Two experiments were conducted, contrasting a minimally guided discovery condition with a variety of instructional conditions. College students interacted with a computer-based tutor that presented algebra-like problems in a novel graphical representation. Although the tutor provided no instruction in a discovery condition, it constrained the…

Our research project aimed at understanding the complexity of the construction of knowledge in a CAS environment. Basing our work on the French instrumental approach, in particular the Task-Technique-Theory (T-T-T) theoretical frame as adapted from Chevallard's Anthropological Theory of Didactics, we were mindful that a careful task design process…

The paper on "Magnetism and Simultaneity" by Adler provides an excellent new thought experiment involving the lack of simultaneity in Einstein's special relativity. Adler uses the lack of simultaneity rather than the Lorentz contraction to derive the formula for the magnetic force on a moving charged particle. Advantages of his derivation are that…

A problem posed in an influential textbook is analysed in more detail than is given there. The textbook answer to the problem has been represented as counterintuitive, as is probably the case; however, it is here shown that it depends critically on an assumption. If this is relaxed, then a wide variety of possible answers is available. All the…

Several jeeps are poised at base camp on the edge of a desert aiming to escort one of them as far as possible into the desert, while the others return to camp. They all have full tanks of gas and share their fuel to maximize penetration. In a friendly desert it is best to leave caches of fuel along the way to help returning jeeps. We solve the…

We describe a simple photographic assignment appropriate for an abstract algebra (or other) course. Students take digital pictures around campus of various examples of symmetry. They then classify these pictures according to which of the 17 plane symmetry groups they belong. (Contains 2 figures.)

A secretary has "n" letters and "n" addressed envelopes. Instead of matching each letter with the corresponding envelope, she inserts the letters in a random manner. What are the chances that every letter will be in the wrong envelope? In this article, the author presents a solution to this problem and discusses possible ways of placing the…

Teachers can use geometric patterns to promote students' understanding of functional relationships. In this article, the authors first look at a problem-solving process that supports the use of figural reasoning to explore and interpret geometric pattern tasks and generalize function rules. Second, the authors discuss a framework for…

As educators and policymakers search for ways to prepare students for the rigors of algebra, teachers in the Helena, Montana, school system are starting early by attempting to nurture students' algebraic-reasoning ability, as well as their basic number skills, in early elementary school, rather than waiting until middle or early high school.…

A visual approach to understanding the chain rule and related derivative formulae, for functions from R to R and from C to C, is presented. This apparently novel approach has been successfully used with several audiences: students first studying calculus, students with some background in linear algebra, students beginning study of functions of a…

The purpose of this research was to investigate student conceptions of the topic of asymptotes of rational functions and to understand the connections that students developed between the closely related notions of asymptotes, continuity, and limits. The participants of the study were university students taking Calculus 2 and were mostly freshmen. …

Community colleges have served as the gateway to educational opportunities for many individuals by providing open access to certificate programs and associate degrees, and--more recently--developmental studies for underprepared students. However, only 29.5% of these students actually receive a degree. Research has been conducted to determine why…

With passage of The No Child Left Behind Act in 2001, the nation's schools have been under pressure to bring all students to a proficient level as defined by their state. The State of California currently uses a status model accountability system to measure whether a student, school, or district is proficient. Other states such as Arizona,…

Despite increasing accountability requirements and a national call for transparency in public policy, Mid-Atlantic Region state education agencies indicate that they have little information about what instructional materials districts adopt. This report describes first-year results of an ongoing project to generate and share information on core…

Planning effective instruction for a classroom full of learners demands that the educator know what works and more importantly what works better for the group of people residing in the educator's classroom today. This action research study tested the efficacy of using the full compliment of assessments included in the curriculum adoption at the…