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Visual perception is a critical factor in mathematical performance. The current study investigated whether form-perception speed underlies the association between visual perception capability and mathematical performance. Visual form perception tasks having different perceptual loads were administered to 162 adults in Experiment 1 and 273 children in Experiment 2. Experiment 1 showed that adult's visual perception capability correlated with mathematical performance, even after controlling for age, gender, nonverbal matrix reasoning, choice reaction time, and mental rotational ability. However, only the correlation modulated by processing speed-and only visual perceptions with lower perceptual loads- predicted the variance of mathematical performance for adults mathematical performance. The findings in children corresponded to those in adults. Thus, form-perception speed modulates the association between visual perception capability and mathematical performance., Author(s): Yiyun Zhang [sup.1] [sup.2] , Shijia Fang [sup.2] , Zhuo Chen [sup.1] , Xinlin Zhou [sup.2] Author Affiliations: (1) https://ror.org/04c3cgg32, grid.440818.1, 0000 0000 8664 1765, School of Psychology, Liaoning [...]

Mathematical abilities are important for children's academic achievement during the primary education phase. Understanding which cognitive factors underlie individual differences in mathematics is essential to obtaining insights into children's mathematical development. This study explored the roles of phonological processing skills and visual-spatial skills in arithmetic, mathematical problem solving, and mathematical reasoning among primary school children. Two hundred and fifty-one primary school children (mean age: 8.31 ± 0.89 years old), including 87 first graders, 83 s graders, and 81 third graders participated in this study. Children's rapid automatized naming was measured using a rapid digit naming task, and phonological awareness was measured with a character rhyming task. Additionally, children's visual perception was measured with a figure matching task, and mental rotation was measured with a 2D/3D mental rotation task. Children's mathematical abilities were measured with three mathematics tests: calculation task, mathematical problem solving task, and mathematical reasoning task. Regression analyses and Bayesian hypothesis testing showed that phonological awareness uniquely contributed to children's mathematical abilities, especially mathematical problem solving. The results suggest that phonological awareness serves as a key precursor of mathematical abilities during the primary education phase., Author(s): Liyan Li [sup.1] , Heyi Zhang [sup.2] , Yue Qi [sup.3] , Xiuya Lei [sup.1] , Xiao Yu [sup.1] , Haoning Liu [sup.1] Author Affiliations: (1) https://ror.org/04xv2pc41, grid.66741.32, 0000 [...]

This study was to investigate how phonological processing and visual-spatial abilities contributed differently to arithmetic operations including addition, subtraction, multiplication, and division. Eighty-eight Chinese fifth graders completed a rapid digit naming task, a character rhyming task, a figure matching task, a 3D mental rotation task, and arithmetic calculation tasks. Results showed that when controlling for age, visual acuity, and nonverbal intelligence, phonological processing accounted for unique variance in both addition and multiplication, whereas visual-spatial processing explained unique variance in subtraction and division. In particular, rapid automatized naming (hereafter, RAN) explained more of the variance in addition than mental rotation did, although both were significantly associated with addition. Mental rotation explained more of the variance in subtraction than phonological awareness and RAN did, although the three skills were all related to subtraction. Importantly, RAN was a unique correlate of multiplication, while mental rotation was a unique correlate of division. These findings highlight different contribution of the phonological processing and visual-spatial skills underlying four arithmetic operations., Author(s): Xiao Yu [sup.1] , Kaichun Liu [sup.2] , Yuhan Wang [sup.2] , Xiujie Yang [sup.2] , Jingyuan Yang [sup.1] Author Affiliations: (1) https://ror.org/04xv2pc41, grid.66741.32, 0000 0001 1456 856X, Department [...]

Numerical skills are a major cognitive foundation of mathematical learning, especially in young children. The role of domain-general factors in explaining individual differences in numerical skills is largely unknown. The current study aims to shed light on the different roles of multiple domain-general abilities, including spatial abilities, visuospatial working memory, executive functions, attention, and fine motor ability in the development of numerical skills. Using an extensive sample of children (n = 339) of varying ages (between 3 and 8 years-old), we examined the influence of domain-general factors on numerical skills (symbolic and non-symbolic). We used factor analysis to cluster tasks into latent variables to analyze the data. Based on the results, we used Structural Equation Modeling with numerical abilities as the outcome variable. Spatial skills directly and strongly affected numerical skills, and executive functions also affected numerical skills. Executive functions made a unique contribution to symbolic numerical skills, dissociating symbolic and non-symbolic numerical skills. These results indicate that multiple factors can affect numerical performance. This conclusion bears significant implications for the early diagnosis of learning disorders and intervention methods for these disorders., Author(s): Sarit Ashkenazi [sup.1] , Anna Adi [sup.1] Author Affiliations: (1) grid.9619.7, 0000 0004 1937 0538, The Seymour Fox School of Education, Learning Disabilities, The Hebrew University of Jerusalem, , [...]

