Current curriculum initiatives (e.g., National Governors Association Center for Best Practices and Council of Chief State School Officers 2010) advocate that models be used in the mathematics classroom. However, despite their apparent promise, there comes a point when models break, a point in the mathematical problem space where the model cannot,…

How do children map number words to the numerical magnitudes they represent? Recent work in adults has shown that two distinct mechanisms--structure mapping and associative mapping--connect number words to nonlinguistic numerical representations (Sullivan, J., 2012). This study investigated the development of number word mappings, and the roles of…

This article reports on students' problem-solving approaches across three representations--number lines, coordinate planes, and function graphs--the axes of which conventional mathematics treats in terms of consistent geometric and numeric coordinations. I consider these representations to be a part of a "hierarchical representational…

Number words are generally used to refer to the exact cardinal value of a set, but cognitive scientists disagree about their meanings. Although most psychological analyses presuppose that numbers have exact semantics ("two" means exactly two), many linguistic accounts propose that numbers have lower-bounded semantics (at least two), and…

Using generating functions, we develop a number of properties of sums of products of Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal and Jacobsthal-Lucas numbers. (Contains 2 tables.)

We examined relations between symbolic and non-symbolic numerical magnitude representations, between whole number and fraction representations, and between these representations and overall mathematics achievement in fifth graders. Fraction and whole number symbolic and non-symbolic numerical magnitude understandings were measured using both…

To become scientifically literate, students need to interpret science concepts using numbers, text, and visuals (Lemke 2004). Scientists use multiple modes to communicate their ideas to each other and the public, including images, text, mathematical notations, symbols, diagrams, charts, and graphs. Several of the science and engineering practices…

In experiments with univalent target stimuli, task-switching costs can be eliminated if participants are unaware of the task rules and apply cue-target-response associations. However, in experiments with bivalent target stimuli, participants show task-switching costs. Participants may exhibit switch costs even when no task rules are provided in…

Math outcomes for students with disabilities, in particular students with emotional disturbance (ED), are bleak, warranting intervention strategies that have research to support their utility. The purpose of this meta-analysis was to examine the literature of math interventions used with students with ED to improve math outcome variables. Our…

This paper presents primary school teachers' perceptions associated with their engagement in a project aimed at building Year 3 and Year 4 students' number fact and computational fluency. Throughout the project, teachers were provided with classroom ideas and suggestions for engaging students in number fact practice and for generating student…

This report summarizes the findings across two case studies conducted as part of WestEd's evaluation of the California Statewide Early Math Initiative (CAEMI). The case studies aimed to understand the planning and implementation of professional learning and coaching in local communities. They also examined changes in educators' math knowledge,…

Practical problems that use mathematical concepts are among the highlights of any mathematics class, for better and for worse. Teachers are thrilled to show applications of new theoretical ideas, whereas most students dread "word problems." This article presents a sequence of three activities designed to get students to think about…

This study investigated the sets of mental computation strategies used by prospective elementary teachers to compute sums and differences of whole numbers. In the context of an intervention designed to improve the number sense of prospective elementary teachers, participants were interviewed pre/post, and their mental computation strategies were…

Mathematical ideas from number theory, group theory, dynamical systems, and computer science have often been used to explain card tricks. Conversely, playing cards have been often used to illustrate the mathematical concepts of probability distributions and group theory. In this paper, we describe how the 21-card trick may be used to illustrate…

Scaffolding is defined as help that can assist students to solve problems or understand concepts that were not able to complete independently. In modern times, the concept of technology-based Scaffolding in learning has developed. In this study an educational game called "Telolet" was designed with the aim of providing Scaffolding to…