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 … Continue Reading ››

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 … Continue Reading ››

When I was child, I loved to solve the brainteasers in logic puzzle magazines. You probably know the type: Ruth, Phyllis, and Joan each bought a different kind of fruit (orange, apple, pear) and a different vegetable (spinach, kale, carrots) at the supermarket. No one bought both an orange and carrots. Ruth didn't buy an apple or kale. … Continue Reading ››

There are certain topics in mathematics education not appropriate for polite discussion. Number bases other than 10 fit this category well, perhaps because of their association with the maligned "new math" of the 1960s. That's a shame because there is a lot to learn from them, especially when presented as interactive puzzles. Below (and here) are … Continue Reading ››

For a while now, I’ve been intrigued by the ways in which the study of geometric transformations can provide students with a very effective introduction to function concepts. Daniel and I have written a couple of articles about this topic, and we created a number of activities to take advantage of what can arguably be … Continue Reading ››

In my prior post, I presented an interactive Web Sketchpad odometer that is a great tool for introducing young learners to place value. Well, technology moves fast these days, and the latest odometers are more powerful than ever. While our prior odometer featured '+' buttons above each digit, our newest innovation in number-tracking technology features … Continue Reading ››

With the World Cup in our hemisphere, and the US squad having started out with a win over Ghana, my thoughts turned to the mathematics of soccer. My friend Henri Picciotto has a nice page about the shooting angle, the angle within which a shot is on goal, so I thought of … Continue Reading ››

Every week, The New York Times challenges its readers to solve a mathematical puzzle in its online Numberplay column. This week's puzzle was proposed by none other than Dan Bennett, a former editor and author at Key Curriculum Press, and his colleague, Avery Pickford. Here is their … Continue Reading ››

In Where Mathematics Comes From, cognitive scientists George Lakoff and Rafael Nuñez assert that our understanding of abstract mathematical concepts relies upon our sensory-motor experiences:

“For the most part, human beings conceptualize abstract concepts in concrete terms, using ideas and modes of reasoning grounded in the sensory-motor system. The mechanism by which the … Continue Reading ››