Exponents As Abbreviations, Exponents As Language All Its Own

Most teachers think that exponents are a way to write 4*4*4*4*4*4*4*4*4*4*4*4 in a convenient way. In other words, most teachers think that, fundamentally, exponents are an abbreviation. This is obviously a legitimate mathematical perspective.

If exponents are abbreviations, then your lessons should give kids a chance to feel the need to abbreviate. My favorite version of this way of thinking about teaching exponents comes from Dan.

You want kids to feel the impulse to abbreviate? Give them big numbers and tell them to write 'em down! It's elegant.

There's an entirely different view about what exponents are. In this world, exponents aren't a notational shortcut anymore than multiplication is a notational shortcut of addition. It's a language to describe a sort of pattern, a special sort of number. Instead of just an abbreviation, exponents are the language of successive grouping.

They're the sort of thing that let us say what we see -- what we should be able to see -- when we look at this picture:

The teaching that follows from this view of exponents is going to look a lot different than Dan's (awesome!) lesson. We're going to need to figure out how to create a need for this language in our students. Creating that need is going to need two things: 1) Helping our kids see what's special about these patterns and 2) Running up against the limits of our language for describing these patterns.

But how can we do this? Stay tuned for some thoughts!

Immediate Feedback Is Too Soon For Feedback

The Quickest Feedback You Can Get, via Frank

(Blogging between classes here, so I've set myself a 7 minute time limit on writing this post. Apologies if it shows!)

(The obvious questions is, can't the post wait? That's a good question.)

What makes for good feedback on a quiz?

The most common answer around the internet is that it all has to do with timing. Immediate, personal feedback is the best, and we should take steps to shrink the amount of time between an assessment and our input. I totally subscribed to this approach for a few years. Frank's brilliant management system made it possible for me to pull this off in my classes.

Lately, I've been questioning the assumption that the best feedback is immediate. For one, there's a sort of emotional investment that a kid has with a quiz right after they've finished it. A lot of the kids who checked their quizzes right after they finished had this sort of "I got this wrong? WHY?????" reaction that I didn't love.

I also suspect that there are learning benefits to delaying the feedback on a task a day or two. Delayed exposure to ideas, and all that.

And the other thing is that, realistically, the only way to pull this off is for kids to be all on their own with an answer key. Maybe you guys are better at making answer keys than I am, but I wasn't sure how much my kids were getting out of them. What they really wanted was a conversation about the problems, but I couldn't give the kids that conversation because their classmates were still taking the quiz.

(Actually, I'm sure you guys are better at making answer keys than I am because I always end up hastily scribbling them on my way down to the photocopier machine.)

(I've just blown past my 7 minute deadline. We're up to 11 minutes now. Oh dear.)

In short, I wanted to give my kids feedback with a little bit of a delay, but not too long of a delay. I wanted kids to be able to reflect on their own work, but I wanted it to be social.

Here's what I do now:

1. Kids take quizzes on Friday.
2. I collect them, photocopy them, take notes and make grades. Digitally scan them for reference.
3. Kids get back their originals on Monday with no notes on them from me. They get back what they handed in.
4. They get into groups, go over their quizzes together, iron out any differences or disagreements. I always make my quizzes a little bit on the long side, so there's usually a question or two they didn't get to on Friday that they can work on if they finish.
5. I have notes about areas of concern. I check in on kids who I had flagged for a conversation. I ask groups questions if there's a disagreement that they're not seeing yet. I'll ask the whole class questions if there are problems with widespread errors.

And the only part of this that I lied about is the part about giving grades, since I'm currently teaching at a school that doesn't give grades. But that's how I would've graded them, if I had to. 

It's entirely possible that this system wouldn't work in a place where grades were given, like where I'd been teaching before this year. Kids would probably go nuts not knowing what their grades were. I'm optimistic that, with enough time, I could have convinced them that this way was better, but dunno. 

If you try this, let me know how it goes!

(Clocking in at 19 minutes. Gah!)

Bear With Me, I'm Trying Really Hard To Understand Dweck's Distinction Between Learning and Performance Goals

In "Motivational Processes Affecting Learning," Carol Dweck distinguishes between performance goals and learning goals, and ties a lot of good behaviors to the having of learning goals and the opposite to the having of performance goals. (link)

A learning goal is defined as a goal "in which individuals seek to increase their competence, to understand or master something new."

A performance goal is defined as a goal "in which individuals seek to gain favorable judgments of their competence or avoid negative judgments of their competence."

