Mathematics anxiety and working memory: Longitudinal associations with mathematical performance in Chinese children

The link between mathematics anxiety and mathematical performance in young children remains inconclusive. The present study examined the longitudinal associations between mathematics anxiety and mathematical performance (calculation and story problem solving) in 246 Chinese children followed from second to third grade. Multiple regression analyses showed that mathematics anxiety made independent contributions to mathematical performance beyond non-verbal intelligence, working memory, number skills, general and test anxieties. However, mathematics anxiety does not affect all children and all kinds of mathematical performance equally. Mathematics anxiety has a more pronounced impact on mathematical problems that require more processing resources, as opposed to simple arithmetic problems and straightforward story problems and children who are higher in working memory are more vulnerable to its deleterious impacts.     

The relationship between math anxiety and math achievement in young children is mediated through working memory,not by number sense,and it is not direct

Math anxiety is considered a predictor of math achievement, although the cognitive mechanism whereby math anxiety impairs math achievement is unclear. The paper presents the results of cross-sectional (N = 241) and longitudinal (N = 369) studies conducted among early school-aged children on the cognitive mechanism whereby math anxiety impairs math achievement. The following hypotheses were tested: (1) math anxiety directly affects math achievement; (2) in accordance with processing efficiency and attentional cognitive theories, math anxiety indirectly affects math achievement through working memory; (3) in accordance with the cognitive deficit model, math anxiety indirectly affects math achievement through number sense. The results mostly confirm the mediating role of working memory and undermine the mediating role of number sense and the direct path in the relationship between math anxiety and math achievement. Because previous studies undertaken in adults show the direct path from math anxiety to math achievement and the role of symbolic number processing in explaining the relationship between the two, the methodological and developmental aspects of the obtained results are discussed in the paper.     

Pathways to reading,mathematics, and science: Examining domain-general correlates in young Chinese children

Early competencies in reading, mathematics, and science are associated with later school achievement and adulthood socioeconomic status. This cross-sectional study examined how fundamental domain-general capacities, including language, spatial, and self-regulatory skills, together relate to competencies in reading, mathematics, and science in young Chinese children. A total of 584 Chinese children aged approximately six years were tested individually on their language (receptive vocabulary), spatial (spatial perception, spatial visualization, and mental rotation), and self-regulation (behavioral regulation and working memory) skills, as well as their academic competencies in reading, mathematics, and science. The results showed that vocabulary, spatial, and self-regulatory skills were all associated with Chinese reading, mathematics, and life sciences, whereas only vocabulary was related to earth and physical sciences. The relation between vocabulary and formal mathematics and that between mental rotation and life sciences were found to be stronger in boys than in girls. The findings suggest that foundational domain-general skills may provide the building blocks for children’s academic competencies.     

Epistemic profiles and self-regulated learning: Examining relations in the context of mathematics problem solving

Relations were examined between epistemic profiles, regulation of cognition, and mathematics problem solving. Two hundred sixty-eight students were sampled from undergraduate mathematics and statistics courses. Students completed inventories reflecting their epistemic profiles and learning strategies, and were profiled as rational, empirical, or both. Based on their profiles, 24 students participated in two problem-solving sessions. Episodes were coded for planning, monitoring, control, use of empirical and rational argumentation, and justification for solutions. For both self-reported metacognitive self-regulation and regulation of cognition during problem solving, students profiled as rational had the highest self-reported mean and actual frequency of regulation of cognition compared to students profiled as predominantly empirical. Moreover, students profiled as predominantly rational correctly solved more problems than the other two groups. Finally, students’ approaches to problem solving were consistent with their epistemic profiles. Relations are discussed in the context of various theoretical frameworks.     

