Inspired by (Blatantly Stealing from?) Dan Meyer

Yesterday a colleague and I were discussing interval and set notation. Should we introduce it in the first year of our integrated math curriculum was our initial question, but we quickly moved into more general questions about these notations, what they’re good for, how much we expose students to them, and how much we expect students to use them.

And then, inspiration struck, and I could hear Dan Meyer in my head saying, “If set notation is the aspirin, what’s the headache?” (If you’ve never heard Dan ask this question before, then start here and read his collection of posts on questions like this.)

So, why do we need set notation in, say, an Algebra 1 or Algebra 2 class? My first response was, “Hey, I’ve seen it come up in some SAT II practice tests.” So we looked, and we found these:

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This problem has issues because it states that x is “equal to” the set instead of being “an element of” the set.

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This problem uses set notation in a way that’s fine but is, in the words of my colleague, just gift wrapping, i.e., completely unnecessary. I’m not sure what the purpose of including the notation in the question is. Are they trying to throw weaker students off? 
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This example uses the set notation just fine, but again, would anything be lost if we didn’t include it? So, “if you want to do well in the SAT” is probably not a sufficient headache to motivate the distribution of this aspirin.

My next idea was that set notation somehow simplifies communication of what we’re dealing with. After thinking about this for a few minutes, though, I don’t actually think that’s true. Whether a student writes “x = 3” or “{3}” makes no difference in a student’s attempt to clearly communicate her solution to an equation, and at the high school level we’re talking about, their ability to communicate this information is far more important to me than their ability to correctly apply a complex idea (sets) to a simple situation (the solution is 3).

Don’t get me wrong – there are ways in which ideas related to sets are very important to understand at this level. We want our students to know that two line segments cannot be called equal, why this is true, and why it matters. I’m just not seeing a reason to go beyond this set-based idea to actually introduce set notation and insist on its use in ways that don’t reinforce the important concepts that underlie the notation.

So what’s the right way to extend Dan Meyer’s headache-and-aspirin analogy? I think it’s to say that we have developed the bad habit of mathematically over-medicating. We all know the stories of people who demand antibiotics to fight off viral infections and what kind of problems this can lead to. I would argue that we do the same thing in high school math. We have taken problems that can be handled with a cup of soup and a good night’s sleep, and we’ve chosen to address them with IV medications (which is sometimes watered down, making it even worse). So I’ll keep looking for the headaches that need to be cured, but I’m going to start being a lot more aware of of the ones that can be taken care of with the remedies we already have available to us.