Early place-value understanding as a precursor for later arithmetic performance--a longitudinal study on numerical development

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.

Transcoding of French numbers for first- and second-language learners in third grade

Transcoding is the process of translating between spoken and written numbers, and it is correlated with other mathematical skills. In the present study, we investigated the link between French number writing of 49 students in the third grade (aged 7-9 years) and their language skills. Transcoding in French is of particular interest because the spoken number language system does not completely correspond to that of the written digits (e.g., quatre-vingt-dix [four-twenty-ten] and 90). We hypothesised that the complex linguistic structure of spoken numbers in French would be challenging for students who are learning to transcode. First and second French-language learners' accuracy and errors were recorded during a writing task of 3- to 7-digit numbers. Children also completed linguistic tests (e.g., receptive vocabulary, receptive syntax). Results showed that first- and second-language learners did not differ in their transcoding accuracy. Number size, decade complexity of stimulus number words in French (i.e., numbers containing a complex decade, operationalized as a number between soixante-dix, 70, and quatre-vingt-dix-neuf, 99), and receptive vocabulary predicted children's French transcoding skills. Students were more likely to produce errors (e.g., 68 or 6018 for 78) when they transcoded complex decade numbers compared with simple decade numbers. When an error was made on the complex decade portion of a number, it was likely a lexical error. In conclusion, third graders, both first- and second-language learners, found complex decade numbers challenging and their performance was related to their general vocabulary skills.

Top-Down Number Reading: Language Affects the Visual Identification of Digit Strings

Reading numbers aloud involves visual processes that analyze the digit string and verbal processes that produce the number words. Cognitive models of number reading assume that information flows from the visual input to the verbal production processes-a feed-forward processing mode in which the verbal production depends on the visual input but not vice versa. Here, I show that information flows also in the opposite direction, from verbal production to the visual input processes. Participants read aloud briefly presented multi-digit strings in Hebrew, in which the order of words is congruent with the order of digits (21 = twenty-and-one), and in Arabic, in which the ones word precedes the tens word (one-and-twenty). The error-by-digit-position curve was affected by language: relative to Hebrew, in Arabic the error rate was slightly lower for the unit digit and slightly higher for the decade digit, indicating that in Arabic the unit digit was processed earlier and the decade digit later, in accord with the Arabic word order. This language-dependent processing order originated in the visual level and was not a verbal confound, because it persisted even when I controlled for the serial position of the decade/unit word in the verbal number by using numbers with 0 (two hundred three/two hundred thirty). I conclude that the visual analyzer's digit scanning order, decade-first or unit-first, is not fixed but affected by the language in which the number is produced-a top-down, verbal-to-visual information flow.

Does Conceptual Transparency in Manipulatives Afford Place-Value Understanding in Children at Risk for Mathematics Learning Disabilities?

We investigated the effect of conceptual transparency in the physical structure of manipulatives on place-value understanding in typically developing children and those at risk for mathematics learning disabilities. Second graders were randomly assigned to one of three manipulatives conditions: (a) attachable beads that did not make the denominations or ones in the denominations transparent, (b) pipe cleaners that made only the denominations transparent, and (c) string beads that made both the denominations and the ones in the denominations transparent. Participants used the manipulatives to represent double- and triple-digit numerals. Statistical analyses indicated that the transparency of the denominations, but not the transparency of the ones in the denominations, is responsible for children's number representation and place-value understanding. Descriptive analyses of their responses revealed that the at-risk children were at a greater disadvantage than their typically developing peers with the attachable beads, failing to use place-value concepts to interpret their representations.

The structure of the notation system in adults' number line estimation: An eye-tracking study

Research on rational numbers suggests that adults experience more difficulties in understanding the numerical magnitude of rational than natural numbers. Within rational numbers, the numerical magnitude of fractions has been found to be more difficult to understand than that of decimals. Using a number line estimation (NLE) task, the current study investigated two sources of difficulty in adults' numerical magnitude understanding: number type (natural vs rational) and structure of the notation system (place-value-based vs non-place-value-based). This within-subjects design led to four conditions: natural numbers (natural/place-value-based), decimals (rational/place-value-based), fractions (rational/non-place-value-based), and separated fractions (natural/non-place-value-based). In addition to percentage absolute error (PAE) and response times, we collected eye-tracking data. Results showed that participants estimated natural and place-value-based notations more accurately than rational and non-place-value-based notations, respectively. Participants were also slower to respond to fractions compared with the three other notations. Consistent with the response time data, eye-tracking data showed that participants spent more time encoding fractions and re-visited them more often than the other notations. Moreover, in general, participants spent more time positioning non-place-value-based than place-value-based notations on the number line. Overall, the present study contends that when both sources of difficulty are present in a notation (i.e., both rational and non-place-value-based), adults understand its numerical magnitude less well than when there is only one source of difficulty present (i.e., either rational or non-place-value-based). When no sources of difficulty are present in a notation (i.e., both natural and place-value-based), adults have the strongest understanding of its numerical magnitude.

