Summer Workshops for Educators

This three-and-a-half-day workshop is designed for middle and high school mathematics teachers who want to make algebra more accessible, richer and more fun. I will present a wealth of visual approaches to the teaching of algebra, including:

• Lab Gear manipulatives for basic symbol manipulation
• geoboard lattices for slope and radicals
• a powerful parallel axes representation for functions
• intelligent use of technology
• three distinct visual paths to the quadratic formula

Topics range from Pre-Algebra to Algebra II. Activities complement the material in any textbook, whether reform or traditional.

Participants will learn techniques to serve the whole range of students by offering

• greater access, because of addressing multiple intelligences;
• greater challenge, because of expecting multi-dimensional understanding; and
• greater variety, because of using manipulative and electronic tools.

In addition, we will work on teacher-level problems rooted in high school subject matter, and strengthen understanding of the underlying mathematics.

High school math offers very few opportunities to work in three dimensions. As a result, our students are often overwhelmed by topics such as solids of revolution in calculus. In this one-day workshop, we will start by learning the basics of creating three-dimensional constructions with Cabri 3D. This software gives students a rich environment to create and explore interesting and unusual mathematical structures, thereby enhancing appreciation of the beauty of mathematics, the ability to visualize in three dimensions, and the generation of conjectures. Because it allows students to look at objects from different points of view in space, and to interact with them in the same style as they can in two-dimensional dynamic geometry environments, it makes some difficult ideas much more accessible. This workshop will prepare participants for multiple three-dimensional applications: transformations; surface area and volume for geometry (including some interesting historical approaches); conic sections, planes, and vectors for precalculus; astronomy and chemistry modeling; and yes, volumes of revolution!

In this two-day workshop for middle and high school teachers, I will present hands-on tools and activities to preview, review or extend key concepts in geometry, as well as some enrichment lessons. This work is intended to complement, not replace, related work in the traditional paper-pencil and compass-straightedge environments.

• The tools include manipulatives (such as pattern blocks and geoboards) and puzzles (such as tangrams, pentominoes, and supertangrams)
• The activities include "walking geometry," "soccer angles," "tile design," and "slicing a cube"
• We will use interactive geometry software to extend these activities, and to work through a challenging and highly motivational construction unit.

I will also present an authentic approach to proof, which tries to navigate a middle course between the too-abstract traditional curriculum and the insufficiently rigorous nature of some reform programs.

These lessons were developed in somewhat heterogeneous classes, and reach a wide range of students. They provide support for the less visual by complementing the drawing and studying of figures, and enrichment for the more talented by offering by offering deep and challenging problems.