The "testing effect" refers to the finding that after an initial study opportunity, testing is more effective for long-term retention than restudying. The testing effect seems robust and is a finding from the field of cognitive science that has important implications for education. However, it is unclear whether this effect also applies…

In an open competition, students were to determine the mass of a metal cylinder hanging on a spring inside a transparent enclosure. With the time for experiments limited to 24 h due to the unexpectedly large number of participants, a few surprisingly accurate results were submitted, the best of them differing by no more than 0.5% from the true…

Children ask questions and learn from the responses they receive; however, little is known about how children learn from listening to others ask questions. Five experiments examined preschoolers' ("N" = 179) ability to solve simple problems using information gathered from listening to question-and-answer exchanges between 2 parties present in the…

Brainstorming is the default method of idea-generation in organisations, and is widely applied in higher education by students, academics and support staff. Its popularity is mainly attributable to an illusory belief that groups working together are more productive than individuals working apart. Shared responsibility, the need for collaboration…

This paper presents the results of a study that aimed at increasing students' problem-solving skills. Polya's (1985) heuristic for problem solving was used and students were required to articulate their thought processes through the use of a structured diary. The diary prompted students to answer questions designed to engage them in the phases of…

Human performance technology (HPT), like other concepts, models, and frameworks that we use to describe the world in which we live and the way we organize ourselves to accomplish valuable activities, is built from paradigms that were fresh and relevant at the time it was conceived and from the fields of study from which it grew. However, when the…

Researchers have speculated that children find it more difficult to acquire conceptual understanding of the inverse relation between multiplication and division than that between addition and subtraction. We reviewed research on children and adults' use of shortcut procedures that make use of the inverse relation on two kinds of problems:…

Characterizing the modes of reasoning typically applied by students to solve different types of chemistry problems is of central importance for the design of instructional strategies that can better support their learning of specific content. Thus, the central goal of this study was to identify dominant modes of reasoning expressed by college…

A visual proof that 1 - (1/2) + (1/4) - (1/8) + ... 1/(1+x[superscript 4]) converges to 2/3.

An idempotent satisfies the equation x[superscript 2] = x. In ordinary arithmetic, this is so easy to solve it's boring. We delight the mathematical palette here, topping idempotents off with modular arithmetic and a series of exercises determining for which n there are more than two idempotents (mod n) and exactly how many there are.

The ability to count has traditionally been considered an important milestone in children's development of number sense. However, using counting (e.g., counting on, counting all) strategies to solve addition problems is not the best way for children to achieve their full mathematical potential and to prepare them to develop more complex and…

This article reports findings from an investigation of precalculus students' approaches to solving novel problems. We characterize the images that students constructed during their solution attempts and describe the degree to which they were successful in imagining how the quantities in a problem's context change together. Our analyses revealed…

The golden ratio, one of the most beautiful numbers in all of mathematics, arises in some surprising places. At first glance, we might expect that a General checking his troops' progress would be nothing more than a basic distance-rate-time problem. However, further exploration reveals a multi-faceted problem, one in which the ratio of rates…

This research analyzes how external representations created by a student, Robert, helped him in building mathematical understanding over a sixteen-year period. Robert (also known as Bobby), was an original participant of the Rutgers longitudinal study where students were encouraged to work on problem-solving tasks with minimum intervention (Maher,…

The Venn diagram is suggested as a graphical solution to conjunction fallacies and a modification of it is suggested to more fully communicate set relations.