Seeing is Believing?

“Proofs Without Words” are proofs based on a visual representation of a theorem which provides a convincing argument about its validity without the need for any accompanying text. The genre has been much enriched by the increased availability of computer animation. This is of course relevant to math education: many of the concepts we teach can be illustrated visually, including with… Continue reading Seeing is Believing?

Transformational Geometry for Teachers

I taught geometry for decades, starting in the 1980’s, and loved it. I’m reasonably good at manipulating algebraic symbols, but I don’t especially enjoy it. In contrast, I am happy to spend plenty of time on visual puzzles, and I am enthusiastic about sharing that passion with colleagues and with students. Early in my high… Continue reading Transformational Geometry for Teachers

Tiling in GeoGebra

In my last two posts, I promoted the idea of using tiling (tessellation) as an interesting context in geometry class, especially for the introduction of some basic ideas of transformational geometry. One reason this works is the connection with art, including the abstract patterns in Islamic art and the mind-bending creations of M.C. Escher.  John Golden is a connoisseur of… Continue reading Tiling in GeoGebra

Virtual Manipulatives: Part 2

In the previous post, I discussed virtual manipulatives in general, and a particular implementation for algebra, using a Google Drawings representation of the Lab Gear. In this post, I will explore GeoGebra as a platform for virtual manipulatives. Pattern Blocks Virtual pattern blocks are not hard to find on the Web. One good implementation is on the Math… Continue reading Virtual Manipulatives: Part 2

Virtual Manipulatives: Part 1

Learning Tools In 1981, after ten years in K-5, I switched to teaching high school math. In some ways, this felt like starting a whole new career: the math was more involved, the relationship with students less like parenting, and tradition weighed a lot more heavily on the profession. Still, in other ways, teaching is… Continue reading Virtual Manipulatives: Part 1

Wallpaper Symmetry Part 2

This post is a continuation of Wallpaper Symmetry Part 1, which you should read first. Both posts are companions to the Wallpapers Catalog on my website. --------------------------------------------------------------------- A long time ago, in my twenties, I attended a lecture about the mathematics of wallpaper designs. The presenter gave an overview of the entire proof that there are only… Continue reading Wallpaper Symmetry Part 2