Monday, December 24, 2012

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 WarsLaser 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.

Friday, December 14, 2012

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

Sunday, December 9, 2012

Making Mathematical Decisions

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


Here are some math mistakes (source: 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.


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. 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?

Monday, December 3, 2012

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.


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

Friday, November 9, 2012

How not to teach it: division by zero

What doesn't work

We all agree that this is unsatisfying:
You aren't allowed to divide by zero, because it's a rule.
And many of us (read: me, yesterday) think that this is better:
What's 10 divided by 2? It's 5, because 10 split up evenly into two groups has 5 in each. 10 divided by 1? 10 in 1 group has 10. But 10 in 0 groups? What would that even mean? 
And, then, having clearly and elegantly explained why dividing by zero would be a very, very silly thing to do, we go back to the day's main topic:
So 2 to the negative 3rd power is 1/8...
Wait, hold on.

I get it: if you think of division as evenly grouping items, then dividing by zero makes no sense. But that's just the normal wear and tear of a mathematical model. We ask kids to believe that exponentiation is like repeated multiplication, and then we ask them to forget that when we introduce negative powers. Multiplication is repeated addition until you throw in "2.3 times 5.1" and then everything goes to hell. We freaking give quadratics imaginary solutions, and our failure to imagine what "0 groups" looks like is stopping us from dividing by zero? Yeah, right.

(Also, wouldn't zero groups have no items in them?)

And another thing: we tell kids that division by zero is undefined. Skeptically, they take out their calculators and punch some keys and get an error message. "Wait! He's right. It gives you an error."

When it comes to dividing by zero, there is a lot wrong with the standard teachery maneuvers:

  • Any sort of "wtf would 0 groups mean" argument does not show that division by 0 is non-sensible. All that it shows is that this particular model of division -- the grouping model -- breaks down for non-integers. That's normal in math. Kids should be regularly creating and discarding conceptual models.
  • "Undefined" is the best we can do? Language matters, and saying that 5/0 is undefined makes it sound like, shoot, well, we were going to get around to it but we just chose to let it slide.
  • The kids are checking their calculators to see if division by zero makes sense. For crying out loud, that's not math. They're wondering whether to believe you or not, because what you're saying doesn't make sense. Hell, everyone knows that 5 divided by 0 is 0. It just makes so much sense...
This works better

March in front of the classroom. "What's 3 divided by 0? Someone tell me NOW," you say.

If your students are a bunch of sissies and nerds they'll shout "You can't divide by zero!"

"Oh don't give me that math teacher stuff. Who says that I can't divide by zero? Give me a real answer."

That's all it takes. Really. They've been waiting in every math class since they were 8 to get this off their chests. 

"3 divided by 0 is 0."

OK, cool. Now we've got something to work with. Ask the class, agree or disagree?

But if 3 divided by 0 is 0, and 5 divided by 0 is 0, then wouldn't this follow?
So 5 = 3, right?

They squirm. They try something else. Maybe 5 divided by 0 is 5? You can handle that too. The point is to lead them to contradiction, and let them grapple with that tension. There are other ways to tug out the contradictions. 

Here's why this is better:
  • We shouldn't be telling kids that the reason that we don't divide by zero is because an intuitively pleasing model fails. That should never stop a good mathematician.
  • Telling kids that division by zero is "undefined" sounds lazy. It's more accurate and informative to say that division by zero leads to contradiction. 
  • How do you help kids see that it leads to contradiction? Take suggestions from the kids of what division by zero should mean, and then let them see the implications. Let them try to make things consistent. Make the choices clear. Sure, we can think of division that way. We'd just have to refine our rule for multiplying fractions. So, what's your new rule for multiplying fractions?
All of this is more authentic than what I used to do. ("What I used to do"? I'm talking about yesterday. Things move fast around here.) The biggest change in what I'm doing this year is slowing down and having kids make arguments in class. Proof and argument is how I'm helping my kids make sense of this stuff. There's no shortcut or substitute.

Tuesday, October 30, 2012

Answering an easier question

There are a wide variety of mathematical errors, and it's worthwhile to try to find patterns and themes that stand behind the particular mistakes.

