Category: Mathematics
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[mathjax]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…
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How notation gets in the way of understanding [mathjax]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…
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[mathjax]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…
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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…
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[mathjax]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…
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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…
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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…
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[mathjax]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…
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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…
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[mathjax]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?…