The Best Blogging of 2012

Here is the best stuff that I saw in 2012:

Fawn Nguyen - Her Google Form for reassessments has changed my life for the better. I really like her Foxy Fives problem set. Her love of the Shell Centre materials has sent me digging through their lessons and tasks. This year, her blog became my favorite resource-sharing blog. And she also has a 180 blog? So prolific, and so good. Her work has pushed me to be more organized, and to plan more carefully.

Christopher Danielson - His hexagon investigation might be my favorite post of the year. He's consistently thoughtful and sharp on twitter. Logarithms are another strength of his work. His online course is an exciting experiment. He wrote -- somewhere -- that he constantly asks his education students to make explicit the pedagogical assumptions of a text, lesson or activity, and that line rings in my head nearly daily. It was an excellent year for his blog, and I can't wait for what comes next.

Justin Lanier - I've always enjoyed Justin's work over at I Choose Math, but it's his 180 blog that really opened up his classroom to me. His Geometry class digs into beautiful investigations, and I loved his simplification of the proof that the square root of 2 is irrational. His work shows how exciting class can be when argument, proof and just plain-old thinking are at the center of things. My attempt to emulate his and Paul Salomon's work has been the biggest change in my tone and style this year.

Dan Meyer - Always Dan Meyer. What can be said about his work that hasn't already been said by so many others? A lot of my thinking starts with, "How does Dan pull that off?" His influence is just remarkable, and I get the sense that a lot of people are watching his career very carefully and taking notes.

Paul Salomon - Over the summer Paul shared his introduction to exponent properties with me. Here's a video of him explaining it. (Ignore the goofy redhead.) I walked away from that conversation realizing that I'd been neglecting proof and argument in class, and that it was time to bring them back to the center of our discussions.

Kate Nowak - She's the one that mentioned that she was bringing her CME texts with her to her new gig and got me interested. She's the most creative activity planner out there. Log Wars. Laser Kids. Line of Best Fit. Her work is just so impossibly and consistently good.

Finally, not really a blogger, but whatever:

CME Project - I bought the teacher's editions over the summer, and their work is just phenomenal. Owning these texts has raised my baseline performance by giving me high-quality material to lean on. The texts have also shown me how to teach so that proof and structure become visible and integral.

Teaching complex numbers

I've been thinking a lot about complex numbers over the past few weeks. I wasn't happy with any of the available introductions to complex numbers. I wanted to put transformations of the plane at the center of it all.

I've been experimenting in the classroom. I started by defining (0, 1) as a transformation as the unique rotation that takes (1, 0) to (0, 1). In other words, a 90 degree rotation. By similar reasoning, (-1, 0) is an 180 degree rotation. And then I asked kids to figure out what (0, 1) applied to (4, 1) would be. (They studied rotations in Geometry, and didn't need much review.)

Then we generalized further: Let (a, b) be the unique transformation that maps (1, 0) to (a, b). It's pretty intuitive that you can accomplish this in the plane with just a rotation and a dilation.

That's where things got interesting for us. This approach is still very much a work in progress, and my Thursday lesson flubbed. It's annoying, because I've got a bunch of kids going across the Atlantic on an exchange program and I didn't get to wrap things up for them. So I wrote this letter to tie together the loose ends.

I want to write more about this later, but at the moment this stands as my clearest statement of how I want to approach introducing complex numbers to kids.

Feedback please?

A Letter to My Algebra 2 Students on Complex Numbers

Making Mathematical Decisions

[The first draft of this post was over at the Global Math Department, where I presented about mathmistakes.org. Listen to the full recording for some analysis about the way users interact with the site.]

Evidence

Here are some math mistakes (source: MathMistakes.org). What do you notice?

To draw things out, kids are doing the following:

• 3 ^ 9 = 27
• 5 ^ 2 = 10
• x ^ 0 = 0
• 100 ^ 1/2 = 50

3 Theories

Why do kids do this? Here are 3 options:

1. They don't understand exponents, and are just guessing.
2. They are reasoning consistently within a model, but a mistaken model. They think that 5^2 means "You have 2 5's. That gives you 10." They think that 2^0 means "You have no 2's. That gives you 0." They think that 100^1/2 means "You have half of an 100, which is 50." 
3. Kids are not reasoning explicitly at all, but rather have a strong intuition that exponentiation should be solved using multiplication.

