At the heart of the recent focus on mathematics has been an increased emphasis on developing students' "number sense." Ironically, although growing as a force in the education literature, number sense has not been clearly defined for teachers. Teachers need specific support in understanding how to develop number sense in students, to…

In constructivist principles, learning is a process in which individuals construct knowledge. Research in Mathematics Education looks for ways to make mathematics education less dry and more attractive. When solving polynomial equations of the first degree, it is very common for teachers to work with the mistaken idea of "changing the…

Cognitively Guided Instruction (CGI) researchers have found that while teachers readily ask initial questions to elicit students' mathematical thinking, they struggle with how to follow up on student ideas. This study examines the classrooms of three teachers who had engaged in algebraic reasoning CGI professional development. We detail teachers'…

In Singapore, 6-12 year-old students are taught to solve algebra word problems with a mix of arithmetic and pre-algebraic strategies; 13-17 year-olds are typically encouraged to replace these strategies with letter-symbolic algebra. We examined whether algebra problem-solving proficiency amongst beginning learners of letter-symbolic algebra is…

This article describes a vision of student use of dynamically connected representations as they investigate mathematical ideas. Although this article is intended to generate discussion about the potential of dynamically connected representations, we situate the discussion within actual descriptions of its use with students in a newly designed…

This research examines students' ability to integrate algebraic variables with arithmetic operations and symbols as a result of the type of instruction they received, and places their work on scales that illustrate its location on the continuum from arithmetic to algebraic reasoning. It presents data from pre- and post-instruction clinical…

An understanding of Algebra I and the role that this foundational course plays as an entry to the college preparatory pathway in secondary education and its influence on mathematical achievement is an integral component for the education of American youth in the global world of science and technology. Achievements in high school curricula are…

In "Beyond Test Scores: Leading Indicators for Education," Foley and colleagues (2008) define "leading indicators" as those that "provide early signals of progress toward academic achievement" (p. 1) and stress that educators "need leading indicators to help them see the direction their efforts are going in and to take corrective action as soon as…

This article presents an overview of some issues that were confronted when delivering an online second Linear Algebra course (assuming a previous Introductory Linear Algebra course) to graduate students enrolled in a Secondary Mathematics Education program. The focus is on performance in one particular aspect of the course: "change of basis" and…

The ability to flexibly solve problems is considered an important outcome for school mathematics and is the focus of this paper. The paper describes the impact of a three-week summer course for students who struggle with algebra. During the course, students regularly compared and contrasted worked examples of algebra problems in order to promote…

"Saxon Math" is a textbook series covering grades K-12 based on incremental development and continual review of mathematical concepts to give students time to learn and practice concepts throughout the year. The series is aligned with standards of the National Council of Teachers of Mathematics (NCTM) and various states, and can be…

This article explores phenomena related to fitting polynomials with data sets with equally spaced x-values.

In the present work we treat the problem of a particle in a uniform magnetic field along the symmetric gauge, so chosen since the wavefunctions present the required cylindrical symmetry. It is our understanding that by means of this work we can make a contribution to the teaching of the present subject, as well as encourage students to use…

As the name of the paper implies, a converse of Fermat's Little Theorem (FLT) is stated and proved. FLT states the following: if p is any prime, and x any integer, then x[superscript p] [equivalent to] x (mod p). There is already a well-known converse of FLT, known as Lehmer's Theorem, which is as follows: if x is an integer coprime with m, such…

This article provides for a fast extremely accurate approach to graphing functions that is based on learning function reference graphs and then applying algebraic transformations to these reference graphs.