The Wolverine Wrangler

Karim: [offering a lesson plan attempting to defang the wolverine, using iPad pricing as a hook for linear equations]

Zeno: What reason is there to think there would be a linear relationship between the storage capacity and the price of an iPad?

Karim: Great question! If you wanted, that could actually be the hook, no? [offering other remarks on expanding mathematical access]

Zeno: [offering nothing; no response]

Zeno isn’t wrong to ask for proof. If you get students in the habit of extrapolating any two points into a linear model, you’re setting them up for a whole lot of pain later. On the other hand, if you insist that middle schoolers justify every linear extrapolation or (for another example) define every polygon as “a simple closed plane curve composed of finitely many straight line segments,” you’re positing mathematics as a 400-pound wolverine with fur like razor wire and teeth like broken glass, which makes you kind of a monster.

Most educators, I think, understand instinctively the tension between access and correctness, the difficulty of extending one while insisting on the other.

There is a demographic, though, that feels little tension along that line. Call them “wolverine wranglers.” These people handle dangerous animals like you and I can’t believe. They’re gifted and there aren’t a lot of them. Their most striking feature, though, is their conviction that wolverines are dangerous and you are not taking that seriously enough. Work up the nerve to approach a wolverine and the nearest wrangler will remind you of all the ways that could go wrong.

I find their motivations mystifying, though, certainly, if I were much good at wrangling wolverines, I would find it tempting to remind people by means both subtle and obvious that they needed my wrangling skills.

About 
I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. More here.

28 Comments

  1. Well, yes, in the case of teaching middle schoolers, the wolverine’s not dangerous. No one gets hurt when your 8th grader’s linear extrapolation of iPad prices is wrong.

    But in many other cases, like safety engineering, or the utterly-wrong risk analyses underlying the mortgage crisis etc., the wolverine *is* dangerous and sometimes people *don’t* take it seriously enough.
    Even pretty-good wolverine-petters thought they were playing with a tame wolverine (they thought their simplistic mathematical *model* of the risk involved was reality), but the risk *in reality* turned into a ginormous Godzilla monster that fetched a few sticks and then suddenly ate them for lunch…?

    Argh, I’m screwing up the metaphor, but basically: the intensely-serious wolverine wranglers you describe may be out of place in the K-12 classroom, but I do wish we had more of them in the *right* places. I don’t think you have to assume they’re intentionally self-advertising — they may just have trouble fighting back instincts (“careful, mate! those data aren’t normally distributed! 2 standard deviations aren’t enough! *whipcrack*”) that are overkill here but do serve a real need elsewhere.

    Anyhow — love your blog and this metaphor especially!

  2. Rob McTaggart

    July 8, 2010 - 7:09 pm -

    As an Australian, my only experience with a wolverine was Hugh Jackman, so I googled it…

    “The name wolverine means glutton. The wolverine doesn’t eat more than he needs. If he kills a caribou or bear he will spray it with musk and bury it. Later he will come back and finish it.”
    – from http://library.thinkquest.org/3500/Wolverine.html

    There you go. After they eat you, they spray you with musk. I don’t know if that helps you with your metaphor…?

    Seriously, I’ve caught myself being a wolverine wrangler on a few occasions, and I doubt you can say you have never thought “C’mon kid, you’re not taking this seriously.”, or just wanted to pull the pen out of their hand and do it for them, showing them every little thing that effects the problem. That’s when you need to pull back and think, ‘oh, yeah – I’m trying to make this _more_ approachable for the student.’

    I’ve found many of your approaches help to do that wonderfully. Keep on keepin’ on, Dan!

  3. Jerzy: But in many other cases, like safety engineering, or the utterly-wrong risk analyses underlying the mortgage crisis etc., the wolverine *is* dangerous and sometimes people *don’t* take it seriously enough.

    Certainly. I hope my admiration for the wranglers is clear, however I qualify that admiration when it comes to teaching students not to fear / hate math. I prefer that kind of relentless attention to detail just about everywhere else.

    Rob McTaggart: Seriously, I’ve caught myself being a wolverine wrangler on a few occasions, and I doubt you can say you have never thought “C’mon kid, you’re not taking this seriously.”

    Definitely, but I don’t think I’ve defined wrangling carefully. My worst moments wrangling are when I draw authority over my students from my mathematical skill. That’s when I find myself smugly satisfied that I have one over on them and I start taking away points for not labeling the axes “x” and “y” or for not writing “x =” next to an answer.

