## Math: The Angry, Injured Wolverine

July 3rd, 2010 by Dan Meyer

[*Virtual Conference on Soft Skills*]

Many of my students have been failing math for as long they’ve been assigned grades. It’s been necessary, then, to disabuse them *as fast as possible* of the negative, self-defeating misconceptions about math they bring to my classroom on the first day of school. Here’s one:

My students give math a wide berth. They treat math like it’s some kind of injured wolverine. Like, if you treat it just-so, if you come to class every day, keep your eyes low, and write your name neatly at the top of your homework, then it might let you sneak past it for a C-. If you attempt to engage the wolverine, though, it might *kill* you. Down that road lies pain, brother.

That’s the misapprehension. The truth is that math is a happy, gregarious, *cuddly* wolverine. It only *looks* scary, and we should heap scorn on parents and teachers who convince students otherwise. (“I tell TF that I always hated math when I was a kid,” said the parent to TF’s slack-jawed math teacher.) In point of fact, math loves to play.

Throw it a stick and it brings two back. Throw it two sticks and it brings four back. Don’t be scared to speculate what’ll happen when you throw those four sticks. Whatever happens will happen and that’ll inform your next hypothesis. In the meantime, you won’t ever find math impatient or angry. It’s always eager to play.

In practice, if we’re graphing 3x + 2y = 12, I’ll ask a student for a solution. The student will reply “I don’t know,” because, well, I mean, look at the *fangs* on that thing.

So I ask for two numbers. Any two numbers. The student replies “one and three.” And we’ll evaluate (1, 3). And it doesn’t work, but that incorrectness is properly perceived as a gift. We put an x on the board at (1, 3) and we express our gratitude to the student for helping us narrow in on a solution by identifying something that *isn’t*.

The student relaxes a bit, realizing that the wolverine isn’t going to retaliate. It’s slobbering but not because it’s mad. It just wants you to throw the stick again.

“Okay, so (1, 3) didn’t work. Find me five pairs that *do* work.”

That’s where I stop and they begin, students of every ability, playing with an animal they previously assumed hostile. And there am I, surprised and grateful that my paid job was to make that introduction.

on 03 Jul 2010 at 7:05 am1 Please Proceed to the Grand Ballroom « Point of Inflection[...] the Main Stairway, through Entrance One by Alpha Lot) to see the first keynote speaker, Dan Meyer!Right this way!Afterwards, check out the rest of the convention center! The convention center page will update [...]

on 03 Jul 2010 at 7:58 am2 Sue VanHattumAhh, lovely!

on 03 Jul 2010 at 8:41 am3 KellyDan, for real, you’re the man. This is pure poetry.

on 03 Jul 2010 at 9:05 am4 Kate NowakI’m glad you went first. I need to get back to work.

on 03 Jul 2010 at 9:21 am5 Jason Dyer(I’m going to try to comment on all the Virtual Conference posts. Here goes.)

You’ve struck upon two threads here —

One is the idea of persistence through failure. On my shortlist for “must know” ed concepts for new teachers is Dweck’s research on fixed versus growth mindsets. People with fixed mindsets assume that one’s skill is at a set ability and no amount of extra work will help; those with a growth mindset assume the problem will fall given enough effort. Surveys of K-2 students indicate they all start with a growth mindset, and only obtain a fixed mindset through culture and peer pressure; i.e. observing the “smart” student who seems to blaze effortlessly through math and inferring one’s own skills will always be inadequate.

To address this in class sometimes I will do low-stakes puzzles that students naturally enjoy enough they are willing to work a little harder than normal, but that aren’t easy enough to fall on the first five tries. Eventually students start to struggle and assume the task is impossible, so I have to throw it back at them and encourage them to push further. When the puzzle is finally broken I open a meta-conversation about the process and why it is the puzzle they solved they presumed was impossible just five minutes before.

The second thread is that in mathematics we are interested just as much in things that are false as things that are true. Witness the excitement over the counterexample to the Hirsch Conjecture which had stood for 53 years. The negative space as an answer can sometimes not only reveal things about the positive space, but provide the answer outright or suggest new conjectures.

