Not realizing that you don't know: Fraction state anxiety is reduced by natural number bias.

BACKGROUND: Research has shown that mathematics anxiety negatively correlates with primary school mathematics performance, including fraction knowledge. However, recently no significant correlation was found between fraction arithmetic performance and state anxiety measured after the fraction task. One possible explanation is the natural number bias (NNB), a tendency to apply natural number reasoning in fraction tasks, even when this is inappropriate. Students with the NNB may not realize they are answering incorrectly. AIMS: The aim is to examine whether a misconception, namely the NNB, can influence students' fraction state anxiety. SAMPLE: The participants were 119 fifth- and sixth-grade students categorized as belonging to an NNB group (n = 60) or a No-NNB group (n = 59), according to their NNB-related answering profile on a fraction arithmetic task. METHODS: Group differences were examined for state anxiety and performance on a fraction and a whole number arithmetic task and self-reported trait mathematics anxiety. RESULTS: The NNB group reported lower fraction state anxiety than the No-NNB group, but there was no significant difference in trait mathematics anxiety. Furthermore, the NNB group reported lower fraction state anxiety than whole number state anxiety, while the opposite was true for the No-NNB group. CONCLUSION: The present study suggests that students' perceptions of their own performance influence their state anxiety responses, and students with a NNB may not be aware of their misconception and poor performance. Not taking into account qualitative differences in low performance, such as misconceptions, may lead to misinterpretations in state anxiety-performance relations.

Mathematical skills of 11-year-old children born very preterm and full-term.

Preterm birth affects the academic development of children, especially in mathematics. Remarkably, only a few studies have measured specific effects of preterm birth on mathematical skills in primary school. The aim of this study was to compare 11-year-old children, with an IQ above 70, born very preterm (N = 64) and full-term (N = 72) on a variety of 5th grade mathematical skills and cognitive abilities important for mathematical learning. The measures were spontaneous focusing on numerosity (SFON), spontaneous focusing on quantitative relations (SFOR), arithmetic fluency, mathematics achievement, number line estimation, rational number magnitude knowledge, mathematics motivation, reading skills, visuospatial processing, executive functions, and naming speed. The children born very preterm and full-term differed in arithmetic fluency, SFON and SFOR. Domain general cognitive abilities did not fully explain the group differences in SFON and SFOR. Retrospective comparisons of the samples at the age of five years showed large group differences in early mathematical skills and cognitive abilities. Despite lower early mathematical skills, the children born very preterm reached peer equivalent performance in many mathematical skills by the age of 11 years. Nevertheless, they appear less likely to focus on implicit mathematical features in their everyday life.

Predicting adaptive expertise with rational number arithmetic.

BACKGROUND: Adaptive 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. AIMS: We 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. SAMPLE: 173 6th and 7th grade students from a school in the southeastern US (51% female) participated in the study. METHODS: At 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. RESULT: Prior 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. CONCLUSION: Results suggest expanding investigations of mathematical development to also explore different features of adaptive expertise as well as spontaneous mathematical focusing tendencies.

Cross-notation knowledge of fractions and decimals.

Understanding fractions and decimals requires not only understanding each notation separately, or within-notation knowledge, but also understanding relations between notations, or cross-notation knowledge. Multiple notations pose a challenge for learners but could also present an opportunity, in that cross-notation knowledge could help learners to achieve a better understanding of rational numbers than could easily be achieved from within-notation knowledge alone. This hypothesis was tested by reanalyzing three published datasets involving fourth- to eighth-grade children from the United States and Finland. All datasets included measures of rational number arithmetic, within-notation magnitude knowledge (e.g., accuracy in comparing fractions vs. fractions and decimals vs. decimals), and cross-notation magnitude knowledge (e.g., accuracy in comparing fractions vs. decimals). Consistent with the hypothesis, cross-notation magnitude knowledge predicted fraction and decimal arithmetic when controlling for within-notation magnitude knowledge. Furthermore, relations between within-notation magnitude knowledge and arithmetic were not notation specific; fraction magnitude knowledge did not predict fraction arithmetic more than decimal arithmetic, and decimal magnitude knowledge did not predict decimal arithmetic more than fraction arithmetic. Implications of the findings for assessing rational number knowledge and learning and teaching about rational numbers are discussed.

Individual differences in fraction arithmetic learning.

Understanding fractions is critical to mathematical development, yet many children struggle with fractions even after years of instruction. Fraction arithmetic is particularly challenging. The present study employed a computational model of fraction arithmetic learning, FARRA (Fraction Arithmetic Reflects Rules and Associations; Braithwaite, Pyke, and Siegler, 2017), to investigate individual differences in children's fraction arithmetic. FARRA predicted four qualitatively distinct patterns of performance, as well as differences in math achievement among the four patterns. These predictions were confirmed in analyses of two datasets using two methods to classify children's performance-a theory-based method and a data-driven method, Latent Profile Analysis. The findings highlight three dimensions of individual differences that may affect learning in fraction arithmetic, and perhaps other domains as well: effective learning after committing errors, behavioral consistency versus variability, and presence or absence of initial bias. Methodological and educational implications of the findings are discussed.

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

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