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…

This study examined repeating and growing pattern knowledge and their associations with procedural and conceptual arithmetic knowledge in a sample of U.S. children (N = 185; M[subscript age] = 79.5 months; 55% female; 88% White) and adults (N = 93; M[subscript age] = 19.5 years; 62% female; 66% White) from 2019 to 2020. Three key findings emerged:…

In the last two decades, studying the possible relations between the history of mathematics and mathematics education has revealed the importance of certain foundational issues concerning the nature, relevance and validity of historical knowledge vis-à-vis problems of teaching and learning, which call for deeper exploration. Reviewing aspects of…

When, how, and why students use conceptual knowledge during math problem solving is not well understood. We propose that when solving routine problems, students are more likely to recruit conceptual knowledge if their procedural knowledge is weak than if it is strong, and that in this context, metacognitive processes, specifically feelings of…

It has become increasingly clear that the early use of decomposition for addition is associated with later mathematical achievement. This study examined how younger children execute a base-10 decomposition strategy to solve complex arithmetic (e.g. two-digit addition). 24 addition problems in two modalities (WA: Written Arithmetic; OA: Oral…

Algebraic thinking and strategy flexibility are essential to advanced mathematical thinking. Early algebra instruction uses 'missing-operand' problems (e.g., x - 7 = 2) solvable via two typical strategies: (1) direct retrieval of arithmetic facts (e.g., 9 - 7 = 2) and (2) performance of the inverse operation (e.g., 2 + 7 = 9). The current study…

In this article we discuss relationships between the cyclic group Z[subscript 12] and Western tonal music that is embedded in a 12-note division of the octave. We then offer several questions inviting students to explore extensions of these relationships to other "n"-note octave divisions. The answers to most questions require only basic…

Research has shown that fraction magnitude and whole number division are important precursors to learning and understanding fractions. Deaf and hard-of-hearing (DHH) students are consistently challenged with learning fractions from K-12 through college. Sixty DHH college students were tested for both their understanding of magnitude between two…

Fraction arithmetic is among the most important and difficult topics children encounter in elementary and middle school mathematics. Braithwaite, Pyke, and Siegler (2017) hypothesized that difficulties learning fraction arithmetic often reflect reliance on associative knowledge--rather than understanding of mathematical concepts and procedures--to…

Fraction arithmetic is among the most important and difficult topics children encounter in elementary and middle school mathematics. Braithwaite, Pyke, and Siegler (2017) hypothesized that difficulties learning fraction arithmetic often reflect reliance on associative knowledge--rather than understanding of mathematical concepts and procedures--to…

Are there differential benefits of training sequential number knowledge versus spatial skills for children's numerical and spatial performance? Three- to five-year-old children (N = 84) participated in 1 session of either sequential training (e.g., what comes before and after the number 5?) or non-numerical spatial training (i.e., decomposition of…

There are many continuous quantitative dimensions in the physical world. Philosophical, psychological and neural work has focused mostly on space and number. However, there are other important continuous dimensions (e.g., time, mass). Moreover, space can be broken down into more specific dimensions (e.g., length, area, density) and number can be…

This study aimed to examine the relations among property strategies, working memory, and multiplication tasks with 101 Chinese fourth-grade students. Two multiplication property strategies (associative and distributive) were compared with no strategy and demonstrated differentiated effects on students' accuracy and reaction time. Associative…

The middle grades are a critical transition period in students' mathematics trajectories, as students move from arithmetic to the more complex and abstract concepts of algebra. Teachers' and parents' judgments of students' math abilities in these years are important to instructional planning and decision making for teachers, and can advise parents…

Mastering single-digit arithmetic during school years is commonly thought to depend upon an increasing reliance on verbally memorized facts. An alternative model, however, posits that fluency in single-digit arithmetic might also be achieved via the increasing use of efficient calculation procedures. To test between these hypotheses, we used a…