The relation to operands (RO) principles describe the relation between operands and answers in arithmetic problems (e.g., the sum is always larger than its positive addends). Despite being a fundamental property of arithmetic, its empirical relation with arithmetic/algebraic problem solving has seldom been investigated. The current longitudinal…

Elementary school children (ages 7-11) struggle to understand mathematical equivalence, a foundational prealgebraic concept. Some manipulations to the learning environment, including well-structured nontraditional arithmetic practice alone, have been shown to improve children's understanding; however, improvements have been modest. The goal of…

In this paper we explore some of factors that affect bilingual students' comprehension of story problems: vocabulary, syntax, cultural relevance and understanding of the word problem genre. In an effort to determine how these factors interact, we asked 18 Spanish/English bilingual children to retell and solve arithmetic story problems in…

Numerical understanding and arithmetic skills are easier to acquire for whole numbers than fractions. The "integrated theory of numerical development" posits that, in addition to these differences, whole numbers and fractions also have important commonalities. In both, students need to learn how to interpret number symbols in terms of…

Decipipes are a representational model that can be used to help students develop conceptual understanding of decimal place value. They provide a non-standard tool for representing length, which in turn can be represented using conventional decimal notation. They are conceptually identical to Linear Arithmetic Blocks. This article reviews theory…

The supposed pre-eminence of an external examination can exert a disproportionate influence on a curriculum and the associated learning and teaching. Teaching can easily subordinate learning and understanding to curriculum coverage if the society develops a culture that appears to make such demands. This study focuses on Tanzania and provides the…

This experiment tested the hypothesis that organizing arithmetic fact practice by equivalent values facilitates children's understanding of math equivalence. Children (M age = 8 years 6 months, N = 104) were randomly assigned to 1 of 3 practice conditions: (a) equivalent values, in which problems were grouped by equivalent sums (e.g., 3 + 4 = 7, 2…

Early measures of mathematics skill and development have focused on early numeracy skills like counting, number identification, and sequencing of numbers. This study attempted to expand early mathematics assessment. Six new measures of early mathematics skill competence were developed and evaluated. Four existing measures also were examined.…

Many students and adults feel that algebra is merely the shuffling of symbols. The three interrelated concepts of variable, expression, and equation are central to beginning algebra, and in recent years, helping students understand the idea of a variable has been emphasized. Although graphing calculators help students solve equations, it is also…

Previous research on children's conceptual and procedural understanding of fractions, and other arithmetic skills, has led to contradictory conclusions. Some research suggests that children learn conceptual knowledge before procedural knowledge, some suggests that they learn procedural knowledge before conceptual knowledge, and other research…

Children's understanding of the inversion concept in multiplication and division problems (i.e., that on problems of the form "d multiplied by e/e" no calculations are required) was investigated. Children in Grades 6, 7, and 8 completed an inversion problem-solving task, an assessment of procedures task, and a factual knowledge task of simple…

Understanding conceptual relationships is an important aspect of learning arithmetic. Most studies of arithmetic, however, do not distinguish between children's understanding of a concept and their ability to identify situations in which it might be relevant. We compared 8- to 9-year-old children's use of a computational shortcut based on the…

Many mathematics educators believe that it is insufficient for students' knowledge of the division of fractions to be limited to the invert-and-multiply algorithm, arguing that students' learning of mathematics must go beyond rote memorization of procedures. Teachers ought to carefully consider what students need to learn beyond the algorithmic…

This article explores how conceptualization of absolute value can start long before it is introduced. The manner in which absolute value is introduced to students in middle school has far-reaching consequences for their future mathematical understanding. It begins to lay the foundation for students' understanding of algebra, which can change…

This article shares an assessment tool that uses student-written word problems to provide meaningful information regarding the depth of mathematical understanding. Students' word problems representing 6 [division] 1/2 were classified using the scoring categories described in the NAEP. By categorizing the problems, students' levels of understanding…