Problem solving, introduced in every grade and practiced throughout elementary school, is an essential foundation on which students build their understanding of mathematical ideas, operations, and relationships. This article discusses 1-2-3 Time, an instructional mode that promotes multimodal math learning in ways that engage and sustain students'…

The nature of mathematical discourse and its influence on the development of students' mathematical understanding has received much attention from researchers in recent years. Engaging students in discursive practices can be difficult; teachers can increase their competence in facilitating discourse through greater awareness of the impact of…

The purpose of this study was to measure the difference in achievement between those students enrolled in a beginning-level, university Algebra course in southeastern Idaho university the spring semester of 2012 who received an Algebra Exponents and Polynomials instructional unit in a traditional face-to-face setting and those students who…

Representational fluency (RF) includes an ability to interpret, create, move within and among, and connect tool-based representations of mathematical objects. Taken as an indicator of conceptual understanding, there is a need to better support school algebra students' RF in learning environments that utilize both computer algebra systems…

The new Common Core State Standards for Mathematics have been formulated to provide students with instruction that will help them acquire a thorough knowledge of math at their grade level, which will in turn enable them to move on to higher mathematics with competence and confidence. "Hands-on Activities for Teaching the Common Core Math…

High academic standards are an important first step in ensuring that all Alaska's students have the tools they need for success. These standards reflect the collaborative work of Alaskan educators and national experts from the nonprofit National Center for the Improvement of Educational Assessment. Further, they are informed by public comments.…

Video podcasts allow students to control when, where, and what they learn, as well as the pace of learning. Considerable research has been conducted in higher education on video podcast use, but not in middle schools (grades six to eight). This study investigated the use of worked example video podcasts in mathematics classrooms with students 11…

In this article we report on the effects of a digital intervention on the development of algebraic expertise of 17-18 year old students in the Netherlands. The question to be answered was whether the intervention would be effective and what factors influenced the outcome. With notions of formative assessment and symbol sense as guiding theoretical…

This protocol guided the review of research that informed the recommendations contained in the What Works Clearinghouse (WWC) practice guide "Improving Mathematical Problem Solving in Grades 4 through 8," published in May 2012. The research review involved the following steps: (1) The research staff searched the professional literature…

To find the number of distinct real roots of the cubic equation (1) x[caret]3 + bx[caret]2 + cx + d = 0, we could attempt to solve the equation. Fortunately, it is easy to tell the number of distinct real roots of (1) without having to solve the equation. The key is the discriminant. The discriminant of (1) appears in Cardan's (or Cardano's) cubic…

In a first calculus course, it is not unusual for students to encounter the theorems which state: If f is an even (odd) differentiable function, then its derivative is odd (even). In our paper, we prove some theorems which show how the symmetry of a continuous function f with respect to (i) the vertical line: x = a or (ii) with respect to the…

This article presents a new approach to generalizing the definition of means. By this approach we easily obtain generalized means which are quite different from standard arithmetic, geometric and harmonic means.

A graphical method is presented for assessing the state of identifiability of the parameters in a linear structural equation model based on the associated directed graph. We do not restrict attention to recursive models. In the recent literature, methods based on graphical models have been presented as a useful tool for assessing the state of…

A colleague was moving, and someone on the professional moving crew, upon hearing she was a mathematician, asked what happens when you repeatedly transfer water back and forth between two classes using a straw. The question is simple to solve if you alternate which glass you transfer from and to, but if more general patters are allowed, some…

Building on the papers in this special issue as well as on our own experience and research, we try to shed light on the construct of "example spaces" and on how it can inform research and practice in the teaching and learning of mathematical concepts. Consistent with our way of working, we delay definition until after appropriate reader experience…