The usual definition of the Riemann integral as a limit of Riemann sums can be strengthened to demand more of the function to be integrated. This super-Riemann integrability has interesting properties and provides an easy proof of a simple change of variables formula and a novel characterization of derivatives. This theory offers teachers and…

Students often complain that the rules of mathematics are being changed. A short conversation between a professor and a class of college algebra students dramatizes this in the realm of complex numbers and the legal realm of speed limits.

In this article, I describe my attempts to use various pedagogical techniques to teach college level mathematics and the particular problems that I ran into with each. Then I explain how I reached the conclusion that there is no best pedagogical approach. Rather each person's teaching methods should fit their personality, their mathematical…

Reasons for using Rolle's Theorem in calculus are discussed, with comparisons to the theorems of Lagrange and Cauchy. (MNS)

Standard calculus textbooks often include a related rates problem involving light cast onto a straight line by a revolving light source. Mathematical aspects to these problems (both in the solution and in the method by which that solution is obtained) are examined. (JN)

Linear ciphers, substitution ciphers, public-key cryptosystems, and trapdoor knapsacks are each discussed. (MNS)

An exhibit at the San Francisco Exploratorium is used to discuss problem solving and illustrate optimization. The solution is discussed in detail. (MNS)

How current calculus textbooks consider the relationship between the tangent line and the derivative are discussed, with three theorems presented. (MNS)

An interesting graphical interpretation of complex roots is presented, since it is probably unfamiliar to many mathematics teachers. (MNS)

Historical problems are presented which can readily be solved by students once some elementary probability concepts are developed. The Duke of Tuscany's Problem; the problem of points; and the question of proportions, divination, and Bertrand's Paradox are included. (MNS)

Discusses (in 10 sections) some problem-solving techniques that are typically encountered in a first course in discrete mathematics. Sections (which include exercises) are titled: draw a figure; search for a pattern; mathematical induction; one-to-one correspondence; recurrence relations; generating functions; calculus; finite differences; and…

Mathematical observations are made about some continuous curves, called transitions, encountered in well-known experiences. The transition parabola, the transition spiral, and the sidestep maneuver are presented. (MNS)

A possible method of derivation of prescriptions for solving problems, found in Babylonian cuneiform texts, is presented. It is a kind of "geometric algebra" based mainly on one figure and the manipulation of or within various areas and segments. (MNS)

Provides examples in which graphs are used in the statements of problems or in their solutions as a means of testing understanding of mathematical concepts. Examples (appropriate for a beginning course in calculus and analytic geometry) include slopes of lines and curves, quadratic formula, properties of the definite integral, and others. (JN)

An approach to polynomial approximations that leads the student to stumble on Taylor's theorem and its proof is presented, with a generalization. (MNS)