Inhibition and cognitive load in fractions and decimals

Children's estimates of equivalent rational number magnitudes are not equal: Evidence from fractions, decimals, percentages, and whole numbers

Integration of rational number knowledge with prior whole number knowledge has been theorized as critical for mathematical success. Fractions, decimals, and percentages are generally assumed to differ in difficulty based on the degree to which their structure is perceptually similar to whole numbers. Specifically, percentages are viewed as most similar to whole numbers with their fixed unstated denominator of 100. Decimals are often assumed to be easier than fractions because their place-value structure is an extension of the base-ten system for whole numbers, unlike fractions, which have a bipartite structure (i.e., a/b). However, there has been no comprehensive investigation of how fraction, decimal, and percentage knowledge compares with whole number knowledge. To assess understanding of the four notations, we measured within-participants number line estimation of equivalent fractions and decimals with shorter string lengths (e.g., 8/10 and 0.8) and longer string lengths (e.g., 80/100 and 0.80), percentages (e.g., 80%), and proportionally equivalent whole numbers on a 0-100 scale (e.g., 80.0). Middle school students (N = 65; 33 female) generally underestimated all formats relative to their actual values (whole numbers: 3% below; percentages: 2%; decimals: 17%; fractions: 5%). Shorter string-length decimals and fractions were estimated as smaller than equivalent longer string-length equivalents. Overall, percentages were estimated similarly to corresponding whole numbers, fractions had modest string-length effects, and decimals were the most underestimated, especially for single-digit decimals. These results highlight the strengths and weaknesses of children's understanding of each notation's magnitudes and challenge the assumption that decimals are easier than fractions.

Relationship Between Inhibitory Control and Arithmetic in Elementary School Children With ADHD: The Mediating Role of Working Memory

OBJECTIVES

To test if inhibitory control was a significant predictor for arithmetic in children with ADHD and if the relationship between inhibitory control and arithmetic was mediated by working memory.

METHODS

Eighty-four children (ADHD, n = 54; Non-ADHD, n = 30) were tested on their interference control, behavioral inhibition, working memory, and arithmetic. Regression analysis was used to test the predictive role of inhibitory control in arithmetic. Moreover, mediation analysis was done to test whether working memory mediated the relationship between inhibitory control and arithmetic memory.

RESULTS

Interference control but not behavioral inhibition was a significant predictor for arithmetic. In addition, interference control had direct and indirect effects via working memory on arithmetic.

CONCLUSIONS

Results demonstrated that inhibitory control contributed to arithmetic in children with ADHD. Furthermore, interference control had direct and indirect effects via working memory on arithmetic, suggesting interventions for arithmetic difficulties should involve training on both inhibition and working memory.

Children's gesture use provides insight into proportional reasoning strategies

Children struggle with proportional reasoning when discrete countable information is available because they over-rely on this numerical information even when it leads to errors. In the current study, we investigated whether different types of gesture can exacerbate or mitigate these errors. Children aged 5-7 years (N = 135) were introduced to equivalent proportions using discrete gestures that highlighted separate parts, continuous gestures that highlighted continuous amounts, or no gesture. After training, children completed a proportional reasoning match-to-sample task where whole number information was occasionally pitted against proportional information. After the task, we measured children's own gesture use. Overall, we did not find condition differences in proportional reasoning; however, children who observed continuous gestures produced more continuous gestures than those who observed discrete gestures (and vice versa for discrete gestures). Moreover, producing fewer discrete gestures and more continuous gestures was associated with lower numerical interference on the match-to-sample task. Lastly, to further investigate individual differences, we found that children's inhibitory control and formal math knowledge were correlated with proportional reasoning in general but not with numerical interference in particular. Taken together, these findings highlight that children's own gestures may be a powerful window into the information they attend to during proportional reasoning.

Mapping components of verbal and visuospatial working memory to mathematical topics in seven- to fifteen-year-olds

BACKGROUND

Developmental research provides considerable evidence of a strong relationship between verbal and visuospatial working memory (WM) and mathematics ability across age groups. However, little is known about how components of WM (i.e., short-term storage, processing speed, the central executive) might relate to mathematics sub-categories and how these change as children develop.

AIMS

This study aimed to identify developmental changes in relationships between components of verbal and visuospatial WM and specific mathematics abilities.

SAMPLE

Children (n = 117) were recruited from four UK schools across three age groups (7-8 years; 9-10 years; and 14-15 years).

METHODS

Children's verbal and visuospatial short-term storage, processing speed, and central executive abilities were assessed. Age-based changes in the contributions from these abilities to performance on mathematics sub-categories were examined.

RESULTS

When WM was examined both as an amalgamation of its component parts, and individually, relationships with mathematics were more evident in younger children compared to the middle and older age groups. However, when unique variance was examined for each WM predictor (controlling for the other components), many of those relationships disappeared. Relationships with processing speed and the central executive were found to be more evident in the older age groups.

CONCLUSIONS

The WM-mathematics relationship changes dependent on age and mathematical sub-component. Overlap in individual WM abilities in younger children, compared to reliance on the central executive and processing speed in older children, suggests a set of fluid resources important in mathematics learning in younger children but separating out as children grow older.

