were all individuals (adult college students) who completed the NLE task in the laboratory between 2014 and 2019 for whom SAT scores were available (n = 227).

RESULTS

We replicated the left digit effect but found its size was not correlated with SAT math score, although it was negatively correlated with SAT verbal score for one NLE task version.

CONCLUSIONS

These findings provide further evidence that individual digits strongly influence estimation performance and suggest that this effect may have different cognitive contributors, and predict different complex skills, than overall NLE accuracy.

The dynamic nature of children's strategy use after receiving accuracy feedback in decimal comparisons

When students start learning decimals, they may incorrectly apply features of their prior numerical knowledge (e.g., whole-number or fraction rules). However, because whole numbers, fractions, and decimals all have their own unique features, these whole-number and fraction strategies do not always lead to correct solutions. We examined whether receiving immediate accuracy feedback while comparing decimal pairs that were either congruent with whole-number rules (e.g., decimals with more digits were larger in magnitude) or incongruent with whole-number rules (e.g., decimals with fewer digits were larger in magnitude) would lead students to change their decimal comparison strategies. We also examined whether students' potential improvement after feedback would generalize to decimal comparisons involving different numbers of digits. We found that sixth- to eighth-grade students' use of the whole-number strategy declined and their use of the normative decimal strategy increased over the course of receiving feedback, whereas no significant strategy change was observed among students who did not receive any feedback. Students who received feedback were also less likely to use a whole-number strategy and more likely to use a decimal strategy in different decimal comparisons in an immediate posttest and a 2-week delayed posttest. Our exploratory analyses found that students' improvement on decimal comparisons did not transfer to decimal arithmetic. Moreover, students' inhibitory control also predicted strategy use in immediate and delayed posttests. Our study provides insights into the mechanisms of rapid strategy change and has implications for designing interventions to improve children's understanding of decimal magnitudes.

Self-Regulation and Mathematics Performance in German and Iranian Students of More and Less Math-Related Fields of Study

Self-regulation is a multidimensional construct that is positively related to academic achievement, such as successful mathematics performance. However, this relation of self-regulation and mathematics performance has mainly been investigated in Western countries with similar cultural contexts, although self-regulation is assumed to be context-sensitive. Therefore, the present study investigated the relation of self-regulation and mathematics performance across two different countries (Germany vs. Iran) in college students. The relation of self-regulation and mathematics performance was expected to be weaker in students of math-related fields, such as Engineering/Informatics, as they are assumed to need less self-regulation to solve the mathematics problems than students of less math-related fields, such as Human Sciences. In total, 122 undergraduate students (German = 60; Iranian = 62) of Human Sciences or Engineering/Informatics participated in this study. We measured self-regulation with the Brief Self-Control Scale (Tangney et al., 2004) and mathematics performance with a complex multiplication test. Results showed that self-regulation did not predict multiplication performance in German or Iranian students, in general. However, when the field of study was considered, self-regulation predicted multiplication performance in the subgroup of German and Iranian students studying Human Sciences within each country. We conclude that cultural context does not seem to play a dominant role in moderating the relation between self-regulation and math performance, however, field of study and more generally familiarity with math may be an important factor to consider in single or cross-cultural studies.

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.

Shifting attention does not influence numerical processing

Many theories of numerical cognition assume that numbers and space share a common representation at the response level. For example, observers are faster to respond to small numbers with their left hand and large numbers with their right hand (the SNARC effect). There is also evidence that viewing numbers can produce spatial shifts of attention, suggesting that attention may play a role in the spatial representation of numbers. In the present study, we assessed whether shifts of attention can influence numerical processing. Participants viewed a leftward or rightward peripheral cue followed by a centrally presented number, then judged whether the number was odd or even. Participants responded faster and made fewer errors when the number magnitude and response side were compatible, revealing a response-based SNARC effect. Participants also responded faster when the cue direction and response side were compatible, revealing a Simon effect. However, participants did not respond faster when the cue direction and number magnitude were compatible. Similar findings were observed when the association between numbers and space was relatively explicit. Moreover, although we failed to observe a response-based SNARC effect when number magnitude was directly relevant to observers' task, we observed a large Simon effect. Together, these findings suggest that although numbers and space share a common representation at the response level, attention does not play a substantial role in the spatial representation of numbers.