It is well known that sports and mathematical abilities are related to spatial abilities, also a relation between sport and mathematical abilities is assumed. However, the relation between all three aspects has not been investigated until now. Therefore, the main goal of the study is to examine the relationship between sport, spatial and mathematical ability in elementary school aged children. 50 boys and 42 girls from third grade solved spatial and sport tasks, which can be differentiated into the following: intrinsic-dynamic, intrinsic-static and extrinsic-static. Furthermore, their performances in mathematical (separated into numerical/arithmetical and geometrical) abilities were analyzed. The results showed significant correlations between the static spatial and sporting activities. This correlation is due to the strong correlation within the group of girls. Furthermore, a good performance in the intrinsic-spatial ability was related to a high geometrical ability of the children, especially in boys. However, the geometrical ability could not be predicted by sport abilities., Author(s): Petra Jansen [sup.1] , Stefanie Pietsch [sup.2] Author Affiliations: (1) grid.7727.5, 0000 0001 2190 5763, Faculty of Human Science, University of Regensburg, , Universitätsstr. 31, 93053, Regensburg, Germany (2) [...]

Mental abacus is mental arithmetic with the help of an imagined abacus. Children skilled in mental abacus have been shown to exhibit top-quality arithmetic abilities. The current study investigated whether children with high-level mental abacus ability could outperform untrained control children in non-symbolic number sense, which is considered to be fundamental for arithmetic development. One hundred and fifty children (75 children skilled in mental abacus and 75 controls) took part in this study. Children skilled in mental abacus completed a mental abacus level test. The two groups of children performed serial cognitive tasks, assessing non-symbolic number comparison, arithmetic, language, spatial processing, visual perception, attention, processing speed, working memory, and general intelligence. Results showed that children skilled in mental abacus had significantly better non-symbolic number sense than the other children after controlling for general intelligence. The significant difference in non-symbolic number sense remained after controlling for age, gender, all types of cognitive processing available, and arithmetic performance. A mediation model showed that non-symbolic number sense partially mediated the group difference in arithmetic development. These findings suggest that children skilled in mental abacus have enhanced non-symbolic number sense and raise the possibility that mental abacus training could directly improve children's non-symbolic numerical skills., Author(s): Jiaxin Cui [sup.1] [sup.2] [sup.3] , Rui Xiao [sup.1] [sup.2] [sup.3] , Mei Ma [sup.1] [sup.2] [sup.3] , Li Yuan [sup.1] [sup.2] [sup.3] , Roi Cohen Kodash [sup.4] , [...]

Less depends upon the choice of words than upon this, that their introduction shall be justified by pregnant theorems. --Carl Friedrich Gauss (1827) on the primacy of numeracy over literacy [...]

In this paper, we focus on the development of geometric cognition. We argue that to understand how geometric cognition has been constituted, one must appreciate not only individual cognitive factors, such as phylogenetically ancient and ontogenetically early core cognitive systems, but also the social history of the spread and use of cognitive artifacts. In particular, we show that the development of Greek mathematics, enshrined in Euclid's Elements, was driven by the use of two tightly intertwined cognitive artifacts: the use of lettered diagrams; and the creation of linguistic formulae (namely non-compositional fixed strings of words used repetitively within authors and between them). Together, these artifacts formed the professional language of geometry. In this respect, the case of Greek geometry clearly shows that explanations of geometric reasoning have to go beyond the confines of methodological individualism to account for how the distributed practice of artifact use has stabilized over time. This practice, as we suggest, has also contributed heavily to the understanding of what mathematical proof is; classically, it has been assumed that proofs are not merely deductively correct but also remain invariant over various individuals sharing the same cognitive practice. Cognitive artifacts in Greek geometry constrained the repertoire of admissible inferential operations, which made these proofs inter-subjectively testable and compelling. By focusing on the cognitive operations on artifacts, we also stress that mental mechanisms that contribute to these operations are still poorly understood, in contrast to those mechanisms which drive symbolic logical inference., Author(s): Mateusz Hohol [sup.1] [sup.2] , Marcin Milkowski [sup.2] Author Affiliations: (1) grid.5522.0, 0000 0001 2162 9631, Copernicus Center for Interdisciplinary Studies, Jagiellonian University, , Kopernika 27, 31-501, Cracow, Poland [...]

This study investigated creative thinking abilities among two groups of 20 children with autism spectrum disorders (ASD) compared to 20 children with typical development ages 9-11. The study compared performance on two different creativity tests: general creativity (Pictorial Multiple Solutions-PMS) test versus mathematical creativity (Creating Equal Number-CEN) test, and investigated relationships between general and mathematical creative thinking across various cognitive measures including non-verbal IQ, verbal and non-verbal working memory and Attention. Results of the study demonstrate significant correlations among the measures of creativity indicating that the PMS and the CEN tasks represent different skills, or perhaps, different domains of creativity. Findings suggest that creativity can be found among individuals with ASD., Author(s): Orit Hetzroni [sup.1] , Hila Agada [sup.1] , Mark Leikin [sup.1] [sup.2] [sup.3] Author Affiliations: (Aff1) 0000 0004 1937 0562, grid.18098.38, Department of Special Education, Faculty of Education, University [...]

