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Introduction: Hand gestures and actions-with-objects (hereafter 'actions') are both forms of movement that can promote learning. However, the two have unique affordances, which means that they have the potential to promote learning in different ways. Here we compare how children learn, and importantly retain, information after performing gestures, actions, or a combination of the two during instruction about mathematical equivalence. We also ask whether individual differences in children's understanding of mathematical equivalence (as assessed by spontaneous gesture before instruction) impacts the effects of gesture- and action-based instruction. Method: Across two studies, racially and ethnically diverse third and fourth-grade students (N=142) were given instruction about how to solve mathematical equivalence problems (eg., 2+9+4=__+4) as part of a pretest-training-posttest design. In Study 1, instruction involved teaching students to produce either actions or gestures. In Study 2, instruction involved teaching students to produce either actions followed by gestures or gestures followed by actions. Across both studies, speech and gesture produced during pretest explanations were coded and analyzed to measure individual differences in pretest understanding. Children completed written posttests immediately after instruction, as well as the following day, and four weeks later, to assess learning, generalization and retention. Results: In Study 1 we find that, regardless of individual differences in pre-test understanding of mathematical equivalence, children learn from both action and gesture, but gesture-based instruction promotes retention better than action-based instruction. In Study 2 we find an influence of individual differences: children who produced relatively few types of problem-solving strategies (as assessed by their pre-test gestures and speech) perform better when they receive action training before gesture training than when they receive gesture training first. In contrast, children who expressed many types of strategies, and thus had a more complex understanding of mathematical equivalence prior to instruction, performed equally with both orders. Discussion: These results demonstrate that action training, followed by gesture, can be a useful stepping-stone in the initial stages of learning mathematical equivalence, and that gesture training can help learners retain what they learn.

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It is well established that gesture facilitates learning, but understanding the best way to harness gesture and how gesture helps learners are still open questions. Here, we consider one of the properties that may make gesture a powerful teaching tool: its temporal alignment with spoken language. Previous work shows that the simultaneity of speech and gesture matters when children receive instruction from a teacher (Congdon et al., 2017). In Study 1, we ask whether simultaneity also matters when children themselves are the ones who produce speech and gesture strategies. Third-graders (N = 75) were taught to produce one strategy in speech and one strategy in gesture for correctly solving mathematical equivalence problems; they were told to produce these strategies either simultaneously (S + G) or sequentially (S➔G; G➔S) during a training session. Learning was assessed immediately after training, at a 24-h follow-up, and at a 4-week follow-up. Children showed evidence of learning and retention across all three conditions. Study 2 was conducted to explore whether it was the special relationship between speech and gesture that helped children learn. Third-graders (N = 87) were taught an action strategy instead of a gesture strategy; all other aspects of the design were the same. Children again learned across all three conditions. But only children who produced simultaneous speech and action retained what they had learned at the follow-up sessions. Results have implications for why gesture is beneficial to learners and, taken in relation to previous literature, reveal differences in the mechanisms by which doing versus seeing gesture facilitates learning.

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Social inequality in mathematical skill is apparent at kindergarten entry and persists during elementary school. To level the playing field, we trained teachers to assess children's numerical and spatial skills every 10 wk. Each assessment provided teachers with information about a child's growth trajectory on each skill, information designed to help them evaluate their students' progress, reflect on past instruction, and strategize for the next phase of instruction. A key constraint is that teachers have limited time to assess individual students. To maximize the information provided by an assessment, we adapted the difficulty of each assessment based on each child's age and accumulated evidence about the child's skills. Children in classrooms of 24 trained teachers scored 0.29 SD higher on numerical skills at posttest than children in 25 randomly assigned control classrooms (P = 0.005). We observed no effect on spatial skills. The intervention also positively influenced children's verbal comprehension skills (0.28 SD higher at posttest, P < 0.001), but did not affect their print-literacy skills. We consider the potential contribution of this approach, in combination with similar regimes of assessment and instruction in elementary schools, to the reduction of social inequality in numerical skill and discuss possible explanations for the absence of an effect on spatial skills.

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Experts claim that individual differences in children's formal understanding of mathematical equivalence have consequences for mathematics achievement; however, evidence is lacking. A prospective, longitudinal study was conducted with a diverse sample of 112 children from a midsized city in the Midwestern United States (Mage [second grade] = 8:1). As hypothesized, understanding of mathematical equivalence in second grade predicted mathematics achievement in third grade, even after controlling for second-grade mathematics achievement, IQ, gender, and socioeconomic status. Most children exhibited poor understanding of mathematical equivalence, but results provide clues about which children are on the path to constructing an understanding and which may need extra support to overcome their misconceptions. Findings suggest that mathematical equivalence may deserve more attention from educators.

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Although children can use social categories to intelligently select informants, children's preference for in-group informants has not been consistently demonstrated across age and context. This research clarifies the extent to which children use social categories to guide learning by presenting participants with a live or video-recorded action demonstration by a linguistic in-group and/or out-group model. Participants' (N = 104) propensity to imitate these actions was assessed. Nineteen-month-olds did not selectively imitate the actions of the in-group model in live contexts, though in-group preferences were found after watching the demonstration on video. Three-year-olds selectively imitated the actions demonstrated by the in-group member regardless of context. These results indicate that in-group preferences have a more nuanced effect on social learning than previous research has indicated.

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Infants' direct interactions with caregivers have been shown to powerfully influence social and cognitive development. In contrast, little is known about the cognitive influence of social contexts beyond the infant's immediate interactions with others, for example, the communities in which infants live. The current study addressed this issue by asking whether neighborhood linguistic diversity predicts infants' propensity to learn from diverse social partners. Data were taken from a series of experiments in which 19-month-old infants from monolingual, English-speaking homes were tested in paradigms that assessed their tendency to imitate the actions of an adult who spoke either English or Spanish. Infants who lived in more linguistically diverse neighborhoods imitated more of the Spanish speaker's actions. This relation was observed in two separate datasets and found to be independent from variation in infants' general imitative abilities, age, median family income and population density. These results provide novel evidence suggesting that infants' social learning is predicted by the diversity of the communities in which they live.

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