Familiar Sequences Are Processed Faster Than Unfamiliar Sequences, Even When They Do Not Match the Count-List.

In order processing, consecutive sequences (e.g., 1-2-3) are generally processed faster than nonconsecutive sequences (e.g., 1-3-5) (also referred to as the reverse distance effect). A common explanation for this effect is that order processing operates via a memory-based associative mechanism whereby consecutive sequences are processed faster because they are more familiar and thus more easily retrieved from memory. Conflicting with this proposal, however, is the finding that this effect is often absent. A possible explanation for these absences is that familiarity may vary both within and across sequence types; therefore, not all consecutive sequences are necessarily more familiar than all nonconsecutive sequences. Accordingly, under this familiarity perspective, familiar sequences should always be processed faster than unfamiliar sequences, but consecutive sequences may not always be processed faster than nonconsecutive sequences. To test this hypothesis in an adult population, we used a comparative judgment approach to measure familiarity at the individual sequence level. Using this measure, we found that although not all participants showed a reverse distance effect, all participants displayed a familiarity effect. Notably, this familiarity effect appeared stronger than the reverse distance effect at both the group and individual level; thus, suggesting the reverse distance effect may be better conceptualized as a specific instance of a more general familiarity effect.

Multiple number-naming associations: How the inversion property affects adults' two-digit number processing.

Some number-naming systems are less transparent than others. For example, in Dutch, 49 is named "negenenveertig," which translates to "nine and forty," i.e., the unit is named first, followed by the decade. This is known as the "inversion property," where the morpho-syntactic representation of the number name is incongruent with its written Arabic form. Number word inversion can hamper children's developing mathematical skills. But little is known about its effects on adults' numeracy, the underlying mechanism, and how a person's bilingual background influences its effects. In the present study, Dutch-English bilingual adults performed an audiovisual matching task, where they heard a number word and simultaneously saw two-digit Arabic symbols and had to determine whether these matched in quantity. We experimentally manipulated the morpho-syntactic structure of the number words to alter their phonological (dis)similarities and numerical congruency with the target Arabic two-digit number. Results showed that morpho-syntactic (in)congruency differentially influenced quantity match and non-match decisions. Although participants were faster when hearing traditional non-transparent Dutch number names, they made more accurate decisions when hearing artificial, but morpho-syntactically transparent number words. This pattern was partly influenced by the participants' bilingual background, i.e., their L2 proficiency in English, which involves more transparent number names. Our findings suggest that, within inversion number-naming systems, multiple associations are formed between two-digit Arabic symbols and number names, which can influence adults' numerical cognition.

Concepts of order: Why is ordinality processed slower and less accurately for non-consecutive sequences?

Both adults and children are slower at judging the ordinality of non-consecutive sequences (e.g., 1-3-5) than consecutive sequences (e.g., 1-2-3). It has been suggested that the processing of non-consecutive sequences is slower because it conflicts with the intuition that only count-list sequences are correctly ordered. An alternative explanation, however, may be that people simply find it difficult to switch between consecutive and non-consecutive concepts of order during order judgement tasks. Therefore, in adult participants, we tested whether presenting consecutive and non-consecutive sequences separately would eliminate this switching demand and thus improve performance. In contrast with this prediction, however, we observed similar patterns of response times independent of whether sequences were presented separately or together (Experiment 1). Furthermore, this pattern of results remained even when we doubled the number of trials and made participants explicitly aware when consecutive and non-consecutive sequences were presented separately (Experiment 2). Overall, these results suggest slower response times for non-consecutive sequences do not result from a cognitive demand of switching between consecutive and non-consecutive concepts of order, at least not in adults.

Miscategorized subset-knowers: Five- and six-knowers can compare only the numbers they know.

