A traditional abstract algebra course typically consists of the professor stating and then proving a sequence of theorems. As an alternative to this classical structure, the students could be expected to discover some of the theorems even before they are motivated by classroom examples. This can be done by using a software system to explore a…

One of the earlier, more challenging concepts in linear algebra at university is that of basis. Students are often taught procedurally how to find a basis for a subspace using matrix manipulation, but may struggle with understanding the construct of basis, making further progress harder. We believe one reason for this is because students have…

The aim of this article is to characterize the 2 x 2 matrices "X" satisfying X[superscript 2] = X + I and obtain some new identities concerning with Fibonacci and Lucas numbers. The recommendations regarding the teaching of the identities given in this article can be presented in two cases. The first is related to the pedagogical aspect. The…

In this note we present a class of functions (f, g) that satisfy "freshman rules" in calculus.

For a given initial speed of water from a spigot or jet, what angle of the jet will maximize the visual impact of the water spray in the fountain? This paper focuses on fountains whose spigots are arranged in circular fashion, and couches the measurement of the visual impact in terms of the surface area and the volume under the fountain's natural…

The fundamental theorem of algebra for the complex numbers states that a polynomial of degree n has n roots, counting multiplicity. This paper explores the "perplex number system" (also called the "hyperbolic number system" and the "spacetime number system") In this system (which has extra roots of +1 besides the usual [plus or minus]1 of the…

This article describes the development of the Precalculus Concept Assessment (PCA) instrument, a 25-item multiple-choice exam. The reasoning abilities and understandings central to precalculus and foundational for beginning calculus were identified and characterized in a series of research studies and are articulated in the PCA Taxonomy. These…

In 1992, Klamkin and Liu proved a very general result in the Extended Euclidean Plane that contains the theorems of Ceva and Menelaus as special cases. In this article, we extend the Klamkin and Liu result to projective planes "PG"(2, F) where F is a field. (Contains 2 figures.)

If you take a circle with a horizontal diameter and mark off any two points on the circumference above the diameter, then these two points together with the end points of the diameter form the vertices of a cyclic quadrilateral with the diameter as one of the sides. We refer to the quadrilaterals in question as diametric. In this note we consider…

Prior research on students' uses of technology in the context of Euclidean geometry has suggested that it can be used to support students' development of formal justifications and proofs. This study examined the ways in which students used a dynamic geometry tool, NonEuclid, as they constructed arguments about geometric objects and relationships…

Mathematical induction is one of the major proof techniques taught to mathematics students in the first years of their undergraduate degrees. In addition to its importance to mathematics, induction is also required for computer science and related disciplines. However, even if the concepts of a proof by induction are taught and understood, many…

The methods for calculating returns on investments are taught to undergraduate level business students. In this paper, the authors demonstrate how such calculations are within the scope of senior school students of mathematics. In providing this demonstration the authors hope to give teachers and students alike an illustration of the power and the…

College students enrolling in the calculus sequence have a wide variance in their preparation and abilities, yet they are usually taught from the same lecture. We describe another pedagogical model of Individualized Additional Instruction (IAI) that assesses each student frequently and prescribes further instruction and homework based on the…

To focus on mathematical reasoning and what makes a good argument, the authors developed an assignment that requires college students to submit a book of mathematical reasoning as an assessment during the semester. To begin, the authors looked for questions and tasks that lend themselves to developing mathematical arguments and justifications and…

Weber (2009) suggested that counterexamples can be generated by a syntactic proof production, apparently contradicting our earlier assertion (Alcock & Inglis, 2008). Here we point out that this ostensible difference is the result of Weber working with theoretical definitions that differ slightly from ours. We defend our approach by arguing that…