Properties of curved functions considered to be parallel over their domains are investigated. Parallel curves in a given plane may appear identical but are actually not superimposable and thus are not congruent. Translational shifted functions in a plane are not parallel curves because the shortest perpendicular distance between them is not…

One typical challenge in algebra education is that many students justify the equivalence of expressions only by referring to transformation rules that they perceive as arbitrary without being able to justify these rules. A good algebraic understanding involves connecting the transformation rules to other characterizations of equivalence of…

This article explains how explorations into the quadratic formula can offer students opportunities to learn about the structure of algebraic expressions. In this article, the author leverages the graphical interpretation of the quadratic formula and describes an activity in which students derive the quadratic formula by quantifying the symmetry of…

Students' solutions of enumerative combinatorial problems may be assessed along two main dimensions: the correctness of the solution and the method of enumeration. This study looks at the second dimension with reference to the Cartesian product of two sets, and at the 'odometer' combinatorial strategy defined by English (1991). Since we are not…

The present work presents a proposal for study and investigation, in the context of the teaching of Mathematics, through the history of linear and recurrent 2nd order sequences, indicated by: Fibonacci, Lucas, Pell, Jacobsthal, Leonardo, Oresme, Mersenne, Padovan, Perrin and Narayana. Undoubtedly, starting from the Fibonacci sequence, representing…

Analysing graphs, formulating covariational relationships, and hypothesizing systems' behaviour have emerged as frequent objectives of contemporary research in physics education. As such, these studies aim to help students achieve these objectives. While a consensus has been reached on the cognitive benefits of emphasizing the structural domain of…

In a series of group activities supplemented with independent explorations and assignments, calculus students investigate functions similar to their own derivatives. Graphical, numerical, and algebraic perspectives are suggested, leading students to develop deep intuition into elementary transcendental functions even as they lay the foundation for…

This study investigates university students' graph interpretation strategies and difficulties in mathematics, physics (kinematics), and contexts other than physics. Eight sets of parallel (isomorphic) mathematics, physics, and other context questions about graphs, which were developed by us, were administered to 385 first-year students at the…

We use eye tracking as a method to examine how different mathematical representations of the same mathematical object are attended to by students. The results of this study show that there is a meaningful difference in the eye movements between formulas and graphs. This difference can be understood in terms of the cultural and social shaping of…

As teachers working with students in entry-level algebra classes, authors Amy Nebesniak and A. Aaron Burgoa realized that their instruction was a major factor in how their students viewed mathematics. They often presented students with abstract formulas that seemed to appear out of thin air. One instance occurred while they were teaching students…

This paper presents the results of a research, which proposes the introduction of the teaching by Research and Study Paths (RSPs) into Argentinean secondary schools within the frame of the Anthropologic Theory of Didactics (ATD). The paths begin with the study of "Q[subscript 0]: How to operate with any curves knowing only its graphic…

In mathematics, as in baseball, the conventional wisdom is to avoid errors at all costs. That advice might be on target in baseball, but in mathematics, avoiding errors is not always a good idea. Sometimes an analysis of errors provides much deeper insights into mathematical ideas. Certain types of errors, rather than something to be eschewed, can…

Many students have trouble grasping algebra. In this book, bestselling authors Judith, Gary, and Erin Muschla offer help for math teachers who must instruct their students (even those who are struggling) about the complexities of algebra. In simple terms, the authors outline 150 classroom-tested lessons, focused on those concepts often most…

If you want your students to graph a cubic polynomial, it is best to give them one with rational roots and critical points. In this paper, we describe completely all such cubics and explain how to generate them.

We explore how curvature and torsion determine the shape of a curve via the Frenet-Serret formulas. The connection is made explicit using the existence of solutions to ordinary differential equations. We use a paperclip as a concrete, visual example and generate its graph in 3-space using a CAS. We also show how certain physical deformations to…