No power: exponential expressions are not processed automatically as such

Little is known about the mental representation of exponential expressions. The present study examined the automatic processing of exponential expressions under the framework of multi-digit numbers, specifically asking which component of the expression (i.e., the base/power) is more salient during this type of processing. In a series of three experiments, participants performed a physical size comparison task. They were presented with pairs of exponential expressions that appeared in frames that differed in their physical sizes. Participants were instructed to ignore the stimuli within the frames and choose the larger frame. In all experiments, the pairs of exponential expressions varied in the numerical values of their base and/or power component. We manipulated the compatibility between the base and the power components, as well as their physical sizes to create a standard versus nonstandard syntax of exponential expressions. Experiments 1 and 3 demonstrate that the physically larger component drives the size congruity effect, which is typically the base but was manipulated here in some cases to be the power. Moreover, Experiments 2 and 3 revealed similar patterns, even when manipulating the compatibility between base and power components. Our findings support componential processing of exponents by demonstrating that participants were drawn to the physically larger component, even though in exponential expressions, the power, which is physically smaller, has the greater mathematical contribution. Thus, revealing that the syntactic structure of an exponential expression is not processed automatically. We discuss these results with regard to multi-digit numbers research.

Multilingual two-digit number naming: The influence of composition rules on language switching

The aim of this study was to examine language switching in a two-digit number naming task. In contrast to single digits, two-digit numbers have a composition rule (i.e., morphological configuration) that may differ between languages. For example, the Arabic number 21 is read with an inverted composition rule in German (unit before decade) and a non-inverted composition rule in English (decade before unit). In the present experiment, one group of German native speakers and one group of Spanish native speakers had to name two-digit numbers in German, English, or Spanish. The results demonstrate a language-switch cost, revealing better performance in language repetition than in language-switch trials. This switch cost was further modulated by repeating or switching the composition rule, since the language repetition benefit (i.e., the switch cost) was reduced in trials with composition-rule switches compared with trials with composition-rule repetitions. This finding indicates that the language in which the number word has to be produced and its composition rule are not switched independently but rather may be integrated into one language schema.

No calculation necessary: Accessing magnitude through decimals and fractions

Research on how humans understand the relative magnitude of symbolic fractions presents a unique case of the symbol-grounding problem with numbers. Specifically, how do people access a holistic sense of rational number magnitude from decimal fractions (e.g. 0.125) and common fractions (e.g. 1/8)? Researchers have previously suggested that people cannot directly access magnitude information from common fraction notation, but instead must use a form of calculation to access this meaning. Questions remain regarding the nature of calculation and whether a division-like conversion to decimals is a necessary process that permits access to fraction magnitudes. To test whether calculation is necessary to access fractions magnitudes, we carried out a series of six parallel experiments in which we examined how adults access the magnitude of rational numbers (decimals and common fractions) under varying task demands. We asked adult participants to indicate which of two fractions was larger in three different conditions: decimal-decimal, fraction-fraction, and mixed decimal-fraction pairs. Across experiments, we manipulated two aspects of the task demands. 1) Response windows were limited to 1, 2 or 5 s, and 2) participants either did or did not have to identify when the two stimuli were the same magnitude (catch trials). Participants were able to successfully complete the task even at a response window of 1 s and showed evidence of holistic magnitude processing. These results indicate that calculation strategies with fractions are not necessary for accessing a sense of a fractions meaning but are strategic routes to magnitude that participants may use when granted sufficient time. We suggest that rapid magnitude processing with fractions and decimals may occur by mapping symbolic components onto common amodal mental representations of rational numbers.

