Tuesday, May 22, 2012

Skills vs. Knowledge

The best way to learn something is to have it clearly explained to you.

The thing that's driving me nuts lately is that so much of what's good about life is inaccessible to you if you believe the things about learning that my students do. In particular, I think you're at a real disadvantage in life if you believe that the best way to learn something is to have it explained to you.  More and more I see this as something that I need to address in class, every day.

The stakes are awfully high here.  Because if you believe that explanations are the key to learning, then the following are natural things to conclude:

  • Learning usually requires a teacher or a tutor
  • Learning usually stops after school because teachers and tutors are hard to come by
  • Learning is easy, but hard to retain.
A word about the last of these. If you think that explanations are the key to learning, then when you go on TED and listen to some guy talk about some tech thinger revolution paradigm whatever, and you understand the words when he speaks them, then you've learned something. That was easy. And then later you forget what he said, because learning is cheap and retention is hard.

That's the worst misconception of all. Because what's really true is that
  • Learning is hard, but it lasts a really long time
  • But learning doesn't require a teacher or a tutor
  • And learning is something that should be part of your entire life
Why is it that almost everybody I meet, ever, thinks that learning is easy, but hard to retain? I think that it has to do with the experience of learning something in our everyday lives.

Say that I want to buy a bag of chips. I don't know how much money to bring with me. I find someone who knows how much chips cost. I ask him how much they cost, I understand his response ("$1.99") and I go to the store. That's information that I soon forget a day or two after I go to the store.

That's quick. That's clean. That's easy. And that's the way that most of us think about how we gain more content knowledge on a daily basis. 

But people also know that not everything is like that. For instance, you can't learn to ride a bike, play violin or build a boat via someone's explanation. For these things, explanations won't work. You have to learn how to do X by doing X. 

But the common perception is that there's a difference between the way that we learn skills and content. Yeah, of course you need to practice shooting a basketball to get better at free throws. But that's because shooting a basketball is a skill.  Knowing who built the Panama Canal is knowledge, and for that you need a good explanation. Like a podcast, or a video, or a smart friend, or a teacher, or a book. 

It's the whole dichotomy between skills and knowledge that needs to be torn down. Because it's this flimsy dichotomy that supports the idea that explanations are the key to learning. And it's that misconception about learning that keeps my students from realizing the potential for learning to change their lives.

Look: not all skills are learned by doing or practicing them. If I know how to cook really well, and you give me a short explanation of how to make a carrot cake, I'll probably be able to pull off the carrot cake recipe without much practice. And not all knowledge is learned via explanation. No matter how clearly you explain some fact about Lie Algebras to me it won't make sense to me, because I know nothing about it.

The real difference isn't between skills and knowledge. It's about easy things and hard things. You want to learn something easy? Explanations are fast and efficient. 

But you want to learn something hard? I'm talking about the sort of things that you don't already know how to do, like juggling fire, like speaking a new language, like learning Algebra, writing poetry -- you know, the cool things. You need to practice these things. A lot. 

Monday, May 14, 2012


I'm not a very fun teacher, yet. My daily shtick is sets of problems that students struggle through. Some of the problems are fun and playful, but most are strictly business.

That's an issue, and I'm working on fixing it. But here's a lesson that's a step in the right direction. Here's how class went today, twice.

Warm Up

Take a penny, find a partner. Flip your coins at the same time. Record the number of Head/Head, Head/Tail and Tail/Tail results.

I combined all of their data onto an Excel spreadsheet and we got something like this:

"Does it make sense that Heads/Tails beat the junk out of the other two? What would happen if we kept on going?"

The kids found the people who thought that the probability would eventually stabilize around 33/33/33 and straightened them out. Then I showed them how to make a tree, because that could be on the Regents. Then we moved to the main show.

Act One

"What are the chances that he gets $10,000 twice? Is is super unlikely? Very likely? Possible but unlikely?"

