The author remarks on the purpose(s) and content of existing geometry courses and offers alternative purposes and goals for geometry courses and some consequences of this alternative point of view. (JG)

A discussion of the use of mathematics problems containing erroneous statements and the use of mutual consultation by students on the solution of mathematics problems. Critical questioning ability of students is enhanced through the utilization of both these procedures, according to the author. (JG)

Non-traditional methods of presenting and solving calculus problems in high school mathematics classes are presented. All problems deal with the principle that the maximum product of two numbers whose sum is constant is obtained if the numbers are equal (i.e., the arithmetic mean of n numbers is greater than or equal to the geometric mean). (JG)

The author presents a variation of a tiling problem where each row or column contains a complete set of elements, each diagonal is free of contiguous like elements, and at least one partition exists. These restrictions limit the number of solutions considerably, and methods of solution are subsequently discussed. (CT)

Discussion is focused on standard teaching procedures that contain the elements of modern mathematics. Of the various methods which have been tested, it is concluded that the solving of problems remains the most effective basic activity. A sample collection of problems is presented to indicate relationships and the principles they illustrate. (RP)

Report of a followup study made of students in the mathematics Advanced Placement classes at a large New York City high school from 1956 to 1965. Data from a questionnaire sent to students in these classes are reported. (FL)