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!

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!

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

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.

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.

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.)

‘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

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?

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.

@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.

… 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?