Number-fact Retrieval Research Articles

Brief memory reactivations induce learning in the numeric domain

Learning of arithmetic facts such as the multiplication table requires time-consuming, repeated practice. In light of evidence indicating that reactivation of encoded memories can modulate learning and memory processes at the synaptic, system and behavioral levels, we asked whether brief memory reactivations can induce human learning in the numeric domain. Adult participants performed a number-fact retrieval task in which they learned arbitrary numeric facts. Following encoding and a baseline test, 3 passive, brief reactivation sessions of only 40 s each were conducted on separate days. Learning was evaluated in a retest session. Results showed reactivations induced learning, with improved performance at retest relative to baseline test. Furthermore, performance was superior compared to a control group performing test-retest sessions without reactivations, who showed significant memory deterioration. A standard practice group completed active-retrieval sessions on 3 separate days, and showed significant learning gains. Interestingly, while these gains were higher than those of the reactivations group, subjects showing reactivation-induced learning were characterized by superior efficiency relative to standard practice subjects, with higher rate of improvement per practice time. A follow-up long-term retention experiment showed that 30 days following initial practice, weekly brief reactivations reduced forgetting, with participants performing superior to controls undergoing the same initial practice without reactivations. Overall, the results demonstrate that brief passive reactivations induce efficient learning and reduce forgetting within a numerical context. Time-efficient practice in the numeric domain carries implications for enhancement of learning strategies in daily-life settings.

Reading and math tests differentially predict number transcoding and number fact speed longitudinally: A random intercept cross-lagged panel approach.

Children’s early math abilities have been claimed to depend differentially on various cognitive competencies (e.g., core number skills, working memory, and general math and reading abilities). Clarifying the relative importance of these different cognitive markers in predicting key early math skills would provide a conceptual framework for understanding development of math competence. To this end, we assessed 267 six-year-olds’ core number (dot enumeration, number comparison) abilities, visuospatial working memory, and standardized math and reading test performance, as well as their number fact retrieval speed and ability to transcode number strings (e.g., “4,073”) at 6, 7, and 8 years. To minimize interpretive difficulties associated with previous analyses, we used a random intercept cross-lagged panel model to characterize the reciprocal relationship between number transcoding ability and number fact speed over time. We tested the hypotheses that general math scores would predict individual differences in number fact speed, while general reading scores would predict individual differences in number transcoding accuracy. Findings showed that both standardized tests and core number measures differentially predict math skills in the early school years. Dot enumeration and general math ability predicted individual differences in number fact speed, while general reading and general math predicted individual differences the ability to transcode number strings over time. We suggest combining standardized assessments of math and reading, together with measures of core numerical ability, at school entry offers important information about the factors associated with key early math skills. (PsycINFO Database Record (c) 2019 APA, all rights reserved)

Data-driven heterogeneity in mathematical learning disabilities based on the triple code model

Many classifications of heterogeneity in mathematical learning disabilities (MLD) have been proposed over the past four decades, however no empirical research has been conducted until recently, and none of the classifications are derived from Triple Code Model (TCM) postulates. The TCM proposes MLD as a heterogeneous disorder, with two distinguishable profiles: a representational subtype and a verbal subtype. A sample of elementary school 3rd to 6th graders was divided into two age cohorts (3rd − 4th grades, and 5th − 6th grades). Using data-driven strategies, based on the cognitive classification variables predicted by the TCM, our sample of children with MLD clustered as expected: a group with representational deficits and a group with number-fact retrieval deficits. In the younger group, a spatial subtype also emerged, while in both cohorts a non-specific cluster was produced whose profile could not be explained by this theoretical approach.

Mathematical skills in children with dyslexia

The mathematical performance of 17 children with developmental dyslexia (DD) was assessed and compared to a control group to examine whether difficulties related to reading and phonological processing affect the development of mathematical skills. The DD group performed worse than the controls on number fact retrieval, multi-step arithmetic problem solving, and multi-digit calculation, whereas their scores on tasks tapping approximate arithmetic and conceptual understanding (i.e., place value, calculation principles) were equal to the controls. In view of the Triple-code model, the findings demonstrate that children with DD have problems with tasks depending on verbal–phonological number codes but have no problems with tasks depending on analogue magnitude representations or visual-Arabic number codes.