The following quote is from Thinking Fast and Thinking Slow:
"I propose a simple account of how we generate intuitive opinions on complex matters. If a satisfactory answer to a hard question is not found quickly, [the intuitive capacity] will find a related question that is easier and will answer it. I call the operation of answering one question in place of another substitution. I also adopt the following terms: the target question is the assessment you intend to produce; the heuristic question is the simpler question that you answer instead."

Not all errors involve some sort of substitution. Sometimes we avoid jumping to a conclusion, we grapple with the difficult problem, but we still make a substantive error. This is just one type of mathematical error.

Questions for homework:
  1. Do you agree with the author of this post? 
  2. Are there other categories of mathematical error that you can identify?
  3. What can teachers do to effect the usage of heuristic questions by their students?
  4. Is this the sort of question the domain of psychologists? Of teachers? Of both? 

Tuesday, October 23, 2012

Encryption and Inverse Functions: First Draft

Here's what kids usually see when they walk in to my room:

Here's what they saw today:

I told them to break the code. It didn't take long, especially because there was a huge hint up there. But the point was that I wanted to talk about codes, encryption and reversible functions today.

After they broke the code, I asked them to explain the encryption process in terms of functions. We ended up with G(a), which takes letters and spits out numbers, and f(n), which takes a number and gives you three more than that.

Then I gave them another encryption.
This time I told them the key. It was g(n) = absolute value(n - 10).

"Wait, it could be two letters."
"It's 'HELLO' but it could've been 'FILLO'."

It's a lousy code, because it's ambiguous. The information about the starting letter is ambiguous. 

The rest of the lesson was sort of lousy, with some good moments. I teachernated that if a function is reversible, then it makes a good code. And another way to say that it's reversible is that it has an inverse function. Most of the rest of class was spent trying to figure out if various functions had inverses. But there were some highlights:
  • "It's only a bad code if you use all the alphabet." We talked about restricting the domain artificially.
  • "So any code that has two different letters with the same number is lousy." Nailed it, kid.
Basically, I'm sold on the idea of using encryption as a context for motivating the distinction between one-to-one and non-one-to-one functions, and I'm also sold that this can motivate functions versus non-functions. (Just try imagining what the inverse of one of those non-invertible functions would look like.)

But I feel like I didn't nail this lesson. The concept seems solid, but I don't think I made it really interesting or especially challenging. Any ideas on how to improve it? I'm giving it another shot next week with 11th graders.

Wednesday, October 10, 2012

"Substituting for x" is a subtle killer

"So I just swap the number and then treat it like arithmetic? Oh, that's easy!"

Here are some common mistakes kids make when evaluating expressions or functions:
  • Able to evaluate forward, but unable to undo the evaluation, even when given something like f(230).
  • They'll swear to you that a^2 is -1 when a is -1, because -1^2 is 1, even though they know that (-1)^2 = 1, and that -1 times -1 is 1.
  • They'll make weird calculation errors when evaluating expressions that they wouldn't make if they were just doing the arithmetic.
I think that if you want to help your kids avoid these mistakes, you're not doing them any favors by talking about swapping, replacing, substituting or blanks. All of this language support a "mystery value" picture of expressions and functions, where variables stand for particular numbers, and every variable is just waiting to be revealed as standing for a particular mystery number.

Instead, it's helpful for kids to think of expressions and functions as operations to be done on any number. Number tricks are a nice way of setting this up, but I think that you can undercut things by talking about swapping/replacing/blanks when dealing with expressions or functions. The reason is (and this is subtle, and possibly wrong) because swapping says "this expression is just about particular numbers."

Better language would be applying the expression/function to a number. This emphasizes that the expressions says something about numbers in general, which can be applied to any particular number. (Evaluating is fairly neutral language, but not if you define evaluating as "substituting.")


There's a further difficulty when teaching function notation that I want to get off my chest. If you introduce function notation with evaluation, and define evaluation as swapping, kids miss out on the subtleties of the notation. Why do they miss out? Because evaluation with swapping is too easy -- you just ignore the random letter before the parentheses, take the number inside the parentheses and swap any variables with that number.

But does that f stand for something? And what are the parentheses doing? What is f equal to? What if you have an f inside the f? Is that like f times f? And what does this have to do with outputs and inputs? Does f stand for the output?

Evaluating functions with swapping doesn't give kids enough friction to force them to notice the weirdness of this notation. And that means that they're missing out on the move from seeing functions as processes to seeing them as mathematical objects, the sorts of things that we can use adjectives and predicates to describe.