Numerical competence in young children and in children with mathematics learning disabilities

A longitudinal study was conducted on 82 children to investigate, firstly the numerical competence of young children and the predictive value of (pre)-numerical tests in kindergarten, and, secondly, whether children's knowledge of the numerical system and representation of the number size is related to their computation and logical knowledge and to their counting skills. In an additional cross-sectional study on 30 children with a clinical diagnosis of mathematical learning disability (MLD) of 8,5 years, age- and ability-matched with 2 × 30 children the same parameters of numerical competence were assessed. The longitudinal data showed individual differences in numerosity, as well as the relationship between a delay in arithmetics in grade l and problems on numerosity in kindergarten. In the cross-sectional results some evidence was found for the independence of numerical abilities in MLD-children. About 13% of them had still severe pre-numerical processing deficits (in number sequence production, cardinality skills and logical knowledge) in grade 3. About 67% had severe difficulties in executing calculation procedures and a lack of conceptual knowledge. A feature of 87% of the MLD-children was severe translation deficits, with a significantly worse knowledge of number words compared with the knowledge of Arab numerals. Finally a severe deficit in subitizing was found to be present in 33% of the MLD children. On a group level the processing deficits were linked to understanding numerosity, since the ability-matched younger children and the MLD-children had the same pre-numerical and numerical profile. Implications for the assessment of mathematical disabilities and the value of TEDI-MATH® as an instrument in this process are discussed.     

First-graders’ knowledge of multiplicative reasoning before formal instruction in this domain

Our study investigated children’s knowledge of multiplicative reasoning (multiplication and division) at the end of Grade 1, just before the start of formal instruction on multiplicative reasoning in Grade 2. A large sample of children (N = 1176) was assessed in a relatively formal test setting, using an online test with 28 multiplicative problems of different types. On average, the children correctly answered more than half (58%) of the problems, including several bare number problems. This indicates that before formal instruction on multiplicative reasoning, children already have a considerable amount of knowledge in this domain, which teachers can build on when teaching them formal multiplication and division. Using analysis of variance and cross-classified multilevel regression analysis, we identified several predictors of children’s pre-instructional multiplicative knowledge. With respect to the characteristics of the multiplicative problems, we found that the problems were easiest to solve when they included a picture involving countable objects, and when the multiplicative situation was of the equal groups semantic structure (e.g., 3 boxes of 4 cookies). Regarding student characteristics, pre-instructional multiplicative knowledge was higher for children with higher-educated parents. Finally, the mathematics textbook used in school appeared to have influenced children’s pre-instructional multiplicative knowledge.     

Mathematics anxiety in young children: Concurrent and longitudinal associations with mathematical performance

This study explored mathematics anxiety in a longitudinal sample of 113 children followed from second to third grade. We examined how mathematics anxiety related to different types of mathematical performance concurrently and longitudinally and whether the relations between mathematics anxiety and mathematical performance differed as a function of working memory. Concurrent analyses indicated that mathematics anxiety represents a unique source of individual differences in children’s calculation skills and mathematical applications, but not in children’s geometric reasoning. Furthermore, we found that higher levels of mathematics anxiety in second grade predicted lower gains in children’s mathematical applications between second and third grade, but only for children with higher levels of working memory. Overall, our results indicate that mathematics anxiety is an important construct to consider when examining sources of individual differences in young children’s mathematical performance. Furthermore, our findings suggest that mathematics anxiety may affect how some children use working memory resources to learn mathematical applications.     

The developmental trajectories of mathematics anxiety: Cognitive,personality, and environmental correlates

Students who are highly anxious about mathematics-related activities generally exhibit lower mathematics achievement and motivation compared to their less anxious counterparts. Despite negative implications of mathematics anxiety (MA) on mathematics learning, there is a paucity of research examining how MA develops over time. Using the Longitudinal Study of American Youth dataset (N = 3116), the present study investigated two main questions regarding the development of MA in secondary school: (1) Is the development of MA characterized by a heterogeneous subset of growth trajectories? (2) How are time-varying personal and environmental factors (e.g., mathematics achievement; perceptions of math teachers) related to specific MA growth trajectories? Student MA was repeatedly assessed in six annual waves spanning across middle and high school. Using growth mixture modeling, we identified four growth trajectories of MA: (1) The non-anxious group that exhibited chronically low MA; (2) The highly anxious group which displayed moderately high MA over time; (3) The resilient group that exhibited high initial MA that steadily decreased over time; and (4) The vulnerable group that reported low initial MA that drastically increased over time. In addition, significant differences in the development of mathematics achievement, personality hardiness, and perceptions of mathematics teachers were found in these four MA groups. Findings highlight heterogeneity in the development of MA, identify middle school as a critical period for MA development, and emphasize the importance of examining developmental changes in cognitive, personality, and environmental factors to help clarify distinct MA trajectories across middle and high school.     

