Let R be a ring with identity. Then {0} and R are the only additive subgroups of R if and only if R is isomorphic (as a ring with identity) to (exactly) one of {0}, Z/pZ for a prime number p. Also, each additive subgroup of R is a one-sided ideal of R if and only if R is isomorphic to (exactly) one of {0}, Z, Z/nZ for an integer n = 2. This note…

In this paper, we propose an analytic tool for describing the mathematics made available to learn in a "textbook lesson". The tool is an adaptation of the Mathematics Discourse in Instruction (MDI) analytic tool that we developed to analyze what is made available to learn in teachers' lessons. Our motivation to adapt the use of the MDI…

The first requirement for an online mathematics homework engine is to encourage students to practice and reinforce their mathematics skills in ways that are as good or better than traditional paper homework. The use of the computer and the internet should not limit the kind or quality of the mathematics that we teach and, if possible, it should…

In the present study, we ask whether and how online assessment can inform teaching about students' understanding of advanced concepts. Our main goal is to illustrate how we study design of tasks that support reliable online formative assessment by automatically analyzing the objects and relations that characterize the students' submissions. We aim…

The set of functions {x[superscript q] | q[element of][real numbers set]} is linearly independent over R (with respect to any open subinterval of (0, 8)). The titular result is a corollary for any integer n = 2 (and the domain [0, 8)). Some more accessible proofs of that result are also given. Let F be a finite field of characteristic p and…

The Gauss-Lucas Theorem is a classical complex analysis result that states the critical points of a single-variable complex polynomial lie inside the closed convex hull of the zeros of the polynomial. Although the result is well-known, it is not typically presented in a first course in complex analysis. The ease with which modern technology allows…

Earlier "Discovery" articles (de Mestre, 1999, 2003, 2006, 2010, 2011) considered patterns from many mathematical situations. This article presents a group of patterns used in 19th century mathematical textbooks. In the days of earlier warfare, cannon balls were stacked in various arrangements depending on the shape of the pile base…

This article's aim is to suggest a supplementary learning environment to understand the hierarchical classification of quadrilaterals for high school or higher degree learners. Three diagonal properties, "being congruent, being perpendicular" and "dividing each other in particular ratio," and all possible combinations of these…

In this article, I discuss how teaching of mathematics can develop and encourage creative thinking, not only in mathematics, but also in general. I begin by discussing creativity as a habit that teachers can help students develop. Then I briefly present the investment theory of creativity. The basic idea of the investment theory is that creative…

The intent of this commentary is to identify elements of fraction sense and note how the research studies provided in this special issue, in related but somewhat different ways, validate the importance of such understandings. Proficiency with fractions serves as a prerequisite for student success in higher level mathematics, as well as serving as…

Spreadsheets provide a rich setting for first-year algebra students to solve problems. Individual spreadsheet cells play the role of variables, and creating algebraic expressions for a spreadsheet to perform a task allows students to achieve a glimpse of how mathematics is used to program a computer and solve problems. Classic optimization…

This paper approaches the concept of the circle through the framework of mathematics-as-becoming. This paper focuses specifically on how a concept can be thought of as a process, and on the implications that this might have for mathematics learning. Contrary to long-standing assumptions about mathematical concepts as ideal, inert, Platonic forms,…

In this article, Monica Housen describes how she uses Guess the Number of . . . , a game that develops estimation skills and persistence to provide a fun, to provide a meaningful experience for her high school students. Each week she displays objects in a clear plastic container, like those for pretzels sold in bulk. Students enter a…

The multiplication principle serves as a cornerstone in enumerative combinatorics. The principle underpins many basic counting formulas and provides students with a critical element of combinatorial justification. Given its importance, the way in which it is presented in textbooks is surprisingly varied. In this paper, we analyze a number of…

Despite its central place in the mathematics curriculum the notion of mathematical proof has failed to permeate the curriculum at all scholastic levels. While the concept of mathematical proof can serve as a vehicle for inculcating mathematical thinking, studies have revealed that students experience serious difficulties with proving that include…