The study of loops and spaces in mathematics has been the subject of much interest among researchers. In Part 1 of "The Theory on Loops and Spaces," published in the "Mathematics Teaching Research Journal," introduced the concept and the basic underlying idea of this theory. This article continues the exploration of this topic…

We are all fascinated by loops and their formation in space. When a line cuts itself, it forms an intersection point and creates a space. This is an experimental study done by analyzing several loops, forming a concrete formulation by visualizing the patterns observed, and then proving the formulations proposed using the known standard…

One way to develop a cross-curricular lesson is to select the most common mathematical formulas used in science and carefully develop and implement tasks that allow students to make connections between the mathematical representations and theoretical/physical science concepts. The slope-intercept formula, which is used to study relationships…

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…

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…

This study explored how students develop meaning of functions by building on their understanding of expressions and equations. A teaching experiment using design research was conducted in a sixth-grade classroom. The data was analyzed using a grounded theory approach to provide explanations about why events occurred within this teaching episode…

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 paper is about the development of a task sequence to help overcome the fragmented understanding of the "function" concept that students often bring with them into the initial stage of upper secondary school level. Our aim is to make the students' use of functions more flexible in certain respects, for example when functions are…

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…

A teaching experiment-using Mathematica to investigate the convergence of sequence of functions visually as a sequence of objects (graphs) converging onto a fixed object (the graph of the limit function)-is here used to analyze how the approach can support the dynamic blending of visual and symbolic representations that has the potential to lead…

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…

Nonroutine function tasks are more challenging than most typical high school mathematics tasks. Nonroutine tasks encourage students to expand their thinking about functions and their approaches to problem solving. As a result, they gain greater appreciation for the power of multiple representations and a richer understanding of functions. This…

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…

This paper provides surface area and volume formulas for surfaces of revolution in R[superscript n]. In addition the authors illustrate how to obtain the formulas for volume and surface areas of revolution about the x- or y-axis in two different ways: a "heuristic" argument and a rigorous calculation using "cylindrical" coordinates. In the last…