Well that wasn’t any good.

Here’s a picture of the main part of my Calculus lesson from today:

One example. That’s it. The central element of my entire lesson was… one example. And not even a task. Just an example. A watch-me-as-I-carefully-walk-through-every-step-of-this-sucker and make-sure-you’re-on-guard-in-case-I-ask-you-any-leading-questions example. Oh my.

Granted, there was more after that example, just not a great deal more. And none of it great. We actually wrapped up the example, started the “next thing,” and quickly abandoned ship after some “Show me on your fingers how you’re doing” feedback from students revealed that all was not well (not by a long shot).

This changing of gears led to a somewhat-useful last 10 minutes of class (thanks in part to Desmos), which in turn led me to wonder: What did the end of class have that the start of class was missing? For one thing, after seeing the first part flop I had to clarify in my mind the bottom line goal for the lesson. I settled on this: If students left my room with the ability to translate verbal and algebraic problem descriptions into graphs, and those graphs into integral expressions, we’d be golden.

Period 3

Well that wasn’t any good, either.

I went overboard Sunday evening creating a slide deck that (I thought) would help me lead students through a carefully crafted conversation on the topic of trigonometric properties and identities. The slide deck was slick as all get out. But the lesson was boring. You could see it on their little compliant faces. They didn’t even complain. They just sat there. Copying a property or two here, sketching a graph or two there, dutifully jotting down an observation or two when I asked, and so on, for the better part of half an hour. Argh!

Inspiration ≠ Incorporation

At this point I have no idea if I’ve painted a clear picture of what took place today in my classroom. Even less so what’s going on in my head right now. If you’re feeling uncomfortable, abandon ship now, ’cause this is about to get even less coherent.

You see, I’ve been struggling with a number of thoughts over the past few months. To name a few:

• Sam Shah is awesome. As some teachers call out worksheets on Twitter, Sam is busy packing one aha-moment after another into carefully crafted mathematical adventures for his students, all on that oldie-but-goodie 8.5 by 11 format. I want to bring these kinds of things into my own classroom!
• Dan Meyer is awesome. As some teachers give the all-call for real world applications over all else, Dan is blasting through pseudocontext and drawing attention to the heart of the matter (engagement), regardless of whether a task is labeled “real world” or “fake world.” I want to make genuine student engagement a central (and regular) feature of my teaching practice!
• Karim Ani and Team Mathalicious are awesome. As some teachers are firing up Khan Academy accounts and printing off hundred-of-the-same worksheets from Google search results, Karim and the crew are laying siege to the notion that math is uninteresting, unengaging, unimportant, or unworthy of our attention. I want to bring Mathalicious-style conversations into my own lessons!
• Jonathan Claydon is awesome. Seriously, have you seen the pictures of his class over at Infinite Sums? His students do stuff. All the time. They’re active, they’re involved. I don’t care what the lesson is, they’re at the heart of it. I want this to be true of my own students!

And I haven’t even mentioned Stadel, Nguyen, Kaplinsky, Vaudrey, Stevens… The list goes on. And while my inspiration grows, my frustration does too, because I can’t find a way to incorporate all of this awesome into a coherent whole in my own teaching world.

That’s really the issue. And I’m just using a frustrating Monday morning to process what I’ve been struggling with for months in the hope that I can make some sense of it all.

The Challenge

(Still with me? Awesome. Hang in there, we’re almost done.)

So let me try to name my struggle, clearly and succinctly, so I can go about the task of moving beyond it. Here goes:

For the past 500 days I’ve been inspired daily (literally, every single day) by what I see in the MTBoS. At the same time, I have yet to find a way to weave that inspiration into my own practice in a coherent, compatible way.

The Way Forward

I don’t know the entire solution, but I know it starts with this: I’m done designing scripted lessons, those awful handouts with eleven-teen examples that we’ll walk through. Together. All of us. At the same pace. (I’ve created enough of those to last a lifetime, and they don’t develop in students any of what I’m after.) I’m done drawing up anything where I can predict with 99%+ accuracy what the students will be thinking at any given point. I’m done throwing together slide decks that demand students focus on the same thing at the same time. I’m done throttling their insights, their noticings, and their wonderings by squeezing out of them a certain style of efficiency that is anything but effective.

Instead, I’ll be spending my time infusing worksheets with aha-moments and did-you-just-see-that mathematical surprises. I’ll be on the lookout for visuals that mess with students minds and spark dozens of questions they actually want to answer. And I’ll expand my teaching skillset so that I can navigate the waters of a class full of students exploring different problems inspired by the same visual. I’ll take risks, push the boundaries of what I’m currently capable of, and through it all develop my ability to orchestrate rich mathematical discussions, whether they’re centered around a thought-journey disguised as a worksheet, a rich and who-cares-if-it-has-no-context problem, an engaging and demanding task, or an honest-to-goodness real-world scenario. And whatever I do, I’ll make sure my students are at the center of it.

In short, I’m done with trying to script their thinking. I’m going all in with prompting them to think. “The script is dead. Long live the prompt!”

A little over a year ago I created a set of Nike+ Running-themed proportional reasoning challenges for my middle school students. The idea originated as a failed attempt at a Three Act task, and eventually turned in to a somewhat-useful website (proportionplay.com). You can read more about the project here.

Even after using these problems occasionally throughout last year, and also in various teacher workshop settings, I was never satisfied with the results. Things usually drifted into uncomfortable arithmetic, and my focus from the word “Go!” was always on the proportional reasoning element.

So after mulling things over during last year’s spring semester—and trying out some of those ideas at the start of this year—I’ve settled on a new approach to the same set of problems. I’m optimistic that this new twist will make the problems much more engaging to my own students (and maybe even to some of yours as well).

