General Mathematics Achievement Research Articles

Mathematics achievement in the last year of primary school. Longitudinal relationship with general cognitive skills and prior mathematics knowledge

The aim of this study was to analyze the joint, relative, and unique predictive value of students’ prior knowledge of mathematics (knowledge of fractions and ability to divide natural numbers) and general cognitive ability (fluid intelligence and working memory) upon general mathematics achievement in the last year of primary school. Seventy-five students participated (M age = 11.2 years old, SD = 0.40). Hierarchical regression analysis showed that the ability to divide and fractions knowledge accounted for 41% of the variance in mathematics achievement, both acting as significant predictors. By incorporating working memory and fluid intelligence into the model, fraction knowledge showed to be no longer a significant predictor. These general cognitive skills explained an additional 8% of the variance in mathematics knowledge, both being significant predictors and contributing to mathematics achievement in a unique way. The implications of these results for mathematics teaching are discussed.

Keep DRAGging ON: Is solving more problems in DragonBox 12+ associated with higher mathematical performance during the COVID‐19 pandemic?

AbstractPrior research has shown that game‐based learning tools, such as DragonBox 12+, support algebraic understanding and that students' in‐game progress positively predicts their later performance. Using data from 253 seventh‐graders (12–13 years old) who played DragonBox as a part of technology intervention, we examined (a) the relations between students' progress within DragonBox and their algebraic knowledge and general mathematics achievement, (b) the moderating effects of students' prior performance on these relations and (c) the potential factors associated with students' in‐game progress. Among students with higher prior algebraic knowledge, higher in‐game progress was related to higher algebraic knowledge after the intervention. Higher in‐game progress was also associated with higher end‐of‐year mathematics achievement, and this association was stronger among students with lower prior mathematics achievement. Students' demographic characteristics, prior knowledge and prior achievement did not significantly predict in‐game progress beyond the number of intervention sessions students completed. These findings advance research on how, for whom and in what contexts game‐based interventions, such as DragonBox, support mathematical learning and have implications for practice using game‐based technologies to supplement instruction. Practitioner notesWhat is already known about this topic DragonBox 12+ may support students' understanding of algebra but the findings are mixed. Students who solve more problems within math games tend to show higher performance after gameplay. Students' engagement with mathematics is often related to their prior math performance. What this paper adds For students with higher prior algebraic knowledge, solving more problems in DragonBox 12+ is related to higher algebraic performance after gameplay. Students who make more in‐game progress also have higher mathematics achievement, especially for students with lower prior achievement. Students who spend more time playing DragonBox 12+ make more in‐game progress; their demographic, prior knowledge and prior achievement are not related to in‐game progress. Implications for practice and/or policy DragonBox 12+ can be beneficial as a supplement to algebra instruction for students with some understanding of algebra. DragonBox 12+ can engage students with mathematics across achievement levels. Dedicating time and encouraging students to play DragonBox 12+ may help them make more in‐game progress, and in turn, support math learning.

Expectancy-value beliefs optimize mathematics achievement and learning strategy use: a bifactor approach (Las creencias sobre los valores y las expectativas optimizan el rendimiento matemático y el uso de estrategias de aprendizaje: un enfoque bifactorial)

ABSTRACT Expectancy-value theory mostly focuses on how expectancy and value beliefs are associated with academic achievement. Both expectancy and value beliefs are multidimensional and can be further decomposed into general and specific factors. However, few studies have explored the associations of these general and specific factors with academic achievement and learning strategy use. This study aimed to investigate the bifactor structure of expectancy and value beliefs and investigate the linkages between general and specific expectancy-value factors and mathematics achievement and learning strategy use. Based on the expectancy and value dimensions measured by the Motivated Strategies for Learning Questionnaire (MSLQ), bifactor models were used to analyse the data collected from 563 senior secondary school students in China. Three key findings emerged: (1) both expectancy and value beliefs could be disentangled into general and specific factors; (2) general factors were significantly associated with mathematics achievement and learning strategy use; and (3) the specific factors were positively associated with mathematics learning strategy use but not mathematics achievement. Methodological and practical implications are discussed.

