Before I go any further, I want to make it clear that the audience for this is people fairly comfortable for mathematics, particularly teachers. I’ll be heading into woods that will likely disorient many students and others who struggle with mathematics. That said, if you want to challenge your thinking, please read on. Earlier today,…
Category: Notation
The Language of Mathematics
Mathematical notation is a language. The study of mathematics is about finding patterns in our universe, but we need a method of communicating those patterns to other people. That’s where notation comes in. Like the languages that we speak, this system of notation did not spring up all at once. It was not the result…
Why 0! = 1 (Set Theory Explanation)
I’ve been seeing a few videos lately explaining why 0!=1. Over my time as a teacher, I’ve seen a lot of various explanations for this equality, and they generally fall into these categories: What I don’t recall seeing is a complete explanation from the perspective of set theory, so I’ll provide that in this article….
PEMDAS (another rant)
It’s been a while since I’ve complained about PEMDAS, so now’s a good time. We make a big deal about pointing out that multiplication is commutative, and that addition is commutative, meaning that it doesn’t matter what order we perform either one in. For instance, 4 + 3 + 5 = (4 + 3) +…
Basic Operators: Addendum
One thing I realized while writing and editing the previous article is the depth of the mismatch between notation for addition and multiplication (on the one hand) and exponentiation (on the other). Students struggle with understanding the notation \(y = b^x\), and one reason that’s been clear to me is that there’s no overt operator….
The Basic Operators
By the time most students graduate from high school in the United States, they have seen the following operators*: Addition, subtraction, negation, multiplication, division, reciprocation, exponentiation, radicals, logarithms, sine, cosine, and tangent. There is a certain symmetry in this list: They are clearly grouped in threes, and I’ve listed them so the traditionally dominant one…
What is Subtraction? (Reflective draft)
Conceptually, subtraction and addition of negatives are two very different processes. Subtraction involves an undoing of addition: It is an inverse function. Addition of negatives involves an extension of the number system to a mirror world. The Exploding Dots model, for instance, relies on this extension. That is, here are two ways we can see…
Logs, Roots, and Fractions
How notation gets in the way of understanding The other day I tweeted this: Objectively, I realize that \(\sqrt2\), \(\log6\), and \(\frac57\) are all specific numbers and that they’re the simplest way to write those specific numbers. But I struggle with convincing my brain of that. And if I struggle, I don’t at all wonder…
How Notation Obscures Patterns
This is another stab for me at what continues to prove to be a complicated topic: How our mess of mathematical notation obfuscates key patterns. This is also a rough draft of thoughts, not meant to be a polished product. In Algebra II recently, the unit is radical notation and rational exponents. That means that…
Arithmetic and Operations
This is a “what-if” document. It’s not intended as a serious suggestion for how we should write mathematical notation or for replacing current notation, but rather an exploration of how things might work if mathematical notation had developed in a different way. Do I want to develop this further? I don’t know yet. But this…