In short, we could be dealing with guessing, reasoning and intuitions. Which of these is right?

I dismiss the first option. If kids were guessing, then why don't they ever add the exponent and the base? Why don't they ever subtract? At best this explanation is incomplete. 

The second option is more attractive. A kid answers exponentiation problems by saying "3^2 is 9, because two 3's make a 9" and extends that incorrectly: "3^0 is 0, because no 3's make a 0." The idea is that kids are reasoning about exponentiation in an explicit way, but that this explicit way is mistaken.

The third option is that kids are not reasoning explicitly. A search for a model would be beside the point -- kids have a strong intuition, in certain contexts, that exponentiation should be treated as multiplication.

How can we distinguish the second and third options?

• If the second option is right, then kids should need to pause to reason before incorrectly evaluating an exponent such as 100^1/2 as 50. If the third option is right, then kids should just be able to shout that answer out very quickly.
• If the second option is right, then kids should be able to explain their reasoning. Relatedly, we'd expect students without any strong model for exponents to be unable to provide any answer at all to unfamiliar exponentiation problems.
• If the second option is right, then kids should operate consistently. They shouldn't sometimes reason according to the model and sometimes not. (Or, alternatively, the fact that they inconsistently apply this model would require explanation, one potentially provided by the third option.)

My Claim

I don't have all the evidence that I need to knock out the second theory, the idea that kids are explicitly reasoning about exponents. But, from what I've seen, kids have answers to the exponents questions WAY too quickly for it to be explicit reasoning. I think that there's something to the theory that this is an intuition.

So what's going on? There is a strong connection between exponentiation and multiplication. Everyone learns this strong connection. And in unfamiliar contexts the brain falls back on the intuitive connections between exponentiation and multiplication, and answers the question "What's the base times the exponent?"

Why? For reasons that I've tried to articulate before, I think that kids sometimes see harder problems as easier ones.

Predictions

How could we prove or disprove this specific idea? Here are some predictions of my lil' theory:

• There should be other strongly connected operations, and we should similar mistakes when we ask kids to do tough things with those operations. A likely suspect would be subtraction of negative numbers, which asks kids to take subtraction into unfamiliar territory. There's even a bit of evidence that they treat that stuff like addition in these contexts.

• We might even find evidence of more of this stuff in the early years of schooling, as kids are just learning their operations. MathMistakes.org could use some Elementary School submissions.
• The best way to support my idea would be to artificially induce the sorts of mistakes that I'm talking about in students. The idea would work like this: I would define a new operation. Kids would show proficiency with it. Then, I'd define another operation in terms of the first. Kids would show proficiency with that one too. The next part is fun. Then I'd ask kids to extend the second operation in an unusual way, and see if they spit out the value of the first operation.

I'd like to see more stuff like this

I took a bunch of examples of student errors, I tried to unify them under some sort of theory that would make sense out of them. I tried to think through the theory to consider its competitors. I'm considering what would count as evidence for and against my theory. I'm trying to find testable predictions of my theory.

In other words, I'm trying to participate in the science of how kids learn stuff. And I think that more teachers should do that. Especially since understanding student errors would be widely valuable outside the classroom, but is easiest to theorize about when in the classroom and in interaction with warm bodies. Especially since digital cameras make it easy to collect lots and lots of evidence of how kids mess stuff up while you're grading. 

Especially because it's fascinating, and I want to know more about it. Come up with a theory and write about it. How do people reach mathematical decisions?

7 Best, 5 Worst

It's time for a quarterly review. Here's the good and the awful from this year's teaching.

7 Best 

1. Paul Salomon's Introduction to Proving Stuff about Exponents - The idea is to use function notation to prove things about exponents without being distracted by the repeated-multiplication model that we (rightly) inculcate our kids with in the early years. Maybe the best thing about this problem is that it really does force everybody in the room to use proof. There's no mushiness, no intuition to fall back on, just cold reason. The second best thing might be getting student work that looks like this:

2. Exponents for Functions - And all it took was a little bit of nudging to get kids to understand why the hell f^-1 should refer to the inverse of f. It was beautiful. Then we drew the analogy farther, figuring out what a rational "power" of composition would have to mean.