The plural counts: Inconsistent grammatical number hinders numerical development in preschoolers - A cross-linguistic study

The role of grammar in numerical development, and particularly the role of grammatical number inflection, has already been well-documented in toddlerhood. It is unclear, however, whether the influence of grammatical language structure further extends to more complex later stages of numerical development. Here, we addressed this question by exploiting differences between Polish, which has a complex grammatical number paradigm, leading to a partially inconsistent mapping between numerical quantities and grammatical number, and German, which has a comparatively easy verbal paradigm: 151 Polish-speaking and 123 German-speaking kindergarten children were tested using a symbolic numerical comparison task. Additionally, counting skills (Give-a-Number and count-list), and mapping between non-symbolic (dot sets) and symbolic representations of numbers, as well as working memory (Corsi blocks and Digit span) were assessed. Based on the Give-a-Number and mapping tasks, the children were divided into subset-knowers, CP-knowers-non-mappers, and CP-knowers-mappers. Linguistic background was related to performance in several ways: Polish-speaking children expectedly progressed to the CP-knowers stage later than German children, despite comparable non-numerical capabilities, and even after this stage was achieved, they fared worse in the numerical comparison task. There were also meaningful differences in spatial-numerical mapping between the Polish and German groups. Our findings are in line with the theory that grammatical number paradigms influence. the development of representations and processing of numbers, not only at the stage of acquiring the meaning of the first number-words but at later stages as well, when dealing with symbolic numbers.

The relevance of basic numerical skills for fraction processing: Evidence from cross-sectional data

Recent research indicated that fraction understanding is an important predictor of later mathematical achievement. In the current study we investigated associations between basic numerical skills and students' fraction processing. We analyzed data of 939 German secondary school students (age range = 11.92 to 18.00 years) and evaluated the determinants of fraction processing considering basic numerical skills as predictors (i.e., number line estimation, basic arithmetic operations, non-symbolic magnitude comparison, etc.). Additionally, we controlled for general cognitive ability, grade level, and sex. We found that multiplication, subtraction, conceptual knowledge, number line estimation, and basic geometry were significantly associated with fraction processing beyond significant associations of general cognitive ability and sex. Moreover, relative weight analysis revealed that addition and approximate arithmetic should also be considered as relevant predictors for fraction processing. The current results provide food for thought that further research should focus on investigating whether recapitulating basic numerical content in secondary school mathematics education can be beneficial for acquiring more complex mathematical concepts such as fractions.

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.

Visual Occipito-Temporal N1 Sensitivity to Digits Across Elementary School

Number processing abilities are important for academic and personal development. The course of initial specialization of ventral occipito-temporal cortex (vOTC) sensitivity to visual number processing is crucial for the acquisition of numeric and arithmetic skills. We examined the visual N1, the electrophysiological correlate of vOTC activation across five time points in kindergarten (T1, mean age 6.60 years), middle and end of first grade (T2, 7.38 years; T3, 7.68 years), second grade (T4, 8.28 years), and fifth grade (T5, 11.40 years). A combination of cross-sectional and longitudinal EEG data of a total of 62 children (35 female) at varying familial risk for dyslexia were available to form groups of 23, 22, 27, 27, and 42 participants for each of the five time points. The children performed a target detection task which included visual presentation of single digits (DIG), false fonts (FF), and letters (LET) to derive measures for coarse (DIG vs. FF) and fine (DIG vs. LET) digit sensitive processing across development. The N1 amplitude analyses indicated coarse and fine sensitivity characterized by a stronger N1 to digits than false fonts across all five time points, and stronger N1 to digits than letters at all but the second (T2) time point. In addition, lower arithmetic skills were associated with stronger coarse N1 digit sensitivity over the left hemisphere in second grade (T4), possibly reflecting allocation of more attentional resources or stronger reliance on the verbal system in children with poorer arithmetic skills. To summarize, our results show persistent visual N1 sensitivity to digits that is already present early on in pre-school and remains stable until fifth grade. This pattern of digit sensitivity development clearly differs from the relatively sharp rise and fall of the visual N1 sensitivity to words or letters between kindergarten and middle of elementary school and suggests unique developmental trajectories for visual processing of written characters that are relevant to numeracy and literacy.

