"You need to be super smart to do well in math!" Young children's field-specific ability beliefs.

Success in science, technology, engineering, and mathematics (STEM) fields is often believed to require intellectual talent ("brilliance"). Given that many cultures associate men more than women with brilliance, this belief poses an obstacle to women's STEM pursuits. Here, we investigated the developmental roots of this phenomenon, focusing specifically on young children's beliefs about math (N = 174 U.S. students in Grades 1-4; 93 girls, 81 boys; 52% White, 17% Asian, 13% Hispanic/Latinx). We found that field-specific ability beliefs (FABs) that associate success in math (vs. reading/writing) with brilliance are already present in early elementary school. We also found that brilliance-oriented FABs about math are negatively associated with elementary school students' (and particularly girls') math motivation-specifically, their math self-efficacy and interest. The early emergence of brilliance-oriented FABs about math and the negative relation between FABs and math motivation underscore the need to understand the sources and long-term effects of these beliefs. RESEARCH HIGHLIGHTS: Field-specific ability beliefs (FABs) are beliefs about the extent to which intellectual talent (or "brilliance") is required for success in a particular field or context. Among adults, brilliance-oriented FABs are an obstacle to diversity in science and technology, but the childhood antecedents of these beliefs are not well understood. The present study (N = 174) found that FABs that associate success in math (vs. reading/writing) with brilliance were already present in Grades 1-4. Brilliance-oriented FABs about math were negatively associated with elementary school students' (and particularly girls') math motivation-specifically, their math self-efficacy and interest.

Children's confidence on mathematical equivalence and fraction problems.

Metacognition is how people think about their own thinking, and it includes children's confidence in their problem-solving solutions. We assessed children's metacognition in two areas of mathematics that are often plagued by misconceptions and often studied separately-mathematical equivalence (e.g., 3 + 4 = 5 +__) and fraction magnitude (e.g., 1/3 = 2/__). Specifically, we quantified children's metacognitive skills across these topics, examined whether these skills are topic-specific, and determined how these skills covary with domain-general executive functioning. A total of 80 elementary school children (Mage = 7.70 years, SD = 0.69) provided trial-by-trial performance data and confidence judgments on equivalence and fraction problems as well as assessments of executive function. Children's metacognitive skills were especially impaired when using incorrect strategies based on misconceptions, and we did not find robust evidence for cross-topic associations. We found modest associations between children's metacognitive skills and their working memory and inhibitory control. Findings have theoretical and practical implications for understanding which children struggle with these important mathematics concepts and why.

Young children interpret number gestures differently than nonsymbolic sets.

Researchers have long been interested in the origins of humans' understanding of symbolic number, focusing primarily on how children learn the meanings of number words (e.g., "one", "two", etc.). However, recent evidence indicates that children learn the meanings of number gestures before learning number words. In the present set of experiments, we ask whether children's early knowledge of number gestures resembles their knowledge of nonsymbolic number. In four experiments, we show that preschool children (n = 139 in total; age M = 4.14 years, SD = 0.71, range = 2.75-6.20) do not view number gestures in the same the way that they view nonsymbolic representations of quantity (i.e., arrays of shapes), which opens the door for the possibility that young children view number gestures as symbolic, as adults and older children do. A video abstract of this article can be viewed at https://youtu.be/WtVziFN1yuI HIGHLIGHTS: Children were more accurate when enumerating briefly-presented number gestures than arrays of shapes, with a shallower decline in accuracy as quantities increased. We replicated this finding with arrays of shapes that were organized into neat, dice-like configurations (compared to the random configurations used in Experiment 1). The advantage in enumerating briefly-presented number gestures was evident before children had learned the cardinal principle. When gestures were digitally altered to pit handshape configuration against number of fingers extended, children overwhelmingly based their responses on handshape configuration.

Longitudinally adaptive assessment and instruction increase numerical skills of preschool children.

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.

Elementary school teachers' math anxiety and students' math learning: A large-scale replication.