Initial acquisition of the first symbolic numbers is measured with the Give a Number (GaN) task. According to the classic method, it is assumed that children who know only 1, 2, 3, or 4 in the GaN task, (termed separately one-, two-, three-, and four-knowers, or collectively subset-knowers) have only a limited conceptual understanding of numbers. On the other hand, it is assumed that children who know larger numbers understand the fundamental properties of numbers (termed cardinality-principle-knowers), even if they do not know all the numbers as measured with the GaN task, that are in their counting list (e.g., five- or six-knowers). We argue that this practice may not be well-established. To validate this categorization method, here, the performances of groups with different GaN performances were measured separately in a symbolic comparison task. It was found that similar to one to four-knowers, five-, six-, and so forth, knowers can compare only the numbers that they know in the GaN task. We conclude that five-, six-, and so forth, knowers are subset-knowers because their conceptual understanding of numbers is fundamentally limited. We argue that knowledge of the cardinality principle should be identified with stricter criteria compared to the current practice in the literature. RESEARCH HIGHLIGHTS: Children who know numbers larger than 4 in the Give a Number (GaN) task are usually assumed to have a fundamental conceptual understanding of numbers. We tested children who know numbers larger than 4 but who do not know all the numbers in their counting list to see whether they compare numbers more similar to children who know only small numbers in the GaN task or to children who have more firm number knowledge. Five-, six-, and so forth, knowers can compare only the numbers they know in the GaN task, similar to the performance of the one, two, three, and four-knowers. We argue that these children have a limited conceptual understanding of numbers and that previous works may have miscategorized them.

Weaker semantic priming effects with number words in the second language of math learning.

Bilinguals' exact number representations result from associations between language-independent Indo-Arabic digits ("5"), two verbal codes ("fünf" and "cinq") and a common, largely overlapping semantic representation. To compare the lexical and semantic access to number representations between two languages, we recruited a sample of balanced highly proficient German-French adult bilinguals. At school, those bilinguals learned mathematics in German for 6 years (LM1) and then switched to French (LM2) in 7th grade (12 years old) until 13th grade. After the brief presentation of primes (51 ms) consisting of Indo-Arabic digits or number words in German or French, an Indo-Arabic digits target had to be read in either German or French in an online study. Stimuli were numbers from 1 to 9, and we varied the absolute distance between primes and targets from 0 (i.e., 1-1) to 3 (1-4; as in Reynvoet et al., 2002). The priming distance effect (PDE) was used to measure the strength of numerical semantic association. We find comparable PDEs with Indo-Arabic digits and German number word primes, independently from the target naming language. However, we did not find a clear PDE with French number word primes, neither when naming targets in German, nor in French. The weaker PDE from LM2 compared to LM1 primes is interpreted as a weaker lexico-semantic association of LM2 number words. These results indicate a critical role of the LM1 and further emphasize the role of language in processing numbers. They might have important implications for designing bilingual school curricula. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Symbolic number ordering strategies and math anxiety.

Math anxiety results in a drop in performance on various math-related tasks, including the symbolic number ordering task in which participants decide whether a triplet of digits is presented in order (e.g. 3-5-7) or not (e.g. 3-7-5). We investigated whether the strategy repertoire and reaction times during a symbolic ordering task were affected by math anxiety. In study 1, participants performed an untimed symbolic number ordering task and indicated the strategy they used on a trial-by-trial basis. The use of the memory retrieval strategy, based on the immediate recognition of the triplet, decreased with high math anxiety, but disappeared when controlling for general anxiety. In the study 2, participants completed a timed version of the number order task. High math-anxious participants used the decomposition strategy (e.g. 5 is larger than 3 and 7 is larger than 5 to decide whether 3-5-7 is in the correct order) more often, and were slower in responding when both memory- and other decomposition strategies were used. Altogether, both studies demonstrate that high-math anxious participants are not only slower to decide whether a number triplet is in the correct order, but also rely more on procedural strategies.

Unraveling the role of math anxiety in students' math performance.

Math anxiety (MA; i.e., feelings of anxiety experienced when being confronted with mathematics) can have negative implications on the mental health and well-being of individuals and is moderately negatively correlated with math achievement. Nevertheless, ambiguity about some aspects related to MA may prevent a fathomed understanding of this systematically observed relationship. The current study set out to bring these aspects together in a comprehensive study. Our first focus of interest was the multi-component structure of MA, whereby we investigated the relationship between state- and trait-MA and math performance (MP) and whether this relation depends on the complexity of a math task. Second, the domain-specificity of MA was considered by examining the contribution of general anxiety (GA) and MA on MP and whether MA also influences the performance in non-math tasks. In this study, 181 secondary school students aged between 16 and 18 years old were randomly presented with four tasks (varying in topic [math/non-math] and complexity [easy/difficult]). The math task was a fraction comparison task and the non-math task was a color comparison task, in which specific indicators were manipulated to develop an easy and difficult version of the tasks. For the first research question, results showed a moderate correlation between state- and trait-MA, which is independent of the complexity of the math task. Regression analyses showed that while state-MA affects MP in the easy math task, it is trait-MA that affects MP in the difficult math task. For the second research question, a high correlation was observed between GA and MA, but regression analyses showed that GA is not related to MP and MA has no predictive value for performance in non-math tasks. Taken together, this study underscores the importance of distinguishing between state and trait-MA in further research and suggests that MA is domain-specific.