A Network Analysis of Children's Emerging Place-Value Concepts

Examining how informal knowledge systems change after formal instruction is imperative to understanding learning processes and conceptual development and to implementing effective educational practices. We used network analyses to determine how the organization of informal knowledge about multidigit numbers in kindergartners (N = 279; mean age = 5.76 years, SD = 0.55; 135 females) supports and is transformed by a year of in-school formal instruction. The results show that in kindergarten, piecemeal knowledge about the surface properties of reading and writing multidigit numbers and the use of base-10 units to determine large quantities are strongly associated with each other and connected in a stringlike manner to other emerging skills. After a year of instruction, each skill becomes connected to the "hub" abilities of reading and writing multidigit numbers, which also become strongly connected to more advanced knowledge of base-10 principles. These findings provide new insights into how partial knowledge provides the backbone on which explicit principles are learned.

Transcoding counts: Longitudinal contribution of number writing to arithmetic in different languages

Number writing involves transcoding from number words (e.g., "thirty-two") to written digit strings (32) and is an important unique predictor of arithmetic. The existing longitudinal evidence about the relation between transcoding and arithmetic is mostly language specific. In languages with number word inversion (e.g., German), the order of tens and units is transposed in spoken number words compared with Arabic numbers. This makes transcoding more challenging than in languages without number word inversion (e.g., English). In the current study, we aimed to understand whether the contribution of number writing to the development of arithmetic is similar in languages with and without number word inversion. German-speaking children (n = 166) and English-speaking children (n = 201) were followed over the first 3 years of primary school. In a series of multiple linear regressions, we tested whether number writing of multi-digit numbers was a significant unique predictor of arithmetic after controlling for well-known non-numerical predictors (nonverbal reasoning and working memory) and numerical predictors (symbolic and nonsymbolic magnitude comparison). Number writing in Grade 1 predicted arithmetic in Grades 1, 2, and 3 over and above the other predictors. Crucially, number writing performance was of comparable importance for arithmetic development in German- and English-speaking children. Our findings extend previous evidence by showing that transcoding predicts the development of arithmetic skills during the first 3 years of primary school in languages with and without number word inversion.

The development of place value concepts: Approximation before principles

Place value concepts were measured longitudinally from kindergarten (2017) to first grade (2018) in a diverse sample (n = 279; Mage  = 5.76 years, SD = 0.55; 135 females; 41% Black, 38% White, 8% Asian, 12% Latino). Children completed three syntactic tasks that required an explicit understanding of base-10 symbols and three approximate tasks that could be completed without this explicit understanding. Approximate performance was significantly better in both age groups. A factor analysis confirmed that syntactic and approximate tasks tapped separate latent variables in kindergarten, but not in first grade. Path analyses indicated that only kindergarten approximate performance predicted overall first-grade place value understanding. These findings suggest that explicit understanding of base-10 principles develops from implicit, partial knowledge of multidigit numbers.

The Tools of Enculturation

We propose an account of cognitive tools that takes into account the process of enculturation by which tools are integrated into our cognitive systems. Drawing on work in cultural evolution and developmental psychology, we argue that cognitive tools are complex entities consisting of physical objects, representational systems, and cognitive practices for the physical manipulation of the tool. We use an extensive case study of spatial navigation to demonstrate the core claims. The account we provide is contrasted with conceptions of cognitive tools that simplify cognition, in particular that they offload cognitive work, or that the tools themselves are temporary developmental scaffolds or props. Enculturation results in transformed cognitive systems, and we can now think and act in new ways with cognitive tools.

This article has two aims: First to present a case for the integration of cognitive tools into our cognitive systems as a process of Enculturation. Second, to present an argument against the simplifying account of cognitive tools as ‘offloading’ cognitive complexity, or ‘outsourcing’ cognitive processing to the tools themselves. We use an extensive case study of spatial navigation to demonstrate the core claims.

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.

Transcoding Errors of Two-Digit Numbers From Arabic Digits Into Verbal Numbers and From Verbal Numbers Into Arabic Digits by Arab First Graders

The study focuses on the effect of the lexical-syntactic structure on the patterns of errors by Arab first graders in tasks involving reading two-digit number and writing two-digit numbers to dictation. Children made few change or omission errors, indicating that they had little problem with the lexical aspects of the counting system. However, they made frequent substitution errors (e.g., 23 for 32), especially in the number reading task, and especially for numbers that depended strongly on the numerical syntactic structure. Such errors were less common for decade numbers and for the 11–19 number range than for other two-digit numbers. The results suggest particular difficulty with the syntactic rather than lexical aspects of the counting system. The syntactic aspects may be particularly difficult for Arabic-speaking children, due to the inversion feature of the Arabic counting system.

