Wednesday, October 30, 2013

My Love Letter To Orpda

This is my love letter to Orpda, an invented number language in Base 5 that I learned about through Christopher Danielson. Half of what I want to say about Orpda is "Go read Danielson's post!"

(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:

Then we draw another red circle and ask kids: "What do we call this many things?"

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!

Random Thoughts

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.

Monday, October 28, 2013

Someone Is Wrong on The Internet; "Bad at Math" Edition

This article -- which is showing up every 15 seconds in my twitter feed right now -- is not very good.

We believe that the idea of “math people” is the most self-destructive idea in America today.
The most? Really? I'm not going to respond to this, but if you want to defend this line I'll see you in the comments.
So why do we focus on math? For one thing, math skills are increasingly important for getting good jobs these days—so believing you can’t learn math is especially self-destructive. 
This line is repeated a few time in the article. The idea that there's a shortage of STEM workers shouldn't be taken for granted.
While American fourth and eighth graders score quite well in international math comparisons—beating countries like Germany, the UK and Sweden—our high-schoolers underperform those countries by a wide margin. This suggests that Americans’ native ability is just as good as anyone’s, but that we fail to capitalize on that ability through hard work.
Yeah, it's either hard work or sub-par teaching or inequality or anything else. Sloppy reasoning.
Different kids with different levels of preparation come into a math class...The well-prepared kids, not realizing that the B students were simply unprepared, assume that they are “math people,” and work hard in the future, cementing their advantage. Thus, people’s belief that math ability can’t change becomes a self-fulfilling prophecy.
Great! So there's no genetic advantage. It's incredibly robust advantage in skills, psyche and mindset.

What sort of comfort is this supposed to provide a kid? "The reason why you suck at math is because your parents didn't do math with you, and because you fell into a self-destructive cycle of behavior. And now you're very far behind."

It's just replacing one deterministic story with another, in my opinion.
A great deal of research has shown that technical skills in areas like software are increasingly making the difference between America’s upper middle class and its working class. While we don’t think education is a cure-all for inequality, we definitely believe that in an increasingly automated workplace, Americans who give up on math are selling themselves short.
Hear that working class? Stop selling yourselves short! You're contributing to income inequality, guys, so cut it out.
We think what many of them are afraid of is “proving” themselves to be genetically inferior by failing to instantly comprehend the equations (when, of course, in reality, even a math professor would have to read closely). So they recoil from anything that looks like math, protesting: “I’m not a math person.”
It's not obvious to me that when kids say that they're not math people that they exclusively mean that they are the sort of people that are bad at it.

Often when kids mention it to me it's in the form of an apology while they're asking me for help. As in, "Hey Mr. P, I'm really sorry but I'm totally not a math person and this isn't making sense to me." In that context it can't possibly mean "I can't get better at high school math." There it's just providing a description, something like "I'm not good at math and I've never really been good at math, and I'm pretty slow when it comes to understanding this stuff." Say what you will about kids having that attitude, but it's not the same as a deterministic view that they can't handle high school math.

And don't some kids just mean "I don't like math" when they say "I'm not a math person"? That's what I mean when I say "I'm not a vanilla person."
We see our country moving away from a culture of hard work toward a culture of belief in genetic determinism.
Not any time soon, you aren't.


I'm pretty confident that the piece is pretty sloppy. I'm less sure about all of this, though...

The core argument of the article is that your level of success in high school math is determined by the amount of work that you put in. This is largely true -- no disagreements from me about the value of hard work. But since high school students are children, what it comes down to is blaming children from not working hard enough. And that seems unfair to me. Your typical 5th Grader can't be expected to break through the sorts of disadvantages that put them behind in math through hard work. That's an unfair burden to place on a child.

I see articles like this all the time. They trumpet hard work as the cure to so many of society's evils, and the key to personal redemption. Once you commit yourself to a mindset of hard work, you've unlocked the secret to success. We lament the death of hard work and rediscover it in science, ignoring the reality that it's part of our society's cultural backbone. And -- here's the clincher -- by putting hard work on a pedestal and acting as if we've just discovered it, we let the culture of hard work off the hook for being part of the very social problems that we're lamenting.

If the culture of hard work has been around for centuries, then how come there's all this inequality? How come students don't realize that hard work will help them get better at math?

