In this note several cubic polynomials and their roots are examined, in particular, how these roots move as some of the coefficients are modified. The results obtained are applied to eigenvalues of matrices. (Contains 8 figures and 1 footnote.)

There has been a broad wave of change in tertiary calculus courses in the past decade. However, the much-needed change in tertiary pre-calculus programmes--aimed at bridging the gap between high-school mathematics and tertiary mathematics--is happening at a far slower pace. Following a discussion on the nature of the gap and the objectives of a…

This paper discusses the curriculum and students' and teachers' conceptions of proof. It goes on to discuss university students' difficulties with proving related to understanding and using definitions and theorems, understanding the structure of a proof, knowing how to read and check proofs, knowing and using relevant concepts, bringing…

This article examines a "calculus graphing schema" and the triad stages of schema development from Action-Process-Object-Schema (APOS) theory. Previously, the authors studied the underlying structures necessary for students to process concepts and enrich their knowledge, thus demonstrating various levels of understanding via the calculus…

CDM, for computational discrete mathematics, is a course that attempts to teach a number of topics in discrete mathematics to computer science majors. The course abandons the classical definition-theorem-proof model, and instead relies heavily on computation as a source of motivation and also for experimentation and illustration. The emphasis on…

The fields of mathematics, science, and engineering are replete with diagrams of many varieties. They range in nature from the Venn diagrams of symbolic logic to the Periodic Chart of the Elements; and from the fault trees of risk assessment to the flow charts used to describe laboratory procedures, industrial processes, and computer programs. All…

The authors give a rigorous treatment of the differentiability of the exponential function that uses only differentiable calculus. It can thus make "early transcendental" courses complete.

This paper examines the mathematical work of the French bishop, Nicole Oresme (c. 1323-1382), and his contributions towards the development of the concept of graphing functions and approaches to investigating infinite series. The historical importance and pedagogical value of his work will be considered in the context of an undergraduate course on…

We provide a geometric proof of the formula for the sine of the sum of two positive angles whose measures sum to less than [pi]/2. (Contains 1 figure.)

Calculus has been witnessing fundamental changes in its curriculum, with an increased emphasis on visualization. This mode for representing mathematical concepts is gaining more strength due to the advances in computer technology and the development of dynamical mathematical software. This paper focuses on the understanding of the function and its…

In this issue's "Classroom Notes" section, the following papers are discussed: (1) "Constructing a line segment whose length is equal to the measure of a given angle" (W. Jacob and T. J. Osler); (2) "Generating functions for the powers of Fibonacci sequences" (D. Terrana and H. Chen); (3) "Evaluation of mean and variance integrals without…

Methods for teaching traditional college mathematics materials in the areas of analysis, vector algebra, etc., are reviewed. Examples of interesting and novel construction problems, logical decisions, generalizations with calculators, and other topics used in an experiment at a technical college in Budapest, Hungary are presented. (MP)

In university mathematics courses, the activity of proof construction can be viewed as a problem-solving task in which the prover is asked to form a logical justification demonstrating that a given statement must be true. The purposes of this paper are to describe some of the different types of reasoning and problem-solving processes used by…

Especially in their first upper-division mathematics courses, students often have trouble with proofs; and sometimes they object, "This is hard. I do not get it. Why am I doing this?" Though symptomatic of emotional reaction to difficulty, at its heart this is a legitimate question and it deserves a legitimate answer. This article offers one such…