The other day, I saw a tweet joking that while calculus teachers insist that \(\frac{dy}{dx}\) is not a fraction, the LaTeX is \frac{dy}{dx}. That reminded me of my longtime complaint that “fraction” as a mathematical term is vaguely defined, so I asked my Twitter followers a simple question: “Which is a fraction?” I provided these…
Category: Pedagogy
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…
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…
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…
Math Education and One True Wayism
This is a common criticism of Common Core (CCSS): It offers these strange new methods that students must use. Except… only the first part of that is true. CCSS does offers some new strategies, but it doesn’t say that students have to use them. This article isn’t a defense of CCSS, by the way. It’s…
Functions and Domains: The Other Shoe
I have taught high school mathematics for nearly a decade. I have a BS in Mathematics. The Algebra II curriculum, which I largely built for my school, is based on “the story of functions”. And yet, it was only the other day that I noticed something that was woefully wrong about the way that I’ve…
The Eye of the Dragon
We obscure the beauty of mathematics when we jump straight to the end. The last few weeks, for our virtual, quarantine edition of Algebra II, we’ve been exploring the unit circle. I’ve been trying to wrap my mind around how to describe the way I see too many mathematical concepts introduced, with a standard presentation…
Should we cancel cancelling “cancel”?
One word is being scapegoated for student misunderstanding. A recent trend in mathematics education is the idea that we should never, ever say the word “cancel”. The argument for the prosecution is reasonable, and I will lay it out first. But I think the problem isn’t the word itself, but rather how we approach language…
Mathematics and Trauma
I teach two sections of Honors Algebra II and two sections of Algebra II. Today, I introduced the concept of finding roots of rational functions. I did this with a discovery-based task. The student response to the task contained a lot of learning for me, and not just in the form of what they wrote…
Adding vs. Multiplying
We tend to act as if multiplying is repeated addition. This misses a key, crucial difference between the two operations: WE CANNOT ADD UNLIKE THINGS. We can absolutely multiply unlike things. Sometimes, the result doesn’t make any real world sense, but we can do it. This is because multiplication doesn’t care about units, and addition…