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 sixthto 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. (C) 2020 Elsevier Inc. All rights reserved.

Individual differences in children's number knowledge arise early and are associated with variation in parents' number talk. However, there exists little experimental evidence of a causal link between parent number talk and children's number knowledge. Parent number talk was manipulated by creating picture books which parents were asked to read with their children every day for 4 weeks.N = 100 two- to four-year olds and their parents were randomly assigned to read either Small Number (1-3), Large Number (4-6), or Control (non-numerical) books. Small Number books were particularly effective in promoting number knowledge relative to the Control books. However, children who began the study further along in their number development also benefited from reading the Large Number Books with their parents.

Elementary school students often lack a conceptual understanding of linear measurement, which is revealed by their poor performance when the object to be measured is not aligned with the start of the ruler. Instead of correctly counting the units that correspond to the object (e.g., inches or centimeters), children often use 1 of 2 incorrect strategies: reading off the number that corresponds to the end of the object (the least-mature, read-off strategy) and counting the hash marks that flank the object (a more mature, but still incorrect, hash-mark strategy). We hypothesized that shifting to a more mature linear measurement strategy would require the ability to inhibit less-mature prepotent responses, such as read-off and hash-mark responses. In the present study, we predicted that children with better inhibitory control would be more likely to improve in their linear measurement strategy use over one year. Participants (n = 317) were in 1st through 3rd grades when they completed a linear measurement task that required measuring objects that were not aligned with the start of the ruler; they also completed an inhibitory control task and control measures (visuospatial working memory, arithmetic calculations, and number line estimation). One year later, they repeated the linear measurement task. Students with higher initial inhibitory control were more likely to adopt a more mature strategy over time. Moreover, inhibitory control was a significant predictor of strategy improvement over and above other cognitive measures, including visuospatial working memory and arithmetic calculation skill.

Children and adults often have difficulties comparing decimal magnitudes. Although individuals attempt to reconcile decimals with prior whole-number and fraction knowledge, conceptual and procedural differences between decimals and prior knowledge of whole numbers and fractions can lead to incorrect strategies. The dynamic strategy choice account has proposed that saliency, recency, prior knowledge, and other factors contribute to strategy use when reasoning about decimals. We experimentally tested this theory using a priming technique to manipulate the saliency of different strategies prior to completion of a decimal magnitude comparison task. We hypothesized that whole-number priming (practicing whole-number comparisons with feedback) would increase whole-number bias in decimal comparisons (treating decimals with more digits as larger in magnitude), whereas fraction priming would increase fraction bias (treating decimals with fewer digits as larger in magnitude). We also explored participants' performance in decimal comparisons after being primed by decimal comparisons with feedback. Sixth to eighth graders (N = 149) and adults (N = 175) were randomly assigned to 1 of 4 priming conditions: whole-number, fraction, decimal, or a control (flanker) task. Participants first completed numerical magnitude comparisons according to their priming condition (or control task) with feedback, then all conditions completed decimal comparisons without feedback. In both children and adults, fraction priming significantly reduced whole-number bias compared with the control. Among children, fraction priming significantly increased fraction bias. Moreover, children's performance in the control and whole-number-priming conditions was characterized by strong whole-number bias, but in the decimal-priming condition, relatively brief feedback substantially improved decimal comparison performance.

When asked to explain their solutions to a problem, children often gesture and, at times, these gestures convey information that is different from the information conveyed in speech. Children who produce these gesture-speech mismatches on a particular task have been found to profit from instruction on that task. We have recently found that some children produce gesture-speech mismatches when identifying numbers at the cusp of their knowledge, for example, a child incorrectly labels a set of two objects with the word three and simultaneously holds up two fingers. These mismatches differ from previously studied mismatches (where the information conveyed in gesture has the potential to be integrated with the information conveyed in speech) in that the gestured response contradicts the spoken response. Here, we ask whether these contradictory number mismatches predict which learners will profit from number-word instruction. We used the Give-a-Number task to measure number knowledge in 47 children (M-age = 4.1 years, SD = 0.58), and used the What's on this Card task to assess whether children produced gesture-speech mismatches above their knower level. Children who were early in their number learning trajectories (one-knowers and two-knowers) were then randomly assigned, within knower level, to one of two training conditions: a Counting condition in which children practiced counting objects; or an Enriched Number Talk condition containing counting, labeling set sizes, spatial alignment of neighboring sets, and comparison of these sets. Controlling for counting ability, we found that children were more likely to learn the meaning of new number words in the Enriched Number Talk condition than in the Counting condition, but only if they had produced gesture-speech mismatches at pretest. The findings suggest that numerical gesture-speech mismatches are a reliable signal that a child is ready to profit from rich number instruction and provide evidence, for the first time, that cardinal number gestures have a role to play in number-learning.