A solid foundation in math is important for children's long-term academic success. Many factors influence children's math learning-including the math content students are taught in school, the quality of their instruction, and the math attitudes of students' teachers. Using a large and diverse sample of first-grade students (n = 551), we conducted a large-scale replication of a previous study (Proceedings of the National Academy of Sciences of the USA, 2010, 1860; n = 117), which found that girls in classes with highly math anxious teachers learned less math during the school year, as compared to girls whose math teachers were less anxious about math. With a larger sample, we found a negative relation between teachers' math anxiety and students' math achievement for both girls and boys, even after accounting for teachers' math ability and children's beginning of year math knowledge, replicating and extending those previous results. Our findings strengthen the support for the hypothesis that teachers' math anxiety is one factor that undermines children's math learning and could push students off-track during their initial exposure to math in early elementary school.

Facilitating young children's numeracy talk in play: The role of parent prompts.

Previous research has demonstrated the contribution of parents' number language to children's own engagement with numbers and later mathematical achievement. Although there is evidence that both the quantity and complexity of parent number talk contribute to children's math learning, it is unclear whether different forms of parents' number talk-statements versus prompts-offer unique contributions to how children engage in math. We examined parent number talk among 50 dyads of parents and 2- to 4-year-olds during pretend play, coding parents' provisions of informative number statements and prompts inviting children to engage in number talk. The total amount (tokens) and diversity (types) of children's number words were analyzed separately. Parents' number utterances, particularly prompts about number, were infrequent. Both parents' number statements and their prompts were uniquely related to children's number word tokens. Only prompts were associated with children's number word types. Follow-up analyses indicated that prompts were associated with lengthier parent-child conversations about number than parent statements and that children used larger number words when responding to parent prompts than when they themselves initiated number talk. These findings highlight the importance of parents' prompts for enhancing the quality of parent-child math exchanges by providing opportunities for children to advance their current use of numerical language. Consequently, parents' use of number-related prompts may play an important role in children's early math engagement.

Causal Effects of Parent Number Talk on Preschoolers' Number Knowledge.

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.

Number gestures predict learning of number words.

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 (Mage  = 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.

Parents' early book reading to children: Relation to children's later language and literacy outcomes controlling for other parent language input.

It is widely believed that reading to preschool children promotes their language and literacy skills. Yet, whether early parent-child book reading is an index of generally rich linguistic input or a unique predictor of later outcomes remains unclear. To address this question, we asked whether naturally occurring parent-child book reading interactions between 1 and 2.5 years-of-age predict elementary school language and literacy outcomes, controlling for the quantity of other talk parents provide their children, family socioeconomic status, and children's own early language skill. We find that the quantity of parent-child book reading interactions predicts children's later receptive vocabulary, reading comprehension, and internal motivation to read (but not decoding, external motivation to read, or math skill), controlling for these other factors. Importantly, we also find that parent language that occurs during book reading interactions is more sophisticated than parent language outside book reading interactions in terms of vocabulary diversity and syntactic complexity.

Parents' Spatial Language Mediates a Sex Difference in Preschoolers' Spatial-Language Use.

Do boys produce more terms than girls to describe the spatial world-that is, dimensional adjectives (e.g., big, little, tall, short), shape terms (e.g., circle, square), and words describing spatial features and properties (e.g., bent, curvy, edge)? If a sex difference in children's spatial-language use exists, is it related to the spatial language that parents use when interacting with children? We longitudinally tracked the development of spatial-language production in children between the ages of 14 and 46 months in a diverse sample of 58 parent-child dyads interacting in their homes. Boys produced and heard more of these three categories of spatial words, which we call "what" spatial types (i.e., unique "what" spatial words), but not more of all other word types, than girls. Mediation analysis revealed that sex differences in children's spatial talk at 34 to 46 months of age were fully mediated by parents' earlier spatial-language use, when children were 14 to 26 months old, time points at which there was no sex difference in children's spatial-language use.

How do generic statements impact performance? Evidence for entity beliefs.

