This document summarizes the results of a dissertation (M. Alford, 1999) that focused on the success of Wake County Public School System (WCPSS), North Carolina, students who failed Algebra I in 1994-1995 and retook it during the 1995 summer session or the 1995-1996 regular school year. Success was defined by three measures of student learning:…

Given two interesecting lines, the "pencil" of this interesection is the family of all lines passing through the point of intersection. Presented is a characterization of all lines both in the plane and lying in opposite sectors of the plane as determined by one of the original intersecting lines. (JP)

Paper shows that an obvious generalization of the subjective metrics model by Bloxom, Horan, Carroll and Chang has a very simple algebraic solution which was previously considered by Meredith in a different context. (Author)

The results of a study of third-graders' performance in solving certain open sentences, and implications for teaching open sentences are discussed. (DT)

The author argues that a knowledge of mathematical structure is only helpful to the high ability student. (MM)

The Shepherd's Principle (to count the number of sheep in a field, count the number of legs and divide by four) is applied to the problem of finding the number of different rectangles on an N by N checkerboard. (MM)