The Math Education Blog
Michael de Villiers teaches courses in mathematics and mathematics education at University of KwaZulu-Natal in South Africa. His website features a wealth of Dynamic Geometry-related books, articles, and sketches. He is the author of the Sketchpad activity module Rethinking Proof with The Geometer’s Sketchpad. This blog post is a condensed version of a longer article, “Conjecturing, Refuting and Proving...
The 17th-century Dutch mathematician Frans van Schooten developed “hands-on manipulatives” centuries before the term became popular in math education circles. Below are two images of ellipse-drawing linkages from van Schooten’s manuscript, Sive de Organica Conicarum Sectionum in Plano Descriptione, Tractatus (A Treatise on Devices for Drawing Conic Sections). Building physical models of these devices isn’t hard....
On November 6 I had the honor of being one of the panelists in a Symposium Honoring Zalman Usiskin, held to honor Zal’s many years of contributions to mathematics education, from his groundbreaking 1971 textbook Geometry: A Transformation Approach (GATA) to his continuing activities today. My panel was supposed to discuss his work on the...
As readers of this blog can probably tell, I like puzzles. I especially enjoy taking ordinary mathematical topics that might not seem puzzle worthy and finding ways to inject some challenge, excitement, and mystery into them. This week, I set my sights on isosceles triangles. It’s common to encounter isosceles triangles as supporting players in geometric proofs,...
When it comes to simultaneous equations, I like to push the bounds of conventional pedagogical wisdom. In an earlier post, I offered a puzzle in which elementary-age students solve for four unknowns given eight equations. Now, I’d like to present a puzzle that might sound even more audacious: Solving for ten unknowns. Oh, and did I mention that the unknowns are...
Dan Anderson commented on my Pentaflake post to observe that the pentaflake can also be created by a random process, sometimes called the Chaos Game. In this game you start with an arbitrary point and dilate it toward a target point that’s randomly chosen from some set of points that you’ve established. You then dilate...
The study of multiples and factors is ripe with opportunities to engage students in intriguing mathematical puzzles. In prior posts (When Factoring Gets Personal, and Open the Safe), I’ve given some examples of what can be done. Now I’d like to introduce you to another puzzle of mine called Mystery Number that focuses on multiples. Here’s how the Web Sketchpad...
A couple of days ago I got an email from my long-time friend Geri, who was spending some quality Sketchpad time with her 12-year-old grandson Niels. Geri emailed me for advice because Neils was having some trouble figuring out how to construct a pentaflake. Neither Geri nor Niels had any idea that I’d never even...
We’ve all seen amazing examples of illusions, but did you know that there is a fertile community of researchers creating new ones? The Best Illusion of the Year contest and website provide a showcase for celebrating illusions. This year’s winner for best illusion was created by Christopher D. Blair, Gideon P. Caplovitz, and Ryan E.B. Mruczek from the University of Nevada,...
From 2009 to 2013, I had the pleasure of working with the elementary teachers at the School for Discovery and Exploration in Brooklyn, New York as they field tested Sketchpad materials for the Dynamic Number project. Although the project has ended, I still keep in touch with a few of the teachers, and they continue to...