Generic statements about the abilities of children's social groups (e.g. 'Girls/Boys are good at this game') negatively impact children's performance - even if the statements are favorable towards children's own social groups. We explored the mechanism by which generic language impairs children's performance. Across three studies, our findings suggest that generic statements influence children's performance by creating an entity belief (i.e. a belief that a fixed ability determines performance). Children who were exposed to a generic statement about their social group's ability performed worse than children in control conditions. This effect hurt children's performance even when the person who made the generic statement was no longer present and a new person not privy to the statement replaced them. However, when children heard a generic statement paired with an effort explanation (i.e. 'Girls/Boys are good at this game because they try really hard when they draw') they performed better than children who heard the generic statement with no explanation (i.e. just 'Girls/Boys are good at this game') and children who heard the generic statement paired with a trait explanation (i.e. 'Girls/Boys are good at this game because they are smart and really good at drawing'). This work uncovers when and how generic statements that refer to the ability of one's social group hinder performance, informing the development of practices to improve student motivation and learning.

Commentary on Leibovich et al.: What next?

The conclusions reached by Leibovich et al. urge the field to regroup and consider new ways of conceptualizing quantitative development. We suggest three potential directions for new research that follow from the authors' extensive review, as well as building on the common ground we can take from decades of research in this area.

On the relationship between math anxiety and math achievement in early elementary school: The role of problem solving strategies.

Even at young ages, children self-report experiencing math anxiety, which negatively relates to their math achievement. Leveraging a large dataset of first and second grade students' math achievement scores, math problem solving strategies, and math attitudes, we explored the possibility that children's math anxiety (i.e., a fear or apprehension about math) negatively relates to their use of more advanced problem solving strategies, which in turn relates to their math achievement. Our results confirm our hypothesis and, moreover, demonstrate that the relation between math anxiety and math problem solving strategies is strongest in children with the highest working memory capacity. Ironically, children who have the highest cognitive capacity avoid using advanced problem solving strategies when they are high in math anxiety and, as a result, underperform in math compared with their lower working memory peers.

A parent-directed language intervention for children of low socioeconomic status: a randomized controlled pilot study.

We designed a parent-directed home-visiting intervention targeting socioeconomic status (SES) disparities in children's early language environments. A randomized controlled trial was used to evaluate whether the intervention improved parents' knowledge of child language development and increased the amount and diversity of parent talk. Twenty-three mother-child dyads (12 experimental, 11 control, aged 1;5-3;0) participated in eight weekly hour-long home-visits. In the experimental group, but not the control group, parent knowledge of language development increased significantly one week and four months after the intervention. In lab-based observations, parent word types and tokens and child word types increased significantly one week, but not four months, post-intervention. In home-based observations, adult word tokens, conversational turn counts, and child vocalization counts increased significantly during the intervention, but not post-intervention. The results demonstrate the malleability of child-directed language behaviors and knowledge of child language development among low-SES parents.

Intergenerational Effects of Parents' Math Anxiety on Children's Math Achievement and Anxiety.

A large field study of children in first and second grade explored how parents' anxiety about math relates to their children's math achievement. The goal of the study was to better understand why some students perform worse in math than others. We tested whether parents' math anxiety predicts their children's math achievement across the school year. We found that when parents are more math anxious, their children learn significantly less math over the school year and have more math anxiety by the school year's end-but only if math-anxious parents report providing frequent help with math homework. Notably, when parents reported helping with math homework less often, children's math achievement and attitudes were not related to parents' math anxiety. Parents' math anxiety did not predict children's reading achievement, which suggests that the effects of parents' math anxiety are specific to children's math achievement. These findings provide evidence of a mechanism for intergenerational transmission of low math achievement and high math anxiety.

The effects of word-learning biases on children's concept of angle.

Despite evidence that young children are sensitive to differences in angle measure, older students frequently struggle to grasp this important mathematical concept. When making judgments about the size of angles, children often rely on erroneous dimensions such as the length of the angles' sides. The present study tested the possibility that this misconception stems from the whole-object word-learning bias by providing a subset of children with a separate label to refer to the whole angle figure. Thirty preschoolers (M = 4.86 years, SD = .53) were tested with a pretest-training-posttest design. At pretest, children showed evidence of the whole-object misconception. After training, children who were given a novel-word label for the whole object improved significantly more than those trained on the meaning of "angle" alone.