If a child has a performance goal, their "satisfaction with outcomes is based on the ability they believe they have displayed."

If a child has a learning goal, their "satisfaction with outcomes is based on the effort they have exerted in pursuit of the goal."

---

Jim: Wait, that's cheating! A learning goal is just a type of performance goal.

Nancy: Oh great, here he goes again...

Jim: Now hold on, I'm being serious. If I have a learning goal, that means that I'm trying to increase my competence. Isn't that, in itself, a performance?

Nancy: We've been through this before, and you've got it all wrong. Learning goals are all about not caring whether other people judge you or not. A learning goal is one where there is no performance.

Jim: You seem to be saying that in order for me to have a performance goal, there needs to be someone actually judging me. I think I have a counterexample to your claim. Sometimes a person will do something just because they want to feel good about themselves.

Nancy: Of course. Like, everybody on Facebook is posting their scores from 2048. That's classic performance goal behavior.

Jim: Yeah, true. But haven't you sometimes played a game like that, even though you never shared your score with anyone? Like, a person is motivated to play 2048 because they're good at 2048, and they want to get a high score because it'll make them feel smart, but not because anyone else is actually going to judge them as smart or talented. It's all about the judgement that they give themselves.

Nancy: That sounds right to me. But that would just be another, more abstract, version of a performance goal. When you play 2048 alone in your room and never share your score, it might be because you want to be gain a favorable judgement...from yourself!

Jim: It's almost like imagining that others could judge you favorably, and that's where the satisfaction comes from.

Nancy: So I agree with that. What's the problem?

Jim: Take a close look at the definition of a learning goal. It's a goal "in which individuals seek to increase their competence, to understand or master something new." Imagine someone who really wants to increase their competence at a sport, like bowling. And they try really, really hard. And then they fail. They don't get any better. Would a person like that be satisfied?

Nancy: Dweck says that he would be satisfied with the effort that he put in to bowling.

Jim: But why would he be satisfied? His goal was to increase his competence, and he failed!

Nancy: Sure, but he failed on his own, it's not like anyone else is judging him for his failure...

Jim: ... but we said that having a performance goal doesn't depend on actually being judged by anyone else. It just has to do with doing the sort of thing that could be judged as a failure by others. What makes this learning goal different than a performance goal?

Nancy: Hmm. I see what you're saying. But there's obviously a difference between the sort of person that is motivated by a performance goal and the sort of person that is just trying to do their best. I mean, can't we imagine a person that would just be satisfied with effort? Someone like our bad bowler, but who isn't trying to achieve any learning in particular? Someone who is just trying to try to learn?

Jim: I think so. But what would we call that objective? It's not a learning goal, exactly. It's like a trying goal? An effort goal?

Nancy: I think maybe it has more to do about character. Our bowler isn't trying to do anything, he's just trying to be a certain sort of person. He wants to be the sort of people that is satisfied with effort, even if he don't meet his goals?

Jim: This is all sounding awfully Stoic. I need to think about it all.

---

This doesn't exactly fit into the dialogue, but here's a bit from a recent article that's related to all this.

This dovetails with new research led by the psychologist Christopher J. Bryan, who finds that for moral behaviors, nouns work better than verbs. To get 3- to 6-year-olds to help with a task, rather than inviting them “to help,” it was 22 to 29 percent more effective to encourage them to “be a helper.” Cheating was cut in half when instead of, “Please don’t cheat,” participants were told, “Please don’t be a cheater.” When our actions become a reflection of our character, we lean more heavily toward the moral and generous choices. Over time it can become part of us.

Maybe we want people to be "hard workers," or "good learners"? Ideas?

A Hastily Composed Rant Against Fancy-Pants Colleges

Apparently, adults are still telling children who will definitely be going to go to college that they should try to get into the best college possible. Moreover, apparently we teachers are actively encouraging our high-flying students to get into the highest-ranked college possible.

Here's my argument that this is an awful thing and that we should stop.

You'll learn more at the top-notch universities.

Let's start with the observation that universities don't typically do anything interesting, teaching-wise. The bread and butter and meat and potatoes and carrots and salmon of your college education is the lecture. And lecture isn't terribly effective, as kids know when they're in high school but seem to forget six months later at good old U of Suck, where all pedagogical sins are forgiven. If you're getting an education in lectures, you're not getting much of an education at all, at least not from your classroom time.

(As long as we're here: forcing students to figure out a way to learn despite your lousy teaching is not a valid way of teaching intellectual independence. Good teaching promotes independence, along with everything else.)

