This article uses dynamic software in Excel to demonstrate several ways in which graphical and numerical approaches can be introduced both to enhance student understanding of l'Hopital's Rule and to explain why the Rule actually works to give the "right" answers. One of the approaches used is to visualize what is happening by examining…

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…

In this paper we present two simulations designed with GeoGebra that illustrate dynamically a key concept in Vector Calculus: line integrals of vector fields, along with other associated mathematical properties and applications. Students are not required to know the GeoGebra environment: a user-friendly interface with buttons, functionalities and…

A video is taken of a road-sign mirage from the passenger seat in a car traveling at constant speed on a highway. The video spans the duration of seeing the mirage of the sign, viewing the vanishing of the mirage as the car approaches, and passing the road sign. The mirage angle, defined as the angle with respect to the horizontal at the moment…

While tutoring his granddaughter in second-year algebra recently, the second author lamented that every textbook he could find expresses the quadratic formula as probably the most common form of the formula. What troubled him is that this form hides the meaning of the various components of the equation. Indeed, the meaning was obscured by the…

It is important to be able to make better informed decisions about issues such as sustainability and climate change that have both personal and global impact as early as possible in life. Primary teachers have a significant role in supporting students' learning and understanding of these concepts. One important teaching skill that needs to be…

The use and understanding of mathematics is a crucial component of the physical sciences. Much work has been done in physics education research and science education more broadly to determine persistent difficulties with mathematics. This work has led to the development of numerous problem solving strategies aimed at helping learners approach…

Fractional approximations of e and p are discovered by searching for repetitions or partial repetitions of digit strings in their expansions in different number bases. The discovery of such fractional approximations is suggested for students and teachers as an entry point into mathematics research.

An introductory course in algebra may cover various types of polynomial manipulation, here we cover the background of these topics and a relatively unknown property of polynomials, namely Viete's relations. We define relevant operations and properties of polynomials over the real numbers as well as their algebraic forms, divisibility properties,…

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.

The paper focuses on four upper secondary students' collaborative small-group mathematical reasoning (MR) with respect to a sinusoidal function. The students were collaboratively engaged in a process of MR regarding the relationships between mathematical theoretical descriptions of parameters in the algebraic expression of the sinusoidal function…

The 0.70 and 0.80 reliability standards, proposed by Jum Nunnally, are widely employed across a spectrum of research domains. Nonetheless, due to their arbitrary nature, both standards fail to satisfy the needs of researchers. This paper presents a set of formulas that can be used to compute minimum reliability standards as a function of a…

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…

The objective of this work was to evaluate the implementation of a metavisual strategy for students to revise and self-regulate concepts arising in a study of a chemical reaction between ions. For this purpose, two chemistry education undergraduate students at a Brazilian public university carried out an investigative activity, involving…

Undergraduate mathematics instructors often report that students make careless errors or have not previously learned key mathematical ideas and strategies. The purpose of this study is to explore evidence of an alternative explanation that at least some of these "errors" may result from students' application of conceptions developed in…