This note generalizes the formula for the triangular number of the sum and product of two natural numbers to similar results for the triangular number of the sum and product of "r" natural numbers. The formula is applied to derive formula for the sum of an odd and an even number of consecutive triangular numbers.

In this article, the author relates how he found "MT202" full of interest and how he was fascinated by the number of threads he found running through it. He then describes how he weaved together these threads. (Contains 1 table.)

The generalization of acquired competencies, specifically flexibility of closure, was the subject of this research. Flexibility of closure was defined as the ability to demonstrate selective attention to a specified set of elements when presented within various settings (the larger the number of settings from which the desired set of elements can…

The purpose of this study was to determine whether cognitive-process approach based social skills program was effective on learning and generalizing three social skills (apologizing, coping with teasing and avoiding inappropriate touching) of the nine students with mental retardation. Social skills program covered dimensions of the cognitive…

The NCTM calls for the use of rich tasks that encourage students to apply their own reasoning to problem situations. When students work through algebraic generalization tasks, their reasoning often elicits a variety of strategies (Lannin 2003; Stacey 1989; Swafford and Langrall 2000). Challenges for teachers include facilitating student awareness…

In this study we examined how two students viewed the general nature of their proportional reasoning errors as they attempted to generalize numeric situations. Using a teaching experiment methodology we studied the reasoning of two students over 18 instructional sessions. One student, Dallas, appeared to recognize that the proportional reasoning…

Twenty-one 4th- and 5th-grade students with learning disabilities and emotional disabilities were assigned at random to a control condition or to an experimental condition in which they were taught, over a 9-week period, a five-step self-determination strategy for solving school- or home-related problems. Maintenance was assessed 3 weeks after the…

The purpose of this study was to determine the effect of systematic instruction with a concrete representation on the acquisition of an algebra skill for students with moderate developmental disabilities. Three high school students with moderate developmental disabilities participated in this study. A multiple probe across participants research…

Research investigating algebra students' abilities to generalize and justify suggests that they experience difficulty in creating and using appropriate generalizations and proofs. Although the field has documented students' errors, less is known about what students do understand to be general and convincing. This study examines the ways in which…

This article focuses on the ways in which four eighth-grade girls, each with varying levels of algebraic understanding, share ideas, debate, and gradually move toward generalizations inherent in the "Painted Cube" problem. The intent of this article is to examine how students move to progressive formalization and to provide insights into the ways…

Computer science education should not be based on short-term developments but on content that is observable in multiple domains of computer science, may be taught at every intellectual level, will be relevant in the longer term, and is related to everyday language and/or thinking. Recently, a catalogue of "central concepts" for computer…

Results indicate that training in semantic awareness increased problem-solving ability for both analytic and synthetic problems. (Author)

Presents a generalization to the classic prisoner problem, which is inherently interesting and has a solution within the reach of most high school mathematics students. Suggests the problem as a way to emphasize to students the final step in a problem-solver's tool kit, considering possible generalizations when a particular problem has been…

The concept of symmetry is fundamental to mathematics. Arguments and proofs based on symmetry are often aesthetically pleasing because they are subtle and succinct and non-standard. This article uses notions of symmetry to approach the solutions to a broad range of mathematical problems. It responds to Krutetskii's criteria for mathematical…

Often after students solve a problem they believe they have accomplished their mission and stop further exploration. The purpose of this article is to discuss ways to encourage students to "look back" so as to maximise their learning opportunities. According to Polya, by "looking back" at a completed solution, by reconsidering and re-examining the…