Dan, for real, you’re the man. This is pure poetry.

(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.

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.

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.

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.

@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!

@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?)

Karim,

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

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.

I wish I could take your math class!

@ 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.

When you put it like that, I want to throw the stick too! Thanks.

Dan, your blogs are a work of art. Keep it up!

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.

Dan:
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.

It 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.