Graded human sensitivity to geometric and topological concepts

In a seminal study, Dehaene et al. (2006) found evidence that adults and children are sensitive to geometric and topological (GT) concepts using a novel odd-one-out task. However, performance on this task could reflect more general cognitive abilities than intuitive knowledge of GT concepts. Here, we developed a new 2-alternative forced choice (2-AFC) version of the original task where chance represents a higher bar to clear (50% vs. 16.67%) and where the role of general cognitive abilities is minimized. Replicating the original finding, American adult participants showed above-chance sensitivity to 41 of the 43 GT concepts tested. Moreover, their performance was not strongly driven by two general cognitive abilities, fluid intelligence and mental rotation, nor was it strongly associated with mathematical achievement as measured by ACT/SAT scores. The performance profile across the 43 concepts as measured by the new 2-AFC task was found to be highly correlated with the profiles as measured using the original odd-one-out task, as an analysis of data sets spanning populations and ages revealed. Most significantly, an aggregation of the 43 concepts into seven classes of GT concepts found evidence for graded sensitivity. Some classes, such as Euclidean geometry and Topology, were found to be more domain-specific: they “popped out” for participants and were judged very quickly and highly accurately. Others, notably Symmetry and Geometric transformations, were found to be more domain-general: better predicted by participants’ general cognitive abilities and mathematical achievement. These results shed light on the graded nature of GT concepts in humans and challenge computational models that emphasize the role of induction.

Taking a Closer Look: The Relationship between Pre-School Domain General Cognition and School Mathematics Achievement When Controlling for Intelligence

Intelligence, as well as working memory and attention, affect the acquisition of mathematical competencies. This paper aimed to examine the influence of working memory and attention when taking different mathematical skills into account as a function of children’s intellectual ability. Overall, intelligence, working memory, attention and numerical skills were assessed twice in 1868 German pre-school children (t1, t2) and again at 2nd grade (t3). We defined three intellectual ability groups based on the results of intellectual assessment at t1 and t2. Group comparisons revealed significant differences between the three intellectual ability groups. Over time, children with low intellectual ability showed the lowest achievement in domain-general and numerical and mathematical skills compared to children of average intellectual ability. The highest achievement on the aforementioned variables was found for children of high intellectual ability. Additionally, path modelling revealed that, depending on the intellectual ability, different models of varying complexity could be generated. These models differed with regard to the relevance of the predictors (t2) and the future mathematical skills (t3). Causes and conclusions of these findings are discussed.

Math anxiety differentially impairs symbolic, but not nonsymbolic, fraction skills across development.

Although important for the acquisition of later math skills, fractions are notoriously difficult. Previous studies have shown that higher math anxiety (MA) is associated with lower performance in symbolic fraction tasks in adults and have suggested that MA may negatively impact the acquisition of fractions in children. However, the effects of MA on fraction skills in school-aged children remain underexplored. We, therefore, investigated the impact of MA on the performance of younger (second and third graders) and older (fifth and sixth graders) children in math fluency (MF), written calculation, fraction knowledge (FK), and symbolic fraction and nonsymbolic ratio processing. On the basis of our prior work suggesting a perceptual foundation for fraction processing, we predicted that symbolic, but not nonsymbolic, math skills (especially fractions) would be impaired by MA. As predicted, higher MA was associated with lower performance in general mathematics achievement and symbolic fraction tasks, but nonsymbolic ratio processing was not affected by MA in either age group. Furthermore, working memory capacity partially mediated the effects of MA on general mathematics achievement, FK, and symbolic fraction processing. These results suggest that understanding the bidirectional interactions between MA and fractions may be important for helping children acquire these critical skills.

Predicting adaptive expertise with rational number arithmetic.