From Non-symbolic to Symbolic Proportions and Back: A Cuisenaire Rod Proportional Reasoning Intervention Enhances Continuous Proportional Reasoning Skills

The persistent educational challenges that fractions pose call for developing novel instructional methods to better prepare students for fraction learning. Here, we examined the effects of a 24-session, Cuisenaire rod intervention on a building block for symbolic fraction knowledge, continuous and discrete non-symbolic proportional reasoning, in children who have yet to receive fraction instruction. Participants were 34 second-graders who attended the intervention (intervention group) and 15 children who did not participate in any sessions (control group). As attendance at the intervention sessions was irregular (median = 15.6 sessions, range = 1-24), we specifically examined the effect of the number of sessions completed on their non-symbolic proportional reasoning. Our results showed that children who attended a larger number of sessions increased their ability to compare non-symbolic continuous proportions. However, contrary to our expectations, they also decreased their ability to compare misleading discretized proportions. In contrast, children in the Control group did not show any change in their performance. These results provide further evidence on the malleability of non-symbolic continuous proportional reasoning and highlight the rigidity of counting knowledge interference on discrete proportional reasoning.

The dynamic nature of children's strategy use after receiving accuracy feedback in decimal comparisons

When students start learning decimals, they may incorrectly apply features of their prior numerical knowledge (e.g., whole-number or fraction rules). However, because whole numbers, fractions, and decimals all have their own unique features, these whole-number and fraction strategies do not always lead to correct solutions. We examined whether receiving immediate accuracy feedback while comparing decimal pairs that were either congruent with whole-number rules (e.g., decimals with more digits were larger in magnitude) or incongruent with whole-number rules (e.g., decimals with fewer digits were larger in magnitude) would lead students to change their decimal comparison strategies. We also examined whether students' potential improvement after feedback would generalize to decimal comparisons involving different numbers of digits. We found that sixth- to eighth-grade students' use of the whole-number strategy declined and their use of the normative decimal strategy increased over the course of receiving feedback, whereas no significant strategy change was observed among students who did not receive any feedback. Students who received feedback were also less likely to use a whole-number strategy and more likely to use a decimal strategy in different decimal comparisons in an immediate posttest and a 2-week delayed posttest. Our exploratory analyses found that students' improvement on decimal comparisons did not transfer to decimal arithmetic. Moreover, students' inhibitory control also predicted strategy use in immediate and delayed posttests. Our study provides insights into the mechanisms of rapid strategy change and has implications for designing interventions to improve children's understanding of decimal magnitudes.

Children's discrete proportional reasoning is related to inhibitory control and enhanced by priming continuous representations

Children can successfully compare continuous proportions as early as 4 years of age, yet they struggle to compare discrete proportions at least to 10 years of age, especially when the discrete information is misleading. This study examined whether inhibitory control contributes to individual differences in discrete proportional reasoning and whether reasoning could be enhanced by priming continuous information. A total of 49 second-graders completed two tasks. In the Hearts and Flowers (H&F) task, a measure of inhibition, children pressed on either the corresponding or opposite side, depending on the identity of the displayed figure. In the Spinners task, a measure of proportional reasoning, children chose the spinner with the proportionally larger red area across continuous and two discrete formats. In the discrete adjacent format, the continuous stimuli were segmented into sections, which could be compatible with the proportional information or misleading; the discrete mixed format interspersed the colored sections from the discrete adjacent conditions. Finally, two priming groups were formed. Children who saw the continuous format immediately before the discrete adjacent format formed the continuous priming group (n = 26). Children who saw the discrete mixed format immediately before the discrete adjacent format formed the discrete priming group (n = 23). Our results showed that children who performed better on the H&F task also had better performance on the discrete counting misleading trials. Furthermore, children in the continuous priming group outperformed children in the discrete priming group, specifically in contexts where discrete information was misleading. These results suggest that children's proportional reasoning may be improved by fostering continuous representations of discrete stimuli and by enhancing inhibitory control skills.

A Study on Congruency Effects and Numerical Distance in Fraction Comparison by Expert Undergraduate Students

School mathematics comprises a diversity of concepts whose cognitive complexity is still poorly understood, a chief example being fractions. These are typically taught in middle school, but many students fail to master them, and misconceptions frequently persist into adulthood. In this study, we investigate fraction comparison, a task that taps into both conceptual and procedural knowledge of fractions, by looking at performance of highly mathematically skilled young adults. Fifty-seven Chilean engineering undergraduate students answered a computerized fraction comparison task, while their answers and response times were recorded. Task items were selected according to a number of mathematically and/or cognitively relevant characteristics: (a) whether the fractions to be compared shared a common component, (b) the numerical distance between fractions, and (c) the applicability of two strategies to answer successfully: a congruency strategy (a fraction is larger if it has larger natural number components than another) and gap thinking (a fraction is larger if it is missing fewer pieces than another to complete the whole). In line with previous research, our data indicated that the congruency strategy is inadequate to describe participants’ performance, as congruent items turned out to be more difficult than incongruent ones when fractions had no common component. Although we hypothesized that this lower performance for congruent items would be explained by the use of gap thinking, this turned out not to be the case: evidence was insufficient to show that the applicability of the gap thinking strategy modulated either participants’ accuracy rates or response times (although individual-level data suggest that there is an effect for response times). When fractions shared a common component, instead, our data display a more complex pattern that expected: an advantage for congruent items is present in the first experimental block but fades as the experiment progresses. Numerical distance had an effect in fraction comparison that was statistically significant for items without common components only. Altogether, our results from experts’ reasoning reveal nuances in the fraction comparison task with respect to previous studies and contribute to future models of reasoning in this task.