  4. Rob McTaggart

    July 8, 2010 - 8:24 pm -

    “…when I draw authority over my students from my mathematical skill”.
    That really applies to all areas of teaching – great point.

  5. But in many other cases, like safety engineering, or the utterly-wrong risk analyses underlying the mortgage crisis etc., the wolverine *is* dangerous and sometimes people *don’t* take it seriously enough.

    Even there, you’re mistaking the wolverine’s allegiance. It’s the rest of the world that’s dangerous – the wolverine is waiting to help you tame it.

    Dangerous is when people face engineering tasks, risk analysis, or economics and they’re afraid to let the wolverine get too involved.

  6. Not only was Zeno “not wrong” to ask for proof, he was absolutely right! However, to be fair, the next line in my response was:

    “Actually, we’re not assuming that there is a linear relationship. We’re just finding out.”

    The premise of the question was not that iPad pricing is linear. In fact, students end up determining that it’s not, which can prompt a very good conversation about why it wouldn’t make sense for Apple to charge a constant amount per additional gig. In other words, by starting with a simple question–again, not an assumption: just a question–we end up getting to the heart of what it means for a relationship to be linear (or not).

    I agree with you, Dan, that there’s often a tension between “access and correctness.” In this case, though, I just don’t think it’s apropos.

  7. @Gilbert: Is “Wolverine Wrangler” a euphemism for a PhD mathematician? =p

    Nah, definitely not. Take a guy like Devlin, who could math me under the table with a few pen strokes, who’s also speaking on NPR or writing columns aimed at de-mystifying mathematics. Or Strogatz in the New York Times, who did the same thing for a stretch.

    The point is that wranglers have no such concern for extending access and many clearly enjoy closing it off.

  8. Zoology aside, I sense that the underlying conversation here isn’t as much about pedagogy as it is the question: What do we intend by teaching math?  For us at Mathalicious, the emphasis is very much on the conversation. Of course, most teachers would probably say the same thing, so really the question is: About what?

    In my experience, at least, this often breaks down pretty well along content lines.  I’m interested to hear what high school teachers have to say, but it seems to me that math conversations in high school tend to be about the mathematics itself: the nature of functions, Leibniz vs. Newton, etc. At the middle school level, though–that transition between concrete and abstract–math conversations are often about something else. They don’t necessarily focus inwardly on the content, but use the content to explore something outside: sports, technology, health and wellness, etc.

    Not to put words in his mouth, but if Dan were to take the place of Zeno, the conversation might have gone something like:

    Me: Let’s calculate the line between (16GB, $499) and (32GB, $599) and see whether (64GB, $699) is on it. 

    Dan: Be careful. In doing that, you risk creating a false construct. 

    Already, you can see that we’re headed in very different directions, both of which are equally valid. He intends a conversation about the math per se, whereas we’re aiming for one primarily about Apple’s marketing, which is then supported by a dialogue about why linear pricing wouldn’t make sense.

    Again, the emphasis is on the “other.” Math is simply the vehicle. (Similarly, the lesson XBOX Xponential is partly about exponential growth, but really about what Moore’s Law implies for video games and, more fundamentally, what it means to be human).

    That said, we all have our biases. It’s what prompts one teacher to choose AP Calc, and another 7th grade SpED. And I’ll be honest; insofar as Dan’s post was about the dangers of extrapolating from two points, I found it amusing that the post itself started with two lines from a longer conversation. Still, that’s just my own bias showing, right?, and missed the point (pardon the pun).  It’s about starting a conversation and, so long as you know where you’re trying to go with it, perhaps everything is fair game.

  9. Some people approve of dilettantism, citizen science, amateurs and hobbyists and others do not.

    You can approach this as a power struggle issue, with those demanding too much rigor seen as oppressors. I always want to ask, “Why so srs?” when this line of thought comes up, though.

    Doing algebra with four year olds, I am always wrangling with this questions. Are we really doing algebra? Is it good enough algebra?

    Most of the time, the questions are purely academic, in the bad sense of the term: “priding themselves on having nothing to do with the reality.”

  10. I understand the discussion point that it’s making math really scary when you ask kids to justify every assumption before moving forward. I get it, and I do agree, to some degree. Although I guess for me, how much kids would need to justify themselves would depend on their age, the rigor of the course, and the particular topic at hand. I may not ask the whole class to justify why it’s linear if that’s not the intention of this lesson, but I certainly might pose that question to a few kids while circulating, to gauge how much they really understand about the problem.