Act Two

"What do you need to know for this problem?" They needed to know how many rows there are -- it's hard to see the structure of the Plinko board from the video. So I gave them this (basically ripping off this):
Plinko Handout

If I were doing this with Algebra 1, I would ask them to figure out the chances that he wins $1000 given the fact that the chip is all the way down where it is at the end of the video. But since this was Algebra 2, and I thought that would feel like a cop out to these kids, so I gave them two versions to play with. One where a couple of rows were removed, so as to avoid dealing with the walls (the calculation is a tad more complex when you have to take them into account), and the other as a replica of the board.

Kids wrote probabilities in the little circles. For some kids we took away a few rows. A bunch of kids came across a cute calculation short cut. The guys who thought they had cute little formulas ("2^13! Right?") figured out that this wasn't that kind of cute. Most of their work looked something like this:


  • How much would you pay for the opportunity to play this game 100 times?
  • What happens if you change the number of rows?
  • What are the chances that you hit the zero chip if you start at top? 
A slight complication: most of the kids chose to tackle the simplified Plinko board rather than the actual one, which means that their probabilities would be a bit too high. So we took a second and talked about what would happen if we added a few (4) more rows, and how much that would knock down the chance of winning. Based on this calculation, along with the estimation, kids agreed that you should be winning about 20% of the time.

Act Three

For me, Act Three is the data. And the data gives a nice twist. Via Bowen Kerins (this whole problem is freaking via Bowen Kerins) there's this site that collects all of the data from The Price is Right. It's great, and sorta crazy. But mostly great. Anyway, here was the important table:

People only win the $10,000 slot 14.1% of the time. But that's too low! Why is that?

Kids nailed it: people are being stupid. So, lesson of the day: If you're ever on The Price is Right, and you play Plinko, drop your pieces in the center slot!

Kids wanted to know what happened in the rest of the video. So I showed it. But first I asked them to calculate what the chances are that he could pull off the $10,000 win twice in a row. This was a good moment, because I got a lot of kids saying wrong things and I know that I need to spend a bit more time on independent events. 

  • I did the Warm Up because I wanted the kids to have a framework for understanding what's going on in the Plinko game.  I can also see you saving that for another day, or doing that at the end of class.
  • Our next step is to handle probability questions with more coins and dice. In each of these contexts we're going to run up to cases where what it's really about is figuring out how many ways something can happen. The value of the Plinko problem is that it provides a really clear visual version of this story: it's more likely to fall in the middle because there are more ways to get there.
  • This was just a fun day. Kids were flipping coins. Kids were rooting for some guy on a game show. Kids were chasing an interesting problem. I need to sit down over the summer and scrap units and rebuild them to be funner.
  • (Update 5/28: For another take on this problem, see this article from Mathematics Teacher.)

Wednesday, May 9, 2012

"Everything is a Remix"

I'm trying to put together something coherent for my students on probability. There are a lot of smart people on the internet, and here's what they've come up with:

All that I have of my own is a couple of very rough ideas for three-act problems on 101qs, but neither seems to be doing particularly well.

My job for the next few nights is to remix this stuff into something coherent. If you've got more resources, I'd love to see them. Toss them into the comments, if you don't mind.


  • I bought 100 dice for about 20 bucks so that we can do some simulations in class.
  • I like Plinko. Here's my Act One, and Bowen's slides from NCTM contain a good idea for Act Two. Here's how he develops it in the PCMI problem sets from 2007:

Monday, May 7, 2012

A lot of little things

Here's a bunch of undeveloped things that I've wanted to get down in writing. Maybe I'll expand these if they seem worth expanding. So, without further ado, here are a bunch of Things.

Thing One

Structural stuff matters when you're learning Algebra. And a great way to see structure is by comparing different particulars for similarities. That's why it's great to do things like pushing kids around to teach the commutative property. But sometimes I find that students have a hard time connecting those abstractions to the problems that are sitting in front of them. So sometimes it strikes me as a good idea to connect things back to the problems that they're going to have to solve, almost all the way, and then just holding up at the last minute.

