## Math: The Angry, Injured Wolverine

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.

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.

1. #### Jason Dyer

July 3, 2010 - 9:21 am -

(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.

2. #### Learning Through Teaching

July 3, 2010 - 9:54 am -

One 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.

3. #### Dan Goldner

July 3, 2010 - 10:31 am -

Loved 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.

4. #### Karim @ Mathalicious

July 3, 2010 - 12:41 pm -

It 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

5. #### Elaine C.

July 3, 2010 - 1:13 pm -

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.

6. #### Joel

July 3, 2010 - 5:42 pm -

A 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)

7. #### Karim @ Mathalicious

July 3, 2010 - 8:05 pm -

Elaine-

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.

8. #### Riley

July 4, 2010 - 4:35 am -

I 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.

9. #### Elaine C.

July 4, 2010 - 7:09 am -

@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!

10. #### Barry Lewis

July 4, 2010 - 7:51 am -

This 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.

11. #### Karim @ Mathalicious

July 4, 2010 - 10:47 am -

@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?)

12. #### Zeno

July 4, 2010 - 11:40 am -

Karim,

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

13. #### Karim @ Mathalicious

July 4, 2010 - 2:31 pm -

@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).

14. #### Ron

July 4, 2010 - 3:14 pm -

Right 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.

15. #### abusch

July 5, 2010 - 3:40 pm -

Dan,
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.

16. #### Jaime Brugueras

July 7, 2010 - 6:24 am -

Dan,

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.

17. #### Laura

July 7, 2010 - 7:57 pm -

@ 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.

18. #### David H. Wilkins

July 8, 2010 - 5:58 am -

I love the wolverine and dog analogies. I love being the beneficiary of you folks and your creativity!

19. #### breetai

July 8, 2010 - 5:50 pm -

> 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.

20. #### Robert Z

July 18, 2010 - 2:19 pm -

I 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.

21. #### Liz

July 20, 2010 - 1:37 pm -

Great 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.

22. #### Shannon

July 26, 2010 - 9:04 am -