Is this a function?

This is clearly a function. Functions are patterns.

But there's more. This is a function:

This isn't:

Functions are like the top graph. Non-functions are like the bottom graph.

What's the difference between the top graph and the bottom graph? One difference, for sure, is that the bottom graph has loops. But why do the loops matter? Can you tell a story that matches the top graph? Can you tell a story that matches the bottom graph?

It's harder to tell a realistic story about the bottom graph. Try making other graphs with the string. What if you make a "C"? A "U"? Are these more like the top graph or the bottom graph?

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Here's another function:

Google Translate is a function.

But people's ears and minds are not functions. To us, the words "I love you, man" can mean an immense variety of things. A single utterance can mean "I love you" or "I don't love you" or "I think you're funny" or "Let's just stay friends," and it all depends on a million difference things like the tone of his voice, the tone of her voice, whether you're at a carnival with friends or on sitting alone on a couch.

Functions are automatic translators. Non-functions aren't always sure how to understand a sentence.

And, by the way, a confused boy, unsure how to interpret "I love you, man" is sort of like an impossible graph, no? He knows what time it is, but he just isn't sure how happy to be.

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But now we're getting a bit melodramatic. Let's tone it down a bit.

With credit to Christopher Danielson, this is a function:

But this isn't:

Functions are reliable machines. Non-functions are unpredictable machines. You always put in the same number of tokens, but tons of different things could happen as a result.

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Now, let t stand for time, and let h stand for happiness. Is t^2 = h a function? Is t = h^2? 

(In the comments, Gregory Taylor rightly points out that the question is whether time is a function of happiness, or happiness is a function of time.)

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And, after all that, is this a function?

If we tell our kids that functions are machines, then the only question a kid can ask himself is "Can you think of this as a machine?" But with a richer set of images to draw on, "Is this a function?" becomes connected to a series of more reliable and helpful questions:

• "Is this more like a slot machine or a change machine?"
• "Is this more like Google translate or a confused boy?"
• "Would its graph look more like a loopless graph or a graph with loops?"

The point here isn't to be precise. When we want precision, we'll use the formal definition. The point is to provide students with a set of images around that formal definition that guides their thinking in helpful ways.

Let's try to find a richer set of images for both functions and non-functions. Let's also be more intentional about bridging the gap between linear, quadratic and exponential things and the sort of semi-arbitrary pairings that we want students to recognize as functions.

Postscript:

This post constitutes my final project for Christopher Danielson's really wonderful functions course. He's going to offer stuff like this in the future, and it will definitely be worth your while. 

There's more to say, but I'll save it for another post or the comments. 

Update (3/31/13): After some helpful criticism on twitter, I edited the post for quality.

Some conversations are more important than others

Parent-Teacher conferences are dumb, right? I've got 5 minutes for each parent, 4.5 hours of conferences, a list of things to do tomorrow that's getting longer and longer as the night goes on. The kids start to bleed into each other. Everyone is doing well. Except the kids who aren't, but how do I say that to the parents in a way that will end up being productive for my relationship with the kid?

Conferences are dumb, but that's not really what this post is about. This post is about the kid that I just got off the phone with.

I met with his parents tonight, and I told them that he seemed unhappy with the way class was going. I said that I wanted to talk with him, and they seemed OK with a phone conversation so I asked for their home number. After conferences I gave him a call.

"Look, you don't seem completely satisfied with how class is going. So I was wondering if you had any feedback or anything."

He did.

He thought that class was kinda boring and awfully repetitive. He wasn't a fan of the worksheets that I give out on most days, and he thought that the Warm Up was becoming a distraction. He wanted more notes, because they keep kids from just being spoon-fed information. He wants the notes to be more step-by-step. He's willing to stop by during lunch to show me what he means. He's worried that we're not going to be prepared for the Regents.

"Can you think of a day that went really well in class, a day where you felt I did a lot of good things?"

He thought Trashketball was good and fun. He thought that a lot of days were good, but that it depended a lot on his mood and how into it he was. He likes when we're reviewing for a test.

More on worksheets: it's really easy to copy the work of others, and when I'm walking around it's hard for me to tell the difference, so it doesn't feel like you really have to do the work. He had an idea for the Warm Up -- what if everyone had a different question, so that you couldn't copy others and it felt like everybody owned their own work?

"OK, but I've got a question. Like, I'd get it if you said that you didn't think that the problems in class were helpful or worthwhile, and that's why you don't do them. But it sort of sounds like you're saying that they might be helpful, but you just aren't motivated to do them."