Early bilingual immersion school program and cognitive development in French-speaking children: Effect of the second language learned (English vs. Dutch) and exposition duration (2 vs. 5 years)

The results of studies targeting cognitive and academic advantages in children frequenting early bilingual immersion school programs (CLIL) have been contradictory. While the impact of the amount of CLIL experience has already been studied, the role of the second language learned has been little studied to account for differences among study findings. The link between executive skills (EF) and scholar abilities (e.g., mathematics) in the CLIL context has also been little investigated. The purpose of the present study was to determine if the impact of CLIL on EF and academic performances varies depending on the immersion language and the duration of CLIL experience. The sample included a total of 230 French-speaking children attending second (141) and fifth (89) grade classes. Within each grade, there were three matched language groups composed of children respectively immersed in English, immersed in Dutch, and non-immersed controls. The children were administered tasks assessing executive functions [alerting, cognitive flexibility, and working memory], as well as arithmetic abilities. In second grade, we detected no difference in EF between the language groups. On the other hand, in fifth grade, the two immersed groups outperformed the non-immersed group on the cognitive flexibility task but did not differ between them. Moreover, only the Dutch immersed group outperformed the control group on the working memory task. Arithmetic performances also differed depending on the language learned; in second grade, Dutch learners performed better than the monolingual group. In fifth grade, Dutch learners outperformed the two other groups. These results suggest that the impact of CLIL on executive skills and arithmetic performances might be modulated by the amount of CLIL experience and the second language learned in immersion.

Developmental fronto-parietal shift of brain activation during mental arithmetic across the lifespan: A registered report protocol

Arithmetic processing is represented in a fronto-parietal network of the brain. However, activation within this network undergoes a shift from domain-general cognitive processing in the frontal cortex towards domain-specific magnitude processing in the parietal cortex. This is at least what is known about development from findings in children and young adults. In this registered report, we set out to replicate the fronto-parietal activation shift for arithmetic processing and explore for the first time how neural development of arithmetic continues during aging. This study focuses on the behavioral and neural correlates of arithmetic and arithmetic complexity across the lifespan, i.e., childhood, where arithmetic is first learned, young adulthood, when arithmetic skills are already established, and old age, when there is lifelong arithmetic experience. Therefore, brain activation during mental arithmetic will be measured in children, young adults, and the elderly using functional near-infrared spectroscopy (fNIRS). Arithmetic complexity will be manipulated by the carry and borrow operations in two-digit addition and subtraction. The findings of this study will inform educational practice, since the carry and borrow operations are considered as obstacles in math achievement, and serve as a basis for developing interventions in the elderly, since arithmetic skills are important for an independent daily life.

Syntactic priming reveals an explicit syntactic representation of multi-digit verbal numbers

When we say or understand verbal numbers, a major challenge to the cognitive system is the need to process the number's syntactic structure. Several studies showed that number syntax is handled by dedicated processes, however, it is still unclear how precisely these processes operate, whether the number's syntactic structure is represented explicitly, and if it is - what this representation looks like. Here, we used a novel experimental paradigm, syntactic priming of numbers, which can examine in detail the syntactic representation of multi-digit verbal numbers. In each trial, the participants - Arabic-Hebrew bilinguals and Hebrew monolinguals - heard a multi-digit number and responded orally with a random number. The syntactic structure of their responses was similar to that of the targets, showing that they represented the verbal number's syntax. This priming effect was genuinely syntactic, and could not be explained as lexical - repeating words from the target; as phonological - responding with words phonologically-similar to the target; or as a numerical distance effect - producing responses numerically close to the target. The syntactic priming effect was stronger for earlier words in the verbal number and weaker for later words, suggesting that the syntactic representation is capped by working-memory limits. We propose that syntactic priming could become a useful method to examine various aspects of the syntactic representation of numbers.

Verbal count sequence knowledge underpins numeral order processing in children

Recent research has suggested that numeral order processing - the speed and accuracy with which individuals can determine whether a set of digits is in numerical order or not - is related to arithmetic and mathematics outcomes. It has therefore been proposed that ordinal relations are a fundamental property of symbolic numeral representations. However, order information is also inherent in the verbal count sequence, and thus verbal count sequence knowledge may instead explain the relationship between performance on numeral order tasks and arithmetic. We explored this question with 62 children aged 6- to 8-years-old. We found that performance on a verbal count sequence knowledge task explained the relationship between numeral order processing and arithmetic. Moreover many children appeared to explicitly base their judgments of numerical order on count sequence information. This suggests that insufficient attention may have been paid to verbal number knowledge in understanding the sources of information that give meaning to numbers.