The purpose of this study was to compare the acute effects of Aerobic Exercise (AE), Resistance Exercise (RE), and a nonexercise (NE) control on measures of academic achievement (AA) and cognition in 10th grade males and females. This study utilized a randomized crossover design. Tenth grade males and females performed three exercise trials (AE, RE, NE) separated by 7 days each. Immediately following exercise, participants completed a 10-question mathematics test, followed by the Stroop test. A repeated measures ANOVA revealed small but insignificant differences in mean math test performance between RE and NE, F(1, 86) = 2.81, p = .098, [[eta].sup.2] = .032, and AE and NE, F(1, 86) = 2.03, p = .158, [[eta].sup.2] = .023. Significant differences were found between RE and NE in the Stroop dot test, F(1, 86) = 4.31, p = .041, [[eta].sup.2] = .048, and between AE and NE in the Stroop dot test, F(1, 86) = 10.402, p = .002, [[eta].sup.2] = .108, and Stroop color test, F(1,86) = 6.85, p = .01, n2 = .074. In conclusion, acute RE and AE did not significantly improve scores on a test of mathematics, but did improve measures of cognition in comparison to an NE control., Current research trends in the area of youth physical activity (PA) have highlighted the relationship between students who meet or exceed activity guidelines with improved academic results (Carlson et al., [...]

Computers have become an important piece of technology in classrooms for implementing academic interventions. Often, students' responses to these interventions are used to help make important educational decisions. Therefore, it is important to consider the effect of these interventions across multiple contexts. For example, previous research has demonstrated that when students practice math fact fluency on the computers gains did not generalize to paper-pencil performance. The current study extended this research by examining the effect of multiple stimulus exemplars on the generalization of computer-based math practice to paper-pencil performance. A total of 57 second-grade students completed fluency drills only on the computer, computer mixed with paper-pencil or with only paper-pencil. Pretest-posttest performance was evaluated using a 3 x 2 doubly multivariate repeated-measures ANOVA, with follow-up univariate analysis to determine whether the interaction between time and treatment type was similar across matched and unmatched treatment modalities. Results from previous research were replicated with a lack of generalization across modalities, but the addition of multiple stimulus exemplars resulted in increased generalization for those students receiving a mix of computer and paper-pencil practice., Author(s): Sara E. House Rich [sup.1] , Gary J. Duhon [sup.1] , James Reynolds [sup.1] Author Affiliations: (1) 0000 0001 0721 7331grid.65519.3eSchool of Applied Health and Educational Psychology, 434 Willard [...]

By the time girls are second graders, they may be exhibiting negative attitudes toward math (Cvencek, Meltzoff, & Greenwald, 2011). McFarland, Benson and McFarland (2011) examined girls' math achievement in [...]

Approximate number sense (ANS) acuity refers to the ability to non-symbolically recognize, estimate and operate upon large numerosities. ANS acuity has been reported to be correlated with math achievement in children and adolescents. However, reports of this relationship in adults have been inconsistent. The present study aimed to resolve the inconsistency in the relationship between adults' ANS acuity and math achievement by contrasting between different kinds of mathematical problem solving. We hypothesized that the correlation between ANS acuity mathematical performance would be stronger when deep quantitative processing is required during problem solving. In Experiment 1, ANS acuity was correlated with Mathematical Reasoning but not Directed Calculation performance. In Experiment 2, ANS acuity was correlated with Two-digit Subtraction (but not Addition) performance only when Regrouping (i.e., borrowing) was required. The results from two experiments demonstrated that ANS acuity was correlated with mathematical performance only when problem solving involved effortful, quantitative processing that goes beyond automatized, routinized arithmetic. In addition, ANS acuity was distinguishable from Area acuity regarding its unique relationship with math achievement, which was unconfounded by the influence of demographic variables and fluid intelligence. Overall, the present results help resolve the inconsistency in previous reports of the correlation between ANS acuity and math achievement in adults., Author(s): Selim Jang[sup.1] , Soohyun Cho[sup.1] Author Affiliations: (1) Department of Psychology, Chung-Ang University, 221 Heukseok-dong, Dongjak-gu, Seoul, Korea Introduction Humans and animals are endowed with the ability to approximately [...]