Automatic place-value activation in magnitude-irrelevant parity judgement

Research on multi-digit number processing suggests that, in Arabic numerals, their place-value magnitude is automatically activated, whenever a magnitude-relevant task was employed. However, so far, it is unknown, whether place-value is also activated when the target task is magnitude-irrelevant. The current study examines this question using the parity congruency effect in two-digit numbers: It describes that responding to decade-digit parity congruent numbers (e.g., 35, 46; same parity of decades and units) is faster than to decade-digit parity incongruent numbers (e.g., 25; 36; different parities of decades and units). Here we investigate the (a-) symmetry of the parity congruency effect; i.e. whether it makes a difference whether participants are assessing the parity of the unit digit or the decade digit. We elaborate, how and why such an asymmetry is related to place-value processing, because the parity of the unit digit only interferes with the parity of the decade digit, while the parity of the decade digit interferes with both the parity of the unit digit and the integrated parity of the whole two-digit number. We observed a significantly larger parity congruency effect in the decade parity decision than in the unit parity decision. This suggests that automatic place-value processing also takes place in a typical parity judgment task, in which magnitude is irrelevant. Finally, because of the cross-lingual design of the study, we can show that these results and their implications were language-independent.

Embodied numerical representations and their association with multi-digit arithmetic performance

There is a well-documented association between fingers and numbers, which was claimed to stem from the use of finger-based strategies for counting and calculating during childhood. Recently, it has been argued that this may lead to a concomitant activation of finger-based alongside other numerical representations when encountering single-digit numbers. Indeed, the occurrence of such a co-activation is supported by observed influences of finger counting habits on different numerical tasks, including single-digit arithmetic problem solving. In this study, we pursued the question whether the influence of finger-based representations on arithmetic generalizes to multi-digit arithmetic by investigating the association between the recognition of canonical and non-canonical finger patterns and multi-digit arithmetic in adults. Results indicated that canonical finger-based numerical representations were significantly associated with addition performance only, whereas non-canonical finger-based representations were associated significantly with all four arithmetic operations. We argue that, because non-canonical patterns do not benefit from the iconicity of canonical patterns, their magnitude may need to be constructed through magnitude manipulation which may in turn increase associations with mental arithmetic. In sum, our findings provide converging evidence for a functional association between finger-based representations and arithmetic performance.

How do we process prices? Electrophysiological evidence of componential analysis

The aim of this study was to evaluate if the processing of prices (e.g., 8 euro) involves separate analysis of the digit (8) and the monetary category (euro). Event-related potentials (ERPs) were recorded when participants performed a price comparison task in which a pair of prices was presented and they selected the one with higher monetary value. There were compatible comparisons where the digit and the monetary category of one price were higher than those of the other price (e.g., 7 euro - 4 cent, 7 > 4 and euro > cent), and incompatible comparisons where the number of one price was larger but the monetary category smaller than those of the other price (e.g., 4 euro - 7 cent, 4 < 7 but euro > cent). Compatibility effect modulated ERPs in the 350-450 ms time window, with more negative amplitudes in incompatible trials relative to compatible trials. This pattern of results suggests that prices were processed in a componential manner, challenging the evidences of the holistic model. The results of this study agree with the general model framework for multi-symbol number comparison: The constituents of a multi-symbol magnitude are processed and compared to each other separately in a componential manner.

Sex hormones and number processing. Progesterone and testosterone relate to hemispheric asymmetries during number comparison

