The Math Education Blog
Today there is no lack of outrage directed at the high-stakes standardized testing that has become so prevalent in the U.S. educational system. A recent opinion piece in The New York Times examines the backlash against the Common Core and lays the blame not on the standards themselves, but rather on the rise in testing that has accompanied their...
In my previous post, I shared Dan Meyer’s analysis of what’s wrong with computer-based mathematics assessments. Dan focuses his critique on the Khan Academy’s eighth-grade online mathematics course, identifying 74% of its assessment questions as focusing on numerical answers or multiple-choice items. This is a far cry from the constructing, analyzing, and arguing tasks advocated by the Smarter...
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...
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...
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...
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 using Sketchpad to...
In this guest post, Nate Burchell describes a sketch he uses with his students to explore parametric functions. In this process students work entirely in a graphical world, manipulating graphs directly rather than by way of equations. (Nate teaches in Seoul, Korea, where I enjoyed his family’s hospitality when I attended ICME in 2012. His...
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 abstract is...