Students have a hard time understanding this; I recently had them working on a counterexample for a research problem and it took multiple explanations for them to realize they were looking for an example that BROKE the problem, not one that made it work.

Also along the road there are formulas that can *only* be resolved by approximation methods, where the “wrong” guesses are slowly refined closer and closer to what resembles the “right” one — just like the 3x + 2y = 12 example. This is one of those understandings further along the trail of mathematics that needs to be brought down to the lower levels; I’m guessing some teachers don’t realize there are equations that cannot be solved explicitly, only approximated.

on 03 Jul 2010 at 9:54 am6 Learning Through TeachingOne thing (of many) I like about this approach is how it rewards curiosity for its own sake. We (parents…adults…teachers) are guilty far too often of passing along baggage and stripping the joy right out of trial-and-error-for-the-sheer-fun-of-it-cavorting-with-curiosity learning.

on 03 Jul 2010 at 10:31 am7 Dan GoldnerLoved this. (Did you see Big Fish?) The virtuosity you’re displaying here is (a) recognizing when kids are seeing fangs, and (b) knowing what stick to throw — what question to ask next — that reveals the beast’s desire to play instead of just making it angrier and scarier. If Jason (and Dweck) are right then that means we can all get better at that.

on 03 Jul 2010 at 12:41 pm8 Karim @ MathaliciousIt might be worth consistently stressing that math is a tool…that we invented…at some point in history…to do something. Getting scared of math is akin to getting scared of a hammer.

At certain times of the year, when things start to get too abstract, or kids start to get down, it could be nice to pull back and revisit the history of mathematics. Some shepherd in Africa who wants to know whether all of his sheep return, but hasn’t invented number yet, so makes little notches in a stick, covers them up “one”-by-“one” when they return. If they’re all covered, all good. If not, sheep are missing. How many? Dunno. Haven’t invented number yet.

Fast forward:

The Pythagoreans looking at plants, etc., and wondering about how the planet was ordered. The Greeks disallowing zero and irrational numbers, because they contradicted their ideas of God. At this point, math isn’t too different from religion or philosophy: the origins of our species’ trying to make *sense* (very literally) of the world.

Fast forward:

The Arabs, the Europeans like Newton and Leibniz. Algebra, calculus. We’re breathing a different kind of life into math now. It’s becoming its own thing, which is good–it’s what allows us to build airplanes, after all–but also a bit risky, insofar as we’re becoming a bit like Dr. Frankenstein.

Fast forward:

Kids in school, totally sweating math. Thinking it’s a wolverine, in Dan’s words. Thinking it’s something that’s been alive forever, a beast that humans have to tame, instead of a little mirror into the evolution of our understanding of “real” and “true” and “reality.”

And so you’ve got a kid who’s freaking out over “solve 4x + 9 = 3x – 17,” and you say:

All of that? We invented it. For a reason. What do you think that reason was? What question were we trying to answer?

It’s just grammar. No big deal, right? And kind of a big deal, too.

Anyway, here’s a lesson about the history of number, in case anyone’s interested:

http://www.mathalicious.com/?p=2651

on 03 Jul 2010 at 1:13 pm9 Elaine C.This is like my big dog story I’ve been telling kids for about 3 yrs now…

I ask my kids if they know any big dogs that will bark when they walk by – as if the dogs want to rip the kids’ faces off. Then I talk about how if they come to the house throough the front door, that dog turns into a teddy bear who wants belly rubs and tennis balls thrown. I end up by explaining that math is that dog – if you try to sneak around, and not learn it, its freaking scary and will tear you to shreds. However, if you take the time to get to know it – walk in the front door, and play with it, it’ll be your friend for life.

Math’s not always easy or fun – just like having a dog isn’t. (You have to pick up after the dog, teach it not to pull on walks, etc.) If you put in the time though, its worth it – just like the fun of a dog is worth the work.

on 03 Jul 2010 at 5:42 pm10 JoelA slobbering wolverine — love it!