I buy what Dweck says, to an extent. (There's other research that complicates her rather clean story, but never mind that for now.) I mean, who's going to disagree with the idea that hard work helps you get better at something? (I've never met anyone who doesn't believe that.) But it's important to recognize that the notion that hard work will solve many of our social issues is an old one in this country. It's part of our culture, and it might even be part of our problems.

Take us home, Shamus Khan!
Which means that these are tremendously unequitable institutions. But instead of explaining their position relative to their social advantage, their membership within an upper class, these elites explain it by their hard work and individual skills.
Check out his book Privilege, for real.

Sunday, October 27, 2013

There's Nothing Wrong With Being Mediocre at Math

I have mixed feelings about this poster. On the one hand, I really appreciate the contempt for laziness. I love the no-nonsense, kick-in-the-ass tone of it. I kind of want this poster on my wall. And there's a ton that I agree with here. "Struggling is good. It means you're learning something." Yeah!


"In order to learn anything you have to WANT TO LEARN ALL OF THE TIME."


Look, I want to learn stuff. I read for fun. I write for fun. I do math for fun. But do I want to do math all the time? Hell no. I spent today reading comic books and hanging out with family. It was great.

I know, guys. I read the fine print on this thing too. I know that our enemy here is "I'm really good at cramming" or "But I'm sooo tired this morning." But that's not what "ALL OF THE TIME" means, and that's up there in caps.

Anyway, that's not the part that really caught my attention.

"Don't be mediocre."

Well, why not? What's wrong with being mediocre at math?

I'm a mediocre runner. I'm a mediocre writer. I was a mediocre Physics student in college. I'm mediocre at most things that I do.

What's wrong with being mediocre?

I promise, I'm not trying to be a pain here, and I don't want to take this all too seriously. But should we really be telling our students that in order to learn something that it needs to be constantly on their minds? And should we really be telling students that there's no room for mediocrity in our classrooms?

We wouldn't tell kids not to be poor, or not to be dumb. So why would we abhor mediocrity?

At the very end of A Connecticut Yankee at King Arthur's Court, the Yankee has massacred an army. He boasts:
"Reflect: we are well equipped, well fortified, we number 54. Fifty-four what? Men? No, minds."
In praising hard work, learning and the intellect, it's possible to elevate the mind and dismiss being normal. We can find a certain kind of communal disgust for just being plain old OK at things, and end up teaching lessons to kids that we don't mean to teach. (Like: "Success in life is about finding something that you're especially good at," something that caused me a ton of psychological stress in college.)

You can enjoy math, even if you aren't great at it, right? And you can learn something, even if you're not obsessed with it, right? You don't need to be a mind, but you can be a man and a woman and you can have a mind and a heart and all those other important organs of life.


One Farting Unicorn

"One farting unicorn can make 7 people pass out. If 86 people pass out, how many unicorns were there?"

OK Michael, don't screw this up. You've got an awesome picture here, now you just need to write a post around it. No biggie.

How often do kids get chances to be creative in math class? To really express their...

Oh my god, stop it. That's awful.

This is what makes teaching such a great profession. Even as we lament the direction of policy stuff that I don't actually understand plus I teach at a private school please stop writing this, it's all awful please just don't do this Michael.


No. You've already ruined this post, which shouldn't be a post and is barely a tweet. You liked something that a kid made in class, you took a picture of it, you press publish. That's it.

Just hit publish, dummy.

Wednesday, October 23, 2013

Squaring and Cubing as the Bedrock of Exponentiation

Can anyone show me an example of exponentiation being introduced to our shortest students as anything by repeated-multiplication?

We know that exponentiation is hard for kids. Part of this is because, well, exponentiation is hard. But take a close look at exponents mistakes, and multiplication often ends up looking like the culprit.


Exponentiation isn't necessarily benefiting from its definition as repeated multiplication. 

Professor Danielson suggested that we remedy the situation by introducing exponents in terms of doublings and triplings and the like:
"What if we think of the powers of 2 not as repeated multiplication, but as the number of doublings?"
Though he recently offered an update...

So what else can we do?

My idea is that we build exponentiation on top of rock-solid concepts of squaring and cubing. We start as young as third or fourth grade, where we offer kids squares of various side-lengths and ask them how many unit squares it's composed of. Then we offer them cubes a side-length and ask them to find the number of unit cubes. Eventually, we call this "squaring" and "cubing" without ever using the power notation. Then, in later work, we'll introduce exponentiation as an extension of the concepts of squaring and cubing.