    The discussion of wranglin’ the wrangler of the wolverine analogy actually, for me, obfuscates the question at hand, which is a pretty straight-forward one (albeit difficult to answer): how to make math accessible to all children?

    Dan, I see parallels between your WCYDWT method and the example you gave in your previous wolverine post, in that they both offer easy entries (“bait”) into a more difficult topic. I think those are good tricks for us to all keep in mind, but I think we’re also constantly discovering (…or at least I hope so…) other ways of making math accessible to all kids: a real-life application hook, some hands-on exploration, etc.

    And, I would hope that we are offering our kids access, in order to build up to (some level of) eventual correctness AND deeper understanding. (Not necessarily a tradeoff.)

  11. Oh, hehe, oops. Sorry. I’m a very-habitual skimmer and don’t always follow all links when reading a blog entry (that would take too freakin’ long on a slow internet connection from a 3G USB card in El Salvador) — had no idea you had linked to the same comic strip.

    Pretty funny. :)

  12. ‘1 Jerzy Well, yes, in the case of teaching middle schoolers, the wolverine’s not dangerous. No one gets hurt when your 8th grader’s linear extrapolation of iPad prices is wrong.”

    with 8th graders being our future mathematicians I would say yes we all get hurt when they learn to fear math.

    http://www.google.com/url?sa=t&source=web&cd=1&ved=0CBgQFjAA&url=http%3A%2F%2Fwww.maa.org%2Fdevlin%2FLockhartsLament.pdf&ei=p3E4TOuSF8WblgeK2MzSBw&usg=AFQjCNG45CnNCuc0qLaC9nI8zPhXObN9JQ&sig2=PFO7VGr44w-kpcJ7iiQLkA

  13. @Mimi, I was just glad to know someone would’ve had my back had I overlooked the xkcd reference.

    @Karim: And I’ll be honest; insofar as Dan’s post was about the dangers of extrapolating from two points, I found it amusing that the post itself started with two lines from a longer conversation.

    My head blew up right there.

  14. For what it’s worth, I’ve been goofing around with Sketchpad for the past few hours, trying to figure out how to calculate the slope between one point. I think the program’s busted. GSP6?

  15. Zeno doesn’t think math is an angry wolverine. He thinks it’s a fragile racehorse (those things get sick and die if you look at them wrong).

    Your students are afraid the math will hurt them. Zeno there seems afraid that if he’s not careful, he’s going to hurt the math.

    Admittedly, we don’t want to burden our students with all the intricacies of the care and feeding of mathematics, at least not until they’re much better at it, but there’s something to be said for taking care of it behind the scenes.

    Most important would be letting the students know that, unlike an animal, no great harm happens to anyone when math goes wrong—you just try again.

  16. I think I’m a wolverine wrangler myself, though in a different discipline (foreign language). But there, the point is that often simplifications that have been used to increase accessibility really make future learning harder to do right.

    I think some of us wranglers see the damage done by false models students build over years as a result of other teachers attempting to increase accessibility, and we wonder if they’re really more accessible in the first place.

    After all, the question “why do you think there should be a linear relationship” is not actually that intimidating, and the answers to it largely lie outside of mathematics. A reasonable student could make a guess based on past experience buying other products, or could try to make a guess about the engineering process etc. A class could quickly google the prices of various drives. None of this strikes me as overly harmful.

  17. @22,

    But he was right about the multiplication and addition thing. Teaching multiplication as repeated addition is just a bad explanation, which leaves you cold when it comes to, say, 2.7*3.6, or 1/2 * 5/3, or pi * e.

    While we’re at it, functions are not machines.

  18. @Dan: Cool cool — I was just surprised in your post by “I find their motivations mystifying,” but now I see you meant it specifically about teaching kids, not about “relentless attention to detail just about everywhere else.”

    @joshg: “It’s the rest of the world that’s dangerous – the wolverine is waiting to help you tame it.” Right. So maybe you *should* pause and play with the wolverine (do advanced statistical analysis); but instead you just pat it on the head (make inappropriate simplifying assumptions about the data) and keep going, and you miss the fact that it’s trying to let you know you’re about to fall off a cliff? It’s hard to get an extended metaphor just right :-P

    @Brendan: “we all get hurt when they learn to fear math” — I definitely agree!

  19. … was just reading up on something about Hardy and Ramanujan and was reminded that in the 19th century, the top category of scorers in Cambridge University’s legendary Mathematical Tripos, one of the most committed programs ever of deciding who’s mathematically best and who’s worst, were called “the wranglers.”

    That reference can’t have been on purpose, can it?