Mathematical Abilities in Elementary School Children with Autism Spectrum Disorder

Although clinical practitioners often express concerns about the mathematical functioning of children with autism spectrum disorder (ASD), the field of mathematics remains a relatively unexplored topic in individuals with ASD. Moreover, research findings are fragmentary and hold inconclusive results. The present study aimed to examine whether grade 1 (aged 6–7 years) to 4 (aged 9–10 years) elementary school children with ASD scored significantly different from age‐adequate norms on mathematics. To this end, a multi‐componential approach of mathematics was used. Four domains of mathematics were assessed in 121 children with ASD: procedural calculation, number fact retrieval, word/language problems, and time‐related competences. All children attended general education classrooms, following the standard curriculum, and were coached by integrated educational services. Children with ASD showed a strength in word/language problems in second and fourth grade. There was evidence of a weakness for procedural calculation in first grade and for time‐related competences in first and third grade. In all other cases, average scores were shown. As such, results revealed a profile of strengths, average abilities, and weaknesses in mathematics and highlighted the importance of focusing on different domains of mathematics. Because a high variability in mathematical performance could be observed, we recommend an individual assessment when considering the mathematical trajectory of children with ASD. Copyright © 2015 John Wiley & Sons, Ltd.

Preschool predictors of mathematics in first grade children with autism spectrum disorder.

Up till now, research evidence on the mathematical abilities of children with autism spectrum disorder (ASD) has been scarce and provided mixed results. The current study examined the predictive value of five early numerical competencies for four domains of mathematics in first grade. Thirty-three high-functioning children with ASD were followed up from preschool to first grade and compared with 54 typically developing children, as well as with normed samples in first grade. Five early numerical competencies were tested in preschool (5-6 years): verbal subitizing, counting, magnitude comparison, estimation, and arithmetic operations. Four domains of mathematics were used as outcome variables in first grade (6-7 years): procedural calculation, number fact retrieval, word/language problems, and time-related competences. Children with ASD showed similar early numerical competencies at preschool age as typically developing children. Moreover, they scored average on number fact retrieval and time-related competences and higher on procedural calculation and word/language problems compared to the normed population in first grade. When predicting first grade mathematics performance in children with ASD, both verbal subitizing and counting seemed to be important to evaluate at preschool age. Verbal subitizing had a higher predictive value in children with ASD than in typically developing children. Whereas verbal subitizing was predictive for procedural calculation, number fact retrieval, and word/language problems, counting was predictive for procedural calculation and, to a lesser extent, number fact retrieval. Implications and directions for future research are discussed.

Understanding and processing numbers among Chinese children.

children learn mathematics using a user-friendly, regular number naming system. Such a number naming system has been suggested to facilitate Chinese children's mathematical development. Recent studies investigated how Chinese children, with such a language advantage, understand and process Arabic numerals. These studies consistently indicate that Chinese children become able to process numerals proficiently and automatically and make sense of the place-value Arabic number system in the first grade, and children who fail to do so are apt to encounter difficulties in later mathematical learning. Moreover, recent findings show that Chinese children with mathematical difficulties have deficits in number-fact retrieval and the place- value concept. These studies prompted the first steps to explore Chinese children's number processing and understanding. Yet, further studies are needed to provide a complete picture of how Chinese children process and understand numerals using a user- friendly number naming system. Keywords: numbers, mathematics, Chinese.