Saturday, October 6, 2012

Productivity Experiments

Everybody with an internet connection is participating in a massive experiment. The experiment goes like this: what is the effect of a distraction machine on the human race? (The Amish are the control group.)

I'm incredibly nervous, all the time, about how effectively I'm doing stuff and getting better at doing stuff. And -- right now -- curbing my internet habits is the major front of that effort.

Here's what I've done so far:

  • Killed Facebook.
  • Got my inbox size down. Way down. I print out messages that I'll need to respond to later and post them on a bulletin board.
  • I've bought a bulletin board, by the way. It's great. I post my monthly budget and emails that I need to respond to. I'm less nervous about losing track of stuff. My mind is more settled.
  • I've set up a filter to eliminate the different between read and unread messages. So far? The results aren't great. I'm still checking my email very often, though. We'll wait and see on this one.
  • I'm pretty excited about this one: I've eliminated Google Reader and replaced it with FeedDemon. There are two reasons why I think this is going to make me a more effective blog consumer. First, my RSS reader is no longer in the browser. That means that I can't access it from any computer other than the one that I leave at home. It also allows me to set up filters so that I can try the read/unread experiment with my blogs also. (You can't do that in Google Reader, I think. And it costs $20 for the license to set that up in FeedDemon.) 
  • I also unsubscribed from blogs that post often enough that I could hope to gain something by checking my reader more than once or twice a day.
Overall, the goal is to start batching my consumption of online stuff.

I think that this stuff matters. A lot of folks recommend subscribing to hundreds of blogs and scanning them quickly to find the important stuff. Same with twitter. (Which I struggle with too.) That might work for some folks, but being distracted doesn't support my goal of being a thoughtful teacher that (eventually) comes up with some really good stuff. So they have to go.

Wednesday, October 3, 2012

Rational Expressions Announces $1 Million Prize for Solution of Teaching Problems

Prizes and contests spur innovation. Think of the Millenium Prize Problems or the X-Prize. The other thing that spurs innovation? Lists of problems. Like Hilbert's. Or Jay-Z's.

In the spirit of all these lists and contests, I'm happy to announce the Really Important Teaching Problems (RITP). These are some of the most difficult, knotty problems that teachers are grappling with in this new century. Work on these problems continues because of their importance and seriousness.

Successful solution of any of the RITP problems will be awarded with the following:
  1. A blog post about your solution
  2. One-million dollars
(At this point it seems necessary to mention the generosity of the Gates Foundation.)

Attempts were made at succinct and direct statements of the problem. Problem were selected with input from our board of advisors. Without any further delay, I present to you the RITP problems:

  1. The (Much) Better Lesson Problem: Is it possible to use the internet to create a free curriculum of the highest quality?
  2. Khan's Conjecture: Can classroom learning be personalized to the stage that it performs as well as a high quality tutor?
  3. Meyer Theory : How close can a classroom teacher get to completely engaging every student with every topic?
  4. The Theoretical Problem: Can good math teaching be well-described, understood, and taught to new teachers?
[The board has approved changes to this list of problems, contingent on convincing arguments being dropped into the comments of this post.]

So, get on it, everybody. I don't want to keep all this money for myself.

Tuesday, September 25, 2012

Life Expectancy, and a lesson that didn't work

Here was something that didn't work with either of my Algebra 2 classes. I'm wondering why it didn't work, and if there's a better draft of this problem to be made.

The lesson was vaguely three-acty.

Act One

How long does the calculator say that a person born in the 90s will live? The 80s? The 1890s? Why the difference?

I asked one class: "When will you have to be born to expect to live to 100?" The other class got "How long can you expect to live if you're born in 2010? 2050?"

Act Two

[Source: CDC]

What do you notice about the table? Take a guess for 1980. My guess is 60 years. You guys like that? Why not?

How good is the following rule: "life expectancy = years * 10"? Is the rule "life expectancy = years * 9" better or worse? How can you find a rule that's better than either of them? What are you changing in the equation?

Can you find an equation that fits it pretty well? How far off would the predictions be?

Act Three

Here's our most recent data. What do your equations predict?

What went wrong?

