We examined 2- and 3-year-old children's ability to use second-order correlation learning-in which a learned correlation between two pairs of features (e.g., A and B, A and C) is generalized to the noncontiguous features (i.e., B and C)-to make causal inferences. Previous findings showed that 20- and 26-month-old children can use second-order correlation learning to learn about static and dynamic features in category and noncategory contexts. The current behavioral study and computational model extend these findings to show that 2- and 3-year-olds can detect the second-order correlation between an object's surface feature and its capacity to activate a novel machine, but only if the children had encoded the first-order correlations on which the second-order correlation was based. These results have implications for children's developing information-processing capacities on their ability to use second-order correlations to infer causal relations in the world. Published by Elsevier Inc.

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.

Minimal group paradigms investigate social preferences arising from mere group membership. We asked whether demand characteristics contribute to children's apparent minimal group bias in a preregistered experiment (N = 160). In a group condition, we attempted to replicate findings of bias following assignment to minimal groups. A second closely matched no-group condition retained potential demand characteristics while removing group assignment. Parallel bias in the no-group condition would suggest that demand characteristics contribute to findings of apparent ingroup bias. Three main findings emerged. First, in the group condition, ingroup preference emerged in one of three bias measures only. Second, this preference emerged even though participants evaluated ingroup/outgroup photos varying in race/ethnicity between trials. Third, the measure that yielded ingroup preferences in the group condition produced no parallel bias in the no group condition, consistent with the view that mere membership in a group, not experimental demand, leads to minimal ingroup preferences. (c) 2020 Elsevier Inc. All rights reserved.

Across two studies (N = 120), we investigated the development of children's ability to calibrate the certainty of verbal testimony with observable data that varied in the degree of predictive causal accu-racy. In Study 1, 4-and 5-year-olds heard a certain explanation or an uncertain explanation about deterministic causal relations. The 5-year-olds made more accurate causal inferences when the infor-mant provided a certain and more calibrated explanation. In Study 2, children heard similar explanations about probabilistic relations, making the uncertain informant more calibrated. The 5-year-olds were more likely to infer the correct causal relations when the informant was uncertain, but only when the explanation was attuned to the stochasticity of the individual causal events (or out-comes that sometimes occur). These findings imply that the capac-ity to integrate, and make efficient inferences, from distinct sources of knowledge emerges during the preschool years. (c) 2021 Elsevier Inc. All rights reserved.

The cognitive complexity of adults' questions, particularly during shared book reading, supports children's developing language skills. Questions can be described as having low cognitive demand (CD; e.g., labeling, matching) or high-CD (e.g., comparing, predicting). Little is known about the relation between different types of parental questioning and children's math abilities. The current study examined the quantity of low- and high-CD and domainspecific math questions that parents posed to their 4-year-old children in three structured activities and how the frequency of those questions relates to children's concurrent math and language skills. Parent-child dyads (n = 121) were observed interacting with a picture book, grocery store toys, and a puzzle for about 5 min each, and children completed math and spatial assessments. Although the frequency with which parents asked questions did not relate to children's outcomes, parents' use of high-CD questions was associated with children's spatial skills, standardized math scores, and vocabulary skills after controlling for parental utterances, child utterances, child age, and family socioeconomic status. However, domain-specific math questions were not related to any child outcomes above and beyond parents' total questions. This study suggests that domain-general questions that vary in CD (low and high) are differentially related to children's math and language abilities, which can inform the ways in which parents engage in early learning opportunities with their children. Published by Elsevier Inc.

Decades of research have established that spatial skills correlate with numerical skills. However, because both spatial and numerical skills are multidimensional, we sought to determine how specific spatial skills relate to specific numeracy skills. We used a cohort-sequential design, assessing a large diverse sample of students (N = 612, initially in pre-kindergarten [pre-K]-3rd grade, 4-9 years of age) at four time points spanning 2 years. We examined how initial levels of five spatial skills (visuospatial working memory [VSWM], mental transformation, mental rotation, proportional reasoning, and analog magnitude system [AMS] acuity) related to initial levels and growth rates in exact and approximate calculation skills, and we further investigated number line estimation as a potential mediator. We found unique patterns of relations between spatial skills and numeracy. Initial levels of mental rotation, proportional reasoning, and AMS acuity related to initial levels of exact calculation skill; initial levels of AMS acuity related to initial levels of approximate calculation; and initial levels of proportional reasoning related to initial levels of number line estimation. VSWM and mental transformation did not relate to numeracy skills after controlling for other spatial skills. Initial levels of number line estimation related to both exact and approximate calculation after controlling for spatial skills. Notably, neither spatial skills nor number line estimation predicted growth in exact or approximate calculation skills. These results indicate that there is specificity in the time-invariant relations between spatial skills and numeracy, and they suggest that researchers and educators should treat spatial skills and numeracy as multidimensional constructs with complex and unique interrelations. (c) 2021 Elsevier Inc. All rights reserved.

Counting problems are difficult for students to solve, and there is a perennial need to investigate ways to help students solve counting problems successfully. One promising avenue for students' successful counting is for them to think judiciously about how they encode outcomes - that is, how they symbolize and represent the outcomes they are trying to count. We provide a detailed case study of two students as they encoded outcomes in their work on several related counting problems within a computational setting. We highlight the role that a computational environment may have played in this encoding activity. We illustrate ways in which by-hand work and computer programming worked together to facilitate the students' successful encoding activity. This case demonstrates ways in which the activity of computation seemed to interact with byhand work to facilitate sophisticated encoding of outcomes.

Combinatorics is an area of mathematics with accessible, rich problems and applications in a variety of fields. Combinatorial proof is an important topic within combinatorics that has received relatively little attention within the mathematics education community, and there is much to investigate about how students reason about and engage with combinatorial proof. In this paper, we use Harel and Sowder's (1998) proof schemes to investigate ways that students may characterize combinatorial proofs as different from other types of proof. We gave five upper-division mathematics students combinatorial-proof tasks and asked them to reflect on their activity and combinatorial proof more generally. We found that the students used several of Harel and Sowder's proof schemes to characterize combinatorial proof, and we discuss whether and how other proof schemes may emerge for students engaging in combinatorial proof. We conclude by discussing implications and avenues for future research.