In this article, we present some mathematical tasks designed to support pre-service teachers (PSTs) in obtaining a broader, more connected understanding of decimal numbers and quantities. We describe classroom scenarios from a content course in which PSTs experienced these cognitively demanding decimal tasks, enacted in ways that maintain high…

Recent work by researchers has focused on synthesizing and elaborating knowledge of students' thinking on particular concepts as core progressions called learning trajectories. Although useful at the level of curriculum development, assessment design, and the articulation of standards, evidence is only beginning to emerge to suggest how learning…

Two aspects of mathematics with which students with mathematics learning difficulty (MLD) often struggle are word problems and number-combination skills. This article describes a math program in which students receive instruction on using algebraic equations to represent the underlying problem structure for three word-problem types. Students also…

The aim of this paper is to investigate the ways in which the number line can function in solving mathematical tasks by first graders (6 year olds). The main research question was whether the number line functioned as an auxiliary means or as an obstacle for these students. Through analysis of the 32 students' answers it appears that the number…

Analysis of the patterns of signs of infinitely differentiable real functions shows that only four patterns are possible if the function is required to exhibit the pattern at all points in its domain and that domain is the set of all real numbers. On the other hand all patterns are possible if the domain is a bounded open interval.

Before they enter preschool, children vary greatly in their numerical and mathematical knowledge, and this knowledge predicts their achievement throughout elementary school (e.g. Duncan et al., 2007; Ginsburg & Russell, 1981). Therefore, it is critical that we look to the home environment for parental inputs that may lead to these early…

Individual differences in inhibitory control have been shown to relate to general mathematics achievement, but whether this relation varies for specific areas within mathematics is a question that remains open. Here, we evaluate if inhibitory processes play a specific role in the particular case of fraction comparison, where learners must ignore…

This research aims to describe and analyze result of applying Problem-Based Learning and Cognitive Conflict Strategy (PBLCCS) in increasing students' Mathematical Critical Thinking (MCT) ability and Mathematical Curiosity Attitude (MCA). Adopting a quasi-experimental method with pretest-posttest control group design and using mixed method with…

Looking for, recognizing, and using underlying mathematical structure is an important aspect of mathematical reasoning. We explore the use of mathematical structure in children's integer strategies by developing and exemplifying the construct of logical necessity. Students in our study used logical necessity to approach and use numbers in a…

"Two and Two Make Zero" considers children's acquisition of numerical concepts from a wide range of perspectives including topics that are often overlooked, most notably: the principal properties of the counting numbers in and of themselves; the role that numerical symbols play in number acquisition; the underlying conceptual structure of number…

To conceive the infinity of integers, one has to realize: (a) the unending possibility of increasing/decreasing numbers (potential infinity), (b) that the cardinality of the set of numbers is greater than that of any finite set (actual infinity), and (c) that the leap from a finite to an infinite set is itself infinite (immeasurable gap). Three…

The view of infinity as a metaphor, a basic premise of modern cognitive theory of embodied knowledge, suggests in particular that there may be alternative ways in which one could formalize mathematical ideas about infinity. We discuss the key ideas about infinitesimals via a proceptual analysis of the meaning of the ellipsis "..." in the real…

Math is used in many occupations. And, experts say, workers with a strong background in mathematics are increasingly in demand. That equals prime opportunity for career-minded math enthusiasts. This article describes how math factors into careers. The first section talks about some of the ways workers use math in the workplace. The second section…

This article explores how a focus on understanding divisibility rules can be used to help deepen students' understanding of multiplication and division with whole numbers. It is based on research with seven Year 7-8 teachers who were observed teaching a group of students a rule for divisibility by nine. As part of the lesson, students were shown a…

Cultural-historical activity theory--with historical roots in dialectical materialism and the social psychology to which it has given rise--has experienced exponential growth in its acceptance by scholars interested in understanding knowing and learning writ large. In education, this theory has constituted something like a well kept secret that is…