Children display an early sensitivity to implicit proportions (e.g., 1 of 5 apples vs. 3 of 4 apples), but have considerable difficulty in learning the explicit, symbolic proportions denoted by fractions (e.g., "1/5" vs. "3/4"). Theoretically, reducing the gap between representations of implicit versus explicit proportions…

Kim and Opfer (2017) found that number-line estimates increased approximately logarithmically with number when an upper bound (e.g., 100 or 1000) was explicitly marked (bounded condition) and when no upper bound was marked (unbounded condition). Using procedural suggestions from Cohen and Ray (2020), we examined whether this logarithmicity might…

Recently, there has been increasing evidence showing that males estimate whole numbers more accurately than females on the number line. However, relatively little is known about what factors contribute to this gender gap. The current study explored potential mediators of the gender difference in number line estimation, including spatial skills and…

Toddlers performed a spatial mapping task in which they were required to learn the location of a hidden object in a vertical array and then transpose this location information 90° to a horizontal array. During the vertical training, they were given (a) no labels, (b) alphabetical labels, or (c) numerical labels for each potential spatial location.…

Accurate monitoring and control are essential for effective self-regulated learning. These metacognitive abilities may be particularly important for developing math skills, such as when children are deciding whether a math task is difficult or whether they made a mistake on a particular item. The present experiments investigate children's ability…

Humans and other animals are able to make rough estimations of quantities using what has been termed the "approximate number system" (ANS). Much evidence suggests that sensitivity to numerosity correlates with symbolic math capacity, leading to the suggestion that the ANS may serve as a start-up tool to develop symbolic math. Many…

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…

Representations of numerical value have been assessed by using bounded (e.g., 0-1,000) and unbounded (e.g., 0-?) number-line tasks, with considerable debate regarding whether 1 or both tasks elicit unique cognitive strategies (e.g., addition or subtraction) and require unique cognitive models. To test this, we examined how well a mixed log-linear…

We tested the hypothesis that encoding the numerical-spatial relations in a number board game is a key process in promoting learning from playing such games. Experiment 1 used a microgenetic design to examine the effects on learning of the type of counting procedure that children use. As predicted, having kindergartners count-on from their current…

Are there differential benefits of training sequential number knowledge versus spatial skills for children's numerical and spatial performance? Three- to five-year-old children (N = 84) participated in 1 session of either sequential training (e.g., what comes before and after the number 5?) or non-numerical spatial training (i.e., decomposition of…

Children's understanding of numbers is often assessed using a number-line task, where the child is shown a line labeled with 0 at one end and a higher number (e.g., 100) at the other end. The child is then asked where on the line some intermediate number (e.g., 70) should go. Performance on this task changes predictably during childhood, and this…

In the number-to-position task, with increasing age and numerical expertise, children's pattern of estimates shifts from a biased (nonlinear) to a formal (linear) mapping. This widely replicated finding concerns symbolic numbers, whereas less is known about other types of quantity estimation. In Experiment 1, Preschool, Grade 1, and Grade 3…

Children's sense of numbers before formal education is thought to rely on an approximate number system based on logarithmically compressed analog magnitudes that increases in resolution throughout childhood. School-age children performing a numerical estimation task have been shown to increasingly rely on a formally appropriate, linear…

Focused on cardinality acquisition (thought to be crucial to correct counting) and on the relation between cardinality and counting. Gathered data with three groups of children three to five years old on four kinds of tasks. Found no empirical evidence for considering counting as a single prerequisite to cardinality. Suggests new approach to…

Preschoolers' performance on two tasks demonstrated that, given a perceptually available set of dolls, they were able to use number words to determine the quantity of a hidden or nonexistent set of items that was in a known ratio to the available set. (MM)