BackgroundAdaptive expertise is a highly valued outcome of mathematics curricula. One aspect of adaptive expertise with rational numbers is adaptive rational number knowledge, which refers to the ability to integrate knowledge of numerical characteristics and relations in solving novel tasks. Even among students with strong conceptual and procedural knowledge of rational numbers, there are substantial individual differences in adaptive rational number knowledge.AimsWe aimed to examine how a wide range of domain‐general and mathematically specific skills and knowledge predicted different aspects of rational number knowledge, including procedural, conceptual, and adaptive rational number knowledge.Sample173 6th and 7th grade students from a school in the southeastern US (51% female) participated in the study.MethodsAt three time points across 1.5 years, we measured students’ domain‐general and domain‐specific skills and knowledge. We used multiple hierarchal regression analysis to examine how these predictors related to rational number knowledge at the third time point.ResultPrior knowledge of rational numbers, general mathematical calculation knowledge, and spontaneous focusing on multiplicative relations (SFOR) tendency uniquely predicted adaptive rational number knowledge, after taking into account domain‐general and mathematically specific skills and knowledge. Although conceptual knowledge of rational numbers and general mathematical achievement also predicted later conceptual and procedural knowledge of rational numbers, SFOR tendency did not.ConclusionResults suggest expanding investigations of mathematical development to also explore different features of adaptive expertise as well as spontaneous mathematical focusing tendencies.

Mathematics vocabulary differentially predicts mathematics achievement in eighth grade higher- versus lower- achieving students: Comparisons across two countries

This study explored the relationship between mathematics vocabulary and mathematics achievement, controlling general vocabulary, for eighth graders (ages 13 to 14) from the United States (US; n = 89) and Turkey (n = 188). The mathematics achievement of Turkish students fell into higher- and lower-achieving groups, with students in the higher-achieving group showing achievement levels like their US peers. For US students and the corresponding higher-achieving Turkish students, mathematics vocabulary predicted mathematics achievement and mediated the relation between general vocabulary and mathematics achievement. For the lower-achieving Turkish students, only general vocabulary predicted mathematics achievement. The pattern of results extends findings from studies of younger students and is interpreted in terms of the domain-general and domain-specific contributions to mathematics achievement.

The Role of the Inhibition of Natural Number Based Reasoning and Strategy Switch Cost in a Fraction Comparison Task

Previous research amply showed the importance of a good fraction understanding but also people’s lack of fraction understanding. It is therefore important to investigate the cognitive processes that underlie reasoning with fractions. The present study investigated the role of inhibition and switch costs in fraction comparison tasks. Participants solved a fraction comparison task that alternated between 4 items congruent and 4 items incongruent with natural number reasoning. This allowed to not only investigate congruency switch effects, but also inhibition, given that inhibition was experimentally increased by the prolonged exposure to incongruent trials. Based on data of seventh graders, the present study showed that inhibition does not only play a role in learners’ general mathematics achievement, but also in specific areas of mathematics, such as fractions. Moreover, a switch cost was found in the lower accuracy rates and higher reaction times needed to correctly solve switch items compared to non-switch items.

A Meta‐Analysis of the Relationship between Mathematics Achievement and Creativity

AbstractIn the current investigation, research studies (n = 30) conducted from 1965 to 2018 were synthesized to examine the mathematics achievement–creativity relationship and the range of factors that could have moderated this relationship. The meta‐analysis included sample size of 11,418 and revealed that there was an overall moderate and statistically significant (r = 0.47, p < .05) positive correlation between mathematics achievement and creativity. The investigation further revealed that the relationship was constant across sample location but stronger when the relationship was examined with mathematical creativity (r = 0.53, p < .05) than with general creativity. Findings from studies conducted after 2000 indicated a stronger relationship between general creativity and mathematics achievement than did findings from studies conducted before 2000. The relationship was the strongest when creativity was measured through the Mathematical Creativity Test (MCT) compared to other measures of general or mathematical creativity. Lastly, the relationship was stronger for elementary and middle school students than for high school students. Possible explanations for these findings in relation to existing theories and the implications for future research are provided.

Academic achievement and schizophrenia: a systematic meta-analysis.

