Okay, first of all, that puppy that’s gonna bite it… he looks exactly like the dog I fostered for 18 months who I’m turning in for advanced service dog training this week. So rest assured I will never distribute an exponent over addition again!

How do you prevent/stop crazy algebra mistakes is a deep question! One way I’ve thought about it that feels productive is to analyze the mistake on three levels: Procedural, Methodological, and Conceptual (got to be part of a project that analyzed hundreds of students’ online work on math homework and that’s where the framework evolved from).

Methodologically, I think a couple of problem-solving strategies are at play. One is checking your work. It’s one thing to hope that (x^2 + 2x + 1)/(x^2 + 3) = (2x + 1)/3 but it’s another thing to leave that as your final answer. Specifically, it’s a kind of “check your work” method that has to do with recognizing that your answer might be fishy, and knowing how to check that two statements are equivalent. Students need to use check-work methods like “graph the two functions” or “plug in some values,” not just re-doing the same faulty algebra again. In terms of knowing that your answer might be fishy, that’s a problem-solving skill all by itself.

Steve (the director of the Math Forum) likes to teach the “Solve a Simpler Problem” strategy with a 3-column chart: What Makes This Problem Hard? How Could I Make It Simpler? and Is That Valid? (another way I label that column is Am I Confident/Concerned?). So for the (x^2 + 2x + 1)/(x^2 + 3), the problem is hard because there are multiple terms in the numerator and denominator. I could simplify it by “canceling” terms that appear in both places. I’m concerned that’s not valid because I don’t think you can subtract from the numerator and denominator, I think you have to divide. Getting used to recognizing your simplifying assumptions and thinking about their validity explicitly is really useful.

Another trick a teacher I know in Delaware came up with is “starfishies” — he had a student up at the front of the room explaining his work, and the kid had done one of those famous mistakes. And the rest of the class couldn’t figure out where he’d gone wrong. Finally, after a lot of unproductive conversation, the kid said, “well I knew I was doing something weird here, but I didn’t think I could do anything else.” And Jesse goes, “when you’re thinking there’s something weird in your work, you should mark it so we can talk about it. Like put a star when something is fishy. A star-fishy!”

Procedurally, these mistakes look like the kind Michael Pershan was tweeting about recently… students have a known procedure and so they wishfully think a harder problem can be turned into a simpler one that they can apply their procedure too. Like, I know how to distribute, so I’ll distribute the exponent! Or I know how to use the commutative property to combine like terms, so I’ll combine this x^2 with the x in sin(x). The problem is probably in knowing when to apply a procedure, and when not too… The other procedural aspect is that learning to turn off those wishful thoughts needs to become routine — there needs to be a kind of fluency where the idea of distributing the exponent doesn’t even pop into your brain. So if students can explain why the thing they want to do doesn’t work, and they can check their work and spot the errors they make when they go into wishful thinking mode, then they definitely need to do some practice at not screwing up (and not screwing the pooch, bunny, or kitty).

Conceptually (I saved the category that’s hardest for me for last) I can see a couple possibilities for some of the famous mistakes. The first (puppy death) is the hardest for me to diagnose conceptually. Is the problem with kids’ understanding of exponents? What is exponentiation? Or is it with the idea that a multiple-term expression can represent a single quantity? (I know that’s genuinely hard for a lot of kids and part of what makes factoring and the distributive property so tricky). Is the antidote noticing the hierarchy of operations — multiplication distributes over division, while exponentiation distributes over multiplication? Is it something to do with reinforcing the idea that (x^2 + 3)^2 = (x^2 + 3)(x^2 + 3) while avoiding the cliche “exponentiation is repeated addition”? Is there a context, like area and algebra tiles, that would somehow reinforce the idea that (x^2 + 3) is a single quantity that’s being squared, while also acknowledging that each term in x^2 + 3 gets multiplied by each other term when we distribute? What breaks mathematically when we think (x^2 + 3)^2 = x^4 + 9? Is it just that we want exponentiation to be equivalent to repeated addition in the situations where that makes sense, and so we define (x^2 + 3)^2 = (x^2 + 3)(x^2 + 3)?

The bunny death poster seems a bit clearer to me. I’ve known lots of students whose concepts of functions and trigonometry were shaky and so they didn’t treat sine, cosine, and tangent as functions. We don’t help them by our efficient convention of leaving out the parentheses and writing sin x instead of sin(x). Sometimes I feel like my coworker who thinks teachers should only allow students to use approved shortcuts after they can fully justify them and execute them perfectly. So you only graduate from writing sin(x) as sin x when you can explain what sin(x) means and show that you never would let the bunny die. But I know lots of students who don’t realize that sine is a function that pairs inputs with outputs, and that sin(x) represents a quantity, specifically the ratio of the hypotenuse to the opposite side of a right triangle with an angle x. And that x is the input quantity that tells us which triangle we use, but isn’t otherwise part of the expression. So in that case, probing students’ understanding of function notation and the meaning of the trig functions, and requiring those pesky parentheses might be the conceptual antidote.

For the kitten death poster, I think the concept (or non-concept) of canceling might be at play. When solving an equation, it’s cool to subtract the same thing from both sides and to divide both sides by the same thing. But when simplifying a division problem, subtracting the same thing from the numerator and denominator does not result in an equivalent division problem. Recognizing that this is not an equation, there is no “both sides” here, and that it is in fact a division problem, is key. As is realizing that it doesn’t preserve equivalence to subtract from both sides, and figuring out what kinds of moves do preserve equivalence. It’s not uncommon for students to make the same mistakes with fractions and ratios, simplifying 2/5 to 1/4 or saying that 3/8 = 11/16 because they added 8 to both sides.

Maybe I should have just made my own blog post out of this, but I really enjoyed the chance to think about diagnosing and addressing 3 famous mistakes, and doing so in the context of the imminent demise of innocent baby animals. Thanks for posting, Bowman!

I just peed my pants a little.