A pretest-posttest cluster-randomized trial involving 31 middle schools and 335 students with disabilities tested the effects of combining explicit and anchored instruction on fraction computation and problem solving. Results of standardized and researcher-developed tests showed that students who were taught with the blended units outscored students in Business As Usual classes. Students made the largest gains in computing with fractions and on problems related to ratios, proportions, and geometry. The findings suggest important implications for the way curriculum is designed for middle school students with disabilities who exhibit low performance in math., The National Assessment of Educational Progress (NAEP; National Center for Education Statistics, 2011) reported that almost two thirds (65%) of students with disabilities in eighth grade scored below the Basic [...]

An overarching question guided this study: What is mathematics difficulty (MD) independent of reading difficulty (RD)? The sample included 203 children whom the researchers followed from kindergarten to third grade. The researchers used latent growth modeling to investigate the relationship between MD and measures of working memory, short-term memory, cognitive processing speed, early numerical skills, and phonological processing, independent of reading. Two main findings emerged: (a) deficits in early numerical skills were a defining feature of MD with or without RD, and (b) early numerical skills and phonological processing influenced growth in mathematics from kindergarten to third grade. These results indicate that future MD research should focus on numerical and language-based skills, including whether the mathematics disorder in MD--specific or otherwise--reflects underlying language and/or numerical deficits., Gerardi, Goette, and Meier (2010) of the Federal Reserve recently found a relationship between mortgage foreclosure in a sample of adults and deficiencies in basic mathematical calculations that most children [...]

It is known that number and space representations are connected to one another in numerical and arithmetic abilities. Numbers are represented using the metaphor of a mental number line, oriented along horizontal and vertical space. This number line also seems to be linked to mental arithmetic, which is based partly on arithmetic fact retrieval. It seems that number representation and mental arithmetic are linked together. The present study tested the effect of spatial contextual congruency between stimulus presentation and response key arrangements in arithmetic fact retrieval, using number-matching and addition verification tasks. For both tasks in Experiment 1, a contextual congruency effect was present horizontally (i.e., horizontal presentation of stimuli and horizontal response key alignments) but not vertically (i.e., vertical presentation of stimuli but horizontal response key alignments). In Experiment 2, both tasks showed a contextual congruency effect for both spatial conditions. Experiment 1 showed that the interference and distance effects were found in the horizontal condition, probably because of the spatial congruency between stimulus presentation and response key arrangements. This spatial congruency could be related to the activation of the horizontal number line. Experiment 2 showed similar interference and distance effects for both spatial conditions, suggesting that the congruency between stimulus presentation and response alignment could facilitate the retrieval of arithmetic facts. This facilitation could be related to the activation of both horizontal and vertical number lines. The results are discussed in light of the possible role of a mental number line in arithmetic fact retrieval., Within the conceptual metaphor theory, there is the general idea that space is often used to convey meaning (Lakoff & Johnson, 1980). The importance of visuospatial metaphor in grasping mathematical [...]

Results from a 2-year longitudinal study of 181 children from 4th through 5th grade are reported. Levels of growth in children's computation, word problem, and estimation skills by means of common fractions were predicted by working memory, attentive classroom behavior, conceptual knowledge about fractions, and simple arithmetic fluency. Comparisons of 55 participants identified as having mathematical difficulties to those without mathematical difficulties revealed that group differences in emerging fraction skills were consistently mediated by attentive classroom behavior and conceptual knowledge about fractions. Neither working memory nor arithmetic fluency mediated group differences in growth in fraction skills. It was also found that the development of basic fraction skills and conceptual knowledge are bidirectional in that conceptual knowledge exerted strong influences on all 3 types of basic fraction skills, and basic fraction skills exerted a more modest influence on subsequent conceptual knowledge. Results are discussed with reference to how the identification of potentially malleable student characteristics that contribute to the difficulties that some students have with fractions informs interventions. Also, results will contribute to a future theoretical account concerning how domain-general and domain-specific factors influence the development of basic fraction skills. Keywords: fraction skills, mathematical learning disabilities, mathematical learning difficulties, mathematical development DOI: 10.1037/a0019824

Young adults were asked to solve two-digit addition problems and to say aloud the result of each calculation step to allow the identification of computation strategies. We manipulated the position of the largest addend (e.g., 25 + 48 vs. 48 + 25) to assess whether strategies are modulated by magnitude characteristics. With some strategies, participants demonstrated a clear preference to take the largest addend as the starting point for the calculation. Hence, rather than applying strategies in an inflexible manner, participants evaluated and compared the operands before proceeding to calculation. Further, mathematically skilled participants tended to use those magnitude-based strategies more often than less skilled ones. The findings demonstrate that magnitude information plays a role in complex arithmetic by guiding the process of strategy selection and possibly more so for mathematically skilled participants. Keywords: approximate number system, mathematical ability, mental arithmetic, number sense, strategy DOI: 10.1037/a0020767

