Competence with rational numbers is essential for mathematics proficiency in secondary mathematics. However, many students struggle with rational number concepts, and students with mathematics difficulties struggle even more. The purpose of this study was to examine the effects of an intervention that incorporated the use of explicit instruction…

Students' understanding of the meaning of the equal sign develops slowly over the primary grades. In addition to updating their representations of equations to recognize that the equal sign represents an equivalence relation rather than signaling an operation, students need to move beyond full computation to efficiently solve equivalence problems.…

The aim in the current study is to examine the conceptualizations of zero in arithmetic operations among students with learning disabilities (LD) and no learning disabilities (N-LD). The similarities and differences in the understandings of students with LD and N-LD of zero in arithmetical operations will be discussed. The study is a multiple case…

We propose to assess conceptual knowledge of mathematical notions by having recourse to isomorphic word problems. We assumed that failing to solve isomorphic problems is an indicator of lack of conceptual knowledge. To reach these conclusions, two experiments were conducted among 4th and 5th grade students. In experiment 1, each student had to…

The purpose of this study is to examine the effects of concrete and technology-assisted learning tools on developing the conception of place value, mathematical achievement and arithmetical performance of primary school 4th graders. The study group was comprised of three different primary schools. There were no group differences prior to…

Understanding the decimal system is challenging, requiring coordination of place-value concepts with features of whole-number and fraction knowledge (Moloney and Stacey 1997). Moreover, the learner must discern if and how previously learned concepts and procedures apply. The process is complex, and misconceptions will naturally arise. In a…

Students' success with fourth-grade content standards builds on mathematical knowledge learned in third grade and creates a conceptual foundation for division standards in subsequent grades that focus on the division algorithm. The division standards in fourth and fifth grade are similar; but in fourth grade, division problem divisors are only one…

Many children fail to master fraction arithmetic even after years of instruction. A recent theory of fraction arithmetic (Braithwaite, Pyke, & Siegler, 2017) hypothesized that this poor learning of fraction arithmetic procedures reflects poor conceptual understanding of them. To test this hypothesis, we performed three experiments examining…

Many children fail to master fraction arithmetic even after years of instruction. A recent theory of fraction arithmetic (Braithwaite, Pyke, & Siegler, in press) hypothesized that this poor learning of fraction arithmetic procedures reflects poor conceptual understanding of them. To test this hypothesis, we performed three experiments…

The purpose of the study was to determine whether individual differences in at-risk 4th graders' language comprehension, nonverbal reasoning, concept formation, working memory, and use of decimal labels (i.e., place value, point, incorrect place value, incorrect fraction, or whole number) are related to their decimal magnitude understanding.…

This paper aims to describe how students develop understanding of one-digit decimals. To achieve the aim, Local Instruction Theory (LIT) about the process of learning decimals and the means designed to support that learning are developed. Along with this idea, the framework of Realistic Mathematics Education (RME) is proposed. Based on the aim,…

The aim of this study was to examine the concept development of decimal numbers in 244 Chinese elementary students in grades 4-6. Three grades of students differed in their intuitive sense of decimals and conceptual understanding of decimals, with more strategic approaches used by older students. Misconceptions regarding the density nature of…

The goals for this study were to investigate how fourth-grade students developed an understanding of the arithmetic properties when instruction promoted mathematical argumentation and to identify the characteristics of students' arguments. Using the emergent perspective as an overarching theoretical perspective helped distinguish between two…

Developmental predictors of children's fraction concepts and procedures at the end of fourth grade were investigated in a 2-year longitudinal study. Participants were 357 children who started the study in third grade. Attentive behavior, language, nonverbal reasoning, number line estimation, calculation fluency, and reading fluency each…

After the onset of formal schooling, little is known about the development of children's understanding of the arithmetic concepts of inversion and associativity. On problems of the form a+b-b (e.g., 3+26-26), if children understand the inversion concept (i.e., that addition and subtraction are inverse operations), then no calculations are needed…