Sixty (35 girls and 25 boys) 9th-grade students' conceptual understanding of the number line was qualitatively assessed through verbal explanations and visual representations. The assessment included an open-ended question focused on students' number line descriptions and the explanations coalesced around six features: sequential ordering (i.e.,…

This study sought to understand how students activate number sense in determining the position of fractions on a number line and identify how the natural number bias and number sense influences students' thinking processes. The study utilized the Cognitive Task Analysis (CTA), involving four fifth-grade elementary students as the research…

This study aimed to evaluate the effectiveness of the Prime Factors Chart (PFC) as an alternative teaching material, which was developed for teaching the concepts within the "factors and multiples" unit in the middle school mathematics curriculum. Twelve middle school mathematics teachers used the PFC to teach concepts, such as prime…

Analogy is a powerful learning mechanism for children to learn novel, abstract concepts from only limited input, yet also requires cognitive supports. My dissertation sought to propose and examine number lines as a mathematical schema of the number system to facilitate both the development of rational number understanding and analogical reasoning.…

This study examines the conceptual understanding of numbers and operations among 372 fifth-grade students, based on their responses to an online three-tier test designed to assess their number sense, declarative knowledge, explanatory knowledge, and confidence. The results showed that most students had low performance and low number sense, with…

The "Reasoning Algebraically about Operations Casebook" was developed as the key resource for participants' Developing Mathematical Ideas seminar experience. The thirty-four cases, written by teachers describing real situations and actual student thinking in their classrooms, provide the basis of each session's investigation into the…

This article highlights how the Mathematics Teaching Practice of "facilitating meaningful mathematical discourse" (NCTM 2014, p. 10) was used to support a class of eighth-grade students as they engaged in recognizing patterns in which inputs are related to an output. As Carpenter, Franke, and Levi (2003) recommended, the authors chose…

The current study assessed whether adding worked examples with self-explanation prompts focused on making connections between mathematical principles, procedures, and concepts of rational numbers to a curriculum focused on invented strategies improves pre-algebra students' fraction number line acuity, rational number concepts and procedures.…

When solving a mathematical problem, students who do not have sufficient conceptual understanding may perform poorly and exhibit misconceptions. This study was aimed to examine students' conceptual understanding and significant misconceptions when solving number sense-related problems. An online three-tier diagnostic test was administered to 125…

In this article, we report the findings of a study conducted with 6 Grade 8 students in the United States. The students posed stories for open number sentences involving addition and subtraction of integers. We analysed the stories posed by the students to build models that describe the conceptual structures behind these posed stories -- the…

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…

The integer is a basic concept in studying arithmetic and algebra. However, students still frequently experience misconceptions, especially in negative integer, count operations. Traditional games are activities that are often carried out by students in coastal areas so that they are relevant to be used as a tool to construct learning trajectories…

The comprehension of the place value concept which is based on the grouping of numbers in accordance with distinct systems is not so easy. One of the sources that may contribute to the comprehension of the place value and making connections with daily life is children's books. The effects of several tools on the comprehension of the place value…

The main purpose of this study is to develop a conceptual understanding of the irrational number of the square root of 2 ([square root]2 ). Participants in the study were 20 ninth-grade male students. Activity Theory was used as a framework to show the development of the conceptual understanding. Since this study was conducted during the COVID-19…

Multiplication and division are vital topics in upper level elementary school. A teacher's pedagogical content knowledge (PCK) influences both instruction and students' learning. However, there is currently little research examining teachers' PCK within this domain, particularly regarding professional education of future teachers. To help address…