To address the ongoing debate about the origins of the size effect (faster comparison time for smaller than larger numbers, given a fixed intrapair distance), an indication of automatic number processing was searched for. Participants performed a quantity comparison task in which they had to decide which of two sketched cups contained more liquid, while ignoring the number superimposed on each cup. In the congruent condition, the larger number appeared on the cup containing more liquid, while in the incongruent condition the larger number appeared on the cup containing less liquid. The size effect was round in a numerical comparison task, while in the quantity comparison task the size congruity effect decreased as the magnitude of the irrelevant numbers increased. Together, these results suggest that the size effect reflects a basic feature of the mental number line. Keywords: size effect, mental number line, size congruity effect, automaticity DOI: 10.1037/a0020464

The purpose of the present study was to establish whether students' self competence beliefs and IQ interact when they need to perform at school and if so, whether this interaction [...]

The gap in achievement between minority and non-minority students has become a national priority. To investigate the relationship between school racial composition and the race-based gaps in mathematic achievement, High School Effectiveness Study data on 3,392 students in 177 schools were analyzed. Multilevel analyses revealed that when at least half of the students in a school are Black or Hispanic, all student achievement is lower (for White as well as for ethnic minority students). Asian students' achievement remains lowered until the percentage of Black and Hispanic students is less than 15%. However, schools that are 30-49% Black and/or Hispanic have more egalitarian achievement between White and Hispanic students. Although there is no one-size-fits-all model, this research does confirm that being in a school with a high a concentration of Black and Hispanic students lessens all students' chances of academic achievement, even for students who otherwise should excel. Keywords Racial differences * Education * Academic achievement * Achievement gap * Minority groups * Race * Mathematics achievement

We review the growing literature on health numeracy, the ability to understand and use numerical information, and its relation to cognition, health behaviors, and medical outcomes. Despite the surfeit of health information from commercial and noncommercial sources, national and international surveys show that many people lack basic numerical skills that are essential to maintain their health and make informed medical decisions. Low numeracy distorts perceptions of risks and benefits of screening, reduces medication compliance, impedes access to treatments, impairs risk communication (limiting prevention efforts among the most vulnerable), and, based on the scant research conducted on outcomes, appears to adversely affect medical outcomes. Low numeracy is also associated with greater susceptibility to extraneous factors (i.e., factors that do not change the objective numerical information). That is, low numeracy increases susceptibility to effects of mood or bow information is presented (e.g., as frequencies vs. percentages) and to biases in judgment and decision making (e.g., framing and ratio bias effects). Much of this research is not grounded in empirically supported theories of numeracy or mathematical cognition, which are crucial for designing evidence-based policies and interventions that are effective in reducing risk and improving medical decision making. To address this gap, we outline four theoretical approaches (psychophysical, computational, standard dual-process, and fuzzy trace theory), review their implications for numeracy, and point to avenues for future research. Keywords: risk perception, risk communication, mathematical cognition, intuition, dual processes DOI: 10.1037/a0017327

Abstract. Little is known about how best to teach mathematics to students with learning disabilities. This study explored the performance and self-reported calculator use of 13 sixth-grade and 15 seventh-grade [...]

The goal of the current research was to explore whether monkeys possess conceptual precursors necessary for understanding zero. We trained rhesus monkeys on a nonsymbolic numerical matching-to-sample task, and on a numerical ordering task. We then introduced nondifferentially reinforced trials that contained empty sets to determine whether monkeys would treat empty sets as numerical values. All monkeys successfully matched and ordered the empty sets without any training. Accuracy showed distance effects, indicating that they treated empty sets as values on a numerical continuum. Keywords: zero, empty sets, numerical cognition, distance effects, numerical continuum

How simple division strategies develop over a short period of time was examined with a microgenetic study. Grade 5 students (mean age = 10 years, 3 months) solved simple division problems in 8 weekly sessions. Performance improved with faster and more accurate responses across the study. Consistent with R. S. Siegler's (1996) overlapping waves model, strategies varied in their use. Direct retrieval increased, retrieval of multiplication facts remained steady, and addition facts, derived facts, and special tricks marginally decreased. Consistent with previous research, multiplication fact retrieval was the most common strategy, although it was slower and more error prone than direct retrieval. Strategy variability within and across individuals was striking across all of the sessions and underscores Siegler's (1996) assertion that development is in a constant transitional state. Keywords: arithmetic, division, strategy, microgenetic Cette etude microgenetique examine comment les strategies de division simple se developpent sur une courte periode. Des eleves de [5.sup.e] annee (age moyen = 10 ans et 3 mois) ont resolu des problemes de division simple durant huit sessions hebdomadaires. La performance s'est amelioree au fil de l'etude, les reponses etant plus rapides et plus precises. Conformement au modele de chevauchement ondulatoire de Siegler (1996), une variation quant a l'utilisation des strategies a ete observee. La recuperation directe a augmente, la recuperation des tables de multiplication est demeuree constante, et le recours aux tables d'addition et derivees, ainsi que les trucs speciaux ont marginalement diminue. Conformement aux recherches anterieures, la recuperation des tables de multiplication a ete la strategie la plus commune, bien qu'elle ait ete plus lente et associee a plus d'erreur que la recuperation directe. La variabilite des strategies intra et inter individuelles etait marquante durant toutes les sessions et appuie la proposition de Siegler (1996) selon laquelle le developpement est constamment dans un etat transitoire. Mots-cles : arithmetique, division, strategie, microgenetique

