Category: Mathematics
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[mathjax]There are two basic conceptual ways that multiplication is explained: Repeated addition and the area model. Many people who are adept at multiplication (including teachers) take for granted what many students have trouble connecting; in the case of multiplication, it’s not immediately obvious why repeated addition and the area of a region ought to result…
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[mathjax]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…
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[mathjax]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…
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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) +…
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[mathjax]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…
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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…
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[mathjax]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…
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[mathjax]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…
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[mathjax]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.…
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[mathjax]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…