(A quarter of what I want to say is "Go read Weltman's ba-na-na post!" That leaves me a quarter to talk about what actually happened in my 4th Grade class.)
(The last quarter is sort of long.)
We start with this:
What's the point of this?
The teaching assumptions here are worth making explicit. Sometimes students think that complex things are simple. Often that's just because they're used to it, not because they actually get it. So, how do you reveal the complexity that kids are just ignoring?
A nice solution is to shift to a similar, but unfamiliar context. Since the kids aren't used to it, the complexity becomes clear. Since it's similar, you can make connection to the familiar context.
(This is basically Danielson's fade-away jump shot. See here, here, here and of course Orpda.)
(That and the hair. The hair is also Danielson's fade-away jump shot.)
So, here's why Orpda is a solution to a teaching problem of mine. I'm hanging out with some very confident 4th Graders. They have a decent sense of whole-number place value. But they can't, for example, explain why the "add a zero when you multiply by 10" thing works. And we're going to put a lot of pressure on their place value understanding when we work with multiplication and division. They need to understand, in a deep way, that place-value involves grouping.
So? We make it unfamiliar, and then we connect it to the familiar.
What to expect from the kids
Here's what my kids came up with for representing the next number in Orpda:
- $ + #
- Invent a new symbol for that many dots.
All of these options were considered. I tried to put pressure on these choices by asking them to represent higher numbers. So I drew 24 circles on the board. I asked the kid who offered @@@@@ how they'd represent this many circles. They folded.
Another kid went up to the board and wrote some sort of multiplication problem using the defined symbols. A second kid offered a new suggestion, one that we hadn't seen yet: #%, or "at-percent."
I put pressure on these using different numbers and I point out confusions with their number systems. They don't come up with place-value, which is interesting. After a day (or two?) of discussion, I suggest that we group stuff.
I happen to throw up a slide with ducks on it as we're discussing grouping, so the kids decide to name this new number a "flock."
Then we count, out-loud. (This is so important!) We count: "At, hashtag, dollar-sign, percent, flock, flock-at, flock-hashtag, ..."
Then, we get stuck. After a little bit of discussion...
... Super Flock!
This is such a rich environment to play around in. Here are a few stray thoughts, or things that I learned as I was doing this:
- Language matters. I didn't properly realize until diving into this lesson how there are two number languages, operating side by side: the written and verbal representations of number. These are obviously deeply related, but the kids literally couldn't figure out how place-value would help them until we had named the number a flock. That was huge, and it sensitized me to the way that our spoken language is deeply connected to our conceptual understanding.
- (I actually screwed this up for a while at the beginning by asking the kids "What should we call five things in Orpda?" As a very sweet 4th Grader explained to me, this was a stupid question because Orpda doesn't have the number five. In fact, that's the whole point of this freaking exercise. Thanks R! Saved my ass there.)
- Once the kids landed on calling the next number a "flock," the question was how do we represent it in writing. A kid helpfully drew a pair of underpants on the board and labelled it "Super Flock", ala Captain Underpants. Ha ha, laugh it up kid.
- There were all sorts of computational problems that I created for kids once we had the number system. These questions all pushed on parallels between Orpda and our number system. So I created problems that required carrying in Orpda, or number patterns like @@, ##, $$, %%, ____.
- We also spent a good chunk of class time thinking about questions such as, "What number is like 99 in Orpda?" Or "What's like multiplying by 10 in Orpda?" Or "What number is like 11 in Orpda?" These were all tough and fun.
- A random interesting note: the move from 2 digits to 3 digits was harder than I thought it would be. I thought that, after we had figured out what number came after %, that it would be easy to figure out what comes after %%. Actually, no. It was only when we explicitly drew parallels to our own number system ("What's the biggest two-digit number in our system?") that we were able to figure that out.
- @!!! is a Super-Duper Flock, in case you were wondering.
Anyway, do Orpda with kids. It's a blast, and it really deepened my kids' ability to talk about place-value. I anticipate coming back to it periodically as the year goes on whenever we need to lean heavily on place-value in a sneakily complex way.