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…

Decimals and the related areas of ratio and proportion are recognised as one of the most challenging and complex areas of mathematics for young children to learn. In this study, four dynamic digital representations were used by Year 4 (9-10 years old) students in a set of video-recorded, individual task-based interviews to explore key concepts of…

This article aims to show some activities based on non-routine problems with different levels of difficulty and from different classes that are integrated into the curriculum, additionally two spaces were opened for students to interact with recreational mathematics activities with the purpose of providing opportunities to all students access the…

Teachers' understanding of the concepts they teach affects the quality of instruction and students' learning. This study used a sample of 303 teachers from across the USA to examine elementary school mathematics teachers' knowledge of key concepts underlying fraction arithmetic. Teachers' explanations were coded based on the accuracy of their…

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…

This study investigated relations between the distribution of practice problems in textbooks and students' learning of decimal arithmetic. In Study 1, we analyzed the distributions of decimal arithmetic practice problems that appeared in 3 leading math textbook series in the United States. Similar imbalances in the relative frequencies of decimal…

Background: Prior research with adults and children suggests that inhibitory control may have a role to play in learning counterintuitive fractions and decimals that are inconsistent with whole number knowledge. However, there is little research to date with primary school-aged children at the early stages of fraction and decimal instruction that…

The number one plays a special role in mathematics because it is the identity element in multiplication and division. The present findings, however, indicate that many middle school students do not demonstrate mathematical flexibility representing one as a fraction. Despite possessing explicit knowledge of fraction forms of one (e.g., 95% of…

Children's understanding of fractions, including their symbols, concepts, and arithmetic procedures, is an important facet of both developmental research on mathematics cognition and mathematics education. Research on infants', children's, and adults' fraction and ratio reasoning allows us to test a range of proposals about the development of…

Existing frameworks of teachers' knowledge required to teach mathematics do not adequately capture the dynamic aspects of knowledge manifested in teaching practice. In this paper, we examine the knowledge demands that arise in situ, in the course of a teacher listening and responding to students' thinking, while teaching the topic of decimal…

Children's failure to reason often leads to their mathematical performance being shaped by spurious associations from problem input and overgeneralization of inapplicable procedures rather than by whether answers and procedures make sense. In particular, imbalanced distributions of problems, particularly in textbooks, lead children to create…

Fractions remain a difficult concept for students at the elementary level. On that ground, prospective teachers need to develop the conceptual knowledge to have a deep understanding of the concept and how the concepts are related to each other. Furthermore, they must be able to explain the concepts through media in the form of concrete objects or…

Most students can follow this simple procedure for division of fractions: "Ours is not to reason why, just invert and multiply." But how many really understand what division of fractions means--especially fraction division with respect to the meaning of the remainder. The purpose of this article is to provide an instructional method as a…

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…