First, the objective stuff:
  • Kids didn't seem into it.
  • Kids didn't know where to jump in.
  • Kids were confused by the idea that it has to be a rule that gives a line.
  • Kids thought it unnatural to make a prediction based only on a few prior data points.
Other issues:
  • It wasn't clear to me or them what they were trying to predict. Since we can't check their actual predictions (cuz they're in the future) we have to just limit the data that we make available to them. This seems to be a limitation of the "data analysis of social stuff" type of problem, and an argument for doing regression problems with stuff that we can actually test in the classroom.
  • We'd done a similar, and superior, problem last week with data from 100m dash times. I wanted kids to end up with actual equations as models, and I don't think this was different enough to necessitate equations. A lot of kids repeated their tricks from last time: averaging the rate of change, coming up with recursive rules instead of closed-form rules. I didn't feel as if anything, other than my insistence, was pushing them towards closed-form equations.
  • Post it as a historical puzzle. Let's say you were in 1960: how far off would your best prediction be for life expectancy in 2010?
  • Find a better hook. I needed something like that life expectancy calculator just to make sure kids knew what "life expectancy" means. 
Help? Anyone?

Friday, September 14, 2012

The hard problem of online learning

Idea: Kids learn things on their own, at their own pace.
Problem: They'll get stuck, and frustrated.

Idea: We'll give kids feedback automatically that doesn't need the attention of a person, so that they can still learn stuff on their own.
Problem: That's complicated.

Idea: We'll use computers.
Problem: Computers are pretty cheap, but the quality of the feedback isn't very good.

(See also: Turing Test)

This is a really tough problem. Nobody has any ideas that seem entirely promising. Maybe we just have to wait for the technology to improve.

Here's my pitch for a shift in the way we think about this problem.

In the world of "giving feedback to kids on math" there are two contestants. There's (1) people and (2) machines, and people are beating the stuffing out of machines. Truth is, machines haven't ever shown that they're up to snuff, as far as quality goes.

But humans have issues too. A single human can only be in one place at a time and can only focus on giving feedback to one kid at a time. We people have all sorts of stuff to do besides slavishly providing kids with quality feedback. I mean, unless you're a teacher. But, then you have to figure out a way to give quality feedback to a few dozen kids at once. Humans are limited in a way that machines aren't.

But humans are making progress. Technology is the key, here. Advances in machines have allowed humans to group together to overcome some of the limitations of being a person. If you mess around with Wikipedia,   a human being will find out and fix it. Closer to our discussion, if you ask a question on Math StackExchange you'll get a good, quality response before long. If you ask a teaching question on twitter, you can also depend on getting a good answer.

The lesson here is that the humans are making progress.

Nobody has figured out a way to create a site where math students can get quality human feedback on any topic they're studying. Nobody has figured out a way to get quality feedback out of machines. Pretty much everybody who's working on online learning is trying to figure out a way to help the machines. In the meantime, sites like StackExchange or Physics Forums keep on building and improving their communities.

I've got no idea how to build a site that has a quality, supportive community of students and adults who will provide quality feedback to K-12 students. But there are places on the web that are like this. And if you can sustain a vibrant online community, then you can start creating more difficult tasks for students online. The rate of innovation in online curricula could speed up quite a bit.

Creating a quality community is a hard problem. Creating a piece of software that can give excellent feedback is an attractive and lucrative problem.

The pitch is: focus on the hard problem, not the attractive one.

Thursday, September 6, 2012

What kids hate about school

I've struggled in the past to figure out a way to start the conversation about classroom culture with kids. Here's what I hit on this year:

Here's what I learned:

  1. Kids hate being bored, love being interested.
  2. Kids hate doing homework.
Full responses below:
Best Worst Classrooms

Next step: Post these notes in a very public place.

Update: Ken Templeton has provided us with worldes, and there was much rejoicing.


Wednesday, September 5, 2012

Reality Checks

I just posted this on Math Mistakes. It comes from a piece of work that I collected at the end of class today.

Collecting that first batch of student work was my favorite part of the day. It turned an impossible job into a difficult one. How can I help them learn if I don't know how they think? What it cost me was 3 minutes at the end of class and 10 minutes of analysis after class. And now I know who I'm (likely) going to need to target, who needs me to call home over the next few days, who seems to have their act together.

Consider this an official call for submissions. Collect work from your kids often, and take a picture of some of the interesting stuff and email it to mathmistakes-at-gmail-dot-com.

