This paper explores the dimensions of knowledge and the ways of thinking (procedural or proceptual thinking) adopted by 157 Lebanese students in different grades when defining and identifying irrational numbers when locating them on the number line. It searches for a relation between the performance of the participants and their different grades.…

This report provides a description of the development process, field testing, and psychometric properties of the fall 2015 grades 3-5 Elementary Mathematics Student Assessment (EMSA), a student mathematics test designed to be administered in a whole-group setting to students in grades 3, 4, and 5. The test was administered to 2,614 participating…

The purpose of this study was to examine patterns of mathematical practices evident during children's interactions with touchscreen mathematics virtual manipulatives. Researchers analyzed 33 Kindergarten children's interactions during activities involving apps featuring mathematical content of early number sense or quantity in base ten, recorded…

Purpose: The present study was designed to examine whether mathematics computer games improved young children's learning of number sense under three different conditions: when used individually, with a peer, and with teacher facilitation. Methodology: This study utilized a mixed methodology, collecting both quantitative and qualitative data. A…

Being able to effectively interpret students' thinking and respond effectively in the moment are important skills that preservice teachers (PSTs) need to learn. This study zooms in on 14 PSTs' planning, teaching, and reflections involved in number string lessons, and investigates to what extent PSTs anticipate their students' strategies and…

Human children possess the ability to approximate numerical quantity nonverbally from a young age. Over the course of early childhood, children develop increasingly precise representations of numerical values, including a symbolic number system that allows them to conceive of numerical information as Arabic numerals or number words. Functional…

Skilled instructors observe consistent errors that young children may make in particular situations and seek ways to help children reform their thinking through conceptual understanding. Such teacher observations as those presented at the beginning of this article, combined with similar observations by teachers in other preschool and kindergarten…

Latin squares form the basis for the recreational puzzles sudoku and KenKen. In this article we show how useful several ideas from number theory are in solving a KenKen puzzle. For example, the simple notion of triangular number is surprisingly effective. We also introduce a variation of KenKen that uses the Gaussian integers in order to…

The purpose of this study is to develop design principles for crafting tasks that will encourage conjecturing and proving in the context of elementary number theory at the undergraduate level. From the analyses of the written work of 46 prospective mathematics teachers on a task designed according to these principles, we think that there is…

This investigation examined the degree to which prospective elementary teachers had developed a meaningful and conceptual understanding of what integers are and explored their development of models for multiplication with integers that are related to everyday activities. Additionally, this study explored how these understandings informed…

This article focuses on how the representational properties of manipulatives affect the strategies children employ in problem solving. Two studies examined the effect of physical materials on 4-7-year-old children's problem solving strategies in a numerical (i.e., additive composition) task. The first study showed how children not only identified…

Using a dog's paw as a basis for numerical representation, sixth grade students explored how to count and regroup using the dog's four digital pads. Teachers can connect these base-4 explorations to the conceptual meaning of place value and regrouping using base-10.

This qualitative research, drawing on the theoretical frameworks by Even (1990, 1993) and Sfard (2007), investigated five high school mathematics teachers' geometric interpretations of complex number multiplication along with the roots of unity. The main finding was that mathematics teachers constructed the modulus, the argument, and the conjugate…

During the past few years, several of the authors have incorporated student problem posing as a regular instructional feature in their classrooms. When they offer their students the opportunity to construct their own problems, particularly during the course of an entire school year, they create many novel tasks. Student-created tasks not only…

Today, almost all introductory physics textbooks include standardized "rules" on how to find the number of significant figures in a calculated value. And yet, 30 years ago these rules were almost nonexistent. Why have we increased the role of significant figures in introductory classes, and should we continue this trend? A look back at…