Working memory and mathematics in primary school children: A meta-analysis

Working memory, including central executive functions (inhibition, shifting and updating) are factors thought to play a central role in mathematical skill development. However, results reported with regard to the associations between mathematics and working memory components are inconsistent. The aim of this meta-analysis is twofold: to investigate the strength of this relation, and to establish whether the variation in the association is caused by tests, sample characteristics and study and other methodological characteristics. Results indicate that all working memory components are associated with mathematical performance, with the highest correlation between mathematics and verbal updating. Variation in the strength of the associations can consistently be explained by the type of mathematics measure used: general tests yield stronger correlations than more specific tests. Furthermore, characteristics of working memory measures, age and sample explain variance in correlations in some analyses. Interpretations of the contribution of moderator variables to various models are discussed.     

Statistical reasoning of middle school children engaged in survey inquiry

The case study examined two groups of grade 7 students as they engaged in four inquiry phases: posing a question and collecting, analyzing, and representing data. Previous studies reported analyses of statistical reasoning on a single inquiry phase. Our goal was to identify the modes of statistical reasoning displayed during group discussions in all phases as children designed and conducted their own inquiry. A content analysis of audio and video recorded discussions yielded 10 statistical reasoning modes: six relate to Garfield and Gal’s [Garfield, J., Gal, I. (1999). Teaching and assessing statistical reasoning. In L. V. Stiff, & F. R. Curcio (Eds.), Developing mathematical reasoning in grades K-12. 1999 Yearbook (pp. 207–219). Reston, VA: National Council of Teachers of Mathematics] statistical reasoning types involved in the collection, analysis, and representation of data and four modes deal with an aspect of inquiry not exclusively focused upon in the literature on statistical reasoning—i.e., the problem-posing phase. Although students’ reasoning reflected an incomplete understanding of statistics they serve as building blocks for instruction.     

Teaching children how to include the inversion principle in their reasoning about quantitative relations

The basis of this intervention study is a distinction between numerical calculus and relational calculus. The former refers to numerical calculations and the latter to the analysis of the quantitative relations in mathematical problems. The inverse relation between addition and subtraction is relevant to both kinds of calculus, but so far research on improving children’s understanding and use of the principle of inversion through interventions has only been applied to the solving of a + b − b = ? sums. The main aim of the intervention described in this article was to study the effects of teaching children about the explicit use of inversion as part of the relational calculus needed to solve inverse addition and subtraction problems using a calculator. The study showed that children taught about relational calculus differed significantly from those who were taught numerical procedures, and also that effects of the intervention were stronger when children were taught about relational calculus with mixtures of indirect and direct word problems than when these two types of problem were given to them in separate blocks.     

Examining the dynamics of mathematics anxiety,perceived cost,and achievement: A control-value theory approach