The value of these sorts of problems on their own is immense. First, they connect well with the array work that the kids are hopefully doing. They also drive at the connection between area and volume, and constitute good multiplication practice. We're also helping set them up for square roots and cube roots.

But I see this as crucial in setting up exponentiation. First, because we'll make sure to connect these sorts of problems with the language of squaring and cubing. (None of my 5th Graders knew why squaring was called squaring, FYI.) The second reason why this is crucial is because, again, we're going to introduce exponentiation as an extension of squaring and cubing. This means that students will have a rock solid example to pivot off of as they learn fourth, fifth and other powers. Third, it avoids using first powers in the early development of the exponentiation concept, which might very well be the seed of many future exponent/multiplication mix ups.

I want to be briefish, so I'll say that I tried this out with a group of 5th Graders this week. They had all seen exponentiation before, but were incredibly confused by it. Many were writing "12" for 3^4. None of them knew how the language of squaring and cubing related to powers.

I started class by asking a variety of squaring and cubing problems. These were of medium difficulty for the kids, but eventually we noticed that a cube is made up of X sheets that have X-squared cubes, leading the kids to suggest a generalization: "a number cubed is always the number squared times that number."

It felt natural to go around and remind the kids that 3^4 was like 3-squared or 3-cubed, but to the fourth power. One thing that was frustrating was that a bunch of kids didn't see why 3^2 should be the same thing as "three squared." But I think that's a frustration that I could avoid if I had the chance to lay the groundwork of that kid's concept of exponents.

So, that's my idea. Thoughts?

Update: After a second round trying to help out some 5th Graders, I realized that it's not enough to just deal with 2D squares and 3D cubes. There needs to be a middle stage where we're saying that "squaring" three means building a square out of a length of three cubes. Otherwise the language of "Three cubed is three squares" doesn't really work.

Also, kids today asked whether there was such a thing as "three triangled," which is a great example of a question as evidence of learning.

Sunday, October 20, 2013

Mastermind Part 3: What's the typical number of guesses?

It became clear from my kids' organized data sets and the ensuing conversation that they believed two things:
  1. The word "average" refers to the most common element in a data set.
  2. The most common element in a data set is the most typical.
I very much wanted to put pressure on the second belief, to set them up for eventually ditching the first belief. But I needed some more data.

The first thing I did was a quick experiment that I took from TERC. I took a bucket of dice, and everybody in the room tried to pull as many dice out of the bucket as possible with one hand. A kid kept track of all the results in a line plot at the board:

(Oh, and then I went. I pulled out 31 dice. Should we count that? No? Then there's a word for that, we call that an outlier. Got that?)

We were lucky enough to have a tie, and a tie between an even number of points, no less. I asked kids what the usual, normal, typical number of dice for one of us to pull out of the bucket was. They said 14, or 13, or between 12-14. (I clarified that I just want to know the most normal number, but that felt artificial. Offering a range of numbers seems completely reasonable for this data set.)

Then I asked the kids what the data looked like. This was important, because I wanted them to pay closer attention to the shape of the data, so that they would start thinking of the data set as a whole. (Thanks again, TERC!) They went for "The Chainsaw," or "The Machete." 

Kids these days...

Anyway, I cooked up a page with some line plots. Their instructions were to find the typical number for each, and to name each line plot:

As they finished, I paired the kids up and pointed out their disagreements, and asked them to discuss their choices. They did this nicely. And then we had a conversation together collecting some of the rules they had come up with for finding the typical number:

Their rules:
  1. The typical number is the one that has the most but it's also surrounded by the most.
  2. If they're all over, pick the one in the middle.
  3. If they are all even, pick the one in the middle.
  4. In a castle situation, pick the one in the middle.
BTW, this is a castle situation:

What we're doing is conditionalizing all this stuff. These notions of middle are pushing us past the mode (which we're going to have to name pretty soon) and into the fuzzy idea of middleness. And we're going to have to distinguish between two very different ways to talk about middleness.