Well, when he's confused by something he tends to shut down. And can he move his seat to the front? That works better in his other classes.

There's a lot that's amazing about this kid. His willingness to talk with a teacher for a while on the phone during his free time. His candor and seriousness is something that most kids can't pull off, and it's clear that he's not satisfied with a class where he's just farting around.

(It's also remarkable that you can get this far into the conversation and still put all the responsibility for motivation at the teacher's feet. And I told him something like that.)

Almost every single one of his critiques resonates with me in some way, and I think this really points out that I've got a lot to learn about creating experiences that work for the 24 different people that occupy Room 312 between 2:05 and 2:48.

But there's a bigger point here, and it's that the way I do customer feedback is all wrong.

I do surveys, anonymous and otherwise. I ask for thoughts. I talk to kids during and after class. But all of these systems have the same flaws: (1) they treat every voice equally and (2) they're quick. What was amazing about my conversation tonight is that it was with someone who is just not clicking with my class, and I got a ton of time with him.

So back to conferences: they suck. But what would've been way better is if I could've had a half hour interview with 9 of the students who are having the worst time in my class. I'm pretty sure that I can identify them. I'm pretty sure that we could make some progress in a longer conversation, or at least hit a mutual understanding and figure out some way to move forward.

Here's my commitment: I'm going to draft a list of my 8 unhappiest students, and I'm going to call them and keep them on the phone for as long as I can. I'm going to figure out what my unhappiest students think about what's going on, and I'll try to just shut up and listen.

Slope and Sunset

I wanted the kids to understand slope as a rate of change, and I also wanted them to use slope to understand something interesting. I also didn't want class to be boring. It went OK, but I still feel as if the payoff is a bit weak, and I'm hoping you all can help out a bit.

Last week we developed a metric for slope using a version of Fawn Nguyen's (version of Malcolm Swan's) slope activity. (Note: talking about sports statistics with these 11 boys helped the idea of defining a metric go down smooth.) I also showed them mountains and asked them to rank those, and we considered the various advantages of measuring steepness as "width divided by height."

Today we were going to study sunset at different times and places, and I just wanted something cool that would get kids ready to think about astronomy and stuff. I ended up with this barely related video:



It's great and beautiful and I showed it to the rest of my classes today too.*

* Most wanted to know what the green glow is, and I don't really have any idea how that stuff works. One kid had a pretty good explanation along the lines of "something something force shield." He also knew about solar storms. I'm getting off track here, but kids are tons of fun 95% of the time.

From there I asked them what they knew about sunset. They knew that it happens, that it gets earlier and later depending on the time of year.*

* Pro tip: When living in NYC, don't assume that the kids know anything about nature.

 I asked them how much it changes per week. Their answers ranged from 1 minute to 7. I asked whether that rate was the same all year long. It took a few tries, but I finally got the question across to everyone, and there was a bit of disagreement.

Using the USNO site I made them a bunch of graphs, and asked them to find the slopes between the points. Here's what they got:

February NYC by mbpershan

We needed to remind these kids how to calculate slope, and they moved pretty slowly, so most of them only got through 3 of the graphs today. Some kids had trouble finding the height at first, reasoning that the highest height on the graph was the height we needed for slope. ("Look, I'm holding this paper 7 feet in the air. Does that mean the paper's got a length of 7 feet?")

I was fairly happy with the way class went, though I was worried by the fastest kid who made it through a bunch of the slopes and told me that he didn't see any patterns or interesting stuff emerging. And while the kids were doing better on slope and making progress on interpreting the numbers as a rate, there were warning signs as we tried to wrap things up. (Warning sides include, boredom, confusion about the questions I was asking, difficulty interpreting the units involved in the rates.)

So, the follow up is tomorrow in class. At the heart of this lesson is a really cool idea: that where you're living on Earth radically impacts the patterns of your life. How can I make this pop, while giving my kids good practice with their skills?

Caveats:

• Yeah, I know that it's a bit false to ask for the "slope" when we've got non-linear patterns. But we talked about it, and we agreed to just search for representative points.
• It seems to me that there aren't a lot of good problems hanging out there for kids who need a bit more practice with the connection between slope and rates. We've talked about speed, and they've looked at graphs. We'll do more of that. But I was searching for something a bit more, I dunno, worldly and interesting.
• This lesson is a spin-off of an Exeter problem.

Oh, and one more thing:

My kids are having a really hard time solving systems of equation through substitution. I tried approaching this really slowly, but we're stumbling on the landing. Any ideas?