The relationships of two measures of quantitative estimation skill, computational estimation and numerosity estimation, were examined with the 16 PF Global Factors corresponding to the Big Five personality dimensions in a sample of 80 college students. Correlations of Computational Estimation with the 16 PF scores confirmed findings from a previous study which used the NEO-Five Factor Inventory as a measure of the Big Five, specifically showing that Computational Estimation correlated significantly with the 16 PF's Independence factor. Numerosity Estimation did not correlate significantly with any of the 16 PF Global Factors or with any of the ability measures (Computational Estimation, SAT Verbal, SAT Mathematics, and self-rated mathematics ability) used in the study.

Young and older adults solved complex addition problems such as 40 + 56. Verbal protocols, solution times, and percentage errors documented strategy repertoire and strategy selection in young and older participants and age-related differences in complex arithmetic performance. Both young and older adults used a set of 9 strategies to solve complex addition problems, although many older adults used a smaller strategy repertoire. The data also showed age-related differences in strategy execution and strategy selection. We discuss general implications of the present data to further our understanding of complex arithmetic and the role of strategic variations during aging.

The mathematical performance of 182 third and fourth graders in 8 different areas of mathematics was examined. The children belonged to 4 achievement groups: children with mathematic difficulties (MD only), children with both mathematic and reading difficulties (MD-RD), children with reading difficulties (RD only), and normally achieving children (control group). Both MD groups performed worse than the normally achieving children in all but 1 area, place value knowledge. The MD-only and the MD-RD children performed equally in all areas of mathematics. The RD-only group performed at the same level as the control group on all areas of mathematics. The study provides further evidence that fact retrieval deficits are a cardinal characteristic of children with MD. The MD children's substantial difficulties with mathematic word problem solving can be attributed to several processes involved in problem solving. Besides poor skills in multidigit calculation, arithmetic fact retrieval, and poor understanding of calculation principles, children with MD might have deficits related to specific problem-solving processes such as establishing a problem representation and developing a solution plan. Keywords: mathematical difficulties, reading difficulties, calculation, mathematic problem solving, telling time

The purpose of this study was to explore patterns of difficulty in 2 domains of mathematical cognition: computation and problem solving. Third graders (n = 924; 47.3% male) were representatively sampled from 89 classrooms; assessed on computation and problem solving; classified as having difficulty with computation, problem solving, both domains, or neither domain; and measured on 9 cognitive dimensions. Difficulty occurred across domains with the same prevalence as difficulty with a single domain; specific difficulty was distributed similarly across domains. Multivariate profile analysis on cognitive dimensions and chi-square tests on demographics showed that specific computational difficulty was associated with strength in language and weaknesses in attentive behavior and processing speed; problem-solving difficulty was associated with deficient language as well as race and poverty. Implications for understanding mathematics competence and for the identification and treatment of mathematics difficulties are discussed. Keywords: calculations, word problems, cognitive predictors, mathematics

Abstract Perceived self-efficacy beliefs have been found to be a strong predictor of mathematical performance while problem posing is considered fundamental in mathematical learning. In this study we examined the […]

Recent findings concerning mathematics assessment indicate that students in Japan consistently score above international averages. Researchers have examined specific mathematics beliefs and instructional strategies associated with mathematics achievement for students in Japan. This study examined relationships among self-beliefs, classroom instructional strategies, and mathematics achievement for a large national sample of students (N = 4,207) from the TIMSS 2003 international sample of fourth graders in Japan. Several significant relationships between mathematics beliefs and test scores were found; a number of classroom teaching strategies were also significantly associated with test scores. However, multiple regression using the complete set of five mathematics beliefs and five instructional strategies explained only 25.1% of the variance in mathematics achievement test scores.