Examining the relevance of basic numerical skills for mathematical achievement in secondary school using a within-task assessment approach

Previous research repeatedly found basic numerical abilities (e.g., magnitude understanding, arithmetic fact knowledge, etc.) to predict young students' current and later arithmetic achievement as assessed by achievement tests - even when controlling for the influence of domain-general abilities (e.g., intelligence, working memory). However, to the best of our knowledge, previous studies hardly addressed this issue in secondary school students. Additionally, they primarily assessed basic numerical abilities in a between-task approach (i.e., using different tasks for different abilities). Finally, their relevance for real-life academic outcomes such as mathematics grades has only rarely been investigated. The present study therefore pursued an approach using one and the same task (i.e., a within-task approach) to reduce confounding effects driven by between-task differences. In particular, we evaluated the relevance of i) number magnitude understanding, ii) arithmetic fact knowledge, and iii) conceptual and procedural knowledge for the mathematics grades of 81 students aged between ten and thirteen (i.e., in Grades 5 and 6) employing the number bisection task. Results indicated that number magnitude understanding, arithmetic fact knowledge, and conceptual and procedural knowledge contributed to explaining mathematics grades even when controlling for domain-general cognitive abilities. Methodological and practical implications of the results are discussed.

Measuring spontaneous and automatic processing of magnitude and parity information of Arabic digits by frequency-tagging EEG

Arabic digits (1–9) are everywhere in our daily lives. These symbols convey various semantic information, and numerate adults can easily extract from them several numerical features such as magnitude and parity. Nonetheless, since most studies used active processing tasks to assess these properties, it remains unclear whether and to what degree the access to magnitude and especially to parity is automatic. Here we investigated with EEG whether spontaneous processing of magnitude or parity can be recorded in a frequency-tagging approach, in which participants are passively stimulated by fast visual sequences of Arabic digits. We assessed automatic magnitude processing by presenting a stream of frequent small digit numbers mixed with deviant large digits (and the reverse) with a sinusoidal contrast modulation at the frequency of 10 Hz. We used the same paradigm to investigate numerical parity processing, contrasting odd digits to even digits. We found significant brain responses at the frequency of the fluctuating change and its harmonics, recorded on electrodes encompassing right occipitoparietal regions, in both conditions. Our findings indicate that both magnitude and parity are spontaneously and unintentionally extracted from Arabic digits, which supports that they are salient semantic features deeply associated to digit symbols in long-term memory.

Neurofunctional plasticity in fraction learning: An fMRI training study

BACKGROUND

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

METHOD

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

RESULTS

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

CONCLUSION

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

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.

Twenty-four or four-and-twenty: Language modulates cross-modal matching for multidigit numbers in children and adults

Does number-word structure have a long-lasting impact on transcoding? Contrary to English, German number words comprise decade-unit inversion (e.g., vierundzwanzig is literally translated as four-and-twenty). To investigate the mental representation of numbers, we tested the effect of visual and linguistic-morphological characteristics on the development of verbal-visual transcoding. In a longitudinal cross-linguistic design, response times (RTs) in a number-matching experiment were analyzed in Grade 2 (119 German-speaking and 179 English-speaking children) and in Grade 3 (131 German-speaking and 160 English-speaking children). To test for long-term effects, the same experiment was given to 38 German-speaking and 42 English-speaking adults. Participants needed to decide whether a spoken number matched a subsequent visual Arabic number. Systematic variation of digits in the nonmatching distractors allowed comparison of three different transcoding accounts (lexicalization, visual, and linguistic-morphological). German speakers were generally slower in rejecting inverted number distractors than English speakers. Across age groups, German speakers were more distracted by Arabic numbers that included the correct unit digit, whereas English speakers showed stronger distraction when the correct decade digit was included. These RT patterns reflect differences in number-word morphology. The individual cost of rejecting an inverted distractor (inversion effect) predicted arithmetic skills in German-speaking second-graders only. The moderate relationship between the efficiency to identify a matching number and arithmetic performance could be observed cross-linguistically in all age groups but was not significant in German-speaking adults. Thus, findings provide consistent evidence of a persistent impact of number-word structure on number processing, whereas the relationship with arithmetic performance was particularly pronounced in young children.

A gifted SNARC? Directional spatial-numerical associations in gifted children with high-level math skills do not differ from controls

The SNARC (Spatial-Numerical Association of Response Codes) effect (i.e., a tendency to associate small/large magnitude numbers with the left/right hand side) is prevalent across the whole lifespan. Because the ability to relate numbers to space has been viewed as a cornerstone in the development of mathematical skills, the relationship between the SNARC effect and math skills has been frequently examined. The results remain largely inconsistent. Studies testing groups of people with very low or very high skill levels in math sometimes found relationships between SNARC and math skills. So far, however, studies testing such extreme math skills level groups were mostly investigating the SNARC effect in individuals revealing math difficulties. Groups with above average math skills remain understudied, especially in regard to children. Here, we investigate the SNARC effect in gifted children, as compared to normally developing children (overall n = 165). Frequentist and Bayesian analysis suggested that the groups did not differ from each other in the SNARC effect. These results are the first to provide evidence for the SNARC effect in a relatively large sample of gifted (and mathematically highly skilled) children. In sum, our study provides another piece of evidence for no direct link between the SNARC effect and mathematical ability in childhood.