Cognitive impairments in childhood are associated with increased risk of schizophrenia in later life, but the extent to which poor academic achievement is associated with the disorder is unclear. Major databases were searched for articles published in English up to 31 December 2019. We conducted random-effects meta-analyses to: (1) compare general academic and mathematics achievement in youth who later developed schizophrenia and those who did not; (2) to examine the association between education level achieved and adult-onset schizophrenia; and, (3) compare general academic achievement in youth at-risk for schizophrenia and typically developing peers. Meta-regression models examined the effects of type of academic assessment, educational system, age at assessment, measurement of educational level attained, school leaving age, and study quality on academic achievement and education level among individuals with schizophrenia. Meta-analyses, comprising data of over four million individuals, found that: (1) by age 16 years, those who later developed schizophrenia had poorer general academic (Cohen's d = -0.29, p ⩽ 0.0001) and mathematics achievement (d = -0.23, p = 0.01) than those who did not; (2) individuals with schizophrenia were less likely to enter higher education (odds ratio = 0.49, p ⩽ 0.0001); and, (3) youth reporting psychotic-like experiences and youth with a family history of schizophrenia had lower general academic achievement (d = -0.54, p ⩽ 0.0001; d = -0.39, p ⩽ 0.0001, respectively). Meta-regression analyses determined no effect modifiers. Despite significant heterogeneity across studies, various routinely collected indices of academic achievement can identify premorbid cognitive dysfunction among individuals who are vulnerable for schizophrenia, potentially aiding the early identification of risk in the population.

Effects of Number Sense-Based Instruction on Sixth-Grade Students’ Self-Efficacy and Performance

In this study, the effects of number sense-based instruction on students’ mathematics-related self-efficacy and performance were studied. Self-efficacy relates to both mathematics and number sense, while student performance consists of their number sense, problem solving, general mathematical achievement, and recognition of mathematics in daily life. This was an experimental study where both quantitative research and pretest/posttest control group designs were utilized. The fifteen-week study was conducted with a group of sixth-grade students attending a public school located in the Cankaya district in Ankara Province, Turkey, including thirteen weeks to implement the instructional plan and two weeks for employing the scales. During this process, while students in the test group were introduced to number sense-based training, the control group adhered to the regular syllabus where teachers based most of their instruction on textbooks. The study, beyond its demonstration of number sense, also evaluates number sense development in the context of an instructional plan designed for long-term implementation and contributes to the study of changes in other variables involved in development. Throughout the study, the Number Sense Test was used to identify the number sense of students. The Mathematics Self-Efficacy Perception Scale was used to assess their self-efficacy. The Number Sense Self-Efficacy Scale was used to identify their number sense self-efficacy. The Mathematics in Daily Life Survey was used to assess their recognition of mathematics in daily life. The Problem-Solving Test was used to assess their achievement in solving problems, and end-of-term grading was used to identify their overall mathematical achievement. Data were analyzed with multivariate analysis of variance (MANOVA) tests using SPSS 15.0 (SPSS Inc., Chicago, Il). The results of the analyses revealed that students who received instruction based on number sense experienced significant improvement in their number sense, their recognition of mathematics in daily life, and their problem-solving achievements compared to students who received only regular instruction. Number sense-based instruction did not create any significant changes in mathematics self-efficacy, number sense self-efficacy, or the overall mathematical achievements of the students.

Children’s number sequences as predictors of later mathematical development

Number sequences are defined in terms of children’s abilities to construct and transform units. Children operating with the initial number sequence (INS) construct units of 1. They construct other numbers as strings of 1s and can count on, by 1s, from one number to a subsequent number. A critical benchmark in children’s further numerical development is the construction of units of units, or composite units. This development corresponds to the tacitly nested number sequence (TNS) wherein numbers are nested within other numbers. We report on a large-scale quantitative study of children’s available number sequences and their relationships with later mathematical development, such as multiplicative reasoning and fractions knowledge. This study involved 5747 children from three cohorts surveyed at the beginning of second grade in 2013, 2014, and 2015. We document strong relationships between children’s early construction of an INS and a TNS and the likelihood of their later development of multiplicative reasoning, a measurement meaning of fractions, and general mathematics achievement, while controlling for rote computational skills. Implications for teacher instruction are discussed.