Concerns about the influence of students’ perceived negative consequences of engagement in a task (i.e., cost) on their emotions, motivation, and cognition have increased in the last decade. The use of longitudinal models is needed to provide new insights into the role of perceived cost in mathematics learning. Grounded in the control-value theory, this study examined cross-lagged relations of mathematics anxiety, perceived cost, and mathematics achievement. The participants (N = 335) reported their mathematics anxiety and perceived cost four times during Grades 7 and 8, and their mathematics grades were attained from their school records. Cross-lagged panel model analysis revealed evidence of a long-term positive reciprocal relationship between mathematics anxiety and effort/emotional cost, a gradually diminished relationship between effort/emotional cost and mathematics performance, and a positive achievement to anxiety link during the transition between grade levels. Moreover, mathematics performance is a distal predictor of mathematics anxiety through effort/emotional cost rather than a proximal predictor or an outcome of anxiety. This study also clarified the distinction in the central role of effort/emotional versus opportunity cost in the interrelatedness of mathematics anxiety and performance, where the latter failed to demonstrate significant paths. Specific timing for interventions was discerned. Early cost prevention interventions along with considerations of academic achievement to alleviate both anxiety and perceived effort/emotional are highlighted as crucial for a positive high school mathematics experience.     

Deductive reasoning: in the eye of the beholder

This study examines ways of approaching deductive reasoning of people involved in mathematics education and/or logic. The data source includes 21 individual semi-structured interviews. The data analysis reveals two different approaches. One approach refers to deductive reasoning as a systematic step-by-step manner for solving problems, both in mathematics and in other domains. The other approach emphasizes formal logic as the essence of the deductive inference, distinguishing between mathematics and other domains in the usability of deductive reasoning. The findings are interpreted in light of theory and practice.

Scientific reasoning in elementary school children: Assessment and relations with cognitive abilities

The primary goal of this study was the broad assessment and modeling of scientific reasoning in elementary school age. One hundred fifty-five fourth graders were tested on 20 recently developed paper-and-pencil items tapping four different components of scientific reasoning (understanding the nature of science, understanding theories, designing experiments, and interpreting data). As confirmed by Rasch analyses, the scientific reasoning items formed a reliable scale. Model comparisons differentiated scientific reasoning as a separate construct from measures of intelligence and reading skills and revealed discriminant validity. Furthermore, we explored the relationship between scientific reasoning and the postulated prerequisites inhibitory control, spatial abilities and problem-solving skills. As shown by correlation and regression analyses, beside general cognitive abilities (intelligence, reading skills) problem-solving skills and spatial abilities predicted performance in scientific reasoning items and thus contributed to explaining individual differences in elementary school children's scientific reasoning competencies.     

How gender stereotypes of students and significant others are related to motivational and affective outcomes in mathematics at the end of secondary school

We examined associations between the explicit mathematics-related gender stereotypes of students, parents, teachers, and classmates and students’ motivational-affective outcomes in mathematics (self-concept, interest, anxiety) at the end of Grade 9. Based on representative data from the German Trends in Student Achievement 2018 study (N = 30,019), results of latent multilevel mixture models show that boys’ and girls’ explicit beliefs in the stereotype favoring their own gender in-group (i.e., boys’/girls’ belief that boys/girls do better at mathematics) were related to higher levels of self-concept and interest and to lower anxiety. Parents’ gender stereotypes showed an incremental association with all three outcomes for girls but only with mathematics self-concept for boys. Gender stereotypes of teachers were not related to students’ outcomes. However, classmates’ stereotypes favoring girls or boys in mathematics were negatively associated with outcomes of the positively stereotyped group. Thus, a male student in a classroom with classmates who share the traditional stereotype that boys do better at mathematics than girls would hold a lower self-concept and interest and higher anxiety level after controlling for the beneficial individual association of himself having the same belief and his motivational and affective outcomes. Similarly, a girl’s motivational-affective outcomes would be more favorable in the same environment characterized by the shared traditional stereotype of mathematics as a male domain after controlling for the negative individual association. Shared stereotypes in the classroom could thus trigger social comparison processes to which students are more susceptible than to stereotypes of their teachers.     

Developing a mathematics curriculum to improve learning behaviors and mathematics competency of children

The authors investigated the effect of a mathematical curriculum (CU) developed based on verbal and practical activities on the mathematical competency (MC) and learning behaviors (LB) of preschool children. In a quasi-experimental design, 60 children (5- to 6-year-old girls) were selected using the accessible sampling method. The children were randomly divided into an experimental group and a control group, and the relevant concepts were taught to the children in both groups. While the control group received the typical kindergarten education based on the usual textbooks and worksheets, the CU was taught to the experimental group. Structural equation modeling was used to model the data and statistical evaluation. The results demonstrated a significant difference between the two groups in MC and LB. The CU significantly improved MC directly, and indirectly through the improvement of LB (i.e., engagement and learning focus, verbal behaviors, and type of activity).     