Time Reading in Middle and Secondary School Students: The Influence of Basic-Numerical Abilities

Time reading skills are central for the management of personal and professional life. However, little is known about the differential influence of basic numerical abilities on analog and digital time reading in general and in middle and secondary school students in particular. The present study investigated the influence of basic numerical skills separately for analog and digital time reading in N = 709 students from 5th to 8th grade. The present findings suggest that the development of time reading skills is not completed by the end of primary school. Results indicated that aspects of magnitude manipulation and arithmetic fact knowledge predicted analog time reading significantly over and above the influence of age. Furthermore, results showed that spatial representations of number magnitude, magnitude manipulation, arithmetic fact knowledge, and conceptual knowledge were significant predictors of digital time reading beyond general cognitive ability and sex. To the best of our knowledge, the present study is the first to show differential effects of basic numerical abilities on analog and digital time reading skills in middle and secondary school students. As time readings skills are crucial for everyday life, these results are highly relevant to better understand basic numerical processes underlying time reading.

Mathematical Difficulties vs. High Achievement: An Analysis of Arithmetical Cognition in Elementary School

This study analyzed the contribution of cognitive processes (planning, attention, simultaneous and successive processing) and domain-specific skills (counting, number processing and conceptual comprehension) to the arithmetic performance achieved in the last three grades (4th, 5th, and 6th) of elementary school. Three groups of students with a different arithmetic achievement level were characterized. The predictive value of the cognitive processes and the math specific skills are explored through diverse covariance and discriminant analyses. Participants were 110 students (M = 10.5 years, SD = 1.17) classified in three groups: mathematical difficulties (MD; n = 26), high achieving (HA; n = 26), and typical achieving (TA; n = 58). Cognitive processes and domain-specific skills were evaluated in two individual sessions at the end of the school year. Nonverbal intelligence was assessed in a final collective session with each class. The mathematical difficulties group's achievement was deficient in simultaneous and successive processing, number processing, and conceptual comprehension compared to the typical achievement group. High achievement children obtained significantly better results than the typical achievement children in simultaneous processing, counting, number processing, and conceptual comprehension. Number processing and conceptual comprehension were the most consistent classifiers, although successive and simultaneous processing, respectively, also contributed to identifying students with mathematical difficulties and high achievement. These findings have practical implications for preventive and intervention proposals linked to the observed profiles.

Arabic number writing in children with developmental dyslexia

Abstract Number transcoding is a basic numerical processing task that demands verbal skills during its execution. The goal of this study was to investigate number transcoding ability in children with developmental dyslexia. Twenty-three children with typical development and twenty-six children with developmental dyslexia participated in this study. Results showed that children with dyslexia show a deficit in phonological processing as well as in number transcoding. Repeated-measures analysis of covariance indicated that the dyslexia group presented performance below the average in the number transcoding. Regression analyses indicated that short-term verbal memory, phoneme deletion, rhyme judgment task and automatized naming was a strong predictor of number transcoding difficulties. Children with dyslexia present number transcoding deficits regardless of age and educational level.

English and Chinese Children's Performance on Numerical Tasks

East Asian pupils have consistently outperformed Western pupils in international comparisons of mathematical performance at both primary and secondary school level. It has sometimes been suggested that a contributory factor is the transparent counting systems of East Asian languages, which may facilitate number representation. The present study compared 35 7-year-old second-year primary school children in Oxford, England and 40 children of similar age in Hong Kong, China on a standardized arithmetic test; on a two-digit number comparison test, including easy, misleading and reversible comparisons; and on a number line task, involving placing numbers in the appropriate position on four number lines: 1-10, 1-20, 1-100, and 1-1000. The Chinese children performed significantly better than the English children on the standardized arithmetic test. They were faster but not significantly more accurate on the Number Comparison and Number Line tasks. There were no interactions between language group and comparison type on the number comparison task, though the performance of both groups was faster on easy pairs than those where there was conflict between the relative magnitudes of the tens and the units. Similarly, there were no interactions between group and number line range, though the performance of both groups was influenced by the range of the number line. The study supports the view that counting systems affect aspects of numerical abilities, but cannot be the full explanation for international differences in mathematics performance.

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.

