A generalization of a well-known result for the arctangent function poses a number of interesting questions concerning the existence of integer solutions of related problems.

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…

The ability to generalize is probably the primary aspect of intelligence. The computer's inability to generalize is the major stumbling block associated with machine intelligence. (Author/CP)

Learners must acquire a general strategic orientation in their problem solving, to determine what counts as relevant information in problematic situations. A dynamic systems approach is called for in determining whether general or context-specific instructions are needed to promote transfer of relevant information to novel conditions. (Author/JDD)

Students often fail to apply knowledge and skills learned in one context to other situations. Although the implicit assumption in educational practice has been that transfer takes care of itself, a knowledge of the mechanisms of transfer can enable educators to help their students integrate general and local knowledge. (TE)

Discusses the impact of computer programing instruction on cognitive skills and suggests two mechanisms of transfer--low road, from practice, and high road, from generalization. Previous research that tried to measure transfer from programing is reviewed, six categories in which transfer might occur are presented, and 71 references are provided.…

Discusses paradigmatic teaching, or teaching for analogical transfer, which requires teaching a basic structure by appropriate examples, as well as teaching its application in various fields and contexts. Examples for problem solving, inductive thinking, and learning to learn are given, and a training program with adult learners is described. (15…

Presents 13 examples in which the intuitive approach to solve the problem is often misleading. Presents analysis of these problems for five different sources of misleading intuitive generators: lack of analysis, unbalanced perception, improper analogy, improper generalization, and misuse of symmetry. (MDH)