Stimulated by a picture of a Sierpinski-pyramid and an article by Stewart, students were asked about a connection between this pyramid and the well-known trinomial formula. The results of all the work done with students are presented in this note. (Contains 8 figures.)

Using Newton's method as an intermediate step, we introduce an iterative method that approximates numerically the solution of f(x) = 0. The method is essentially a leap-frog Newton's method. The order of convergence of the proposed method at a simple root is cubic and the computational efficiency in general is less, but close to that of Newton's…

This paper reports the results of an exploratory study of the perceptions of and approaches to mathematical proof of undergraduates enrolled in lecture-based and problem-based "transition to proof" courses. While the students in the lecture-based course demonstrated conceptions of proof that reflect those reported in the research literature as…

A computer-based college course covering the exact and complete theory of logical inference is described. Student's performance and their evaluation of the course are discussed. (SD)

The "word problem" is stated for a given collection. Facts regarding Dehn's Algorithm, definition of Thue systems, a rewriting system, lemmas and corollaries are provided. The situation is examined where the monoid presented by a finite Thue system is a group. (DC)

As students are first learning to construct mathematical proofs, it is often helpful for them to have the opportunity to see and evaluate proofs that others have written. In fact, several textbooks designed for use in a transition or bridge course include a few exercises in which students are given a proposed proof and asked to determine if it…

Many undergraduate students have difficulty with the transition to proof-based courses in mathematics. This paper discusses students' beliefs about proof and justification in mathematics just prior to entry into such courses. The paper is based on in-depth interviews with students. The data suggests that some students have beliefs that may in part…

Twelve unusual problems involving divisibility of the binomial coefficients are represented in this article. The problems are listed in "The Problems" section. All twelve problems have short solutions which are listed in "The Solutions" section. These problems could be assigned to students in any course in which the binomial theorem and Pascal's…

A first-year seminar general education course provides a good opportunity to search for mathematical topics associated with the popular culture represented in the course's required films and readings. We discuss mathematical connections to several books, including "Life of Pi" and "The Curious Incident of the Dog in the Night-Time," and to the…

This paper contains an elementary and short proof for the case that the underlying distribution function F is discrete, and then extends the result to the general F. In other proofs underlying iid sequences of random variables with continuous distributions are considered to be the "ideal" case. In this paper discretization of the underlying iid…

The author arranges 26 current states of mathematical knowledge (in relation to solving a problem) in an informal taxonomy and comments on them. (MNS)

In this paper the notion of "procept" (in the sense of Gray & Tall, 1994) is extended to advanced mathematics by considering mathematical proof as "formal procept". The statement of a theorem as a symbol may theoretically evoke the proof deduction as a process that may contain sequential procedures and require the synthesis…

Criteria for a reasonable axiomatic system are discussed. A discussion of the historical attempts to prove the independence of Euclids parallel postulate introduces non-Euclidean geometries. Poincare's model for a non-Euclidean geometry is defined and analyzed. (LS)

Provides four examples for testing the validity of logical arguments by using the method of truth trees. (YP)

Lists five criteria in the acceptance of mathematical theorems, such as understanding, significance, compatibility, reputation, and convincing argument. Concludes that social and language factors are involved in the process of the acceptance. (YP)