Like many visual stimuli, multi-digit numbers are of a hierarchical nature, with whole number magnitudes depending on individual digit magnitudes. Accordingly, multi-digit numbers can be processed in a holistic (whole number magnitudes) or decomposed manner (digit magnitudes). The compatibility effect during number comparison serves as an indicator of decomposed processing. It is characterized by impaired performance for items where the larger number contains the smaller unit-digit. We were recently able to demonstrate, that the compatibility effect indeed depends on an individual's tendency to process visual hierarchical stimuli on a global or local level. Accordingly, factors affecting global-local processing, should also affect number magnitude processing, i.e. the compatibility effect. Among these factors are hemispheric asymmetries, sex differences and sex hormones (estradiol, progesterone, testosterone). In the present study 39 men and 37 naturally cycling women in their luteal cycle phase completed a number comparison task with stimuli randomly presented to the left and right hemifield. As in previous studies, we observed a larger compatibility effect in the right hemifield (left hemisphere) than in the left hemifield (right hemisphere) and in men than in women. However, this is the first study to evaluate the effects of sex hormones on hemispheric asymmetries during number comparison. We found progesterone to relate to increased hemispheric asymmetries in men, but decreased hemispheric asymmetries in women. Additionally, testosterone was negatively related to hemispheric asymmetries in women's compatibility effect in reaction times. These results add to the growing evidence that sex hormones relate to hemispheric asymmetries in cognitive functions.

Oscillatory EEG Changes During Arithmetic Learning in Children

Most studies have investigated brain activation changes after the course of arithmetic learning, and the question remains whether these changes are detectable during the course of learning, i.e., before memory consolidation. Twenty-four fifth graders solved multiplication problems while ongoing electroencephalography (EEG) was recorded. The arithmetic training revealed reduced errors together with a power increase in theta (4-7 Hz) but not in lower alpha (8-10 Hz) or upper alpha (10-13 Hz) bands. We conclude that increases in theta power subserved a shift from slow, procedural strategies to more efficient, automated procedural and retrieval strategies, which led to more efficient performance.

Sex Differences in Number Magnitude Processing Strategies Are Mediated by Spatial Navigation Strategies: Evidence From the Unit-Decade Compatibility Effect

The hybrid model of number magnitude processing suggests that multi-digit numbers are simultaneously processed holistically (whole number magnitudes) and in a decomposed manner (digit magnitudes). Thus, individual tendencies and situational factors may affect which type of processing becomes dominant in a certain individual in a given situation. The unit-decade compatibility effect has been described as indicative of stronger decomposed number processing. This effect occurs during the comparison of two-digit numbers. Compatible items in which the larger number contains the larger unit digit are easier to solve than incompatible items in which the larger number contains the smaller unit digit. We have previously described women show a larger compatibility effect than men. Furthermore, the compatibility effect is modulated by situational factors like the vertical spacing of the presented numbers. However, it has not been addressed whether situational factors and sex affect the unit-decade compatibility effect interactively. We have also demonstrated that the unit-decade compatibility effects relates to global-local processing, which in turn also affects spatial processing strategies. However, a link between spatial processing strategies and the unit-decade compatibility effect has not yet been established. In the present study we investigate, whether sex differences in the unit-decade compatibility effect (i) depend on the vertical spacing between numbers, (ii) are mediated via sex hormone levels of participants, and (iii) relate to sex differences in spatial processing strategies. 42 men and 41 women completed a two-digit number comparison task as well as a spatial navigation task. The number comparison task modulates compatibility and vertical spacing in a 2 × 2 design. The results confirm a larger compatibility effect in women compared to men and with dense compared to sparse spacing. However, no interactive effect was observed, suggesting that these factors modulate number magnitude processing independently. The progesterone/testosterone ratio was related to the compatibility effect, but did not mediate the sex difference in the compatibility effect. Furthermore, spatial processing strategies were related to the compatibility effect and did mediate the sex difference in the compatibility effect. Participants with a stronger focus on landmarks in the spatial navigation task showed a larger compatibility effect.

Extended mathematical cognition: external representations with non-derived content

Vehicle externalism maintains that the vehicles of our mental representations can be located outside of the head, that is, they need not be instantiated by neurons located inside the brain of the cogniser. But some disagree, insisting that 'non-derived', or 'original', content is the mark of the cognitive and that only biologically instantiated representational vehicles can have non-derived content, while the contents of all extra-neural representational vehicles are derived and thus lie outside the scope of the cognitive. In this paper we develop one aspect of Menary's vehicle externalist theory of cognitive integration-the process of enculturation-to respond to this longstanding objection. We offer examples of how expert mathematicians introduce new symbols to represent new mathematical possibilities that are not yet understood, and we argue that these new symbols have genuine non-derived content, that is, content that is not dependent on an act of interpretation by a cognitive agent and that does not derive from conventional associations, as many linguistic representations do.

