The development of the idea that "The essence of mathematics lies in its freedom," a quotation from Cantor, is discussed. Several examples are given of relative truth, and the problem of consistency is discussed. Mathematics and its relationship to the physical world is also explored. (MNS)

A special summer course for mathematically gifted eighth graders used a problem-centered approach to develop abilities to hypothesize and carry out informal deductive proofs. Students responded well to challenging problems and worked best independently rather than in groups. (Author/CL)

This textbook is a teaching manual for arithmetic, algebra, and geometry. Historical sketches on the subjects are included. Each topic is discussed in detail, and the teacher is advised along what lines to work, and how the work may be successfully accomplished.

The usual Euclidean metric definition is modified; new figures and measures are defined and analyzed using this different distance function. (DT)

A geometric construction for a line parallel to a given line through a given point is explained and analyzed. (DT)

The content of geometry in new mathematics books and in secondary school mathematics courses is discussed, and the suggestion is made that appropriate doses'' of formal reasoning be included at appropriate levels. (DT)

Discusses ways in which geometry can become the perfect answer for preholiday mathematics lessons. (Author)

The author presents research implications in the areas of geometric capabilities of elementary school students, instructional variables related to geometry achievement, transfer effects, and some findings concerning teachers and geometry. (JG)