One of the challenges that I see with students learning mathematics is their confusion with what qualifies as the content of mathematics and the language of mathematics. In a famous and enduring article, “Relational Understanding and Instrumental Understanding”, Richard Skemp pointed out that teaching concepts instead of procedures will be difficult if students think that…
Author: Clio
Enough is Enough: The Hoarding of the Ultrarich
Many years ago, I saw a documentary on a tribe somewhere in the third world. This tribe’s diet relied heavily on taro roots, a vegetable similar to a potato. After the harvest, the tribal elder, an old man, would take the largest roots and put them in storage. Supposedly, this was in case there was…
Fat Jokes about People We Dislike are Still Fat Jokes
Last weekend, New Jersey Governor Chris Christie had a Marie Antoinette moment: Having closed the state’s beaches to the general population, he and his family relaxed on the sand. Aerial photographs caught him in the act, and illogical explanations followed. What also followed? “Beached whale” jokes. Chris Christie is undeniably obese. He has long struggled…
Three and a half methods for finding square roots
The easiest way to find a square root in this technological age is to use a calculator. That’s a fine method if what you want to do is simply calculate a square root. However, if what you want to do is understand what a square root is, here are some methods for finding the value…
Why We’re all Fragile Snowflakes
I’m a fragile snowflake. So are you. I don’t mean this ironically, and I don’t mean it in the same sense as “We’re all racist.” I mean that, in the eyes of someone else, someone who doesn’t know you, you’re too sensitive about something. So am I. Most of my friends and social media contacts…
De Gua and the Pythagoreans
The Pythagorean Theorem states that, given a right triangle, the areas of squares placed along the two legs will have the same area as a square placed on the hypotenuse. This is normally written as \(a^2 + b^2 = c^2\), where \(a\) and \(b\) are the leg lengths and \(c\) is the hypotenuse length. De…
Write This Down Now
I am able to communicate with you right now because of one of the oldest and greatest inventions of humanity: Writing. Prior to writing, humans had to keep their histories alive through spoken stories, which are malleable and unreliable. We had to learn how to do things directly, as apprentices to mentors; if there was…
Pascal, Pacioli, Probability, and Problem-Based Learning
I’m currently reading Howard Eves’s Great Moments in Mathematics After 1650 (1983, Mathematical Association of America), a chronological collection of lectures. The first lecture in this volume (the second of two) is on the development of probability as a formal field of mathematics as it was driven by Pascal and Fermat, with regards to a specific problem…
The Fourth Dimension (Thoughts)
I’ve had two recent thoughts about the fourth dimension. The first relates to Euler’s Formula, which says that the difference between the sum of the vertices and faces of a convex polyhedron and its edges is always 2 (that is, \(v + f – e = 2\)). The Number Devil presents this slightly differently: The sum…
Moment
I used to go by a different name. It’s been so long that I’d nearly forgotten. The other day, I was out somewhere when someone said the name. I turned because it seemed like someone was speaking to me, and they were. I didn’t recognize it at first. It took my mind a few…