One last time: collect information at the end of as many classes at possible over the first two weeks. Try to hold in the laughter when your colleagues start complaining about the surprises on those first batch of end-of-chapter tests.

Tuesday, September 4, 2012

First First Day

I have two first days. First the freshmen come, then the big boys. It's nice for me, because I get to adjust some of my shtick for the second round.

Assigning seats is one of these little things that caused me a ton of stress last year, because I always left it for the last minute and I always screwed things up like forgetting to give a kid a seat or giving a kid four seats. Anyway, here's how I fixed it this year:

Plus, the coordinate plane is in the curriculum.

As long as I'm posting, here's some other assorted stuff from today:

The above is the white board in the computer lab before I applied an hour and a bottle of rubbing alcohol to it. The board hadn't been cleaned in (get this) 5+ years, and it was tough work to get that ink out.

Here's my math classroom. Because of various weirdnesses about our school there are lots of times when students are in the classroom without an adult, and we've gotten ourselves a reputation for not being able to have nice things because the kids can rip them apart. 

This year I want to try out the opposite approach. I'm going to try to keep filling the room with nice (but inexpensive...) stuff, like plants, storage boxes where I can keep materials, posters and calendars. Basically, I want to mark my territory.

We solved these problems today, all of which were happily stolen from Park Math:
Handout p2d1

Which already yielded this happy misunderstanding:

Sunday, September 2, 2012

Post-mortem on a mistake I made

James Tanton's latest newsletter is phenomenal. But I got stuck on this part:
What was driving me nuts was that I thought that this argument was too loose. So I tweeted my question to the author:

Then, I got a series of very helpful tweets from Justin Lanier:

But he wasn't answering my question! He didn't understand me. I had to restate the problem I was having. Justin patiently repeated his argument. Why didn't he get it? How could I explain my issue better?

Then, James Tanton offered some help as well, giving a nearly identical argument as Justin's:
That's when it hit me.

All of a sudden everything that Justin and James Tanton had said made perfect sense. My mistake was clear. I saw where I went wrong. I had messed up an argument, and not an especially tricky one. Besides, this was high school math -- the thing that I'm supposed to be teaching. I felt embarrassed.

Here are some teaching lessons that I want to take away from this experience, assuming that what I experienced is true of others too:
  1. When someone understands most of something, they're equipped to turn a misunderstanding into an objection, and it's much harder to convince a person that their objection is wrong than it is to correct a misunderstanding. The only thing that worked for me was (a) having a second person explain it to me (b) after a break from thinking about the problem. Which, by the way, means that
  2. kids shouldn't be forced to work through problems in order, unless that sequence is necessary. Moving between problems often helps when you're in a rut.
  3. Doing and thinking about math during the year is an important part of teaching. Moments like these remind me a lot of what it felt like to be clueless during high school and college in my math and science classes.
  4. This post is still true. My insecurities about learning could easily bleed into my teaching, but I shouldn't let them.

There was a more self-indulgent post that I wanted to write. I'll throw it in the afterword, though it probably belongs elsewhere.

Here's what I remember about math and science classes in high school and college:
  • Asking dumb questions that everybody else understood in the back of Algebra.
  • Finishing last on Calculus tests.
  • Having to go to office hours every night for Multi-Variable Calculus in college.
  • Trying to understand how my friends got their answers for Mechanics.
  • Not understanding any math lecture that I went to in college, ever. 
I'm slow. Some people are sharp, quick-witted, and that's just not the sort of thing that has ever really been true about me. The kind of difficult I had above is the kind of difficulty I've been having my whole life, as far as learning stuff goes.

To turn it back to students, some of them are sharp, some of them are slow. And I think it's important to remember that nearly every aspect of school celebrates the quick and sharp intelligence over the slow one. Let's not pretend that waiting 30 seconds before taking an answer to a question is enough (though it helps) to even the playing field. If you're slow you tend to do worse on timed tests and on homework. Your sharper classmates will solve more problems than you during class. If you're slow then you finish class and your notebook seems foreign.

Though, maybe being sharp and quick is part of what it means to be good at math. Thoughts?