To plant the seeds of this, and to tie up the Mastermind stuff, I showed the kids this bit of research:

The idea that an average could be 4.340 was absolutely baffling to them. But some of them had already suggested that, when there is a Twin Tower situation (Good Lord these kids...) that the typical answer should still be between them, so that we'd get half responses. And what if there were some more data points a bit to the right of the Twin Towers? Wouldn't that suggest that it's typical to be a bit to the right of the towers?

It was a good, difficult conversation for the kids. 

And, for me? I had hated on the mode from my high school perspective. "Why do kids need to know mode? What's the purpose of mode, ever?" I loudly questioned. 

Well, now I get it.

Mastermind, Part 2: Refining our Notion of Equality

We had finished up our puzzles, but we still had a question: How many turns does a winning Mastermind Puzzle usually take?

I pooled data from the kids' Mastermind Puzzles and from the boards that Anna's 6th Grade class (totally coincidentally) happened to be making, and made sure that they knew what data means. (Turns out? Cell phone plans have forever changed kids' awareness of the word "data." Hmm.)

Then I asked them to organize all of the data any way that they saw fit. (ala TERC. They have kids doing this with counting the contents of boxes of raisins.)

This, frankly, was a bit open for their comfort and the kids weren't so into it. There wasn't a whole lot to talk about in the different ways they organized things, though I tried. And my attempts to spin a conversation about what constitutes the "typical" number from the data set were foiled by the widespread agreement that the typical number of turns is 5.


But I noticed two really interesting things from their work, which you can see in the pictures above:

  1. Almost all the kids wrote something along the lines of 7=1 or 6=3 in their work.
  2. Almost all the kids said that the average was 5, because it was the most common result.
I decided to poke at equality. I asked them what "equals" means. The kids offered two definitions, and we put them both on the board:

  1. "Equals means 'the answer is, duh duh duh DUH!'" The vast majority of the class agreed with this.
  2. "Equals means the same." There was only tepid acknowledgement of this meaning.
I tried to put pressure on them by pointing out that 2=1 or 7=1 is a weird thing to say in some contexts, but the defenders shrugged me off. So I upped my game, and I wrote 3 = 2 + 1 on the board. Does this mean "3, the answer is 2 + 1"? I made some kids nervous, but then B offered that this was really just asking what 2+1 was.

But 2 + 1 = 3 + 0 was much more problematic for them. How do you read this as offering an answer to some question? A few kids tried to suggest that these were answers to the question "Is 2 + 1 the same as 3 + 0?" but I didn't like that because that's not what they said "equals" meant.

I argued that equals means the same amount, not "the answer is...", and by the end of class the kids were vocally agreeing with me, so I guess that worked? I got a better sense of where we stood the next day, when (after looking at their homework) I realized that they didn't know what an equation was either.

This, of course, is super-related to what "equals" means. The idea that an equation is any problem is supported by the notion that equals just marks the answer. I'll spare you the details, but we sorted this out too. 

Next up, putting pressure on their notion of average.

Mastermind, Part 1: Solving and Making Puzzles

Mastermind is a great game. We spent a bunch of time with it in 4th Grade recently, and it was great fun. This is the first of a series of three posts where I get to self-indulgently record all of that.


We started by just playing Mastermind together against the computer. We spent a day basically just doing that, justifying our moves, imposing a grid on the board to make the discussion easier. We used a random number to decide who was going to pick the next move. It was a great deal of fun.

Also, isn't it wild and disturbing that when we need a fair way to pick someone in our class we use randomness? That's actually pretty messed up, now that I think about it...

We start the next day with some more games. The week before had been spent using multiplication to tackle some combinatorics questions (e.g. with Scrabble pieces, with restaurant menus) and I asked the kids some questions in between rounds. ("How many codes could it be once we know all the colors?")

Half way through the period, I ask kids to ask some interesting mathematical questions about the game. We got some good stuff out of the kids:

  • Are there more combinations with duplicates? How many more?
  • How many codes are there?
  • There are always four rows left when we finish a game.
After explaining to M that this third thing wasn't really a question, we discussed it for a bit. It was a good observation, a really good observation, and it gave me an idea for a nice transition to our data unit.

But more about that later. First: a Mastermind Puzzle is a board that is solvable in just one move. I showed the kids a board that I made. Then I asked them to make their own. Which is actually really hard for anyone, 4th Graders very much included. Some of their boards worked, but a lot of their puzzles relied on slippery reasoning. 

