The other day I saw this TikTok by the inimitable Howie Hua: This is a common topic: What is the long division algorithm about anyway? I’ve likely even seen Hua talking about it in the past, but this time, something clicked in my mind. I don’t personally recall struggling with long division. I suppose there…
Category: Mathematics
Math: Estimating Roots
A few years ago, I developed a quick algorithm for approximating square roots. I’ve since come across a more effective, albeit slightly more complicated, algorithm. Both of these are meant to create quick approximations in an era where calculators can come up with more precise values far quicker than humans. My first method If you’re…
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) +…
Set Theory in Elementary School
Note: This is not a polished edit, just some somewhat disorganized thoughts. Hopefully, I’ll write something more organized later. For a long time, I thought I understood set theory. Then, a few years ago, I realized I had somehow messed up what is a fairly rudimentary concept: That sets, by standard definition, do not have…
Coloring Vertices
The other day, I came across this problem on Twitter: How many distinct ways are there to color the vertices of a cube, such that exactly four are one color and the other four are a second color? I played around a bit first. My first task was to list all the possible combinations, and…
On Cognitive Load Theory and Story Problems
I’m currently reading “Sweller’s Cognitive Load Theory in Action” by Oliver Lovell, specifically the section on reducing extraneous load during education (ca. p. 32; I’ve got the e-book). This leads me think about story problems, such as those on the SAT, which often contain information that’s irrelevant to the problem. For example: Carrie invites some…
What is a fraction?
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…
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…