Algebraic competence is a major determinant of college access and career prospects, and equal sign knowledge is taken to be foundational to algebra knowledge. However, few studies have documented a causal effect of early equal sign knowledge on later algebra skill. This study assessed whether second-grade students' equal sign knowledge…

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…

The main goal of this study was to analyze the contribution of predictors of both domain-general (working memory, processing speed, and receptive vocabulary) and domain-specific variable (estimation and magnitude comparison) processes to informal mathematical performance (numbering, comparison, calculation, and understanding of concepts) in…

Difficulties in mathematics are argued to stem from impairments of a specialized system of numerical magnitude representation. This study investigates whether different measures of numerical magnitude understanding in kindergarten uniquely predict mathematical achievement concurrently and 6 months later, and also examines the relative explanatory…

Many people find that solving story problems is challenging. A key source of the difficulty is symbolizing problem situations. In two experiments, we tested whether diagrams promoted successful symbolization of two-operator story problems, and we tested whether diagrams that depicted the integration of operations were more beneficial than diagrams…

Existing research suggests that young children can develop a partial understanding of the equal sign as an operator rather than as a relational symbol of equivalence. This partial understanding can be the result of overemphasis on canonical equation syntaxes of the type a + b = c in elementary school mathematics. This paper presents an examination…

We present a technique that examines handwritten equations from a student's solution to an engineering problem and from this estimates the correctness of the work. More specifically, we demonstrate that lexical properties of the equations correlate with the grade a human grader would assign. We characterize these properties with a set of features…

The development of (early) numerical cognition builds on children's ability to understand and manipulate quantities and numbers. However, previous research did not find conclusive evidence on the role of symbolic and non-symbolic skills in the development of (early) numerical cognition. The aim of the current study was to clarify the relation…

Purpose: Music could be a challenger for mathematics and a potential candidate for the title "The Universal Language." This paper aims to discuss the primary objectives of engaging with music, including the therapeutic benefits. Similarities, between mathematics and music and how studying one might enhance one's abilities of the other…

Numbers and variables often follow the same principles of arithmetic operations, yet numbers can be computed to a value whereas variables cannot. We examined the effect of symbols--numbers versus variables--on middle school students' problem-solving behaviors in a dynamic algebra notation system by presenting problems in numbers (e.g., 3+5-3) or…

Mathematics educators view the equals sign as a bidirectional relation symbol, but the authors have observed that students might not have such flexibility in their understanding of the equals sign. The authors have observed that students have a hard time viewing mathematical properties (for example, log (MN) = log (M) log (N)) with bidirectional…

Many primary school students have difficulties understanding mathematical equivalence with considerably poorer performance in some countries than in others. However, students' formal understanding of equivalence has significant and long-lasting effects, as it predicts arithmetic and algebra achievement throughout school years. Currently, little is…

This qualitative study examines the semiotic meanings communicated through multiple representations in the teaching of an advanced undergraduate course in biology. Scientific discourse is intrinsically multimodal in nature; in other words, science is discussed and researched using multiple modes that convey meaning (Lemke, 1998). Therefore, to…

To efficiently solve mathematical expressions and equations, students need to notice the systemic structure of mathematical expressions (e.g., inverse relation between 3 and 3 in 3 + 5 - 3). We examined how symbols--specifically variables versus numbers--and students' algebraic knowledge impacted seventh graders' problem-solving strategies and use…

In this article, we question the prevalent assumption that teaching and learning mathematics should always entail movement from the concrete to the abstract. Such a view leads to reported difficulties in students moving from manipulatives and models to more symbolic work, moves that many students never make, with all the implications this has for…