Dan Meyer wrote about the activity on his blog, and our resident wordsmith dropped a marbleslides post on the official Desmos blog.

[Psst: If you haven’t played yet, and want to… Go here.]

In reflecting on what I love about marbleslides, my thoughts go back to the session Katie Reneau and I co-presented at this year’s CMC North and South conferences. We presented four principles that we’ve found effective in task selection and implementation with our own students.

1. Use tasks that build capacity for SMP 3.
2. Use tasks that are accessible and extendible.
3. Use tasks that encourage iteration.
4. Use tasks that have room for multiple approaches.

There’s not much in the way of SMP 3 (constructing viable arguments…), but marbleslides is an absolute gold mine when it comes to the other three principles.

Accessible and Extendible

The activity kicks off with a challenge that “even your baby cousin” could tackle. Provided he or she is willing to hit the “launch” button.

From there, the challenges grow. Marbleslides adds one layer at a time, providing students with digital sandboxes in which to explore constants and coefficients, translations and dilations, concavity and restrictions.

As students build their skills, Marbleslides pulls back the supports. Students are asked to find (and reflect on) solutions to increasingly complex challenges.

Iteration

In our conference session, Katie and I used an Open Middle problem to illustrate the meaning (and value) of iteration in math tasks. We both love problems that offer students some early reward, and them immediately invite them to dig deeper, work smarter, etc.

For Marbleslides, imagine this sequence:

• A student builds a solution, clicks launch. No stars. (Bummer.)
• But she tries again. And gets a star. (:internal-fist-pump:)
• She adjusts her graph (or graphs), and gets two more stars. (Nice!)
• So close… Just one more… A few more tweaks (and attempts), and she’s like:

• But then a classmate throws down a challenge: Five parabolas? Betchacant with just three.
• And the tweaking, the improving—the iterating—continues.

Multiple Approaches

Something else I love about marbleslides? Students can legitimately take alternate paths to the solution. Years ago I began asking students (almost like a broken record), “Great! Can you solve it another way?”

For the record, this prompting works out much better when there’s more than one mathematically meaningful approach.

And on most marbleslides levels (especially later in the activities), I’m pretty sure there are. From different paths, to different functions, to minimizing equations, to minimizing the time it takes to get all four stars… There are a lot of options for pressing deeper on any part of these tasks.

Invitation

If you haven’t tried out marbleslides yet, pick your flavor (lines, parabolas, exponentials, rationals, periodics), and let me know what you think.

If you’re already tried it, do me a favor: Save a GIF of your favorite solution (yours, a colleague’s, or a student’s) and drop it in the comments.

I’ve presented the lesson in a variety of workshop settings (each time it was well received), but this week marks the first time I used it with my own students. It was an absolute blast.

The full lesson is available here. I’d love to know what you think, as well as what you’d add or tweak (especially if you test it out with your own students).

Post Script

You can get more lessons on the Lessons Page.

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!”

I received a reply, but I’m more interested in a tangential question @NatBanting asked in the middle of the exchange:

@mjfenton Curious: what makes a task “rich” in your eyes? I’m trying to synthesize my view.

— Nat Banting (@NatBanting) August 16, 2013

I’ve been throwing that phrase (“rich tasks”) around a lot more lately, in tweets, blog posts, workshops, and conversations at school and at home (my wife is either amazingly longsuffering or genuinely interested—maybe both?—when it comes to talking about math education). So what exactly is a “rich task”? What do other people have in mind when they use the phrase? What do I mean when I use it?

I told Nat that I’m still working that out in my own mind, but I think it’s time to clarify what I mean, if for no other reason than to have a better sense of what I’m looking for when I try to find, adapt, and/or create rich tasks for my own students.

Here’s What I Do Mean

A rich mathematical task is one that…

1. Has a low floor and a high ceiling. The first of many ideas I’ve stolen from others, this one from so many sources I don’t even know who to credit anymore. Most recently, Dan Meyer has me thinking about this in his Makeover Monday series. The bottom line: everyone can start, no one can truly say they’ve exhausted the problem’s potential (at least not in a 50-minute period).
2. Has multiple entry points, invites use of multiple representations. Student A starts by exploring numerically, Student B begins by investigating graphically, Student C jumps in by reasoning algebraically, and I don’t have to tell two of them that their approach is a dead end because—even if they don’t always make it—there is fruitful territory a little further down the path in any of their approaches.
3. Has multiple solution paths, provides opportunity for rich discussion. If there’s only one way to solve the task students lose out on the rich discussion of making connections between various approaches and teachers lose the opportunity to build a mathematically coherent, concrete-to-abstract storyline as they orchestrate these discussions.
4. Integrates multiple topics. I owe a lot of what I’m thinking here to a single word Daniel Schneider used in a post about assessment. After my initial foray into standards based grading left me dissatisfied with an overly fractured curriculum, I’m now placing a high priority (philosophically, at least) on tasks and assessments that bring multiple topics together. A rich task, in my estimation, should demand that students wrestle with multiple topics from multiple domains (if I can use the term in the CCSSM sense).
5. Engages student interest, is mathematically/cognitively challenging. I’m a little mixed up here, because I believe engaging students’ interest is massively important, but I’m not ready to throw away tasks that fail to generate buzz among students if I know they nevertheless provide great opportunities for exploration and discussion.

Here’s What I Don’t Mean

To clarify what I think a rich task is, I’ll share a few thoughts on what I think a rich task is not:

1. A well-crafted, but constrained guided-discovery activity. I value this kind of activity, but when students are guided along a specific path to a specific goal, it pushes the lesson into a different category for me.
2. A thoughtfully constructed lecture or an engaging presentation. If the instructor is doing the heavy lifting during class, I would say the students are not engaged in a rich mathematical task. I’m not opposed to heavy lifting, especially in preparation outside of class, but students need to play an active, central role in exploring/solving/reporting if I’m going to use the “rich task” label for an activity.
3. A challenging problem for which students already have a tried-and-true method. I have a large stack of started-but-not-finished books, some related to math and education, others not. George Polya’s How To Solve It is on the list, though I’ve read enough of it to be provoked and inspired, particularly by the distinction Polya provides between a problem (solution/method not known) and an exercise (solution/method already known).

I’ll Close with a Link…

Incidentally, after typing out the post, I Googled “what is a rich mathematical task” and found this.

…And an Invitation

I would love to hear what you think are the key characteristics of a rich mathematical task. Drop a line in the comments or send me a line on Twitter.