This article describes the instructional process of helping students visualize irrational numbers. Students learn to create a spiral, called "the wheel of Theodorus," which demonstrates irrational and rational lengths. Examples of student work help the reader appreciate the delightful possibilities of this project. (Contains 4 figures.)

Under what conditions can a true "science of mental life" arise from psychological investigations? Can psychology formulate scientific laws of a general nature, comparable in soundness to the laws of physics? I argue that the search for such laws must return to the forefront of psychological and developmental research, an enterprise that requires…

In this article, the author offers two well-known mathematical images--that of a dot moving around a circle; and that of the tens chart--and considers their power for developing mathematical thinking. In his opinion, these images each contain the essence of a particular topic of mathematics. They are contrasting images in the sense that they deal…

This article concludes the serialization of the Royal Statistical Society's Schools Lecture for 2004, on "Lies and statistics".

In some elementary courses, it is shown that square root of 2 is irrational. It is also shown that the roots like square root of 3, cube root of 2, etc., are irrational. Much less often, it is shown that the number "e," the base of the natural logarithm, is irrational, even though a proof is available that uses only elementary calculus. In this…

Three elementary-school students were exposed to Fractions 1, an instructional program for fraction problem-solving skills based on "direct instruction" principles. Fraction performance increases could reasonably be attributed to the instructional program. Improvements were durable across a short follow-up period and transfer occurred to similiar…

Three proofs for the problem show there exist irrational numbers a and b such that a to the b power is rational'' are presented and discussed. (DT)

This study was supported by a grant from the Carnegie Corporation to Educational Testing Service. (MH)