Wish I could’ve used that when I attempted to teach 8th graders Algebra last week… But even though I was out of my element it was hard to get the students to realize they knew more than I did and there was no shame in TRYING. It’s like they don’t want to just toss around some numbers, run a few plays to see what works — if they can’t score a touchdown every play (without breaking a sweat) they’re not interested…

(sorry for mixing metaphors)

on 03 Jul 2010 at 8:05 pm11 Karim @ MathaliciousElaine-

Your dog anecdote is a really lovely metaphor for math. I’m curious: when you teach, what’s the “front door?” It seems like many students view learning math as the dog itself, not the means of taming it.

on 04 Jul 2010 at 4:35 am12 RileyI also have my students find points that aren’t on the graph, but for a totally different reason: I found that they didn’t really understand why particular points ARE on a graph. They could make a table of x and f(x) and graph it, but there was a disconnect somewhere because literally zero students could find f(2) given the graph of f(x). Thinking about points that aren’t in the solution helped them form a more complete idea of what the solution WAS.

I wish now that I had thought of this other effect that Dan presents in this post – it effectively removes all judgement from the work that students do. We’re not trying to find “right” answers or even “wrong” answers, we’re just asking questions and finding information. My presentation to my class did not include this note about our process, and I think I really missed an opportunity! Thanks for the great post, Dan.

on 04 Jul 2010 at 7:09 am13 Elaine C.@Karim – The front door is ‘trying to learn’ math. If you try, you’ll learn. Some people have easy to enter front doors – close to the sidewalk, no steps, etc. Others have harder to enter front doors – stairs, long paths, overgrown sidewalks. But everyone has a door. (That’s why different teaching tricks work with different students – they have different front paths/doors!)

The important thing to remember is that every kid has a front door – they can ALL learn!

on 04 Jul 2010 at 7:51 am14 Barry LewisThis is really strong and really beautiful. As Kelly says, this is poetry. You dead on gauge the climate of stress and mathphobia that seems to influence the classrooms of many of today’s algebra customers and you graciously invite each of those kids into the gentle, trail-discovery process that math is and had to have been for everyone whose name winds up in the sidebars and indices of math textbooks.

See what happens when instead of throwing sticks for the wolverine, we throw sticks at the wolverine, via @misscalul8 — Boo JCPenney. Not lovin your style. http://twitpic.com/21whru .

Enter the (Pissed-Off) Wolverine.

Thanks Dan, Riley, and misscalul8.

on 04 Jul 2010 at 10:47 am15 Karim @ Mathalicious@Riley — When you have your students find points that *aren’t* on the graph, do you find that it’s more effective when you put it in context? For instance, the 16GB iPad costs $499, and the 32GB iPad costs $599…

slope: $/GB = $6.25/GB

slope effect: 16GB = $100, 32GB = $200

y-intercept (i.e. the “base” price of the iPad): $299

Now that we have the equation, P = 6.25g + 299, we can ask whether the 64GB iPad ($699) is on the line. If not, is it more or less than we expect? How much more/less? As a percent? What does that tell us about Apple’s pricing scheme? If you worked for their marketing department, what would you charge?

All sorts of good stuff in there, and with a simple visit to the Apple site.

(You could extend this in any number of ways:

Evaluating equations: how much should a 1TB iPad cost?

Solving: if you spent $1450, how many gigs did you get?

Why we need two points to form a line: if all we know is the price of a 16GB iPad, how much are we paying per gig?)

on 04 Jul 2010 at 10:48 am16 Karim @ MathaliciousOops. $399.

on 04 Jul 2010 at 11:40 am17 ZenoKarim,

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

on 04 Jul 2010 at 2:31 pm18 Karim @ Mathalicious@Zeno

Great question! If you wanted, that could actually be the hook, no?

Actually, we’re not assuming that there is a linear relationship. We’re just finding out. In this case, though, when a student goes home and his mom asks what he learned in math, he doesn’t respond “slope” or “y-intercept” but rather:

“I learned how Apple prices its products.” (I know, I know. But we can dream, right?). At any rate, in order to make that claim, how much math did he have to do first?