Anisotropic representations of visual space modulate visual numerosity estimation.

Humans can estimate the number of visually displayed items without counting. This capacity of numerosity perception has often been attributed to a dedicated system to estimate numerosity, or alternatively to the exploitation of various stimulus features, such as density, convex hull, the size of items, and occupancy area. The distribution of the presented items is usually not varied with eccentricity in the visual field. However, our visual fields are highly asymmetric. To date, it is unclear how inhomogeneities of the visual field impact numerosity perception. Besides eccentricity, a pronounced asymmetry is the radial-tangential anisotropy. For example, in crowding, radially placed flankers interfere more strongly with target perception than tangentially placed flankers. Similarly, in redundancy masking, the number of perceived items in repeating patterns is reduced when the items are arranged radially but not when they are arranged tangentially. Here, we investigated whether numerosity perception is subject to the radial-tangential anisotropy of spatial vision to shed light on the underlying topology of numerosity perception. In Experiment 1, observers were presented with varying numbers of discs, predominantly arranged radially or tangentially, and asked to report their perceived number. In Experiment 2, observers were presented with the same displays as in Experiment 1, and were asked to encircle items that were perceived as a group. We found that numerosity estimation depended on the arrangement of discs, suggesting a radial-tangential anisotropy of numerosity perception. Grouping among discs did not seem to explain our results. We suggest that the topology of spatial vision modulates numerosity estimation and that asymmetries of visual space should be taken into account when investigating numerosity estimation.

Teasing apart the unique contributions of cognitive and affective predictors of math performance.

Math permeates everyday life, and math skills are linked to general educational attainment, income, career choice, likelihood of full-time employment, and health and financial decision making. Thus, researchers have attempted to understand factors predicting math performance in order to identify ways of supporting math development. Work examining individual differences in math performance typically focuses on either cognitive predictors, including inhibitory control and the approximate number system (ANS; a nonsymbolic numerical comparison system), or affective predictors, like math anxiety. Studies with children suggest that these factors are interrelated, warranting examination of whether and how each uniquely and independently contributes to math performance in adulthood. Here, we examined how inhibitory control, the ANS, and math anxiety predicted college students' math performance (n = 122, mean age = 19.70 years). Using structural equation modeling, we find that although inhibitory control and the ANS were closely related to each other, they did not predict math performance above and beyond the effects of the other while also controlling for math anxiety. Instead, math anxiety was the only unique predictor of math performance. These findings contradict previous results in children and reinforce the need to consider affective factors in our discussions and interventions for supporting math performance in college students.

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.

Weighted numbers.

Clarke and Beck (C&B) discuss in their sections on congruency and confounds (sects. 3 and 4) literature that has challenged the claim that the approximate number system (ANS) represents numerical content. We argue that the propositions put forward by these studies aren't that far from the indirect model of number perception suggested by C&B.

Automatic integration of numerical formats examined with frequency-tagged EEG.

How humans integrate and abstract numerical information across different formats is one of the most debated questions in human cognition. We addressed the neuronal signatures of the numerical integration using an EEG technique tagged at the frequency of visual stimulation. In an oddball design, participants were stimulated with standard sequences of numbers (< 5) depicted in single (digits, dots, number words) or mixed notation (dots-digits, number words-dots, digits-number words), presented at 10 Hz. Periodically, a deviant stimulus (> 5) was inserted at 1.25 Hz. We observed significant oddball amplitudes for all single notations, showing for the first time using this EEG technique, that the magnitude information is spontaneously and unintentionally abstracted, irrespectively of the numerical format. Significant amplitudes were also observed for digits-number words and number words-dots, but not for digits-dots, suggesting an automatic integration across some numerical formats. These results imply that direct and indirect neuro-cognitive links exist across the different numerical formats.

Symbolic Number Ordering and its Underlying Strategies Examined Through Self-Reports.