Using psychometric technology in educational assessment: The case of a schema-based isomorphic approach to the automatic generation of quantitative reasoning items

This article deals with the investigation of the psychometric quality and constructs validity of algebra word problems generated by means of a schema-based version of the automatic min–max approach. Based on review of the research literature in algebra word problem solving and automatic item generation this new approach is introduced as a theory-based top–down method of automatic item generation featuring a quality control framework aimed to minimize the construct unrelated variance in the item parameters. The first study deals with the evaluation of an initial set of items. The results are replicated in the second study using a larger item set which also allows the investigation of the construct representation of the generated item. Since construct unrelated variance components (e.g. reading comprehension) have been controlled for in the item generation phase the results revealed some interesting insights into the cognitive processes of the actual mathematization phase of algebra word problem solving. The third study investigated the nomothetic span is using hierarchical confirmatory factor analysis. The results argue for the convergent and discriminant validity of the automatically generated items. Taken together, the results indicate that the automatic generation of construct valid algebra word problems at a high psychometric level is viable. The discussion is thus concerned with the implications of this new approach to item generation for theory development and evaluation as well as practical benefits for educational assessment and the development of intelligent tutoring systems.     

Taking the relational structure of fractions seriously: Relational reasoning predicts fraction knowledge in elementary school children

Understanding and using symbolic fractions in mathematics is critical for access to advanced STEM concepts. However, children and adults consistently struggle with fractions. Here, we take a novel perspective on symbolic fractions, considering them within the framework of relational structures in cognitive psychology, such as those studied in analogy research. We tested the hypothesis that relational reasoning ability is important for reasoning about fractions by examining the relation between scores on a domain-general test of relational reasoning (TORR Jr.) and a test of fraction knowledge consisting of various types of fraction problems in 194 s grade and 145 fifth grade students. We found that relational reasoning was a significant predictor of fractions knowledge, even when controlling for non-verbal IQ and fractions magnitude processing for both grades. The effects of relational reasoning also remained significant when controlling for overall mathematics knowledge and skill for second graders but was attenuated for fifth graders. These findings suggest that this important subdomain of mathematical cognition is integrally tied to relational reasoning and opens the possibility that instruction targeting relational reasoning may prove to be a viable avenue for improving children’s fractions skills.     

Achievement goals,academic self-concept,and school grades in mathematics: Longitudinal reciprocal relations in above average ability secondary school students

This study examined the longitudinal reciprocal relations between academic self-concept, achievement goals (i.e., performance-approach, performance-avoidance, and mastery), and achievement (i.e., self-reported grades) in mathematics. The research aim was twofold. First, we examined the confound hypothesis, which states that performance-approach goals do not feature any incremental validity in predicting achievement over and above students' competence perceptions (i.e., academic self-concept). In addition, we expanded research on the confound hypothesis by also investigating performance-avoidance and mastery goals. Second, we investigated the predictive validity of all three achievement goals for changes in academic self-concept. Seven hundred sixty-nine students (50.78% female) attending the highest track of the German three-tier secondary school system participated in three waves of measurement in Grades 5, 6, and 8. Our findings confirmed the confound hypothesis: Performance-approach goals did not explain achievement over and above academic self-concept. The same findings applied to performance-avoidance and mastery goals. Furthermore, performance-approach goals were positively related to academic self-concept changes, whereas performance-avoidance goals showed a negative relation to academic self-concept changes over time. Mastery goals were not associated to changes in academic self-concept. Academic self-concept and achievement showed positive reciprocal relations. To conclude, our results point to complex relations between achievement goals, academic self-concept, and academic achievement over time.