(Oh, great username, by the way).

on 04 Jul 2010 at 3:14 pm19 RonRight on! I need to figure out how to use your metaphor as a constant presence in class….

Jason – so right about Dweck, a must read for every educator. It is so frustrating that students can have a growth mindset on the practice field (thanks Joel) but a fixed mindset in the classroom. I think that exists because they have been conditioned for so many years by simple math in elementary school and once confronted by something harder (Algebra) they fall back on the old notion that not everyone enjoys math like the teacher does and suddenly “this is stupid” becomes the new mindset.

on 05 Jul 2010 at 3:40 pm20 abuschDan,

Just thought I’d give you the courtesy of letting you know I plan to steal/plagiarize/and generally rip off this metaphor for use in my class. Thanks for taking the time and effort to write this blog. I find it challenging and encouraging all at the same time.

on 06 Jul 2010 at 1:36 pm21 cynthiaI wish I could take your math class!

on 07 Jul 2010 at 3:22 am22 Detta gäller inte bara för matte « marten@work[...] en parentes. På tal om vad som är fel med skolan. Läs detta inlägg om hur elever ser på matte. Jag tror väldigt många elever ser på all skolundervisning på det [...]

on 07 Jul 2010 at 5:09 am23 Dispelling the Great Math Myth: Effort Trumps Ability in Mathematics Achievement « I Speak Math[...] 2) Face Their Fear – Focus on small topics that they have had trouble with in the past. Do something different. Do an investigation, make index cards, play a game. Show them that math is accessible. I love Dan’s post regarding this. [...]

on 07 Jul 2010 at 6:24 am24 Jaime BruguerasDan,

My first comment to you after almost 6 months of reading your posts. I used to be a math teacher, and like you, believe the system needs to change. This short explanation of exploring curiosity is a great way to help teachers understand your approach.

On the first day of class, I would tell my students to remove the “I hate math card” they have on their brains. That they are going to learn to solve real-world problems with math’s help. Anyone that thinks math is boring, hard, or useless is going to change their way of thinking. But first, you need to give it a chance and remove the misconception from your brain on day one. At the end of the semester, feel free to put it back in (won’t happen).

Great work, let me know any day you are in Chicago, would love to meet and chat with you.

on 07 Jul 2010 at 7:57 pm25 Laura@ Barry….

That twitter link is perfect. I was in DSW this week and saw a man weave through two aisles to go to the giant clearance chart they have. They literally post 30 specific prices and then post corresponding prices with a 30%, 40%, 50% (yes, 50%), and 70% discount. Yuck!

I’ve lived off the clearance rack my whole life. Calculating sale prices in the stores is how I learned percentages. Mama taught me well. I’m lightning fast, and I don’t feel like I’ve gotten a bargain if it’s already on the tag for me.

Take that JCP.

And Dan, well said, yet again. I think I may have already said that above. But I forget. SO….kudos.

on 08 Jul 2010 at 5:58 am26 David H. WilkinsI love the wolverine and dog analogies. I love being the beneficiary of you folks and your creativity!

on 08 Jul 2010 at 9:37 am27 StephanieWhen you put it like that, I want to throw the stick too! Thanks.

on 08 Jul 2010 at 3:28 pm28 dy/dan » Blog Archive » The Wolverine Wrangler[...] 8th, 2010 by Dan Meyer Karim: [offering a lesson plan attempting to defang the wolverine, using iPad pricing as a hook for [...]

on 08 Jul 2010 at 5:50 pm29 breetai> The negative space as an answer can sometimes not only reveal things about the positive space…

Another useful aspect of negative space, equally revealing: http://bit.ly/cs7CrO

Or like sculpture, where you chip away all the marble that isn’t “The David” to ultimately reveal it.

on 09 Jul 2010 at 8:02 pm30 Björn KarlssonDan, your blogs are a work of art. Keep it up!

on 18 Jul 2010 at 2:19 pm31 Robert ZI think the thing that most gets me, is when students enter the classroom and their parents have told them, “It’s alright. It’s just math. You don’t have to understand it. I don’t understand it either.” I believe that your friendly approach and creative problem solving technique are wonderful ways to have compassion to kids being afraid of math, but also help them see that they can understand it, and that it is just a cuddly wolverine.