Symbolic number ordering has been related to arithmetic fluency; however, the nature of this relation remains unclear. Here we investigate whether the implementation of strategies can explain the relation between number ordering and arithmetic fluency. In the first study, participants (N = 16) performed a symbolic number ordering task (i.e., "is a triplet of digits presented in order or not?") and verbally reported the strategy they used after each trial. The analysis of the verbal responses led to the identification of three main strategies: memory retrieval, triplet decomposition, and arithmetic operation. All the remaining strategies were grouped in the fourth category "other". In the second study, participants were presented with a description of the four strategies. Afterwards, they (N = 61) judged the order of triplets of digits as fast and as accurately as possible and, after each trial, they indicated the implemented strategy by selecting one of the four pre-determined strategies. Participants also completed a standardized test to assess their arithmetic fluency. Memory retrieval strategy was used more often for ordered trials than for non-ordered trials and more for consecutive than non-consecutive triplets. Reaction times on trials solved by memory retrieval were related to the participants' arithmetic fluency score. For the first time, we provide evidence that the relation between symbolic number ordering and arithmetic fluency is related to faster execution of memory retrieval strategies.

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

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

Numerals do not need numerosities: robust evidence for distinct numerical representations for symbolic and non-symbolic numbers.

In numerical cognition research, it has traditionally been argued that the processing of symbolic numerals (e.g., digits) is identical to the processing of the non-symbolic numerosities (e.g., dot arrays), because both number formats are represented in one common magnitude system-the Approximate Number System (ANS). In this study, we abandon this deeply rooted assumption and investigate whether the processing of numerals and numerosities can be dissociated, using an audio-visual paradigm in combination with various experimental manipulations. In Experiment 1, participants performed four comparison tasks with large symbolic and non-symbolic numbers: (1) number word-digit (2) tones-dots, (3) number word-dots, (4) tones-digit. In Experiment 2, we manipulated the number range (small vs. large) and the presentation modality (visual-auditory vs. auditory-visual). Results demonstrated ratio effects (i.e., the signature of ANS being addressed) in all tasks containing numerosities, but not in the task containing numerals only. Additionally, a cognitive cost was observed when participants had to integrate symbolic and non-symbolic numbers. Therefore, these results provide robust (i.e., independent of presentation modality or number range) evidence for distinct processing of numerals and numerosities, and argue for the existence of two independent number processing systems.

Probing the Relationship Between Home Numeracy and Children's Mathematical Skills: A Systematic Review.

The concept of home numeracy has been defined as parent-child interactions with numerical content. This concept started to receive increasing attention since the last decade. Most of the studies indicated that the more parents and their children engage in numerical experiences, the better children perform in mathematical tasks. However, there are also contrasting results indicating that home numeracy does not play a role or that there is a negative association between the parent-child interactions and children's mathematics performance. To shed light on these discrepancies, a systematic review searching for available articles examining the relationship between home numeracy and mathematical skills was conducted. Thirty-seven articles were retained and a p-curve analysis showed a true positive association between home numeracy and children's mathematical skills. A more qualitative investigation of the articles revealed five common findings: (1) Advanced home numeracy interactions but not basic ones are associated with children's mathematical skills. (2) Most participants in the studies were mothers, however, when both parents participated and were compared, only mothers' reports of formal home numeracy activities (i.e., explicit numeracy teaching) were linked to children's mathematical skill. (3) Formal home numeracy activities have been investigated more commonly than informal home numeracy activities (i.e., implicit numeracy teaching). (4) The number of studies that have used questionnaires to assess home numeracy is larger compared with the ones that have used observations. (5) The majority of the studies measured children's mathematical skills with comprehensive tests that index mathematical ability with one composite score rather than with specific numerical tasks. These five common findings might explain the contradictory results regarding the relationship between home numeracy and mathematical skills. Therefore, more research is necessary to draw quantitative conclusions about these five points.

Take it of your shoulders: Providing scaffolds leads to better performance on mathematical word problems in secondary school children with developmental coordination disorder.

BACKGROUND: Many children with developmental coordination disorder (DCD) have mathematical problems which are more pronounced for mathematical skills that also require executive functions. Although empirical evidence is missing, math and special education need teachers of children with DCD report difficulties with mathematical word problem solving that can be remediated by providing the children with scaffolds cueing the intermediate steps. AIMS: This study aims to find empirical evidence for the effectivity of such additional support. In addition, we want to investigate whether the difficulties are due to inefficient arithmetic or executive functioning skills. METHODS AND PROCEDURES: A DCD and a control group solved word problems with and without scaffolds and conducted a series of tasks measuring calculation and executive skills. OUTCOMES AND RESULTS: Performance improves when scaffolds are presented to children with DCD. Children with DCD and control children differ on executive functioning tasks but perform similarly on arithmetic tests. CONCLUSIONS AND IMPLICATIONS: Providing scaffolds for word problem solving is effective in children with DCD. Scaffolds possibly reduce the required cognitive load, making the problem solvable for DCD children that have reduced executive functioning skills.