If you're being lectured to, then who cares where you go for your lectures? It just doesn't make sense, as far as learning goes.

So what are you paying for?

The big bill at a college is for everything besides the classroom. The costs of research. The costs of school spirit. The costs of food, board, and Spring Fling.

More charitably, you're getting as many clubs and hobbies as you can imagine. There's symphony and pop symphony and jazz orchestra and quartet and other quartet and chamber music and I'm clearly exhausting my knowledge of music but the point is that you have a lot of things you get to be involved in, OK? Lots of chances to play oboe, or plays sports, or basically express yourself and chase your interests in a social manner.

(I know this isn't news to you, but the children, they don't quite get it yet.)

If you want to pay for that, then go ahead, but I watch my college friends drop their hobbies right after school. If you want to grow your spirit, you need to figure out how to fit art into your life without the ready-made structure and convenience of campus.

Look, people care about what college you went to. They just do, right? Because it signals to employers that you're someone smart and capable. Are you really more capable? Well, it doesn't matter if it's true, because people treat it like it's true. Going to a fancy college helps you lead a comfortable life. It makes it easier to get your dream job, or even a regular job. You have the right to pursue comfort.

This is the argument I hear the most. It's the argument my family and mentors offered me when I was in high school. It's what I hear from college advisers, kids and teachers.

Let's consider an analogy.

Let's consider beauty.

Now, being beautiful helps you get ahead in this world. It's true! People judge you as relatively competent. If you're beautiful, it's more likely that you'll get the job, that you'll be paid what you deserve to be paid plus a little bit more. In short, being beautiful helps you lead a comfortable life, economically speaking.

Would we advise our friends and students to pursue beauty? To pay thousands and thousands for surgery, to take out loans for our wardrobes?

"No!" we would say. "That's all wrong. It's not comfort that makes a life meaningful and worthwhile. I mean, look, comfort is important, but at what cost? Why pursue comfort if it's just built on something as meaningless as your appearance. Besides, beauty can fade, and then what is your life built on? If your entire life is built around beauty, then even as your beauty fades you'll be forced to endlessly chase it, to create a delusional sense of your own attractiveness well-past it's due date. What's left then?"

"Besides, what about unattractive people? People will just have a worse life because they happened to be ugly? That's hardly fair, and it's downright wrong. I don't want to live in a world where we punish people for their appearance."

Every time an Honors student applies to Harvard, a child is going under the knife. Can we please stop pretending like this is a good idea?

A Very Special Episode of Yarmulke Tales

I've included a lot of obscenities in this post, because that's how it happened.

The subway doors opened at 42nd Street and in walks this guy with a Department of Environmental Conservation t-shirt and lots of chains, lots of keys rattling around his pants. I'm sitting, he ends up standing right in front of me, but only for a second. He takes a look down at me and starts pushing people out of his way.

"Excuse me," he says, "I don't want to stand next to a Jew."

At this point I'm thinking, hey, that right there is some extra-ordinarily polite Jew hating.

Our t-shirt man clarifies: "I hate Jews. They fucking crucified God. Christ killers! Christ killers, I do not stand next to a Jew. No. Fuck you, fuck Jews. Heil Hitler! See you in Heaven, yeah, fuck you Jew. See you in Hell. I'll stick a cross up your ass. You're not a minority, you're the majority."

I mean dude's pointing at me while he's shouting at me from half-way across the car. I start chuckling which in retrospect isn't the best way to assure that crosses and my ass keep their distance.

"I should cut your heart out. I should fucking cut your heart out. I won't, I won't because I'm a Christian. I hope the Arabs kill you. Heil Hitler! Heil Hitler! etc."

When I changed jobs this school year I also changed commutes. I went from walking to work through the lovely Jewish/Dominican ghettos in which I reside to riding the subway for an hour everyday. And by my count this is like the fourth, or maybe fifth time that something like this has happened on the subway this year. There was the guy who threw a penny at me. There was that guy who just started yelling "Shalom" at me. And the lady who screamed "HE'S A JEW! HE'S A JEW! HE'S GOING TO KILL YOU!" while I was riding the C. (I was not.)

You might be thinking, hey, I didn't know that anti-Semitism is a problem in America. There's a good reason why this sort of thing isn't on your radar: it's really, really great to be a Jew in America.

Being a Jew in this country is a remarkably sweet deal, basically because we get to be well-educated white people. We weren't white when we got here, of course. It took a while to get things going, a story you can read about in "How Did Jews Become White Folks?"

