Learning mathematics involves abstract thinking and the use of symbols. To be successful in a mathematics classroom, students must master what symbols mean and how they work. Much prior work has focused on the nature of the external representations used in mathematics instruction, including manipulatives and symbolic representations. In this work,…

Although the roles of symbolic numerical magnitude processing (SNMP) and working memory (WM) in mathematics performance are well acknowledged, studies examining their joint effects are few. Here, we investigated the profiles of SNMP (1- and 2-digit comparison) and WM (verbal, visual and central executive) among Norwegian first graders (N = 256),…

This study examined the effects of training involving spatial versus nonspatial representations of numerical magnitude for promoting arithmetic fluency. The key goal was to advance theoretical understanding of the relation between spatial and math learning, while simultaneously laying the groundwork for the development of future educational…

This study is focused on the relational thinking of first-grade students following their first 3 months of instruction from the Measure Up (MU) curriculum, an adaptation of the El'konin-Davydov curriculum. Following Davydov's outline of instructional material, the MU first-grade materials were designed to have students identify the quantitative…

Children struggle with exact, symbolic ratio reasoning, but prior research demonstrates children show surprising intuition when making approximate, nonsymbolic ratio judgments. In the current experiment, eighty-five 6- to 8-year-old children made approximate ratio judgments with dot arrays and numerals. Children were adept at approximate ratio…

How do children develop associations among number symbols? For Grade 1 children (n = 66, M = 78 months), sequence knowledge (i.e., identify missing numbers) and number comparison (i.e., choose larger number) predicted addition, both concurrently and indirectly at the end of Grade 1. Number ordering (i.e., touch numbers in order) did not predict…

This article focuses on how Swedish Year One students (age 7-8) make meaning when working with images in mathematics textbooks. Images include textbook images, but also students' self-drawn images used as support for calculation. The focus was (1) what the images in the exercises were designed to offer (the designed affordance), and (2) what the…

This research focuses on ways in which balance scales mediate students' relational understandings of the equal sign. Participants included 21 Kindergarten-Grade 2 students who took part in an early algebra classroom intervention focused in part on developing a relational understanding of the equal sign through the use of balance scales. Students…

This research shares progressions in thinking about equations and the equal sign observed in ten students who took part in an early algebra classroom intervention across Kindergarten and first grade. We report on data from task-based interviews conducted prior to the intervention and at the conclusion of each school year that elicited students'…

There are two lines of discussion regarding the function of the Approximate Number System (ANS). The first line focuses on the extent to which visual cues affect the estimation of numerosity. The second line investigates the extent to which ANS precision is associated with symbolic math performance. The current study combined these two lines of…

Research has shown that two different, though related, ways of representing magnitude play foundational roles in the development of numerical and mathematical skills: a nonverbal approximate number system and an exact symbolic number system. While there have been numerous studies suggesting that the two systems are important predictors of math…

To understand mathematics, children must interpret mathematics symbols. In this study, we designed a brief assessment of mathematics symbols for children in the elementary grades. For each of 23 symbols, children identified the symbol, provided the meaning of the symbol, and used the symbol. We assessed 297 children in Grades 1, 3, and 5. Internal…

Many studies conducted in the last decades report that very few elementary-grades students understand the meaning of = in mathematics, and that upper-elementary students do not understand it any better than the lower-elementary students do. Using data from thousands of Florida elementary students, we show that students in Florida perform ten to…

Recent studies suggest that the relation between nonsymbolic magnitude processing skills and math competence is mediated by symbolic number processing. However, less is known about whether mapping between nonsymbolic and symbolic magnitude representations also mediates that relation, and whether the mediating role of symbolic number processing is…

Math relies on mastery and integration of a wide range of simpler numerical processes and concepts. Recent work has identified several numerical competencies that predict variation in math ability. We examined the unique relations between eight basic numerical skills and early arithmetic ability in a large sample (N = 1391) of children across…