Parents' Talk About Letters With Their Young Children.

A literacy-related activity that occurs in children's homes-talk about letters in everyday conversations-was examined using data from 50 children who were visited every 4 months between 14 and 50 months. Parents talked about some letters, including those that are common in English words and the first letter of their children's names, especially often. Parents' focus on the child's initial was especially strong in families of higher socioeconomic status, and the extent to which parents talked about the child's initial during the later sessions of the study was related to the children's kindergarten reading skill. Conversations that included the child's initial were longer than those that did not, and parents presented a variety of information about this letter.

Approximate number word knowledge before the cardinal principle.

Approximate number word knowledge-understanding the relation between the count words and the approximate magnitudes of sets-is a critical piece of knowledge that predicts later math achievement. However, researchers disagree about when children first show evidence of approximate number word knowledge-before, or only after, they have learned the cardinal principle. In two studies, children who had not yet learned the cardinal principle (subset-knowers) produced sets in response to number words (verbal comprehension task) and produced number words in response to set sizes (verbal production task). As evidence of approximate number word knowledge, we examined whether children's numerical responses increased with increasing numerosity of the stimulus. In Study 1, subset-knowers (ages 3.0-4.2 years) showed approximate number word knowledge above their knower-level on both tasks, but this effect did not extend to numbers above 4. In Study 2, we collected data from a broader age range of subset-knowers (ages 3.1-5.6 years). In this sample, children showed approximate number word knowledge on the verbal production task even when only examining set sizes above 4. Across studies, children's age predicted approximate number word knowledge (above 4) on the verbal production task when controlling for their knower-level, study (1 or 2), and parents' education, none of which predicted approximation ability. Thus, children can develop approximate knowledge of number words up to 10 before learning the cardinal principle. Furthermore, approximate number word knowledge increases with age and might not be closely related to the development of exact number word knowledge.

A tale of two hands: children's early gesture use in narrative production predicts later narrative structure in speech.

Speakers of all ages spontaneously gesture as they talk. These gestures predict children's milestones in vocabulary and sentence structure. We ask whether gesture serves a similar role in the development of narrative skill. Children were asked to retell a story conveyed in a wordless cartoon at age five and then again at six, seven, and eight. Children's narrative structure in speech improved across these ages. At age five, many of the children expressed a character's viewpoint in gesture, and these children were more likely to tell better-structured stories at the later ages than children who did not produce character-viewpoint gestures at age five. In contrast, framing narratives from a character's perspective in speech at age five did not predict later narrative structure in speech. Gesture thus continues to act as a harbinger of change even as it assumes new roles in relation to discourse.

Overturning Children's Misconceptions about Ruler Measurement: The Power of Disconfirming Evidence.

Children have persistent difficulty with foundational measurement concepts, which may be linked to the instruction they receive. Here, we focus on testing various ways to support their understanding that rulers comprise spatial interval units. We examined whether evidence-based learning tools-disconfirming evidence and/or structural alignment-enhance their understanding of ruler units. Disconfirming evidence, in this context, involves having children count the spatial interval units under an object that is not aligned with the origin of a ruler. Structural alignment, in this context, involves highlighting what a ruler unit is by overlaying plastic unit chips on top of ruler units when an object is aligned with the origin of a ruler. In three experiments employing a pre-test/training/post-test design, a total of 120 second graders were randomly assigned to one of six training conditions (two training conditions per experiment). The training conditions included different evidence-based learning principles or "business-as-usual" instruction (control), with equal allocation to each (N = 20 for each condition). In each experiment, children who did not perform above chance level on the pre-test were selected to continue with training, which resulted in a total of 88 students for the analysis of improvement. The children showed significant improvement in training conditions that included disconfirming evidence, but not in the structural alignment or control conditions. However, an exploratory analysis suggests that improvement occurred more rapidly and was retained better when structural alignment was combined with disconfirming evidence compared to disconfirming evidence alone.