Children’s reasoning about decimals and its relation to fraction learning and mathematics achievement.

Reasoning about numerical magnitudes is a key aspect of mathematics learning. Most research examining the relation of magnitude understanding to general mathematics achievement has focused on whole number and fraction magnitudes. The present longitudinal study (N = 435) used a 3-step latent class analysis to examine reasoning about magnitudes on a decimal comparison task in 4th grade, before systematic decimals instruction. Three classes of response patterns were identified, indicating empirically distinct levels of decimal magnitude understanding. Class 1 students consistently gave correct responses, suggesting that they understood decimal properties even before systematic decimal instruction. Class 2 students were accurate when a 0 immediately followed the decimal, but were inaccurate when a zero was added to the end of the decimal string, suggesting a partial understanding of place value; their performance was also negatively influenced by a whole number bias. Class 3 students showed misunderstanding of both place value and a whole number bias. Class membership accurately predicted 6th grade mathematics achievement, after controlling for whole number and fraction magnitude understanding as well as demographic and cognitive factors. Taken together, the findings suggest students may benefit from instruction that emphasizes decimal properties earlier in school. (PsycINFO Database Record (c) 2019 APA, all rights reserved)

Who uses more strategies? Linking mathematics anxiety to adults’ strategy variability and performance on fraction magnitude tasks

ABSTRACTAdults use a variety of strategies to reason about fraction magnitudes, and this variability is adaptive. In two studies, we examined the relationships between mathematics anxiety, working memory, strategy variability and performance on two fraction tasks: fraction magnitude comparison and estimation. Adults with higher mathematics anxiety had lower accuracy on the comparison task and greater percentage absolute error (PAE) on the estimation task. Unexpectedly, mathematics anxiety was not related to variable strategy use. However, variable strategy use was linked to more accurate magnitude comparisons, especially among adults with lower working memory performance or those who use mathematics less frequently, as well as lower PAE on the estimation task. These findings shed light on the role of strategy variability in fraction problem solving and demonstrate a link between mathematics anxiety and fraction magnitude reasoning, a key predictor of general mathematics achievement.

Cognitive predictors of children's development in mathematics achievement: A latent growth modeling approach.

Research has identified various domain-general and domain-specific cognitive abilities as predictors of children's individual differences in mathematics achievement. However, research into the predictors of children's individual growth rates, namely between-person differences in within-person change in mathematics achievement is scarce. We assessed 334 children's domain-general and mathematics-specific early cognitive abilities and their general mathematics achievement longitudinally across four time-points within the first and second grades of primary school. As expected, a constellation of multiple cognitive abilities contributed to the children's starting level of mathematical success. Specifically, latent growth modeling revealed that WM abilities, IQ, counting skills, nonsymbolic and symbolic approximate arithmetic and comparison skills explained individual differences in the children's initial status on a curriculum-based general mathematics achievement test. Surprisingly, however, only one out of all the assessed cognitive abilities was a unique predictor of the children's individual growth rates in mathematics achievement: their performance in the symbolic approximate addition task. In this task, children were asked to estimate the sum of two large numbers and decide if this estimated sum was smaller or larger compared to a third number. Our findings demonstrate the importance of multiple domain-general and mathematics-specific cognitive skills for identifying children at risk of struggling with mathematics and highlight the significance of early approximate arithmetic skills for the development of one's mathematical success. We argue the need for more research focus on explaining children's individual growth rates in mathematics achievement.

Fraction magnitude understanding and its unique role in predicting general mathematics achievement at two early stages of fraction instruction.