Friday, August 24, 2012

White Paper on Problem Solving: Struggle without frustration

I’m pretty confident that if I just give my kids a bunch of really hard problems to solve, the following will happen:
  • First, the class erupts with demands for help.
  • Then, the class collectively gives up. Two kids keep on working, because they’re those kids.
  • Students complain to their parents, who complain to the school, who pass it on to me. (“They say you’re not teaching them anything.”)
  • I fax (fax!) everybody in school a long memo detailing where our profession has gone wrong and how to steer it right.
  • I get fired and lose all my clients save for one hot-headed wide receiver.
  • Together we teach each other the importance of trust, love and commitment in both personal and professional relationships.

And I certainly don’t want that to happen.

In the previous post I explained how, last year, I responded to the pressure of keeping my kids un-frustrated by making my problem sets easier. In the first post, I explained why I shouldn't have done that. In this installment I want to come up with some strategies for helping my kids feel comfortable with struggle. I’ll keep my eye on the comments; if you’ve got something good, I’ll toss it into the post.

But first, yet another picture of a cat with a Rubik's cube. Seriously, how many of these are there on the internet?

Never mind. Stupid question.

How to keep kids from getting frustrated* by difficult problems:

* Frustration is sometimes OK, but is just as often unproductive for a student. For the rest of this post, you can assume I'm talking about the unproductive stuff.

Let's start with a distinction. When a kid gets frustrated in an unproductive way while working on math, her frustration comes in two flavors:
  • social
  • intrinsic
Social frustration comes from feeling as if she's inadequate relative to her peers or relative to the expectations of others. It's real, but it comes from her understanding of other people's views of her. Intrinsic frustration is everything else. It's the stuff that would cause a person to walk away from a problem even in a closed room, with nobody watching. (I don't really know where "privately feeling stupid" fits. But whatever.)

Here are ideas for minimizing the social pressures:
  • Be explicit: One day last year, during a quiz, a kid pointed at me and said, "Mr. P, you put a problem on this that we've never seen before!" And I was, like, yeah that's what I was trying to do. But that was actually a really good moment, when the class came to understand what I was about. I should've talked about that in the first week: "Yo, kids, I know this class is different. But it works and you'll still have support, and it'll be OK." That sort of thing.
  • Find unfinished answers interesting: Last year we never had conversations about the kids' work with the whole group. This year I'll bring to the fore not just correct answers, not just finished answers, but approaches and ideas from unfinished problems. I'll intentionally spread the wealth, so that we're talking about everybody's work, eventually. And we won't do the embarrassing "So where did he go wrong?" questions, at least not at first. Instead we'll celebrate the process by asking, "How could we finish it off using his approach?" We won't force everyone to go through the ringer, at least not at first.
  • Try to build team mentality: This is a bigger classroom management puzzle for me, but I'm going to start by throwing in questions to the problem set that say "Look around. Does anybody need help? Take five and see if you can be of service." 
And here are some ideas for minimizing the intrinsic pressures:
  • Mix up easy (but cool) and hard problems: When things are going well, when you're in the zone, you're in a state of flow. Flow is what keeps most of us coming back for more, even when the going gets rough. By mixing up the satisfying questions with the knotty ones, I'm betting that I can get more kids working for longer. (Also, a really good problem has easy, cool and hard aspects all wrapped up in one neat package. Keep an eye out for those.)
  • Be more interesting: Whenever a kid gives up on a problem, part of the problem is motivation. If the problem was SUPER interesting, he would probably keep chugging away. So I need to do a better job finding more interesting problems and more interesting hooks into those problems. 
  • Interrupt more often: There were times last year when kids would be working on problem sets (i.e. worksheets) for 20-30 minutes without serious interruption. That's great, but one way to release the pressure of frustrated students is to pause and take a deep breath. I'll interrupt them more often when I sense that people are struggling. We'll talk about the problems, get some approaches and strategies up on the board, discuss next steps and then send them off for another 5-10 minutes.
  • Refer to common problem solving strategies: I'm betting that if we have a repertoire of habits of mind/strategies that we can all talk about, things will go down easier. I think we'll probably have a poster up in the classroom (like Daniel Schneider's) that has a list of things to try when you're stuck. We'll start with three: Guess/Check/Generalize, Tinker and Find an Easier Problem.
I fully expect the comments to be awesome on this post. 

