The present study investigated the development of fraction comparison strategies through a longitudinal analysis of children's responses to a fraction comparison task in 4th through 6th grades (N ϭ 394). Participants were asked to choose the larger value for 24 fraction pairs blocked by fraction type. Latent class analysis of performance over item blocks showed that most children initially exhibited a "whole number bias," indicating that larger numbers in numerators and denominators produce larger fraction values. However, some children instead chose fractions with smaller numerators and denominators, demonstrating a partial understanding that smaller numbers can yield larger fractions. Latent transition analysis showed that most children eventually adopted normative comparison strategies. Children who exhibited a partial understanding by choosing fractions with smaller numbers were more likely to adopt normative comparison strategies earlier than those with larger number biases. Controlling for general math achievement and other cognitive abilities, whole number line estimation accuracy predicted the probability of transitioning to normative comparison strategies. Exploratory factor analyses showed that over time, children appeared to increasingly represent fractions as discrete magnitudes when simpler strategies were unavailable. These results support the integrated theory of numerical development, which posits that an understanding of numbers as magnitudes unifies the process of learning whole numbers and fractions. The findings contrast with conceptual change theories, which propose that children must move from a view of numbers as counting units to a new view that accommodates fractions to overcome whole number bias.

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Does knowing when mental arithmetic judgments are right—and when they are wrong—lead to more accurate judgments over time? We hypothesize that the successful detection of errors (and avoidance of false alarms) may contribute to the development of mental arithmetic performance. Insight into error detection abilities can be gained by examining the “calibration” of mental arithmetic judgments—that is, the alignment between confidence in judgments and the accuracy of those judgments. Calibration may be viewed as a measure of metacognitive monitoring ability. We conducted a developmental longitudinal investigation of the relationship between the calibration of children's mental arithmetic judgments and their performance on a mental arithmetic task. Annually between Grades 5 and 8, children completed a problem verification task in which they rapidly judged the accuracy of arithmetic expressions (e.g., 25+50 = 75) and rated their confidence in each judgment. Results showed that calibration was strongly related to concurrent mental arithmetic performance, that calibration continued to develop even as mental arithmetic accuracy approached ceiling, that poor calibration distinguished children with mathematics learning disability from both low and typically achieving children, and that better calibration in Grade 5 predicted larger gains in mental arithmetic accuracy between Grades 5 and 8. We propose that good calibration supports the implementation of cognitive control, leading to long-term improvement in mental arithmetic accuracy. Because mental arithmetic “fluency” is critical for higher-level mathematics competence, calibration of confidence in mental arithmetic judgments may represent a novel and important developmental predictor of future mathematics performance.

Advocates of the arts agree that the K‐12 curriculum should include dedicated time for arts instruction. Some have argued further that knowledge and skills acquired through the arts transfer to nonarts domains. Others claim that evidence of this kind of transfer is limited and instead argue that the arts cultivate valuable dispositions that help students succeed both in and outside of school. Another potential benefit of the arts has received little attention, however. Arts integration—the use of the arts as a teaching methodology throughout the curriculum—may improve long‐term retention of content. A variety of long‐term memory effects well known in cognitive psychology are reviewed, and it is argued that arts integration naturally takes advantage of these effects while promoting student motivation. This review of findings and applications provides an example of how existing research from neuroscience and cognitive science can inform the work of practicing educators.