I like this because (1) it worked* and (2) I've got a good, quick conceptual story to tell kids when they're stuck that emphasizes structure. I just get to ()(x + 5) with the student, turn that to ()x + ()5, and then fill in the parentheses that we just drew with a little "x + 1".

*About 2/3 of the class could just do things like (x+3)(x^2 - 3x + 6) after our unit. We devoted no whole-period class time to this.

At the same time, I feel like I'm providing students with something that's basically a procedure. And I worry that I'm falling back into a game where I'm trying to provide students with a better procedure. But I think that if my students can make the connection between ()(x - 5) and the correct distribution, that's enough conceptual understanding for the moment. Maybe.

Thing Two

Here's another thing that basically worked. How do you get kids to avoid just crossing out random stuff in a fraction and calling that simplified? Chastise them for it? Urge them to evaluate the expressions all the time? Reps?

Here are a few observations:
  • For Heaven's sake, don't use the word "canceling" in the classroom. Ever. The language that you use in the classroom matters, and the word "canceling" means something like removing, expelling, kicking out. And that's not what we're doing. 
  • So what are we doing? We're essentially factoring out a "1." But that doesn't have a fun name.
  • So call it a "Special 1," and then ask kids to find it. It's challenging, and it's the fun part of simplifying anyway. 

Thing Three

By the way, if you want any of these files, they're all on Google docs. Google Drive has made it ridiculously easy to share my curriculum with my students and anybody else who wants it. 

What's the advantage of Google Drive over Dropbox or any other cloud service? With Google Drive I have a folder on my desktop that is synced with the web. Anything that I alter or add to my desktop folder gets automatically added to the web. Maybe other services have this feature, but I wasn't able to figure it out with Skydrive or Dropbox.

Here's my curriculum. Chill out. It's my second year. I write it hours* before I use it. The stuff in the second semester is better than the stuff in the first semester. And I haven't done much curriculum writing for Geometry yet.

* Minutes.

There are things there that aren't my own, that I don't have the rights to redistribute. Sorry. I'll worry about that when you, my faithful readers, are more numerous.

Things Four through Seven

  • Clinometers are a great project for Algebra 2. Easy to pull off. Dice are fun for probability. Post-its make great instant histograms. Google Maps is pretty good for getting distance from you to a thing of your choice. Astral Weeks is a great album; I'd take it over Moondance any day.
  • Some things I've learned about making problem-solving work: I either need an answer key available for the kids, or I need to check in with each kid or group, or I need to devote a ton of time in whole-group to talking about the problems and developing a way to check the process. Those are my tools for giving feedback. The choice differs, depending on the day.
  • Also, giving "Leveled Problems" is something I stole from Dan Meyer, and I like it. It helps kids see that there's a hierarchy of complexity to the problems, it gives us a better vocabulary for discussing issues ("I'm fine with Level 3, but...") and it's good for giving kids a sense of how much they're progressing as they master different sorts of problems. I do that whenever we take on something that's algebraically complex. When I remember.
Finally, the last thing: 

I don't think that I really care about whether my kids know math. I mean, I do. But not in the way that  some other math teachers care. Unless you can point me to some evidence, I don't buy the idea that there are Mathematical Habits of Mind that are transferable to non-mathematical contexts. I have a hard time saying that anything past Algebra 1 is really helpful on anything resembling a regular basis.* I agree with those of you who point out that math is like comic books, history, novels, music, or any of the other parts of life that are fun and amazing. At the same time, what about the parts of math that just aren't that interesting to me?

*"Dammit, doesn't anyone here know how to factor a fourth-degree polynomial!" I wish.

What motivates me is making sure that my kids have good feelings about learning. Kids should leave school with the firm belief that learning is something that makes life better. Not math. They can forget math, for all I care, I think. But I want them to leave school with a respect for real learning and all it entails. They should know, from experience, that deep conceptual understanding out-flanks the sort of flimsy procedural knowledge that hucksters try to sell on the cheap. They should know how to learn something new, and they should believe that there are times when doing just that can make their lives better.

That's it for now. As always, comments are open.