Coming up next I'll agonize over whether this sort of problem solving should be an everyday part of my class or whether it belongs alongside other ways of doing things. I'll also post a sample problem set for the first week of class.

Thursday, August 23, 2012

White Paper on Problem Solving: What I did last year

In the first part of this series I defined problem solving as struggling over difficult problems, and I tried to work out what the pedagogical benefits of struggle are. In this post I want to turn a critical eye to the way that I pulled this off last year. Hopefully, I'll end up with a better strategy for this coming year.

But first, here's another cat with a Rubik's cube.

Cats: the silent killers.

What I did last year:
Last fall, two things happened that lead to a change in my classroom.
  1. I noticed that students enjoyed working on my “Warm Up” problems.
  2. I read the book “Drive” by Dan Pink.

What happened as a result is that I put solving problems at the center of my classroom. I felt empowered by how my students enjoyed spending more class time solving problems, and because I read “Drive” I had a framework for understanding why they liked it. I was giving them more autonomy, and people like having choices about how they work.

Here is the sort of thing that I was putting in front of my students with regularity last year:

How proud should I feel of this work?  There are certainly some things that are going right here.
  • This problem set starts with a “Warm Up” section that brings together different problems that connect to the new problem being solved in the “Important Stuff.”
  • The problem set is designed so that students can work on it on their own, and with groups. Kids prefer that to a lecture or an activity that occurs with the whole group. I’ll get more face time with students who are struggling with these ideas, which is also a good thing.
  • The problem set also starts with a question that is pretty concrete and easy, so that the tricky, abstract stuff is built on top of a firm starting point.
But there are also some serious problems lurking under the surface. In particular, I’m doing much of the intellectual work for the students just by sequencing these questions the way that I do. Students knew to expect that there would be a relatively straight line running through the problem set leading to an insight. The result was that a major component of a student’s struggle over a problem was an attempt to figure out the connection between sequences of questions.

Take question 4 in the above document. Just from the sequence of questions students are likely to infer that the way to find the angles whose sine is 0.43 by using the arcsine function to find one angle, and then to find the other using the unit circle.  This kills any chance of multiple approaches to the problem, reduces the difficulty of the problem by many factors, and doesn’t give students a chance to search their memories for a helpful approach.

(I also think that, while it’s admirable that the “Warm Up” contains problems from different topics that connect to the new one, it’s probably a mistake to corner them off into their own section.)

If you see more issues with my approach, please give a shout in the comments.

There’s a reason why I made this problem set the way that I did, though. I was nervous that students wouldn’t be able to figure out the actual new problem of the day on their own. And that’s true. Kids would have gotten frustrated if I just gave them the difficult problem without any context. But the solution that I came up with was to make the problem much, much easier. I need to figure out different ways to keep kids from getting frustrated by difficult problems.

That’s what I’ll write about in the next post. 

Wednesday, August 22, 2012

There is no evidence for the usefulness of math in non-mathematical contexts

"Learning mathematics forces one to learn how to think very logically and to solve problems using that skill. It also teaches one to be precise in thoughts and words. Practice doing that is obviously very useful in many different areas of life." - The Math Forum
Let's make a bold claim. I'm going to claim that there is no evidence that learning math makes you better at other things unless those other things are
  1. math
  2. things that use math, (i.e. they're just math)
I'd love for someone to convince me that I'm wrong on this. Do mathematical habits or skills transfer to other contexts? Drop in the comments if you've got a good argument or some evidence.


Here are some other folks who make claims about the usefulness of math.
"What of the claim that mathematics sharpens our minds and makes us more intellectually adept as individuals and a citizen body? It’s true that mathematics requires mental exertion. But there’s no evidence that being able to prove (x² + y²)² = (x² - y²)² + (2xy)² leads to more credible political opinions or social analysis." -Andrew Hacker, "Is Algebra Necessary?"
The argument for algebra rests on the transfer from math to other areas of life, something that has never been proven despite the claims of people such as University of Virginia cognitive scientist Daniel Willingham. -- Roger Schank, "No Algebra Isn't Necessary"

White Paper on Problem Solving: The Why

I'm putting together a short series of posts on problem-solving to get myself ready for the new year. In particular, there are a bunch of changes that I want to make in my classroom and I want to make sure those are properly justified and motivated.