Identifying Subtypes Among Children With Developmental Coordination Disorder and Mathematical Learning Disabilities, Using Model-Based Clustering

A relationship between motor and mathematical skills has been shown by previous research. However, the question of whether subtypes can be differentiated within developmental coordination disorder (DCD) and/or mathematical learning disability (MLD) remains unresolved. In a sample of children with and without DCD and/or MLD, a data-driven model-based clustering was used to identify subgroups of individuals with relatively homogeneous profiles on measures associated with motor and mathematical skills. One subgroup of children with motor problems was found based on motor variables. Based on mathematical variables, two clinical clusters were found: a subtype with number fact retrieval problems and a subtype with procedural calculation problems. Clustering with motor and mathematical skills revealed two clinical clusters: a cluster with number fact retrieval as well as procedural calculation problems and below average motor and visual-motor integration skills. A second cluster of children had only procedural calculation and visual-motor problems. Our results raise questions about the usefulness of placing children who have below average mathematical skills into a single diagnostic category. Furthermore, we inform ongoing debates about the overlap between DCD and MLD, as below average motor skills were found in both MLD subgroups, although a different motor profile is linked to a different mathematical profile.

Behind mathematical learning disabilities: What about visual perception and motor skills?

In a sample of 39 children with mathematical learning disabilities (MLD) and 106 typically developing controls belonging to three control groups of three different ages, we found that visual perception, motor skills and visual-motor integration explained a substantial proportion of the variance in either number fact retrieval or procedural calculation. Furthermore, children with MLD performed significantly worse on visual perception, motor skills and visual-motor integration in comparison with age-matched control children. A mild developmental delay in visual perception, visual-motor integration and (fine) motor coordination and a severe delay in motor skills were found in children with MLD. However, not all children with MLD have problems on these domains. They seem to be a heterogeneous group, not only with respect to their mathematical profile but also with respect to their visual perceptual, motor and visual-motor integration skills. Diagnostic implications are discussed.

Mathematical problems in children with developmental coordination disorder

Developmental coordination disorder (DCD) is a heterogeneous disorder, which is often co-morbid with learning disabilities. However, mathematical problems have rarely been studied in DCD. The aim of this study was to investigate the mathematical problems in children with various degrees of motor problems. Specifically, this study explored if the development of mathematical skills in children with DCD is delayed or deficient. Children with DCD performed significantly worse for number fact retrieval and procedural calculation in comparison with age-matched control children. Moreover, children with mild DCD differed significantly from children with severe DCD on both number fact retrieval and procedural calculation. In addition, we found a developmental delay of 1 year for number fact retrieval in children with mild DCD and a developmental delay of 2 years in children with severe DCD. No evidence for a mathematical deficit was found. Diagnostic implications are discussed.

Semantic priming with product verification but not production.

Campbell and Reynvoet (2009) found that time to name a single-digit target was about 8 ms faster if preceded by a near prime (±1) compared to a far prime (at least ±3) when prime-digit pairs were interleaved with number comparisons (9↑3; name larger) and not when they were interleaved with multiplication problems (9×3; state product). This is consistent with the claim by previous researchers that magnitude comparison can enable a semantic pathway for digit naming whereas number-fact retrieval can inhibit it. To pursue this, the current study compared priming in the context of multiplication production (9×3=?) versus multiplication verification (e.g., 9×3=24, true or false). Multiplication production, but not verification, may inhibit semantic digit naming to reduce naming-related interference with verbal number production. Indeed, semantic priming of digit naming occurred only in verification and not production blocks. This supports the conclusion that multiplication production can inhibit semantic mediation of digit naming, which is enabled in other number processing tasks (e.g., comparison, verification) that do not compete with naming for verbal number production processes.

The cognitive profile of Chinese children with mathematics difficulties

This study examined how four domain-specific skills (arithmetic procedural skills, number fact retrieval, place value concept, and number sense) and two domain-general processing skills (working memory and processing speed) may account for Chinese children’s mathematics learning difficulties. Children with mathematics difficulties (MD) of two age groups (7–8 and 9–11 years) were compared with age-matched typically achieving children. For both age groups, children with MD performed significantly worse than their age-matched controls on all of the domain-specific and domain-general measures. Further analyses revealed that the MD children with literacy difficulties (MD/RD group) performed the worst on all of the measures, whereas the MD-only group was significantly outperformed by the controls on the four domain-specific measures and verbal working memory. Stepwise discriminant analyses showed that both number fact retrieval and place value concept were significant factors differentiating the MD and non-MD children. To conclude, deficits in domain-specific skills, especially those of number fact retrieval and place value understanding, characterize the profile of Chinese children with MD.

