Spurred by Kilpatrick's (1987) "Where do good problems come from?", this study explores problem-posing triggers of experienced problem posers for mathematics competitions. Triggers are conceived as instances of noticing, where an impulse draws a poser's attention and "triggers off" a mathematical re-action, one of the outcomes…

One of the most salient arguments in favor of a semiotic approach, put forward on various occasions among others by Luis Radford, claims that semiotics is most appropriate to treat the interaction between socio-cultural and objective aspects of knowledge problems. But if we want to take such claims seriously, we have to undertake revisions of our…

Choosing challenging problems to facilitate student learning of new mathematics is a difficult task so by using open-ended problems with prospective mathematics teachers, the potential solution characteristics were studied. It is shown that considering the conceptual diversity of solutions helps students develop a better sense of different…

We analyze probability content within middle grades (6, 7, and 8) mathematics textbooks from a historical perspective. Two series, one popular and the other alternative, from four recent eras of mathematics education (New Math, Back to Basics, Problem Solving, and Standards) were analyzed using the Mathematical Tasks Framework (Stein, Smith,…

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…

A humorous approach to "mathematical applications" focuses on the activities of the housewife in doing domestic chores. Six "lessons" touch on the following topics: algebra, direct and indirect proportions, arithmetic progressions, trigonometry, and non-Euclidean geometry. (MP)

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…

This is the sixth in a series of ten self-contained units designed for use by students in ninth grade general mathematics classes. This unit is divided into eight sections dealing with different types of mathematical thinking. Some topics presented include functions, permutations and combinations, symmetry, inductive thinking, and logic. Though…

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)