A method based on oblique projection is presented for construction of sundials. The derived formulas are classical, but usage of vectors and projections renders a coherent presentation rather than a number of special cases. The presented work is aimed to be useful for those taking a beginning module on vector algebra.

Continuing our critique of the classical derivation of the formula for the area of a disk, we focus on the limiting processes in geometry. Evidence suggests that intuitive approaches in arguing about infinity, when geometric configurations are involved, are inadequate, and could easily lead to erroneous conclusions. We expose weaknesses and…

Understanding how one representation connects to another and how the essential ideas in that relationship are generalized can result in a mathematical theorem or a formula. In this article, the authors demonstrate this process by connecting a vector cross product in algebraic form to a geometric representation and applying a key mathematical idea…

The purpose of this article is to present and discuss two recommended sequences of learning the areas of polygons, starting from the area of a rectangle. Exploring the algebraic derivations of the two sequences reveals that both are valid teaching progressions for introducing the area formula for various polygons. Further, it is suggested that…

This study is concerned with the reasoning that undergraduates apply when deciding whether a prompt is an example or non-example of the subspace concept. A qualitative analysis of written responses of 438 students revealed five unconventional tacit models that govern their reasoning. The models account for whether a prompt is a subset of a vector…

By using high-leverage models to connect student learning experiences to overarching concepts in mathematics, teachers can anchor learning in ways that allow students to make sense of content on the basis of their own prior experiences. A rectangular area model can be used as a tool for understanding problems that involve multiplicative reasoning.…

In this paper, we reconstruct matrices from their minors, and give explicit formulas for the reconstruction of matrices of orders 2 × 3, 2 × 4, 2 × n, 3 × 6 and m × n. We also formulate the Plücker relations, which are the conditions of the existence of a matrix related to its given minors.

One does not have to teach for very long to see students applying the wrong formula in the wrong situation (e.g., Kirshner and Awtry 2004; Tan-Sisman and Aksu 2016). Students can become overreliant on the power of the formula instead of thinking about the relationships it describes. It is not surprising that students can see formulas as a way to…

In this note, we revisit the problem of polynomial interpolation and explicitly construct two polynomials in n of degree k + 1, P[subscript k](n) and Q[subscript k](n), such that P[subscript k](n) = Q[subscript k](n) = f[subscript k](n) for n = 1, 2,… , k, where f[subscript k](1), f[subscript k](2),… , f[subscript k](k) are k arbitrarily chosen…

In this study, a description is provided for the development of two undergraduate students' geometric reasoning about the derivative of a complex-valued function with the aid of "Geometer's Sketchpad" ("GSP") during an interview sequence designed to help them characterize the derivative geometrically. Specifically, a particular…

We study here an aspect of an infinite set "P" of multivariate polynomials, the elements of which are associated with the arithmetic-geometric-mean inequality. In particular, we show in this article that there exist infinite subsets of probability "P" for which every element may be expressed as a finite sum of squares of real…

Generalizing is a foundational mathematical practice for the algebra classroom. It entails an act of abstraction and forms the core of algebraic thinking. Kinach (2014) describes two kinds of generalization--by analogy and by extension. This article illustrates how exploration of fractals provides ample opportunity for generalizations of both…

Admittedly, the study of Complex Analysis (CA) requires of the student considerable mental effort characterized by the mobilization of a related thought to the complex mathematical concepts. Thus, with the aid of the dynamic system Geogebra, we discuss in this paper a particular concept in CA. In fact, the notion of winding number v[f(gamma),P] =…

An algorithm is presented for factorising a quadratic expression to facilitate instruction and learning. It appeals to elementary geometry which may provide better insights to some students or teachers. There have been many methods for factorising a quadratic expression described in school text books. However, students often seem to struggle with…

The article begins with a well-known property regarding tangent lines to a cubic polynomial that has distinct, real zeros. We were then able to generalize this property to any polynomial with distinct, real zeros. We also considered a certain family of cubics with two fixed zeros and one variable zero, and explored the loci of centroids of…