The (Old) Running Game

The old me would start the conversation with an image like this:

Then I’d ask students what they noticed about the images. There would often be a wide range of answers (“I see orange!” “Hey, the phone battery is draining!”), but eventually we’d have covered the basics: “After 10 minutes, some mystery man (or woman) had run 1.88 km. This person apparently kept running for another 10 minutes, and then for another 5. We don’t know the total distance traveled at either of those latter points.”

Next up: Find what’s missing!

This is where we’d wander off track from my primary goal (develop proportional reasoning!) and end up stuck in some tedious arithmetic (with the tediousness increasing slightly each day in the series of 20 challenges). There’s nothing wrong with having (or building) proficiency with decimal arithmetic. However, my goal here was to emphasize and develop proportional reasoning. The doubling, the halving, the combining of these pieces to make five halves of the original… Students would stare into the blinding light of ninety-four hundredths and somehow lose track of what they were after in the first place.

Good conversations often ensued, but I was after great conversations. Something had to change.

The (New) Running Game

Here’s how we roll now: I still begin with an image…

And we still do the whole noticing thing (at least on the first couple of challenges). But then I invite (encourage? demand?) students to cut some corners.

• 21 minutes and 5 seconds? Nah! That’s basically just 21 minutes.
• Double that? 42 minutes. Boom! We have our first answer: “This runner probably took about (hugely important word for the rest of our running game conversations) 42 minutes to run 8 km.”
• Back to 4 km in 21 minutes. Half of 21? Let’s call it 10. So 2 km in about 10 minutes (maybe a little more).
• So 42 plus 10… That’s 52 minutes.
• But maybe our runner is no Madison Bumgarner. Maybe fatigue actually affects our faithful jogging friend. Let’s hedge our bet. Maybe… 54 minutes?

And then, the reveal:

And it turns out we’re wrong. Totally and completely wrong. But now the fun begins, because we can ask:

• What happened?!
• Why were we so off?
• Did our arithmetic betray us? (No. We took some liberties with rounding so the arithmetic would be easy-peasy.)
• So what gives?
• Did this guy (or gal) speed up? Or did he/she slow down? (To me, this conversation alone is worth all the screenshots I took while running with a phone in my hand.)

Maybe we made some assumptions about our runner. Maybe we need to blow up some of those assumptions so our future “off-ness” isn’t so pronounced. (And isn’t that one of the best parts of modeling? Figuring out the subtleties of the situation… Testing and either retaining or tossing out our assumptions…)

So we play again:

And this time we come to the table with some additional information. This dude slows down (and seriously) as the seconds tick by. Let’s keep that in mind as we think through the next round:

• 1.89 miles… That’s pretty close to 2 miles in 16 minutes
• That’s 1 mile every 8 minutes
• 24 minutes… Let’s chalk that up as 3 miles
• 32 minutes… That would be 4 miles
• What about 36 minutes? That’s another 4 minutes (half of 8!) so I’m thinking an extra 0.5 mile. All told: 4.5 miles.

Of course, that would be if our runner followed a predictable pattern. Now the fun comes in. This is where every single student in the class could end up with a unique and not-unjustifiable answer. How do you want to hedge your bet in the end? Will the runner slow down? Have we already accounted for that with our rounding up to 2 miles? (Or have we gone the wrong way?!) And so the customization of the answers begins (ideally with much arguing and hoping and anticipating).

Me? I’m thinking we should tone down the distances… 2.7 miles in 24 minutes, 4 miles in 36 minutes. So, how did I do?

Now when you throw that next image on the board, and one kid in the back slams his fist on the table in defeat and the other pumps her fist in the air in triumph… You know you’ve got them.

The Road Ahead

The challenges are all still available at proportionplay.com. And I’ve replaced the original (and terrible) handout with this shiny new one. I think the new handout does a reasonably good job supporting my updated approach to these tasks by emphasizing the reasoning involved in the entire process. That—combined with my wide open invitation to estimate since the runner is almost certainly not a same-pace-all-day robot—should shift the focus from arithmetic to proportional reasoning.

Post. Script.

The last two challenges are both borderline terrible. Or wonderful. Or both. Here’s a preview of Day 20:

Post^2. Script.

Another favorite aspect of these problems? Whether you round or not, there are several ways to tackle each challenge:

Disclaimer: The whiteboard image is from one of the courses I teach in the grad math/science department at Fresno Pacific University.

I don’t know why, but there’s something about that noise (in a math problem, no less) that simultaneous makes me giggle and fires up the I-need-to-know-what-was-said corner of my brain.

So I made this:

Since I made the video ((For the record, that was two months ago. This post has been sitting in draft purgatory for long enough. So it’s time to drop this in the urgent bin and get it out the door.)) a few things have happened in the world of college basketball.

• UCLA won
• UCLA won
• UCLA lost
• I’m sorry, did they keep playing the rest of the games after UCLA lost?
• Nobody won. The bracket challenge, I mean. I think someone won the tournament. I’m not really sure.
• Warren Buffett and Quicken picked up about a billion new home loan leads.

At any rate, I’m not entirely satisfied with the result. I mean, I was really hopeful UCLA could make it to the Elite Eight I could turn this into an engaging lesson hook, but the first group I tried this on kind of just stared at the screen after the Act 1 video ended.

So I’d love some feedback, either in general, or in response to some of these:

1. Is there any potential here? (I think there is…)
2. What bits have I got right, if any?
3. How would you hook students with the “what are the chances…” question here?
4. Am I missing the point? Is there a better question to ask besides “what are the chances…”?

Thanks in advance for your thoughts!

P.S. If you’re interested in a link to Act 1 and Act 3, you’re welcome. And here’s a notebook full of some links and images I gathered but never used.