This article is a discovery-based instructional resource for faculty to use as a capstone course or exploratory project for undergraduates who are familiar with (but not necessarily fluent in) calculus, linear algebra, complex variables, and geometry. The explorations flow from a simple change in sign: in the complex numbers a + bi, where…

Mathematical investigations pursued by two gifted students are presented. One is written by a boy age 9 with his follow-up at 11; he studied patterns to derive the factors for the difference of two squares. Another boy of 11 discovered formulas for slopes of tangent lines and areas under curves. (LS)

Presented are four mathematical discoveries made by students on an arithmetical function using the Fibonacci sequence. Discussed is the nature of the role of the teacher in directing the students' discovery activities. (KR)

Presented is an example that allows students to investigate trigonometric ideas in a simple and practical way. (PK)

In this chapter, a college teacher describes the use of personal projects in a class on mathematics teaching for preservice teacher-education students. The personal project required students to learn something new or to do something they had never done before, and to document their learning through a journal or other reflective record. In…

This updated handbook is designed as a guide for teachers and students in choosing and developing mathematics projects, from simple demonstrations of mathematical problems or principles that the teacher has assigned as classroom learning experiences to complex, sophisticated exhibits, intended for entries in fairs and competitions. The use of…