Prior studies on fraction magnitude understanding focused mainly on students with relatively sufficient formal instruction on fractions whose fraction magnitude understanding is relatively mature. This study fills a research gap by investigating fraction magnitude understanding in the early stages of fraction instruction. It extends previous findings to children with limited and primary formal fraction instruction. Thirty-five fourth graders with limited fraction instruction and forty fourth graders with primary fraction instruction were recruited from a Chinese primary school. Children's fraction magnitude understanding was assessed with a fraction number line estimation task. Approximate number system (ANS) acuity was assessed with a dot discrimination task. Whole number knowledge was assessed with a whole number line estimation task. General reading and mathematics achievements were collected concurrently and 1year later. In children with limited fraction instruction, fraction representation was linear and fraction magnitude understanding was concurrently related to both ANS and whole number knowledge. In children with primary fraction instruction, fraction magnitude understanding appeared to (marginally) significantly predict general mathematics achievement 1year later. Fraction magnitude understanding emerged early during formal instruction of fractions. ANS and whole number knowledge were related to fraction magnitude understanding when children first began to learn about fractions in school. The predictive value of fraction magnitude understanding is likely constrained by its sophistication level.

Erratum to: The effect of inhibitory control on general mathematics achievement and fraction comparison in middle school children

Individual differences in inhibitory control have been shown to relate to general mathematics achievement, but whether this relation varies for specific areas within mathematics is a question that remains open. Here, we evaluate if inhibitory processes play a specific role in the particular case of fraction comparison, where learners must ignore the potentially misleading information provided by the natural numbers composing fractions (e.g. 2/3 > 4/7 despite 2 < 4 and 3 < 7). To do this, we presented a sample of Chilean children (N = 450) from 5th, 6th, and 7th grade with a numerical comparison task tapping inhibitory and other processes. Results showed that both general math achievement and accuracy in comparing fractions were significantly predicted by inhibition. The former association, however, turned out to mediate the latter one. Another process, related to visual priming, predicted children’s likelihood to answer fraction comparison items focusing exclusively on the fraction components. This relation was, furthermore, not mediated by general math achievement. Altogether, these findings shed light on the mental processes underlying the early stages of the learning of fractions.

Number sense in the transition from natural to rational numbers.

Rational numbers are of critical importance both in mathematics and in other fields of science. However, they form a stumbling block for learners. One widely known source of the difficulty learners have with rational numbers is the natural number bias, that is the tendency to (inappropriately) apply natural number properties in rational number tasks. Still, it has been shown that a good understanding of natural numbers is highly predictive for mathematics achievement in general, and for performance on rational number tasks in particular. In this study, we further investigated the relation between learners' natural and rational number knowledge, specifically in cases where a natural number bias may lead to errors. Participants were 140 sixth graders from six different primary schools. Participants completed a symbolic and a non-symbolic natural number comparison task, a number line estimation task, and a rational number sense test. Learners' natural number knowledge was found to be a good predictor of their rational number knowledge. However, after first controlling for learners' general mathematics achievement, their natural number knowledge only predicted the subaspect of operations with rational numbers. The results of this study suggest that the relation between learners' natural and rational number knowledge can largely be explained by their relation with learners' general mathematics achievement.

The role of physical digit representation and numerical magnitude representation in children’s multiplication fact retrieval

Arithmetic facts, in particular multiplication tables, are thought to be stored in long-term memory and to be interference prone. At least two representations underpinning these arithmetic facts have been suggested: a physical representation of the digits and a numerical magnitude representation. We hypothesized that both representations are possible sources of interference that could explain individual differences in multiplication fact performance and/or in strategy use. We investigated the specificity of these interferences on arithmetic fact retrieval and explored the relation between interference and performance on the different arithmetic operations and on general mathematics achievement. Participants were 79 fourth-grade children (Mage=9.6 years) who completed a products comparison and a multiplication production task with verbal strategy reports. Performances on a speeded calculation test including the four operations and on a general mathematics achievement test were also collected. Only the interference coming from physical representations was a significant predictor of the performance across multiplications. However, both the magnitude and physical representations were unique predictors of individual differences in multiplication. The frequency of the retrieval strategy across multiplication problems and across individuals was determined only by the physical representation, which therefore is suggested as being responsible for memory storage issues. Interestingly, this impact of physical representation was not observed when predicting performance on subtraction or on general mathematical achievement. In contrast, the impact of the numerical magnitude representation was more general in that it was observed across all arithmetic operations and in general mathematics achievement.