Infants who receive better counting input at home tend to become toddlers with better number knowledge in preschool. However, for many children, in-person counting experience is not always available, despite educational media becoming increasingly prevalent. Might virtual counting experience benefit the young mind? Using a novel online looking…

Understanding of the complement principle has been proposed as closely related to computational skills, but few studies have investigated their interrelations. The present longitudinal study attempted to clarify the picture by examining their potential cross-lagged relation. Fourth graders (n = 221) in Hong Kong received 3 cognitive assessments at…

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…

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…

From his first attempts to explain attachment phenomena in the 1940s through his "Attachment and Loss" trilogy (Bowlby, 1969/1982, 1973, 1980), John Bowlby reformulated the theoretical underpinnings of attachment theory several times. He initially attempted to explain attachment phenomena in psychoanalytic terms. Then he invoked…

The aim of the present study was to examine the kinds of developmental profiles of arithmetic fluency skills that can be identified across Grades 1-9 (ages 7-16) in a large Finnish sample (n = 2,518). The study also examined whether membership in the developmental profiles could be predicted using a comprehensive set of kindergarten-age factors,…

Researchers are often inclined to test agreement or discrepancy hypotheses using difference scores. This commentary explains 2 mathematical-statistical principles underlying associations with difference scores and 2 conceptual-interpretation problems that make difference scores inappropriate for testing such hypotheses. The commentary provides…

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…

While much research has focused on understanding the process by which young children learn to count, little work has explored the effects of direct instruction on this process. In the current study, we explored the impacts of training children in an explicit counting procedure on two distinct cardinality tasks. Two- to 5-year-old children first…

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.…

Elementary school students often lack a conceptual understanding of linear measurement, which is revealed by their poor performance when the object to be measured is not aligned with the start of the ruler. Instead of correctly counting the units that correspond to the object (e.g., inches or centimeters), children often use 1 of 2 incorrect…

When proportional information is pit against whole number numerical information, children often attend to the whole number information at the expense of proportional information (e.g., indicating 4/9 is greater than 3/5 because 4 > 3). In the current study, we presented younger (3- to 4-year-olds) and older (5- to 6-year-olds) children a task…

Correct counting respects the stable order principle whereby the count terms are recited in a fixed order every time. The 4 experiments reported here tested whether precounting infants recognize and prefer correct stable-ordered counting. The authors introduced a novel preference paradigm in which infants could freely press two buttons to activate…

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…