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…
Author: Clio
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…
Being a Son, Being a Father
As I write this, it’s Father’s Day. I wish a Happy Father’s Day to every decent person who identifies as a father figure in somebody’s life, be it as a parent, a teacher, a mentor, or a caregiver. This article is dedicated to you. This is the ninth Father’s Day I have been a fatherless…
Geometry for multiplication, division, and roots
Contemporary plane geometry of the sort taught in the standard American high school is most heavily informed by two books and a third mathematician. The first of these is Euclid’s Elements, which is so conceptually tied to planar geometry that it is typically referred to as Euclidean geometry. However, it is only part of the…
Factoring and long division
This morning, I’ve been watching YouTube videos. I started with Tarleen Kaur’s video on Middle Term Splitting. What I find interesting about Kaur’s Chapter to Chapter videos is that, because she’s a student in India, her methods are often different from those I’m familiar with. That’s the case in this video as well. I haven’t…
Positive numbers and absolute value
They say that when you’re a hammer, everything looks like a nail. Since I’m currently thinking about conceptual vs procedural teaching, I’m noticing examples. Here’s a good definition of absolute value: “the magnitude of a real number without regard to its sign; the actual magnitude of a numerical value or measurement, irrespective of its relation…
The smallest angle
I have been thinking about procedural vs conceptual thinking, which Skemp’s seminal article refers to as relational vs instructional. One of the questions on this year’s geometry final asks: Given a triangle ABC with sides AB = 5, BC = 6, and AC = 7, what is the smallest angle? (Edit for clarity: The question is simply…
Bill Maher: Typical White Male Liberal
On Friday’s episode of Real Time, Bill Maher provided a clinic on some of the worst white male verbal behaviors, and in how not to take full responsibility. In two segments, he illustrated deflection, microaggression, defensiveness, gaslighting, and tone-deafness. This was in the form of an apology. On the previous episode, Maher referred to himself…