Category: Mathematics
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Another Rant on “Mathematical” Puzzles A few months ago, I complained about those internet memes which claim to be mathematics. My complaint about them is about the presentation, not the underlying problems: “Only 1% of people get this right!” The questions are framed to encourage people to feel stupid about math. Please don’t feel stupid about math.…
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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…
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One word is being scapegoated for student misunderstanding. [mathjax]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…
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A tale of carts and horses. [mathjax]My child is in fifth grade. Last Friday, we received the second volume of his mathematics workbook. It starts with multiplication of fractions; the next chapter is division of fractions. Addition (and subtraction) of fractions with different denominators is in the first volume. I believe this order persists partly…
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Mathematics isn’t about finding answers. It’s about asking questions. As a mathematician, here’s a question I usually find boring: What’s the answer? Consider this manifestation of a sort of meme that wanders the internet: [mathjax]The most likely intended answer is 73, just to get that out of the way. I’ll call this “Witch”. This is…
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[mathjax]This is one of the most common mistakes students make when adding fractions: \[\frac34 + \frac67=\frac9{11}\] For a long time, I thought that students did this solely based on confusion with multiplication. While we teach addition of integers before multiplication, addition of fractions is a more complicated process. To multiply fractions, multiply the numerators and…
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[mathjax]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…
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[mathjax]I used to hate logarithms. They were hopelessly confusing. Sort of like this: https://www.smbc-comics.com/comic/operations This is the third year now that I’ve been teaching Algebra II. Each year, my understanding of logarithms increases, and my love increases in kind. One reason I disliked logarithms is because of the way in which we tend to compartmentalize…
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Mathematics is beautiful. Mathematical notation, meanwhile, is a horrid mess second only to English itself for its arbitrariness. For instance, basic arithmetic operators have three levels: Addition, Multiplication, and Exponentiation. Addition notation is tidy: We add forward (a + b) or backward (a – b). We call these “addition” and “subtraction” for historical reasons, but…
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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…