Automatic processing of 2-digit numbers was demonstrated using the size congruency effect (SiCE). The SiCE indicates the processing of the irrelevant (numerical) dimension when 2 digits differing both numerically and physically are compared on the relevant (physical) dimension. The SiCE was affected by the compatibility between unit and decade digits but was unaffected by the global magnitude of the numbers. Together these results suggest automatic processing of the magnitudes of the components of the 2-digit numbers but not of whole numbers. Finally, the SiCE was affected more by the magnitude of the decade digits compared with the unit digits, indicating that the syntactic roles of the digits were represented. The implications of these results for understanding the numerical representations are discussed. Keywords: automaticity, numerical processing, two-digit numbers, size congruency effect

A current debate is whether increased looking time in infancy is related to violation of expectations. In this study, 6- to 9-month-old infants' brain activity was analyzed during presentation of correct and incorrect solutions to simple arithmetic equations [(e.g., presentation of 1 + 1; one doll on a TV monitor, with another doll added from behind a screen, followed by a solution of 2 (correct) or 1 (incorrect)]. Infants looked longer at incorrect solutions than at correct ones. Event-related potentials, time-locked to the presentation of the solution, also differed between conditions, with greater negative activity for the incorrect solution condition. Spectral analysis showed a similar pattern to that of adults observing correct and incorrect arithmetical equations. These findings show (i) that the brain network involved in error detection can be identified in infancy and (ii) that this network can support an association between looking time and violation of expectations. early arithmetic | event-related potential | error detection | executive attention | self-regulation

The aim of this study was to test the hypothesis of a complex encoding of numbers according to which each numerical processing requires a specific representational format for input. In three experiments, adult participants were given two numbers presented successively on screen through a self-presentation procedure after being asked to add, to subtract, or to compare them. We considered the self-presentation time of the first number as reflecting the complexity of the encoding for a given planned processing. In line with Dehaene's triple-code model, self-presentation times were longer for additions and subtractions than for comparisons with two-digit numbers but longer for subtractions than for additions and comparisons with one-digit numbers. The implications of these results for different theories of number processing are discussed.

To purchase or authenticate to the full-text of this article, please visit this link: http://dx.doi.org/10.1111/j.1467-8624.2006.00902.x Byline: Matthew Saxton (1), Kadir Cakir (2) Abstract: Factors affecting performance on base-10 tasks were investigated in a series of four studies with a total of 453 children aged 5-7 years. Training in counting-on was found to enhance child performance on base-10 tasks (Studies 2, 3, and 4), while prior knowledge of counting-on (Study 1), trading (Studies 1 and 3), and partitioning (Studies 1 and 4) were associated with enhanced base-10 performance. It emerged that procedural knowledge of counting-on, trading, and partitioning can lead to improvements in procedural knowledge of the base-10 system. The findings lend support to the model of iterative development of conceptual and procedural knowledge advanced by Rittle-Johnson et al. (2001). Author Affiliation: (1)Institute of Education, University of London (2)University of Mersin Article note: Correspondence concerning this article should be addressed to Matthew Saxton, School of Psychology and Human Development, Institute of Education, University of London, 25 Woburn Square, London, WC1H 0AA. Electronic mail may be sent to M.Saxton@ioe.ac.uk.

L. L. Bonatti, M. Pena, M. Nespor, and J. Mehler (2006) argued that P. Perruchet, M. D. Tyler, N. Galland, and R. Peereman (2004) confused the notions of segmentation and generalization by ignoring the evidence for generalization in M. Pena, L. L. Bonatti, M. Nespor, and J. Mehler (2002). In this reply, the authors reformulate and complement their initial arguments, showing that their way of dealing with segmentation and generalization is not due to confusion or ignorance but rather to the fact that the tests used in Pena et al. make it likely that neither segmentation nor generalization were captured in their experiments. Finally, the authors address the challenge posed by Pena et al. of accounting for the whole pattern of their results without invoking rule-based, algebraic-like computations. Keywords: statistical computation, rule-based computation, associative learning, language

The identical elements (IE) model of arithmetic fact representation (Rickard, 2005; Rickard & Bourne, 1996) was developed and tested with multiplication and division. In Experiment 1, we demonstrated that the model also applies to addition and subtraction by examining transfer of response time (RT) savings between prime and probe problems tested in the same block of trials. As is predicted by the IE model, there were equivalent probe RT savings for addition with identical repetition (prime 6 + 9 [right arrow] probe 6 + 9) or an order change (9 + 6 [right arrow] 6 + 9), but much greater savings for subtraction with identical repetition (15 - 6 [right arrow] 15 - 6) than with an order change (15 - 9 [right arrow] 15 - 6), and no savings with an operation change (15 - 9 [right arrow] 6 + 9 or 6 + 9 [right arrow] 15 - 6). In Experiment 2, we examined transfer in simple multiplication and division and demonstrated symmetrical transfer between operations. Cross-operation RT savings were eliminated, however, when the RT analysis included only trials on which both the prime and the probe problems were reportedly solved by direct retrieval. An IE model extended to accommodate savings associated with procedural strategies provides a coherent account of facilitative transfer effects in simple arithmetic.