As with most chicken-and-egg problems, it is hard to know which came first. Did Jews and other Euro-ethnics become white because they became middle-class? That is, did money whiten? Or did being incorporated into an expanded version of whiteness open up the economic doors to middle-class status? Clearly, both tendencies were at work.

These days, in this country, Jew is just another sort of white, and anti-Semitism has receded because you don't mess with white folks. After crazy dude got off the subway the lady sitting next to me reassured me that the guy was just some whack-job. But what makes him crazy, instead of hateful, evil? What makes him nuts is going after me, an upstanding member of the dominant class.

Jews are safe, rich even. Anti-Semitism is not part of the regular life of a Jew in this country, you don't have to gird yourself against it the way you do if you're black or brown...

...unless you go around wearing a yarmulke. Because then? Then all the whack-jobs come out. People stop you in the streets start telling you about their churches. They shout at you, threaten you bodily. They call you names, and who cares if they're crazy because crazy stabs just the same.

I mention all of this not to reinforce the notion of Jews as victims, because we're not worth your concern in America.* But I hope I'm not flattering myself in thinking that my experiences are instructive. I mean, for one you've got the sheer power of whiteness to admire. There are Jews walking around New York really everywhere, and they basically experience none of this stuff because they don't wear a yarmulke and can just slip in and disappear from view. There's all this hate out there for Jewish folks, but whiteness guards you against it in this country pretty much all the time. That's potent stuff.

* OK, one quick plug for anti-Semitism. Jews were knocked around Europe for a couple millenia, and the Holocaust was like a year ago. You think it just went away? Come on.

But the yarmulke reveals all truths, and it's a reminder that hate or prejudice doesn't have to be visible to exist. Check out Bonilla-Silva who showed that white students know how to express racially appropriate opinions on surveys, but revealed their prejudices more fully during subsequent interviews. And I expect that it's the same with Jews and other groups. Circumstance might drive expressions of prejudice away from public view, but it doesn't take much to reveal them.*

* As long as I've got your attention: "Negroes Are Anti-Semitic Because They're Anti-White." Food for thought.

This is only somewhat related, but a third grade student was singing a song in class today, and I asked him what he was singing. "I learned it in church," he said. He smiled. "You wouldn't know it because you don't go to church because you're..." and he runs up to me and actually starts rubbing me yarmulke with his palm.

I don't know how to wrap this up, but anyway it's besides the point. I've got to get moving, there's cleaning to be done, Passover starts on Monday and I've got to get ready.

Open Thread: The Changing Demographics of Teaching

It's conference time, which means that if you're connected to teacher-twitter you're going to see a lot of statistics being tossed around. It's always best to deal with the actual data and reports when trying to figure out what to believe. (For instance, last year I caught Treisman getting a little slippery with his slides.)

Are we not math teachers? Let's dig through some data, and shout out if you find something interesting or noteworthy. Make a graph and I'll toss it into the post. Ask a question, and we'll see if we can find the answer in some of the reports.

And here are links to some reports and data to think about.

Fast Facts about Trends in the Profession
NCEI: Profile of Teachers in the US
The Condition of Education: 2011

I'll start: it seems to me that we have fewer young teachers than we did in 2008. It looks as if the youth and inexperience of the profession peaked in 2008 and has been aging (slightly) since. Truth?

The Sticky, The Woodsicle, The Leg, and The Dogoala

Let's not bury the lede: yes, those are chocolate-filled blueberries and cheese-filled raspberries. A kid brought them in, we ate them, and they're gross.

OK but why? That's a slightly longer story, and I recommend that you start it over at Danielson's place. Basically, I gave kids an object to build a measurement system around. They named it, created a "made-up-of" (e.g. an inch) and a "made-out-of" unit (e.g. a yard). They made a measuring tape, they measured stuff with their units. 

Then we made square units to measure area with, and then cubic units to measure volume. Along the way a ton of good math happened -- fractions, scaling area, units, multiplication, measurement, volume concepts -- and a good time was definitely had by all.

Important questions for us were:

• How should we make a helpful tape measure?
• What's the best way to create a unit of area?
• How come so many of the square small units fit into the square major units?
• Ditto with cubes.

Important observations for me:

• There was a strong desire to measure area in terms of the physical objects that were their measures of length.
• Pretty much everyone had a moment while they were making a square or cubic unit where they went "Wait this is totally way too big." 
• The kids couldn't keep "square" and "cubic" straight. That's a distinction they don't really make in their regular language, and I've seen the same thing with older kids calling a sphere a circle.
• Measurement is simply a great context for fraction work.