Magnitude processing of written number words is influenced by task, rather than notation

The extent to which task and notation influence the processing of numerical magnitude is under theoretical and empirical debate. To date, behavioural studies have yielded a mixed body of evidence. Using the case of written number words in English and Chinese, we re-examined this issue. Thirty-nine bilingual participants who showed a balanced profile of language dominance in English and Chinese completed three tasks of numerical processing (Magnitude Comparison, Numerical Matching, and Language Matching) with pure English, pure Chinese, and mixed notation number words. We conducted frequentist and Bayesian statistics on the data. Magnitude processing, as indexed by the numerical distance effect (NDE), was found to be dependent on task. Specifically, the NDE occurred in all notation conditions in the Magnitude Comparison Task and mixed notation trials in the Numerical Matching Task only. However, the data indicated that magnitude processing was independent of notation. Task and notation had an interactive influence on overall speed of processing, where participants responded to Chinese number words significantly faster than other notations for the Magnitude Comparison and Numerical Matching Tasks only. Finally, Bayesian analyses indicated that task and notation do not interact to affect magnitude processing. Specifically, the Bayes Factor and posterior model probabilities of the Bayesian ANOVA yielded strongest support for the model with three main effects (Task, Notation, Numerical Distance) and two two-way interactions (Task × Numerical Distance, Task × Notation). These findings highlight the critical role of task in numerical magnitude processing, provide support for a notation-independent account of magnitude processing, and suggest that linguistic/orthographic factors, combined with task, may interact to affect overall speed of processing.

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.

Low to No Effect: Application of tRNS During Two-Digit Addition

Transcranial electric stimulation such as transcranial random noise stimulation (tRNS) and transcranial direct current stimulation (tDCS) have been used to investigate structure-function relationships in numerical cognition. Recently, tRNS was suggested to be more effective than tDCS. However, so far there is no evidence on the differential impact of tDCS and tRNS on numerical cognition using the same experimental paradigm. In the present study, we used a two-digit addition paradigm for which significant-albeit small-effects of tDCS were observed previously to evaluate the impact of parietal and frontal tRNS on specific numerical effects. While previous studies reported a modulation of numerical effects of this task through tDCS applied to parietal areas, we did not observe any effect of parietal tRNS on performance in two-digit addition. These findings suggest that tRNS seemed to influence concurrent mental arithmetic less than tDCS at least when applied over the IPS. These generally small to absent effects of tES on actual arithmetic performance in the current addition paradigm are in line with the results of a recent meta-analysis indicating that influences of tES may be more pronounced in training paradigms.

The neural correlates of arithmetic difficulty depend on mathematical ability: evidence from combined fNIRS and ERP

Mathematical abilities are essential for an individual, as they predict career prospects among many other abilities. However, little is known about whether neural correlates of arithmetic problem difficulty differ between individuals with high and low math ability. For instance, the difficulty of two-digit addition and subtraction increases whenever a carry or borrow operation is required. Therefore, we systematically investigated the spatial and temporal neural correlates of the carry and borrow effects for high and low performers in a written production paradigm using combined functional near-infrared spectroscopy (fNIRS) and event-related potential (ERP) measurements. Effects of arithmetic difficulty interacted with an individual's math ability. High performers showed increased frontal activation especially in the left inferior frontal gyrus associated with the carry and borrow effects, whereas low performers did not. Furthermore, high and low performers even differed in their early processing of the borrow effect, as reflected by differences in slow waves at 1000-1500 ms at frontal sites. We conclude that the processing of arithmetic difficulty relies on an individual's mathematical ability, and suggest that individual differences should be taken into account when investigating mental arithmetic in an ecologically valid assessment.

Spontaneous focusing on numerosity in preschool as a predictor of mathematical skills and knowledge in the fifth grade

Previous studies in a variety of countries have shown that there are substantial individual differences in children's spontaneous focusing on numerosity (SFON), and these differences are positively related to the development of early numerical skills in preschool and primary school. A total of 74 5-year-olds participated in a 7-year follow-up study, in which we explored whether SFON measured with very small numerosities at 5 years of age predicts mathematical skills and knowledge, math motivation, and reading in fifth grade at 11 years of age. Results show that preschool SFON is a unique predictor of arithmetic fluency and number line estimation but not of rational number knowledge, mathematical achievement, math motivation, or reading. These results hold even after taking into account age, IQ, working memory, digit naming, and cardinality skills. The results of the current study further the understanding of how preschool SFON tendency plays a role in the development of different formal mathematical skills over an extended period of time.

