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.

Integers are sometimes used in physics problems to simplify the mathematics so the arithmetic does not distract students from the physics concepts. This is particularly important in exams where students should not have to spend a lot of time using their calculators. Common uses of integers in physics problems include integer solutions to…

Designers try to "enlist" whatever they can to help themselves arrive at high quality, novel and original designs. When stimuli are used for this purpose, usually provided at the onset of the design process, these stimuli, or sources, may have one of two effects: they may enhance the design search and contribute to a high-quality, creative design,…

In this article, the authors examine the mathematics identity development of six Black male students over the course of a 4-year The Algebra Project Cohort Model (APCM) initiative. Mathematics identity here is defined as participation through interactions and positioning of self and others. Data collection included nearly 450 minutes of video…

While problem solving as an instructional technique is widely advocated, educators are often challenged in effectively assessing student skill in this area. Students failing to solve a problem might fail in any of several aspects of the effort. The purpose of this research was to validate a scaffolded technique for assessing problem solving in…

Learning to teach is a complex matter, and many different models of pre-service teacher education have been used to support novice teachers' preparation for the classroom. More recently there have been calls for a focus on core high-leverage teaching practices and for novice teachers to engage in representations, decompositions, and approximations…