3. Encryption and Inverse Functions - Not huge, but it gave me a language for talking about invertibility. Plus, it was a ton of fun. ("Can you give us, like, enough time to actually figure the code out?")

4. Swap and Solve with Equations - My kids were struggling with equations. They could handle anything that you could undo the steps on, but that thing don't work if you've got variables on both side of an equation. I wanted to share with them the "you've got equal weights on a balanced scale" thing, but I couldn't make it snappy. 

This was a blast. I gave everyone an index card with a number on it, and they had to write an equation that had that number as its solution. Then, they gave their equation to a pal and asked them to solve it.

Why did this work? Because if you want to stump your friend you need to write a hard equation. And once some jerk reveals what makes x + 30 - 20 + 4 - 7 + 1 = 10 a pretty easy problem ("Oh, come on Mr. P, you gave it away!") you have to up your game. To use fancy man language, there was a load of intellectual need in that room.

5. 100m Dash/Stratos Space Jump - We used the 100m dash to talk about linear regression, and the Space Jump to break it. Both of these problems fundamentally worked as contexts for using the line of best fit to make predictions.

6. Height v. Shoe Size - I love making graphs on the white board. This was a particularly fun way to introduce two-variable data to my Algebra students. They put the post-its at their height and shoe size. Hey, look, there's a trend there. And we can talk about outliers too. The next day I took this picture and abstracted everything but the datapoints, leaving a scatterplot. (Explicitly imitating this guy.)

7. Constructing Number Tricks - This was pretty similar to my swap and solve activity with equations, and it worked in a similar way. Kids like coming up with their own things.

5 Worst 

1. Guess-Check-Generalize - This was a boatload of frustration for me. Guess-Check was an easy sell for me; I'm still looking for buyers on Generalize. I tried lots of problems, drawn from CME and Park Math, and they did hook kids in, but every time that I brought in any abstractions I lost the crowd. My one minor success was with this pretty on-the-nose worksheet. Next time I teach this I'm going to try that sort of on-the-nose stuff earlier, and I might also wait until all my kids are extremely comfortable solving equations to attempt teaching this strategy.

2. Life Expectancy - I blogged about this guy already, but it bears repeating: this was a huge disaster lesson for me.

3. Graphs of Inverse Functions - No idea how to teach this. I'm, like, 1 for 6 in attempts to teach this thing, and I'm pretty sure that the one win was a fluke. Maybe the issue is that I just find it really cool that the graphs of a function and its inverse reflect across y = x, and I expect kids to find it as cool as I do. That very well might be the problem, since I tend to teach this by asking kids to graph and bunch of functions and their inverses and keep an eye out for something cool.

Or maybe the issue is that they're not comfortable with technology and graphing interesting functions is cumbersome? Whatever it is, I don't know how to make what really should be a cool idea pop for students.

4. Defining New Symbols - So promising! I love the problems, some of my kids love the problems, and it seems like a great way to practice evaluating expressions while also ramping-up the sophistication for the stronger kids.

It was way too hard for the kids just getting used to variables and expressions, and my attempts at explaining this stuff were just met with blank stares. (We lost a day to me trying, like, three different ways of explaining this to a eerily quiet room.) I love this idea, but I'm not yet sure how to make it work.

5. Percentage/Fractions - Don't know how to teach 'em, especially quickly, especially to Algebra students who have never quite gotten them and need to know them for more advanced topics. I tried a bunch of stuff, and it all kind of failed. The one thing that I'm feeling better about is division by a fraction, which I'm pretty sure that I know how to teach now.* The issue is everything else.

* Next year you can be sure that I'm going to draw out the distinction between two different division models very early. Is 10/2 = 5 because 10 split up into 2 even groups would have 5 members each, or because there are 5 groups of 2 in 10? Only one of these models really works well for 10/0.5.

Bonus: Solving Equations, in General - I don't know how long it takes most teachers to get kids up to speed on solving linear equations, but holy cow it took me a while. We've got to speed things up, I think.

Soapbox

I wouldn't mind seeing your "X Best and Y Worst" post. I think that would be fun.