Observing the way a watering hose is wrapped on a rotating cylinder, a catenary shape is detected for the suspended portion of the hose. We calculate the angle [alpha] the tangent to the curve at the highest point of the suspended portion makes with the horizontal. By means of this quantity, the length of the suspended and dragged portions of the…

Proportional reasoning is an important cornerstone in children's mathematical development. This sort of reasoning has been shown to develop across the early years of schooling (ages 8 to 10) through the middle years (ages 11-14). In the early years, children tend to use additive reasoning to generate solutions to problems, while later comparisons…

Vocabulary presents a challenge to students in content area classes, making it difficult to understand new concepts and make connections to background knowledge. This article describes the 5 Cs, a tool developed to help content area teachers consider vocabulary as part of lesson planning. By selecting a set of key words for instruction, teachers…

This study investigated the development of the concept of variables in middle grades mathematics textbooks during four eras of mathematics education in the United States. Findings revealed that each of the middle grades mathematics curricula examined used variables, but in varied proportions and levels of complexity. Formal definitions for…

Learning in the cellular phone environment enables utilizing the multiple functions of the cellular phone, such as mobility, availability, interactivity, verbal and voice communication, taking pictures or recording audio and video, measuring time and transferring information. These functions together with mathematics-designated cellular phone…

Although number sense is difficult to define, it involves having a good intuition about numbers and their relationships, including the ability to have a "feel" for the relative size of numbers and to make reasonable estimations. Students with good number sense typically recognise the relative magnitude of numbers, appreciate the effect…

Integral calculus is generally regarded as a fundamental but advanced aspect of mathematics, and it is not generally studied until students are aged about fifteen or older. Understanding the transformative potential of information and communication technology, this project undertook an investigation in four Australian schools to train students…

"Probability is a difficult concept to teach, because children and adults find it counterintuitive." This is impetus to consider the detailed planning of a set of lessons with a "mixed", in many senses, group of fourth graders. Can the use of prior experience, and the knowledge associated with that experience, make probability a concept that is…

Learners are all different and teachers are all different, so why do we often ignore this reality when trying to explain, or demystify, some aspect of mathematics in the classroom? Enabling learning can be challenging, demanding of creativity, and needy of alternatives if understanding is the real goal. Here the authors offer ideas that are aimed…

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…

Students often have difficulty understanding fractions, in general, and understanding how to multiply fractions, in particular. To move past this potential problem area, students need to develop a deeper understanding of multiplication and connect the ideas to fractions. In this article, the authors share their insights into teaching fraction…

Social scientists use adjacency tables to discover influence networks within and among groups. Building on work by Moler and Morrison, we use ordered pairs from the components of the first and second singular vectors of adjacency matrices as tools to distinguish these groups and to identify particularly strong or weak individuals.

The Intermediate Value Theorem (IVT) is not often proved in Calculus I classes because many teachers and students see the theorem as obvious and its proof as impenetrable. This article addresses those two misconceptions, showing "how" the IVT can be proved in Calc I... and "why" it "should" be.

In this article, I summarize how I present the Rayleigh-Ritz procedure, in a second-semester differential equations course that I teach, and describe some benefits that I believe my students have derived from exposure to this topic. (Contains 3 figures.)

Decipipes are a representational model that can be used to help students develop conceptual understanding of decimal place value. They provide a non-standard tool for representing length, which in turn can be represented using conventional decimal notation. They are conceptually identical to Linear Arithmetic Blocks. This article reviews theory…