The method of "Double False Position" is an arithmetic approach to solving linear equations that pre-dates current algebraic methods by more than 3,000 years. The method applies to problems that, in algebraic notation, would be expressed as y = L(x), where L(x) is a linear function of x. Double False Position works by evaluating the described…

It is a routine matter for undergraduates to find eigenvalues and eigenvectors of a given matrix. But the converse problem of finding a matrix with prescribed eigenvalues and eigenvectors is rarely discussed in elementary texts on linear algebra. This problem is related to the "spectral" decomposition of a matrix and has important technical…

A peer mentoring program has been implemented to support a group of at-risk students enrolled in two sections of an elementary algebra course at an urban community college. Peer mentors were recruited from advanced mathematics classes and trained to provide individualized tutoring and mentoring support to at-risk students. The results show that…

In this paper we present a versatile technique to solve several types of solvable quintic equations. In the technique described here, the given quintic is first converted to a sextic equation by adding a root, and the resulting sextic equation is decomposed into two cubic polynomials as factors in a novel fashion. The resultant cubic equations are…

Hill ciphers are linear codes that use as input a "plaintext" vector [p-right arrow above] of size n, which is encrypted with an invertible n x n matrix E to produce a "ciphertext" vector [c-right arrow above] = E [middle dot] [p-right arrow above]. Informally, a near-field is a triple [left angle bracket]N; +, *[right angle bracket] that…

One of the main problems in undergraduate research in pure mathematics is that of determining a problem that is, at once, interesting to and capable of solution by a student who has completed only the calculus sequence. It is also desirable that the problem should present something new, since novelty and originality greatly increase the enthusiasm…

In this note the authors investigate ways to shorten the amount of work involved in using the Rational Roots Theorem to find the rational roots of a polynomial with integral coefficients. The first result is a proof of a fact that we had long suspected, but were never able to find the statement of in any of the college algebra textbooks we had…

It is in the field of numerical analysis that this "easy-looking" function, also known as the Runge function, exhibits a behavior so idiosyncratic that it is mentioned even in most undergraduate textbooks. In spite of the fact that the function is infinitely differentiable, the common procedure of (uniformly) interpolating it with polynomials that…

A novel approach for teaching interpolation in the introductory course in numerical analysis is presented. The interpolation problem is viewed as a problem in linear algebra, whence the various forms of interpolating polynomial are seen as different choices of a basis to the subspace of polynomials of the corresponding degree. This approach…

Discusses common misconceptions in algebra, including: grouping symbols and order of operations; equations, inequalities, and expressions; and domains. Offers recommendations to help students avoid these mistakes. (MKR)

Explores an unexpected connection between a function, its inverse, and the arithmetic mean, algebraically and graphically. (MM)

Technological tools have the potential to offer students the possibility to represent information and relationships embedded in problems and concepts in ways that involve numerical, algebraic, geometric, and visual approaches. In this paper, the authors present and discuss an example in which an initial representation of a mathematical object…

Not too many years ago the author started out to study the students' own perspectives on their experience with the intent to answer the question, "How do college remedial math students define success, and what are they striving for in their math classes?" The author did not quite manage to answer such a grand question, but did gain some very…

The advent of powerful Computer Algebra Systems (CAS) continues to dramatically affect curricula, pedagogy, and epistemology in secondary and college algebra classrooms. However, epistemological and pedagogical research regarding the role and effectiveness of CAS in the learning of algebra lags behind. This paper investigates concerns regarding…

Probably the one "new" mathematical topic that is most responsible for modernizing courses in college algebra and precalculus over the last few years is the idea of fitting a function to a set of data in the sense of a least squares fit. Whether it be simple linear regression or nonlinear regression, this topic opens the door to applying the…