But first, look at this cat:

Such beautiful creatures. But they'll burn you if you're not careful. Anyway,

What do I mean by “problem solving”?
For me it means that students are regularly asked to make progress on questions that they have never been told how to answer. This isn’t an air-tight definition, but it will do for now.

There are a lot of supposed benefits of a problem-solving approach to learning math. Here are a few that come to mind:

  • It’s truer to the work that mathematicians do
  • It’s more fun for students
  • It develops habits of mind that are transferable

I think I agree with the first two ideas, and I’m skeptical of the third, but that's all sort of beside the point. My core responsibility in the classroom is to teach these kids a bunch of skills and concepts in a way that compares favorably to the way they’re learning them next door. If the most effective way to teach is lecturing and drilling, I will teach that way, even if it’s boring and unlike the way that mathematicians work.

The good news is that fun, truth and effective learning coincide in this case.  I think.

I want my students to solve difficult problems in class because I believe it’s the most effective way for them to learn and remember the content. Here are my pedagogical assumptions:

  1. Difficult tasks help organize knowledge: When a person is faced with a difficult task, they search their memory for a way to accomplish the task. They think about the tools that they have and how well they fit the task at hand. This search reinforces a person’s understanding of their tools and how they are used. In math, the tools are the sorts of things we want kids to know: procedures, skills, concepts and habits of mind.
  2. Organization takes the form of connections between topics: It's a pretty solid result that novices organize knowledge by topic, and experts organize them by their underlying structure. A difficult problem doesn’t cue students into the tools that they’ll need to use, and so anything might be relevant. As students attempt difficult problems they need to start organizing what they know into more useful clusters than “first semester” or “lines stuff.” Instead, when presented with an equation they’ll start thinking about what tools they have for solving equations. When presented with a challenging proof they’ll need to think about other problems that they’ve proven in the past and decide which ones are relevant for the current puzzle.
  3. Students will fail often:  Some studies have shown that knowledge sticks better after a person has taken a difficult test and failed. This makes a certain amount of sense – the brain is most attentive when we know we’re missing something. The right answers come in a problem-solving class, but they will always follow failure.
  4. Different approaches invite justification:  It’s helpful for learning to have different approaches to discuss. Multiple approaches create the need for explanation, and explanation and justification also help students organize and remember their mathematical knowledge. When solving a good problem, students will almost never have just one approach. The teacher can skillfully select multiple approaches to bring to the fore.
  5. The mind remembers stories very well:  "If you want to make something memorable, you first have to make it meaningful." But how do you make it meaningful? Stories that connect with the rest of the things that you know can do this. As Dan Meyer has put it, good math stories come in three acts. First comes the hook, where the problem is posed. Then comes the development, where students struggle with the math and run into trouble. Then comes the resolution, which we might talk about as a whole group, or students might discover on their own. The daily story telling comes easily: “Today we tried to do this, and we ran into trouble. Then we discovered X, and then we were able to solve the problem. But what about Y? See you guys tomorrow.” 
These are the things that I think are true. Where possible, I’ve pointed to evidence supporting my assumptions. But they’re empirical assumptions, and I’d feel better if I had more evidence supporting them. Where did I go wrong? Lemme know in the comments.

Coming up in this series I'll point to things that I was doing wrong last year and how I think that I can fix them.

Wednesday, August 1, 2012

Pop Quizzes and Probability

Here's a fun probability problem that worked pretty well in class, and is pretty easy to pull off:

1.  Tell everyone to clear their desks. It's a pop quiz. Use your serious face.
2.  Hand them this:

3.  Read the questions out loud. It gets kind of fun. 
4.  What's the question that everyone wants to know? In my classes, it was "Wait, did I pass?" It was a smooth transition from that question to the question I asked: "Well, what are the chances that you passed?"

Then we're in familiar territory. Take guesses, write them on the board. Take ideas from the whole group. Circulate, offer suggestions, ask questions, give hints as you see fit.

The kids won't let you forget to read out the answers before the end of class.


There was a pretty interesting design question that came up as I was planning this activity: what should the quiz questions be? In order to get random guessing you just need a set of problems that kids couldn't possibly know the answers to. Here were some alternatives that I came up with when planning this problem:

In the end,  I wanted a set of questions that kids could think -- just for a second -- that I actually expected them to know. So I went with some Linear Algebra pulled from a test I took in college. But something else might work for your students.