But there was a lot of hard thinking going on, and the kids got to be creative. So that was still worth it, I think.

Next up, pooling all of our Mastermind puzzles to ask some questions about the whole set of them.

Sunday, October 13, 2013

Explanations and Justifications

This student says that a proof is "something that clearly demonstrates that something has to be true." So a proof is all about guaranteeing truth. In other words, it's a form of justification.

Here are the rest of the class' takes on proof:

  • "A statement that solves a mathematical (mostly Geometric) problem."
  • "Take an example of a problem, explain it to show how whatever you are trying to prove must be true."
  • "A mix of words and equation or pictures that convincingly show some problem or statement to be true."
  • "A proof is a way to guarantee that something is true for all situations."
  • "A proof is something that is uncontestible."
  • "Solid evidence."
  • "Undeniable evidence that can't be proven otherwise."

And then there's one kid who wrote the following:
  • "A proof is a statement of why things work the way that they do, and is also backed up by evidence."
Do you see how that's different? All the other kids were talking about proof as justification, but this kid here is talking about proof as explanation. 

I think that proof as justification is more concrete. Students get the hang of that whenever there's a controversy or doubt, and any good problem can create a good deal of controversy or doubt. It's relatively easy for me to provoke a need for a justifying proof with kids of almost any age.

But Michael Serra told us in the comments that we ought to "focus on proof as a means of 'explaining why' rather than a way of convincing someone the given conjecture is true." And that makes perfect sense. If you only see proof as a justification, then proofs of things that you already believe to be true would seem to be strange, unnatural exercises. There's very little reason to value multiple proofs if the only purpose of proof is to justify a claim.

These kids, for the most part, see proof as a means to justification, and it's making it difficult to motivate all sorts of things in class. For instance, last week we were looking at proofs without words of the Pythagorean Theorem. We did an easy one on Monday. On Tuesday we did a harder one. A kid said, "I don't see the point of this. Wasn't yesterday's proof easier? Shouldn't we just be looking for the simplest proof of something?"

He's working with proof as justification. I need to take him to proof as explanation. I don't know how. Ideas?

Sunday, October 6, 2013

Carol Dweck vs. The American Dream

Americans believe in hard work. You can see this in the poll above. The Council on Foreign Relations reports that American's belief in the power of hard work is relatively unchanged since 25 years ago:

Our culture stands in contrast to that belonging to other nations. Michele Lamont, in her study of French and American upper-middle-class workers (which I haven't read, just to be clear) argues that crucial to French conceptions of moral character is intelligence, while crucial to American moral characterizations is ambition and hard work:

So: Americans believe in the power of hard work to grant economic achievement, even in the face of widespread evidence that this is not the case.

Enter Dweck:

She explains that people with fixed mindsets about X won't work hard if the task is seen to demand X.

Take another quote from her writing:
We often hear these days that we've produced a generation of young people who can't get through the day without an award. They expect success because they're special, not because they've worked hard.
Is there a tension here? Dweck shows that having a fixed mindset about intelligence is extremely prevalent. [citation needed] She also argues that having a fixed mindset causes you to discard hard work, since "risk and effort are two things that might reveal your inadequacies and show that you were not up to the task." Despite this, there's a robust observation that Americans (incorrectly!) believe that hard work is a key to economic achievement.

How do we square Dweck's research about intelligence with the American obsession with hard work? I see only a few possibilities:
  • Americans don't see intelligence as a path toward economic success.
  • "Intelligence" doesn't mean one thing to Americans. Dweck's research is mainly limited to school learning. Do Americans have a fixed mindset about school intelligence, but not about "practical" intelligence?
What do you guys think about this? Do you see the tension that I'm pointing to? Or am I totally off?

I think that there's a certain irony in Carol Dweck's work. She has pointed to a fixed mindset of intelligence as important in determining how much hard work you're willing to put in to your work. She consistently laments the lack of hard work that results from seeing intelligence as a trait, and that this is the difference maker in academic achievement.

So you might've believed that Americans are, on the whole, an entirely lazy bunch. But, to the contrary, Americans are relatively hard working, and extra-ordinarily obsessed with the notion of hard work in determining economic success. 

If that hard work is an indicator of an incremental mindset, then intelligence must be one of the few areas relevant to professional life that Americans actually do have a fixed mindset about.