Children's ability to estimate fractions on a number line is strongly related to algebra and overall high school math achievement, and number line training leads to better fraction magnitude comparisons compared with area model training. Here, we asked whether unidimensionality is necessary for the number line to promote fraction magnitude concepts and whether left-to-right orientation and labeled endpoints are sufficient. We randomly assigned second and third-graders (N = 148) to one of four 15-min one-on-one, experimenter-led trainings. Three number line trainings had identical scripts, where the experimenter taught children to segment and shade the number line along the horizontal dimension. The number line conditions varied only in the vertical dimension of the training number line: pure unidimensional number line (17.5 cm horizontal line), hybrid unidimensional number line (17.5 x 0.6 cm rectangle), and square number line (17.5 x 17.5 cm). In the area model condition, children were taught to segment and shade a square (17.5 x 17.5 cm) along both dimensions. The conditions significantly differed in posttest fraction magnitude comparison accuracy (a transfer task), controlling for pretest accuracy, reading achievement, and age. In preregistered analyses, the hybrid unidimensional number line condition significantly outperformed the square area model condition and the square number line condition. In exploratory analyses accounting for training protocol fidelity, these results held and the pure unidimensional number line also outperformed the area model condition on fraction magnitude comparisons. We argue that unidimensionality is a critical feature of the number line for promoting fraction magnitude concepts because it aligns with a key concept that real numbers, including fractions, can be ordered along a single dimension. (C) 2019 Elsevier Inc. All rights reserved.

Proportional reasoning is an important skill that relates to fraction learning and math achievement. Because both proportional and analogical reasoning involve comparing relations, we hypothesized that supports for analogical reasoning (multiple exemplars and labels) would help children match discrete proportions. Fourth and 5th graders (N = 119) completed a 16-item proportional equivalence choice task in a 2 (exemplars: one, two) x 3 (script type: juice mixing narrative, novel adjectives, no labels) x 2 (trial type: part-foil, whole-foil) mixed-effects design. The juice mixing script included common labels and a story paradigm, whereas the novel adjectives script only utilized common labels. The least-informative no-labels script served as a baseline. Results showed a significant three-way interaction between exemplars, script type, and trial type. Viewing two exemplars led to no significant differences across the other factors. However, when viewing one exemplar, children exhibited a part-matching bias with novel adjectives and no labels, but not the juice narrative. In sum, multiple exemplars and the juice narrative reduced children's part-matching bias; we conclude that supports for analogical reasoning can aid proportional reasoning as well.

In a previous study, parent-child praise was observed in natural interactions at home when children were 1, 2, and 3 years of age. Children who received a relatively high proportion of process praise (e.g., praise for effort and strategies) showed stronger incremental motivational frameworks, including a belief that intelligence can be developed and a greater desire for challenge, when they were in 2nd or 3rd grade (Gunderson et al., 2013). The current study examines these same children's (n = 53) academic achievement 1 to 2 years later, in 4th grade. Results provide the first evidence that process praise to toddlers predicts children's academic achievement (in math and reading comprehension) 7 years later, in elementary school, via their incremental motivational frameworks. Further analysis of these motivational frameworks shows that process praise had its effect on fourth grade achievement through children's trait beliefs (e.g., believing that intelligence is fixed vs. malleable), rather than through their learning goals (e.g., preference for easy vs. challenging tasks). Implications for the socialization of motivation are discussed.

School-entry math achievement is a strong predictor of math achievement through high school. We asked whether reciprocal relations among math achievement, math anxiety, and entity motivational frameworks (believing that ability is fixed and a focus on performance) can help explain these persistent individual differences. We assessed 1st and 2nd graders' (N=634) math achievement, motivational frameworks, and math anxiety 2 times, 6months apart. Cross-lagged path analyses showed reciprocal relations between math anxiety and math achievement and between motivational frameworks and math achievement. Entity motivational frameworks predicted higher math anxiety. High math achievement was a particularly strong predictor of lower math anxiety and less entity-oriented motivational frameworks. We concluded that reciprocal effects are already present in the first 2 years of formal schooling, with math achievement and attitudes feeding off one another to produce either a vicious or virtuous cycle. Improving both math performance and math attitudes may set children onto a long-lasting, positive trajectory in math.

Learning the cardinal principle (the last word reached when counting a set represents the size of the whole set) is a major milestone in early mathematics. But researchers disagree about the relationship between cardinal principle knowledge and other concepts, including how counting implements the successor function (for each number word N representing a cardinal value, the next word in the count list represents the cardinal value N + 1) and exact ordering (cardinal values can be ordered such that each is one more than the value before it and one less than the value after it). No studies have investigated acquisition of the successor principle and exact ordering over time, and in relation to cardinal principle knowledge. An open question thus remains: Is the cardinal principle a gatekeeper concept children must acquire before learning about succession and exact ordering, or can these concepts develop separately? Preschoolers (N = 127) who knew the cardinal principle (CP-knowers) or who knew the cardinal meanings of number words up to three or four (3-4-knowers) completed succession and exact ordering tasks at pretest and posttest. In between, children completed one of two trainings: counting only versus counting, cardinal labeling, and comparison. CP-knowers started out better than 3-4-knowers on succession and exact ordering. Controlling for this disparity, we found that CP-knowers improved over time on succession and exact ordering; 3-4-knowers did not. Improvement did not differ between the two training conditions. We conclude that children can learn the cardinal principle without understanding succession or exact ordering and hypothesize that children must understand the cardinal principle before learning these concepts.