Selective effects of number processing context on number naming

Naming of Arabic digits can utilise semantic or asemantic pathways but neuropsychological evidence suggests that number-fact retrieval (2×3 = ?) can inhibit the semantic route for digit naming. Consistent with this, Campbell and Metcalfe (in press) demonstrated with neurologically intact participants that Arabic digit naming time was about 15 ms slower when performed in the context of number-fact retrieval (multiplication) than in the context of a task requiring semantic processing (magnitude comparison). Experiment 1 here tested whether this context effect generalised to naming a feature incidentally associated with an Arabic digit (font colour). Experiment 2 tested if the effect generalised to a salient semantic feature associated with Arabic digits (parity). In Experiment 1, the context effect on digit naming was greater than on naming the font colour of digits. Experiment 2 demonstrated that the effect occurred selectively for stating the digit name and did not generalise to parity naming. The results suggest that the effect of context (multiplication vs. comparison) on numeral naming arises in a dedicated digit naming circuit and reinforce the view that number processing context can reconfigure the functional architecture for number naming.

Context-dependent semantic priming in number naming.

Previous research has shown that time to name single-digit Arabic numbers is about 15 ms slower when naming trials are interleaved with simple multiplication (e.g., state product of 2 x 3) than when naming digits is interleaved with magnitude comparison (e.g., state larger; 2 upward arrow 3). To explain this phenomenon, J. I. D. Campbell and A. W. S. Metcalfe (2008) proposed that the comparison context enables both semantic and asemantic pathways for digit naming but that number-fact retrieval inhibits the semantic route and slows digit naming relative to the comparison context. To test this hypothesis, the authors modified the naming context paradigm by introducing a semantic priming manipulation. They replicated the digit-naming response time advantage for comparison relative to the multiplication context and observed semantic priming only in the comparison context. In comparison blocks, digit naming was 8 ms faster immediately after naming near digit primes (+/-1) compared to far primes (>or=3), but in multiplication blocks there was no priming. The results reinforce the theory that number-fact retrieval can inhibit the semantic route for digit naming (L. Cohen & S. Dehaene, 1995) and thereby reconfigure the cognitive architecture for naming digits.

Arabic digit naming speed: Task context and redundancy gain

There is evidence for both semantic and asemantic routes for naming Arabic digits, but neuropsychological dissociations suggest that number-fact retrieval (2 × 3 = 6) can inhibit the semantic route for digit naming. Here, we tested the hypothesis that such inhibition should slow digit naming, based on the principle that reduced access to multiple routes would counteract redundancy gain (the response time advantage expected from parallel retrieval pathways). Participants named two single digit numbers and then performed simple addition or magnitude comparison (Experiment 1), multiplication or magnitude comparison (Experiment 2), and multiplication or subtraction (Experiment 3) on the same or on a different pair of digits. Addition and multiplication were expected to inhibit the semantic route, whereas comparison and subtraction should enable the semantic route. Digit naming time was approximately 15 ms slower when participants subsequently performed addition or multiplication relative to comparison or subtraction, regardless of whether or not the same digit pair was involved. A letter naming control condition in Experiment 3 demonstrated that the effect was specific to digit naming. Number fact retrieval apparently can inhibit Arabic digit naming processes.

