Every semester, the authors encounter students who are attracted to the visual and spatial aspects of geometry. They have other students who consider geometry to be challenging for the very same reasons. Students are confounded not only by the fact that geometry relies on visual interpretations but also because it has a language of its own and…

This practice guide presents five recommendations intended to help educators improve students' understanding of, and problem-solving success with, fractions. Recommendations progress from proposals for how to build rudimentary understanding of fractions in young children; to ideas for helping older children understand the meaning of fractions and…

Being able to dive and breathe underwater has been a challenge for thousands of years. In 1980, Fuji Systems of Tokyo developed a series of prototype gills for divers as a way of demonstrating just how good its membranes are. Even though gill technology has not yet reached the point where recipients can efficiently use implants to dive underwater,…

In the author's ongoing attempt to design creative challenges that are relevant, based on observational drawing, and that place students squarely in the creative problem-solving driver's seat, she chose cars and engines as subjects. After all, they have a parking lot full of them and their administrative staff members, who park side-by-side, were…

Learning how to problem solve is one of the key developmental milestones in early childhood. Children's problem-solving skills represent a key feature in the development of social competence. Problem solving allows children to stay calm during difficult situations, repair social relations quickly, and get their needs met in ways that are safe and…

Mathematics educators promote student engagement in the Process Standards and create problem-solving tasks and facilitate discussions to help their students develop strengths in explaining their methods, using and interpreting multiple representations, and making connections between topics. They are excited and encouraged when they see students…

An important component of intermediate and college algebra courses involves teaching students methods to factor a trinomial with integer coefficients over the integers. The aim of this article is to present a theoretical justification of that which is often taught, but really never explained as to why it works. The theory is presented, and a…

The Take Action Project (TAP) was created to help middle school students take informed and effective action on science-related issues. The seven steps of TAP ask students to (1) choose a science-related problem of interest to them, (2) research their problem, (3) select an action to take on the problem, (4) plan that action, (5) take action, (6)…

While some disasters involving engineered structures are due to events in nature (e.g., tornadoes, hurricanes, earthquakes), others may be caused by inadequate materials, design flaws, and poor maintenance. These catastrophes result in the loss of human lives and cost billions of dollars. In the set of lessons described here, students design a…

In a recent paper (Denny 2002 Eur. J. Phys. 23 449-58), entitled "The pendulum clock: a venerable dynamical system", Denny showed that in a first approximation the steady-state motion of a weight-driven pendulum clock is shown to be a stable limit cycle. He placed the problem in a historical context and obtained an approximate solution using the…

This module was initially developed for a course in applications of mathematics in biology. The objective of this lesson is to investigate how the allele and genotypic frequencies associated with a particular gene might evolve over successive generations. The lesson will discuss how the Hardy-Weinberg model provides a basis for comparison when…

Many classes of differential equations are shown to be open to solution through a method involving a combination of a direct integration approach with suitably modified Picard iterative procedures. The classes of differential equations considered include typical initial value, boundary value and eigenvalue problems arising in physics and…

Mathematics may be inconceivable without its diagrams and symbols--its representations. Mathematical representations help individuals organize their thinking; they bring a visual component to abstract ideas and serve as templates for computation with understanding. But the inevitability of representations is no guarantee that they are used…

This material has been used twice as an out-of-class project in a mathematical modeling class, the first elective course for mathematics majors. The only prerequisites for this course were differential and integral calculus, but all students had been exposed to differential equations, and the project was assigned during discussions about solving…

Insects are fascinating creatures--especially when you and your students get up close and personal with them! To that end, the authors facilitated an inquiry-based investigation with an emphasis on identification of the different types of insects found in the school yard, their characteristics, their habitat, and what they eat, while engaging the…