The Dead Puppy Theorem and Its Corollaries

To preface, I normally celebrate mistakes in my classroom as a part of the learning process. But there are some things that really speed the progress of my receding hairline, a large of percentage of which involve bad algebra. I saw this from @lustomatical and thought YES. This is what I need to get my kids to stop distributing powers over terms that are added, and “canceling” things willy nilly, and not respecting the trig functions as operations. Let’s concentrate on the Calculus! So I made these posters for my classroom:

Enjoy. I know my students will, and it will actually give us a funny and memorable way to talk about and avoid these common algebra mistakes.

The other thing that I showed them today to get them to stop just playing around with letters while doing Algebra is the following, which I believe I picked up at a summer workshop:

They literally laughed out loud at this. I said (in a funny, not mean and not sarcastic way), “You think that’s funny?!?!? This is the kind of stuff you guys do on quizzes. When I am correcting your work I sit and laugh and laugh and laugh at the crazy things that you do! No more crazy algebra!”

How do we stop/prevent crazy algebra mistakes besides carefully and repeatedly addressing them when they happen? Any ideas?

Posted on November 6, 2012, in Calculus, Math Ed, Teaching and tagged , . Bookmark the permalink. 16 Comments.

  1. 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 love the “Solve a Simpler Problem” strategy. I’m definitely using that!

      As far as “puppy death” goes, that conceptual confusion is far and away the clearest to me, since it’s the one I see and work with constantly. Students at the younger levels need a lot more experience — and a lot more DEEP experience with the distributive property to grasp it conceptually. I find that my students who needed — and took — a lot of time to really “wallow” in distributive property opportunities have developed a really sturdy and rich understanding of distribution in its many variations and guises, and they are not so easily confused or set astray. Meanwhile, I find that my more advanced students who never really bothered to wallow in the distributive property (and who were always quick to jump onto the next bright shiny object) tend to have the most wobbly relationships with distribution and tend to make these mistakes often because they never learned to see anything deep or valuable about the distributive property.

      I feel a little bit better to think that maybe I have done my small part to help avoid “puppy death” in my quadrant of the galaxy.

      – Elizabeth (aka @cheesemonkeysf)

  2. To go along with some of your points, Max, I think students don’t often realize why you CAN change something like 2/3 into 4/6 (like when you need a common denominator). So, when I teach things like converting from degrees to radians we go through the whole thing like:
    360 deg = 2pi rad, since they represent the same quantity in different ways (in the same way that some people call me Dave and some David, but I’m still me, so they are equal)
    Then you do the algebra of dividing both sides by 360 deg to get:
    1 = 2pi rad/ 360 deg. Then since we can always multiply by 1 we can always multiply by that fraction. Sometimes it’s useful like if we have 12 deg and want it in radians, sometimes it’s weird like if we have pi radians (you CAN multiply by the fraction, but your new units are rad^2/deg which is weird and not so helpful).

    • David – The idea of substitution / the substitution property of equality is another huge conceptual building block that gets short shrift in the younger grades. This comes at the expense of conceptual learning in the later courses like Algebra 2/Precalculus. I’ve come to realize this from watching how incredibly concrete middle schoolers are as thinkers. And this is as true on the mathematical/algebraic side of things as it is on the language arts side. The structural functions of metaphor and metonymy are totally baffling to middle school students, even though they use both (and substitution) multiple times every hour! If you don’t believe me, read their texts. :)

      So many of these modes of abstraction get taught badly or not at all, and almost never in any kind of meaningful cross-curricular way. Yet it would be so beneficial to students to be able to relate different modes and styles of abstraction that they are wrestling with so they can see the connections rather than feeling stove-piped into unrelated courses.

      – Elizabeth (aka @cheesemonkeysf)

  3. I just peed my pants a little.

  4. I think that students really see the silliness of their mistakes when they’re working on whiteboards. If everyone else is in agreement, they’re probably doing it right. So then the question is, what did I do different? They’re sorta “shamed” into fixing the mistake. Not so different from the killing puppies poster (which is awesome)!

  5. Ha, I am totally stealing the dead puppy theorem. One of those simple ways to “call out” a common mistake. I think if you call out those mistakes in such a ridiculous way (like this), not only does it reinforce that it is a mistake, but it also takes away some of the stigma for making the mistake.

  6. This is awesome! I just showed my students your posters today (I made sure to do the “in a funny voice”, not in a mean or sarcastic voice), but I didn’t get as much laughter as I thought I would. I was very, very careful to explain to them that I didn’t condone the death of puppies, in fact, that I wanted to *avoid* the death of puppies, but I think they think I’m really, really weird. (Which they already knew, so I’m not sure why they acted surprised.)

  7. I love this! My AP Calculus students think it is hilarious and they call each other on it all the time! Tank you so much for sharing!

  8. I’ve been telling my students that these mistakes kill unicorns. “But unicorns don’t exist!” they say – to which I reply “That’s because of all the mistakes!”

  9. I LOVE this! I am stealing for use in my class next year. I get so tired of the SAME mistakes all the time. Thank you for sharing!

  10. I call the 1st one the cardinal sin of algebra …. my students always laugh

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