Judging the order of numbers relies on familiarity rather than activating the mental number line.

A series of effects characterises the processing of symbolic numbers (i.e., distance effect, size effect, SNARC effect, size congruency effect). The combination of these effects supports the view that numbers are represented on a compressed and spatially oriented mental number line (MNL) as well as the presence of an interaction between numerical and other magnitude representations. However, when individuals process the order of digits, response times are faster when the distance between digits is small (e.g., 1-2-3) compared to large (e.g., 1-3-5; i.e., reversed distance effect), suggesting that the processing of magnitude and order may be distinct. Here, we investigated whether the effects related to the MNL also emerge in the processing of symbolic number ordering. In Experiment 1, participants judged whether three digits were presented in order while spatial distance, numerical distance, numerical size, and the side of presentation were manipulated. Participants were faster in determining the ascending order of small triplets compared to large ones (i.e., size effect) and faster when the numerical distance between digits was small (i.e., reversed distance effect). In Experiment 2, we explored the size effect across all possible consecutive triplets between 1 and 9 and the effect that physical size has on order processing. Participants showed faster reactions times only for the triplet 1-2-3 compared to the other triplets, and the effect of physical magnitude was negligible. Symbolic order processing lacks the signatures of the MNL and suggests the presence of a familiarity effect related to well-known consecutive triplets in the long-term memory.

Symbolic estrangement or symbolic integration of numerals with quantities: Methodological pitfalls and a possible solution.

Previous studies, which examined whether symbolic and non-symbolic quantity representations are processed by two independent systems or by one common system, reached contradicting findings, possibly due to methodological differences. Indeed, some researchers advocate the two systems approach, based on the presence of notation-specific switch cost in conditions where adults have to compare pairs of symbolic and non-symbolic quantities, in combination with the absence of such a cost in conditions containing quantities of the same notation. However, other researchers used matching instructions, and reported a facilitation in the mixed notation conditions, suggesting that the two systems are automatically integrated. In the current study, we conducted three experiments, in which we examined the existence of two separate quantity systems, but we used various experimental manipulations (e.g., task instructions, presentation order) to unravel the previous inconsistent findings. In Experiment 1, we investigated the role of task instructions by presenting participants with pure and mixed notation trials with both comparison and matching tasks. In Experiment 2, we tested the role of blocked and randomized presentation order for the pure and mixed trials. Our data showed that cost for switching between the symbolic and non-symbolic quantities is present, but is prone to a certain methodological drawback: when the differences between the processing times for two sequentially presented stimuli of different notations are not taken into account, this masks the cost for switching between the two systems. To overcome this problem, in Experiment 3 we used an audio-visual paradigm. Overall, our results provide further evidence for the existence of distinct quantity representations, independently of task instructions or presentation order. Additionally, considering this methodological pitfall we argue that the audio-visual paradigm is better suited when investigating the integration between symbolic and non- symbolic quantities.

Frequency of Home Numeracy Activities Is Differentially Related to Basic Number Processing and Calculation Skills in Kindergartners.

Home numeracy has been shown to play an important role in children's mathematical performance. However, findings are inconsistent as to which home numeracy activities are related to which mathematical skills. The present study disentangled between various mathematical abilities that were previously masked by the use of composite scores of mathematical achievement. Our aim was to shed light on the specific associations between home numeracy and various mathematical abilities. The relationships between kindergartners' home numeracy activities, their basic number processing and calculation skills were investigated. Participants were 128 kindergartners (Mage = 5.43 years, SD = 0.29, range: 4.88-6.02 years) and their parents. The children completed non-symbolic and symbolic comparison tasks, non-symbolic and symbolic number line estimation tasks, mapping tasks (enumeration and connecting), and two calculation tasks. Their parents completed a home numeracy questionnaire. Results indicated small but significant associations between formal home numeracy activities that involved more explicit teaching efforts (i.e., identifying numerals, counting) and children's enumeration skills. There was no correlation between formal home numeracy activities and non-symbolic number processing. Informal home numeracy activities that involved more implicit teaching attempts, such as "playing games" and "using numbers in daily life," were (weakly) correlated with calculation and symbolic number line estimation, respectively. The present findings suggest that disentangling between various basic number processing and calculation skills in children might unravel specific relations with both formal and informal home numeracy activities. This might explain earlier reported contradictory findings on the association between home numeracy and mathematical abilities.