Place-value computation in children with mathematics difficulties

Recent research has provided initial evidence that children with math difficulties (MD) experience problems in processing place-value information in basic numerical tasks. However, it remains unclear whether these problems generalize to basic arithmetic operations. For instance, multi-digit addition problems with carryover specifically require the computation of place-value information. Yet little is known about the carry effect in children with MD. Therefore, the current study investigated whether problems in processing place-value information among third-grade children with MD (n = 29 9-year-olds) compared with an age-matched control group (n = 50) generalize to two-digit addition. The results indicate an increased carry effect for response latencies and error rates in children with MD. These findings suggest that deficits in processing place-value information among children with MD generalize to place-value computations in multi-digit arithmetic. Potential contributions of strategy use and working memory for difficulties in processing place-value information are discussed.

A Mathematical Model of How People Solve Most Variants of the Number-Line Task

Current understanding of the development of quantity representations is based primarily on performance in the number-line task. We posit that the data from number-line tasks reflect the observer's underlying representation of quantity, together with the cognitive strategies and skills required to equate line length and quantity. Here, we specify a unified theory linking the underlying psychological representation of quantity and the associated strategies in four variations of the number-line task: the production and estimation variations of the bounded and unbounded number-line tasks. Comparison of performance in the bounded and unbounded number-line tasks provides a unique and direct way to assess the role of strategy in number-line completion. Each task produces a distinct pattern of data, yet each pattern is hypothesized to arise, at least in part, from the same underlying psychological representation of quantity. Our model predicts that the estimated biases from each task should be equivalent if the different completion strategies are modeled appropriately and no other influences are at play. We test this equivalence hypothesis in two experiments. The data reveal all variations of the number-line task produce equivalent biases except for one: the estimation variation of the bounded number-line task. We discuss the important implications of these findings.

Attention allows the SNARC effect to operate on multiple number lines

To investigate whether participants can activate only one spatially oriented number line at a time or multiple number lines simultaneously, they were asked to solve a unit magnitude comparison task (unit smaller/larger than 5) and a parity judgment task (even/odd) on two-digit numbers. In both these primary tasks, decades were irrelevant. After some of the primary task trials (randomly), participants were asked to additionally solve a secondary task based on the previously presented number. In Experiment 1, they had to decide whether the two-digit number presented for the primary task was larger or smaller than 50. Thus, for the secondary task decades were relevant. In contrast, in Experiment 2, the secondary task was a color judgment task, which means decades were irrelevant. In Experiment 1, decades' and units' magnitudes influenced the spatial association of numbers separately. In contrast, in Experiment 2, only the units were spatially associated with magnitude. It was concluded that multiple number lines (one for units and one for decades) can be activated if attention is focused on multiple, separate magnitude attributes.

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.

Task instructions modulate unit-decade binding in two-digit number representation

Previous studies have found decomposed processes, as well as holistic processes, in the representation of two-digit numbers. The present study investigated the influence of task instruction on such processes. Participants completed both magnitude and parity tasks in one of three instructional conditions, where they were asked to either consider two-digit numbers as a whole or to focus on one specific digit. In two experiments, we found that when participants were asked to consider the two digits as an integrated number, they always exhibited a unit-decade compatibility effect, indicating a failure of selective attention on the digit relevant to the given task. However, the mere presence of the neighboring digit is not a sufficient condition for the compatibility effect: when participants were explicitly asked to process a specific digit, their success/failure to selectively ignore the irrelevant digit depended on task requirements. Further, computer mouse tracking indicated that the locus of the compatibility effect was related to late response-related processing. The results signify the deep involvement of top-down processes in unit-decade binding for two-digit number representation.

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.