Business and economics majors (N = 146) were tested on the D'Amore Test of Elementary Arithmetic, which employs third-grade test items from 1932. Only 40% of the subjects passed the test by answering 10 out of 10 items correctly. Self-predicted scores were a good predictor of actual scores, but performance was not associated with demographic variables, grades in calculus courses, liking for science or computers, or mathematics anxiety. Scores decreased over the subjects' initial years on campus. The hardest test item, with an error rate of 23%, required the subject to evaluate (36 x 7) + (33 x 7). The results are similar to those of Standing in 2006, despite methodological changes intended to maximize performance.

Byline: Helen Cruz (1) Keywords: philosophy of mathematics; cognitive science; animal cognition; cognitive archaeology; evolution Abstract: In the past decades, recent paradigm shifts in ethology, psychology, and the social sciences have given rise to various new disciplines like cognitive ethology and evolutionary psychology. These disciplines use concepts and theories of evolutionary biology to understand and explain the design, function and origin of the brain. I shall argue that there are several good reasons why this approach could also apply to human mathematical abilities. I will review evidence from various disciplines (cognitive ethology, cognitive psychology, cognitive archaeology and neuropsychology) that suggests that the human capacity for mathematics is a category-specific domain of knowledge, hard-wired in the brain, which can be explained as the result of natural selection. Author Affiliation: (1) Centrum voor Logica en Wetenschapsfilosofie, Vrije Universiteit Brussel, Pleinlaan 2, B-1050, Brussels, Belgium Article History: Registration Date: 08/11/2004

The size congruity effect is the interesting result that comparisons of the sizes of the physical formats in which numerals appear are affected by the numerical magnitudes of the respective numerals. We demonstrated that separating the physical and the numerical attributes in space leaves the effect unchanged. We then applied the spatially separated version to two-digit numerals and showed the effect to be comparable to that obtained with single numerals. We showed further that the effect is sensitive to the relative salience of the numeric and physical dimensions, to the extent that when the latter is the more salient dimension, a reverse effect obtains by which physical size interferes with number comparison. The results can be explained by a relative speed of processing account, but they are also compatible with an attention account that does not appeal to the notion of automaticity.

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 [micro] (reflecting shift) and [tau] (reflecting skew) confirmed that retrieval trials contributed predominantly to [micro], 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 [micro] 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.

We used the choice/no-choice methodology in two experiments to examine patterns of strategy selection and execution in groups of undergraduates. Comparisons between choice and no-choice trials revealed three groups. Some participants (good retrievers) were consistently able to use retrieval to solve almost all arithmetic problems. Other participants (perfectionists) successfully used retrieval substantially less often in choice-allowed trials than when strategy choices were prohibited. Not-sogood retrievers retrieved correct answers less often than the other participants in both the choiceallowed and no-choice conditions. No group differences emerged with respect to time needed to search and access answers from long-term memory; however, not-so-good retrievers were consistently slower than the other subgroups at executing fact-retrieval processes that are peripheral to memory search and access. Theoretical models of simple arithmetic, such as the Strategy Choice and Discovery Simulation (Shrager & Siegler, 1998), should be updated to include the existence of both perfectionist and not-so-good retriever adults.

In this study, we examined how math identity moderates women's response to gender-related stereotypes in the domain of mathematics. Male and female college students with varying degrees of math identification took a challenging math test with a gender-related stereotype either activated (i.e., stereotype threat) or nullified. Consistent with previous research, women performed worse than men in the stereotype threat condition, but equal to men in the stereotype nullification condition when performance was adjusted for math SAT scores. Moreover, when faced with stereotype threat, high math-identified women discounted the validity of the test more than did less math-identified women or men in general. We discuss potential benefits and drawbacks of a discounting strategy for women who are highly identified with math. KEY WORDS: stereotype threat; gender; identity; mathematics.

This article considers 3 claims that cognitive sex differences account for the differential representation of men and women in high-level careers in mathematics and science: (a) males are more focused on objects from the beginning of life and therefore are predisposed to better learning about mechanical systems; (b) males have a profile of spatial and numerical abilities producing greater aptitude for mathematics; and (c) males are more variable in their cognitive abilities and therefore predominate at the upper reaches of mathematical talent. Research on cognitive development in human infants, preschool children, and students at all levels fails to support these claims. Instead, it provides evidence that mathematical and scientific reasoning develop from a set of biologically based cognitive capacities that males and females share. These capacities lead men and women to develop equal talent for mathematics and science. Keywords: sex differences, mathematics, science, cognitive development, spatial reasoning

The number skills of groups of 7- to 9-year-old children with specific language impairment (SLI) attending mainstream or special schools were compared with an age and nonverbal reasoning matched group (age control [AC]) and with a younger group matched on oral language comprehension. The SLI groups performed below the AC group on every skill. They also showed lower working memory functioning and had received lower levels of instruction. Nonverbal reasoning, working memory functioning, language comprehension, and instruction accounted for individual variation in number skills to differing extents depending on the skill. These factors did not explain the differences between SLI and AC groups on most skills. Keywords: number development, working memory, specific language impairment, instruction