on 19 Jul 2010 at 10:52 pm32 Despairing vs. Working: Learning Classroom Management and Learning Math « Research in Practice[...] association needs to be disrupted, and as quickly as possible. I know you have all already read Dan’s lyrical description of the problem and one part of how to take it on. For now, what I want to call attention to is the mechanism by which this association renders it [...]

on 20 Jul 2010 at 1:37 pm33 LizGreat post. I’m right there with the students (scared of the angry wolverine) and you had me a tad panicked at the prospect of solving the math problem, and yet, I was calm and less afraid by the end.

What’s great about this is that it can apply to anything you teach that a child might be afraid to learn. You show them it’s not a scary monster, and they can warm up and work out the problem.

on 20 Jul 2010 at 2:16 pm34 MichelleI agree, great post.

on 26 Jul 2010 at 9:04 am35 ShannonDan:

Clearly, we all love the wolverine comparison–it’s so apt, we can’t believe we didn’t think of it, too :) As I’m a humanities teacher, though, I find this comparison most applicable to poetry. So many students come in with negative poetry experiences, based on years of forced sing-songy imitation or death marches through the symbolism valley. With your permission, I’d love to excerpt your piece for my 8th graders next year.

on 26 Jul 2010 at 9:35 am36 Dan MeyerWith my blessing. Everything on this domain is licensed under the Creative Commons — Attribution license.

on 08 Aug 2010 at 3:29 pm37 micheleIt has been wonderful reading all your responses, your enlightenment. I’m a maths tutor, the students I get all have steel gates which are definitely slammed shut in regards to maths. When I play with them I find a wealth of number knowledge and thinking power, and it takes time and empathy to circumnavigate their pain. This may sound dramatic but the pain and disillusionment is real . Your skill and creativeness is inspiring and I shall add them to my toolbox. I often find, me believing in them as mathematicians is the first step in the healing. You are all obviously believers.

on 02 Sep 2010 at 5:37 am38 Beating up on the wolverine… « Quantum Progress[...] 2, 2010 by quantumprogress My intro physics kids fear math—Dan Meyer’s angry wolverine metaphor describes their view of the subject perfectly. This partly because even though they’re better [...]

on 19 Dec 2010 at 6:24 pm39 half year recap and competition « A Brand New Line[...] think I am doing a good job making math less scary. Dan Meyer call it the wolverine which I like a lot. I think many of my students are less timid with the wolverine then they [...]

on 16 Jan 2011 at 6:42 pm40 Surprising Things My Students Did This Week « shifting phases[...] In AC Circuits, I gave the students a list of all the controls on their scope. Mission: find out what they do. They got them all except for the difference between AC and DC coupling (notoriously difficult for beginners to understand). By the end of the class, not only had most people applied the idea to a measurement, but several students proposed alternate ways to do it: “Couldn’t you just adjust the volts/div?” “Couldn’t you just adjust the vertical position?” Understand, too, that this was not coming from the advanced students who’ve had a scope in their basement for years — it was the students who until last week were treating their oscilloscope like an angry, injured wolverine. [...]

on 03 Feb 2011 at 6:53 pm41 Try Questions, More Questions | Math Teacher Life[...] Meyer’s elegant solution to student negativity, where he encourages students to consider that a wolverine can be more [...]

on 19 Jun 2011 at 10:41 am42 Moving: It’s like the mathematical process… | Irrational Cube[...] lack of control. It’s not just a question of being bad at math, but being afraid of the mathematical wolverine – the new place outside of one’s comfort zone; an unexplored, dangerous [...]

on 29 Oct 2011 at 3:56 pm43 LizVery Creative!

on 16 Jul 2012 at 9:43 am44 dy/dan » Blog Archive » [LOA] The Ladder of Abstraction, Part One Of Probably A Lot[...] I've done an excellent job pigeonholing myself as some kind of zealot for applied mathematics but some of my favorite experiences in the classroom haven't involved any applied context at all. Common to all of them (and common to my applied math [...]