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…
Category: Mathematics
The Logarithmic Rules
In this item, I will show how the basic logarithmic rules, including the Change of Base formula, follow from this equivalency: \[\log_b m = n \Leftrightarrow b^n = m\] For the ease of reading, I’ll generally use the natural base (\(e\)) and the natural logarithm (\(\ln\)). However, everything here applies to all valid bases (\(b…
Thoughts on Memorization (Facebook repost)
There are two basic forms of “memorization”: (a) Rote, through the repetition of the specifically, often decontextualized, data and (b) Habituation, through the repetition of acts that involve the information to be “memorized”. When we think of school, we tend to think of the first kind, because that’s the more efficient in the short-term, but…
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…
Marty’s Viral Pizza Load
This is making its internet rounds again: These days, I see it in math teacher forums clucking about our colleagues. There is often a microaggressive wink-wink against elementary school teachers or misogynistic pronoun use where the brilliant student is “he” and the teacher is “she”. First, as a real thing that a real teacher wrote…
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…
In (Partial) Defense of Butterflies
Teaching students how to add fractions can be a real struggle. A big part of this is that we tend to get conceptually complicated about what fractions are. And a big part of this is because fractions can be conceptually complicated. I teach Algebra II. (For an elementary teachers reading this, though, hold on: I’m not…
What’s the Deal with Logarithms?
I’m going to talk about logs here. I have more to say later, but this is a basic intro sketch. First I’m going to talk about the stuff of elementary school. When it comes to mathematics, most people find comfort in elementary school mathematics. So, consider the humble number line: We want to move along…
A Hodgepodge of Inconsistencies
Mathematical terminology and notation through a linguistic lens Introduction The first time I attended graduate school was for Linguistics. My first year, I taught English as a Second Language. My most resistant students were Mathematics majors, because many of them held the opinion that mathematics is a universal language. Why bother getting fluent in English?…
Keeping, Changing, Flipping
Consider the following task: \[1.\quad \text{Simplify the expression }\frac{3}{4}\div\frac{2}{5}.\] It is very common for students to struggle with this sort of task. A common teaching approach is “Keep Change Flip,” but too often that’s presented as a mechanical trick without any deeper understanding of why it works. In proper mathematical language, “Keep Change Flip” translates…