Individual differences influence two-digit number processing, but not their analog magnitude processing: a large-scale online study

Symbolic magnitude comparison is one of the most well-studied cognitive processes in research on numerical cognition. However, while the cognitive mechanisms of symbolic magnitude processing have been intensively studied, previous studies have paid less attention to individual differences influencing symbolic magnitude comparison. Employing a two-digit number comparison task in an online setting, we replicated previous effects, including the distance effect, the unit-decade compatibility effect, and the effect of cognitive control on the adaptation to filler items, in a large-scale study in 452 adults. Additionally, we observed that the most influential individual differences were participants' first language, time spent playing computer games and gender, followed by reported alcohol consumption, age and mathematical ability. Participants who used a first language with a left-to-right reading/writing direction were faster than those who read and wrote in the right-to-left direction. Reported playing time for computer games was correlated with faster reaction times. Female participants showed slower reaction times and a larger unit-decade compatibility effect than male participants. Participants who reported never consuming alcohol showed overall slower response times than others. Older participants were slower, but more accurate. Finally, higher grades in mathematics were associated with faster reaction times. We conclude that typical experiments on numerical cognition that employ a keyboard as an input device can also be run in an online setting. Moreover, while individual differences have no influence on domain-specific magnitude processing-apart from age, which increases the decade distance effect-they generally influence performance on a two-digit number comparison task.

Longitudinal development of subtraction performance in elementary school

A major goal of education in elementary mathematics is the mastery of arithmetic operations. However, research on subtraction is rather scarce, probably because subtraction is often implicitly assumed to be cognitively similar to addition, its mathematical inverse. To evaluate this assumption, we examined the relation between the borrow effect in subtraction and the carry effect in addition, and the developmental trajectory of the borrow effect in children using a choice reaction paradigm in a longitudinal study. In contrast to the carry effect in adults, carry and borrow effects in children were found to be categorical rather than continuous. From grades 3 to 4, children became more proficient in two-digit subtraction in general, but not in performing the borrow operation in particular. Thus, we observed no specific developmental progress in place-value computation, but a general improvement in subtraction procedures. Statement of contribution What is already known on this subject? The borrow operation increases difficulty in two-digit subtraction in adults. The carry effect in addition, as the inverse operation of borrowing, comprises categorical and continuous processing characteristics. What does this study add? In contrast to the carry effect in adults, the borrow and carry effects are categorical in elementary school children. Children generally improve in subtraction performance from grades 3 to 4 but do not progress in place-value computation in particular.

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

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

Remediation for Students With Mathematics Difficulties: An Intervention Study in Middle Schools

As empirical studies have consistently shown, low achievement in mathematics at the secondary level can often be traced to deficits in the understanding of certain basic arithmetic concepts taught in primary school. The present intervention study in middle schools evaluated whether such learning deficits can be reduced effectively and whether the type of instruction influences students' progress. The sample consisted of 123 students in 34 classes, split among one control group and two intervention groups: (a) small group instruction and (b) independent work partially integrated into regular classrooms. Over a period of 14 weeks, students were taught basic concepts, such as place value and basic operations. In addition, they practiced fact retrieval and counting (in groups). Multilevel regression analyses demonstrated that the interventions can be used to reduce given deficits.

Old/New Effect of Digital Memory Retrieval in Chinese Dyscalculia

This study reports the neurophysiological and behavioral correlates of digital memory retrieval features in Chinese individuals with and without dyscalculia. A total of 18 children with dyscalculia (ages 11.5-13.5) and 18 controls were tested, and their event-related potentials were digitally recorded simultaneously with behavior measurement. Behavioral data showed that the dyscalculia group had lower hit rates and higher false rates than the control group. The electroencephalography results showed that both groups had a significant old/new effect and that this effect was greater in the control group. In the 300 to 400 ms processing stages, both groups showed significant differences in digital memory retrieval in the frontal regions. In the 400 to 500 and 500 to 600 ms epochs, the old/new effect in the control group was significantly greater than it was in the dyscalculia group at the frontal, central, and parietal regions. In the 600 to 700 ms processing stages, both groups showed significant differences in digital memory retrieval in the frontal, central, parietal, and occipital regions. These results suggest that individuals with dyscalculia exhibit impaired digital memory retrieval. Extraction failure may be an important cause of calculation difficulties.

Place-value understanding in number line estimation predicts future arithmetic performance

In multi-digit numbers, the value of each digit is determined by its position within the digit string. Children's understanding of this place-value structure constitutes a building block for later arithmetic skills. We investigated whether a number line estimation task can provide an assessment of place-value understanding in first grade. We hypothesized that estimating the position of two-digit numbers requires place-value understanding. Therefore, we fitted a linear function to children's estimates of two-digit numbers and considered the resulting slope as a measure of children's place-value understanding. We observed a significant correlation between this slope and children's performance in a transcoding task known to require place-value understanding. Additionally, the slope for two-digit numbers assessed at the beginning of grade 1 predicted children's arithmetic performance at the end of grade 1. These results indicate that the number line estimation task may indeed constitute a valid measure for first-graders' place-value understanding. Moreover, these findings are hard to reconcile with the view that number line estimation directly assesses a spatial representation of numbers. Instead, our results suggest that numerical processes involved in performing the task (such as place-value understanding) may drive the association between number line estimation and arithmetic performance.