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.

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.

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.

Children's ability to place fractions on a number line strongly correlates with math achievement. But does the number line play a causal role in fraction learning or does it simply index more advanced fraction knowledge? The number line may be a particularly effective representation for fraction learning because its properties align with the desired mental representation and take advantage of preexisting spatial numeric biases. Using a pretest-training-posttest design, we examined second and third graders' fraction learning in 3 conditions: number line training, area model training, and a non-numerical control. Children who received number line training improved at representing fractions with number lines, and children who received area model training improved at representing fractions with area models. Critically, only the number line training led to transfer to an untrained fraction magnitude comparison task. We conclude that the number line plays a causal role in children's fraction magnitude understanding, and is more beneficial than the widely used area model.