A list of heuristics for elementary school children is provided. Two examples which are viewed to demonstrate the conscientious and creative use of problem-solving strategies by fourth graders are detailed. Suggestions for teachers on how to help pupils develop their problem-solving abilities are included. (MP)

Calculators are seen to shift the student focus in problem-solving situations from "how to do it" to "what to do," by keeping computation from standing in the way when pupils write or solve problems. (MP)

Incorporating problem solving as an integral part of mathematics programs in the classroom is discussed. The teacher's focus is seen as stretching pupil thinking. (MP)

Children's difficulties remembering correct algorithms for operations with fractions are traced to difficulties in the instructional sequence. Many teachers "teach mistakes" by failing to provide sufficient referents for their pupils. (MP)

Having pupils create their own mathematical-application word problems is promoted as a powerful teaching strategy. Teachers are encouraged to make creative pupil applications an integral component of day-to-day learning in mathematics. Research is cited that strongly favors the approach presented. (MP)

Ways are identified in which typical textbook word problems can be used to better fulfill their role in the development of mathematical concepts and a sound instructional program of mathematical problem solving. (MP)

Two aspects of instructional practice in mathematics are discussed: the process of developing instructional activities and the importance of engaging students in mathematical discussion. The National Council of Teachers of Mathematics (NCTM) Standards are evaluated based on this discussion. (CW)

The psychological views as well as the important factors and variables of problem solving in mathematics are discussed. (JT)

The use of the term "problem solving" as a goal for the organization of mathematics instruction is criticized as an ambiguous slogan that is building a false bandwagon of supporters. Some possible routes for more carefully defining this goal are suggested. (MP)

Four types of experiences in problem solving are identified, and steps children must learn to do problem solving are presented. (MP)

Article is based on the keynote address of the 1987 annual meeting of the National Council of Teachers of Mathematics. Suggests that George Polya left a legacy which has not only touched knowledge and abilities in mathematics and problem solving but also had significant effects on thoughts about teaching. (PK)