Calculation latency: The μ of memory and the τ of transformation

Does numeral format (e.g., 4 + 8 vs. four + eight) affect calculation per se? University students (N = 47) solved single-digit addition problems presented as Arabic digits or English words and reported their strategies (memory retrieval or procedures such as counting or transformation). Decomposition of the response time (RT) distributions into mu (reflecting shift) and tau (reflecting skew) confirmed that retrieval trials contributed predominantly to mu whereas procedure trials contributed predominantly to tau. The format x problem size RT interaction (i.e., greater word-format RT costs for large problems than for small problems)was associated entirely with mu and not with tau. Reported use of procedures presented a corresponding format x size interaction. Together, these results indicate that, relative to the well-practiced digit format, the unfamiliar word format disrupts number-fact retrieval and promotes use of procedural strategies.

Fan effects reveal position-specific numerical concepts.

We examined associative interference (i.e., fan) effects to determine if the numerical concepts mediating numerical fact retrieval can be position specific. Participants memorized 15 "diamond arithmetic" facts (e.g., 3 [symbol: see text] 5 = 97; Whalen, 1997) involving five different operands. Ignoring operand position (i.e., left vs. right), each operand appeared equally often across items (i.e., position-independent fan was constant). In contrast, position-specific fan values varied across items (e.g., 2 appeared five times as the left operand, but once in the right position; 3 appeared four times as the left operand, and twice as the right operand, etc). Thus, we expect a fan effect only if the concepts activated by arithmetic operands are position specific. Results indicated that position-specific fan was an excellent predictor of trials to criterion, error rate, and latency. The concepts mediating number fact retrieval can be position specific.

Discussion

Noël et al. (Noël, M.-P., Fias, W., Brsybaert, M., 1997. About the influence of the presentation format on arithmetical-fact retrieval processes. Cognition 63, 335-374) examined the simple-multiplication errors of 24 Dutch- and 24 French-speaking adults for evidence that number reading interferes with language-specific, number-fact retrieval processes. They concluded that arithmetic memory is not influenced by reading-based interference and is based on a notation and language-independent mental representation. Alternative analyses of their error data, however, provide strong evidence that arithmetic performance is subject to reading-based interference and provide some support for the language-specificity of number-fact memory.

Reading-based interference in cognitive arithmetic.

Do number-fact retrieval (4.6 =?) and numeral reading (e.g., transcoding 46 into "forty six") access the same retrieval structures? Data from the present experiment suggest that they do. Under instructions to respond quickly, adults performed simple multiplication problems oriented horizontally or vertically with operands presented simultaneously or with a 500 ms preview of one operand. Errors involving congruent operand intrusions (e.g., 2.8 = 24 or 9.6 = 36) were most frequent when conditions afforded the left-to-right encoding sequence that is standard for reading multi-digit numbers. Ninety percent of such intrusions involved the correct answer to another simple multiplication problem (e.g., 4.8 = 28) rather than a miscellaneous answer (e.g., 4.8 = 38). This high rate of arithmetically related intrusions suggests that numeral reading and verbal production of number facts involve a shared representational system.

Architectures for numerical cognition

Current theories of numerical cognition differ in assumptions about the componential architecture of number processing and about the extent of notation-specific processes. To investigate these issues, 64 adult subjects were tested on simple addition and multiplication problems presented in Arabic digit or English number-word format. Overall, response times and error rates were much higher with the word format, but more importantly, presentation format interacted with arithmetic operation and problem size. Operation errors (2 + 4 = 8), operand-naming errors (2 + 8 = 8), and operand-intrusion errors (9 × 6 = 36) were each characterized by a different format x operation interaction, and analysis of inter-trial error priming showed selective interference from preceding trials as a function of number format. These types of format-specific retrieval interference and operation-specific effects of format are problematic for models that hypothesize notation-independent memory processes for arithmetic. Furthermore, analyses of operand-naming errors, operand-intrusion errors, and other operand-priming effects, revealed strong interactions of number reading and number-fact retrieval processes; processes that are typically posited to be functionally independent. The results suggest a complex encoding architecture that incorporates notation-dependent activation of addition and multiplication facts, as well as interpenetration of number reading and number-fact retrieval processes.