In designing learning environments that support student learning, there are many instructional design decisions. These include when and how to provide examples, verbal explanations, feedback, and other scaffolding features. In this paper, the authors investigate instructional guidance as it relates to Cognitive Tutor, an intelligent tutoring…

Many students enter college inadequately prepared to take college-level math courses like College Algebra, and as a consequence must take one or more developmental math courses. It is becoming increasingly common for colleges to offer math courses online, including various developmental math courses including Intermediate Algebra. Many…

Students at a large urban community college enrolled in fourteen sections of a developmental algebra class. While cognitive variables are often used to place students, affective characteristics may also influence their success. To explore the impact of affective variables, students took ACT's Engage survey measuring motivation, academic-related…

This five-year evaluation examined the effectiveness of a promising middle-school mathematics intervention funded through an Investing in Innovation (i3) development grant. Evaluation objectives were to: (1) study the impact of an intervention aimed at increasing the academic achievement of students in Algebra I--a gate-keeping course--as measured…

This investigation utilized the qualitative case study method. Seventy-one College Algebra students were given a mathematical processing instrument. This testing device measured a student's preference for visual thinking. Two students were purposefully selected using the instrument. The visual mathematical learner (VL) was discussed in this…

This article reports on the results of research, the objective of which was to document and analyze the manner in which students relate different representations when solving problems. A total of 20 students attending their first year of university studies took part in the study. In order to design the problem, the underlying information in each…

In this article, the author explores her personal views on mathematics and how she solves problems. She experiences some personal revelations while attending a "Mathsjam". Organised by some of the more prominent names in the popularisation of mathematics, "Mathsjam" was billed as the UK's first Gathering for Gardner, an event…

It is a routine matter for undergraduates to find eigenvalues and eigenvectors of a given matrix. But the converse problem of finding a matrix with prescribed eigenvalues and eigenvectors is rarely discussed in elementary texts on linear algebra. This problem is related to the "spectral" decomposition of a matrix and has important technical…

Much of the neuroimaging research has focused on how mathematical operations are performed. Although this body of research has provided insight for the refinement of pedagogy, there are very few neuroimaging studies on how mathematical operations should be taught. In this article, we describe the teaching of algebra in Singapore schools and the…

Coherence is the core principle underlying standards-based educational reforms. Assessments aligned with content standards are designed to guide instruction and raise achievement. The authors investigate the coherence of standards-based reform's key instruments using the Surveys of Enacted Curriculum. Analyzing 138 standards-assessment pairs…

For almost all students, what happens when they push buttons on their calculators is essentially magic, and the techniques used are seemingly pure wizardry. In this article, the author draws back the curtain to expose some of the mathematics behind computational wizardry and introduces some fundamental ideas that are accessible to precalculus…

This paper presents the development process of a "praxeology" (theory-of-practice) for supporting the teaching of proofs in a CAS environment. The characteristics of the praxeology were elaborated within the frame of a professional development course for teaching analytic geometry with CAS. The theoretical framework draws on Chevallard's…

The purpose of the study reported in this article is to examine the impact of curriculum on instruction. Over a three-year period, we observed 579 algebra-related lessons in grades 6-8. Approximately half the lessons were taught in schools that had adopted a "Standards"-based mathematics curriculum called the Connected Mathematics Program (CMP),…

Freely rising air bubbles in water sometimes assume the shape of a spherical cap, a shape also known as the "big bubble". Is it possible to find some objective function involving a combination of a bubble's attributes for which the big bubble is the optimal shape? Following the basic idea of the definite integral, we define a bubble's surface as…

A plane algebraic curve, the complete set of solutions to a polynomial equation: f(x, y) = 0, can in many cases be drawn using parametric equations: x = x(t), y = y(t). Using algebra, attempting to parametrize by means of rational functions of t, one discovers quickly that it is not the degree of f but the "relative degree," that describes how…