Proliferation of mathematical knowledge prohibits individual reduplication of all results and causes mathematicians to accept much on faith. To convey the spirit of the mathematical community, and provide maximum content coverage, instruction should allow students to accept some theorems without proof. Students would then find mathematics less…

An anecdote shows why children studying mathematics should be given freedom to explore, generalize, classify, and originate proofs. (RS)

After describing desirable characteristics of a student's first encounter with proof, and listing skills prerequisite for the proposed lesson (knowledge of some operational systems), the author describes in detail an introductory session in which students prove consequences of commutativity and associativity for a system supposedly encountered on…

Pedagogical issue related to the teaching of elementary school mathematics are discussed. The authors argue that mathematics instruction should focus on intellectual challenges and problem solving. Four examples at different elementary levels are presented. nSD)