Today, however, I shifted back to a handful of “let’s walk through these together” exercises for the first 15 minutes or so of class. But I added a twist…

At the beginning of each exercise, I asked students to rate their understanding of the problem we were about to attack:

• Beginning (B)
• Developing (D)
• Proficient (P)

These are the same descriptions I use on the proficiency scale for my SBG assessments, so students are familiar with them.

Then, after walking through the problem as a class, I asked them to rate their understanding again.

My goal? To push my students a little further down the road of reflecting on their understanding. In particular, I wanted them to have a sense of what they need to work on prior to our assessments at the end of the week.

I’m hopeful that the 10-second pause on each problem gives them some valuable insight, and possibly some more informed motivation for what comes next. Better yet, maybe this is something they’ll apply on their own initiative in another situation, whether in my classroom or another one.

And if you make a mistake, it can be an absolute bear to track it down. Is your issue differentiation? Evaluating a function? Or is your weak spot related to what’s happening visually in linear approximation?

Here’s a problem from last year’s AP Calculus review workbook:

There’s a lot of great work on the page.

• Take the derivative? Check.
• Find the slope of the tangent line? Check?
• Find the equation of the tangent line? Check.

But that’s where things fall apart.

Now, imagine you’re a calculus student. You’ve been hammering away at this thing for several minutes. Maybe you don’t even remember what the problem’s asking for in the first place. You get to this point, you scan the original problem, see an input value of 4.2, and presto-whammo, plug it in and out comes 0.4. “Great news, everyone! That’s on the list! Well done, folks. On to the next problem!”

Only, that answer is entirely wrong. So how do you debrief this problem with a student, small group, or class, so they can see the source of the error? For problems with even a sliver of something graphical, Desmos has become my go-to tool for helping students find and fix their errors.

Here’s what I built with a pair of students last year (with a link to the live graph here):

Students can’t use Desmos on the AP exam (for now, anyway), so I’m not trying to permanently sidestep what they ultimately must be able to do sans technology (or with a device from that “other” graphing calculator company). But what we can do in class with Desmos is build a better visual/conceptual sense of what’s happening in this problem so they’ll be more prepared for something similar in the future.

Here’s a short list of what this Desmos graph did for us in this scenario:

1. We offloaded the algebraic and numerical work of finding the derivative (and evaluating it for a particular x-value) and built the tangent line in a matter of seconds (rather than minutes). Mentally, we’re still fresh, and ready to focus on what the problem is really asking us to do: compare the function and the tangent line.
2. We gave things specific names so we could call on them in our time of need. Okay, that may sound a little dramatic. But think about why we even bother with function notation. Why give a function a name? Well, why did your parents give you a name? So they could call on you! (“Alfred! Get down here and pick up your comic books!”) So then, why do we give functions names? Because function notation is on the Chapter 8 test in Algebra 2? No! We give functions names so we can call on them. So they can do our bidding. If you don’t name it, it’s difficult to put a function to work for you. Give it a name? Now our wish is its command. (It sounds a little bit like we’re going to take over the world, with math as our trusty sidekick.)
3. We gave things specific names so we could keep clear in our own minds the various moving pieces in the problem. There’s a function. We called that f. There’s a tangent line. We called that l (or t, or whatever). As we approach the end of the problem, and we start looking for the error, it’s easier to avoid simply evaluating f(4.2) or l(4.2), because we know we’re dealing with both f and and l. (How could we forget?! We named them! We practically gave birth to them.)
4. We visualized the error with a beautiful little orange bar, and in doing so imprinted on our minds (for future problems) what linear approximation error looks like.
5. We dropped a slider in so we could answer a hundred related problems in a matter of seconds, further clarifying for the confused student (or teacher; I was terrified of linear approximation my first two years of teaching Calculus) what the problem is really about, and all with a nifty, dynamic burst of compare-and-contrast.

Do you need Desmos to teach this stuff? Maybe not. But given the option, I’ll use it every time. Last year in my classroom, we got a lot more out of a five minute Desmos-supported conversation than we did in a whole day of business-as-usual notes and examples.

So this year? We went straight to Desmos and put the emphasis right where it belongs.

P.S. I love GIFs.

Tomorrow’s Game Plan

When I taught this lesson three years ago, it lasted 40:01. I’m not kidding. (By the way, that link is a whole other coming-over-from-the-dark-side post that I don’t have time for right now…) Two years ago and last year? Basically the exact same lesson. Six problems. Work through ’em as a class. Everyone stick together. Eyes on me. And so on. Bell rings. Head home. Practice. Try not to cry.

A quick preview:

And if you’re curious, a closer look, compliments of your friend the PDF.

If you read last night’s post, possibly the only thing that I made clear was my desire to shift away from “all eyes on me” instruction where it makes sense. (Which, as it happens, is in quite a few places.) With that in mind, here’s my plan for tomorrow:

1. Use Desmos, Keynote, and MathType to build a set of cards containing: graphs (12), integral expressions (12), and directions (1)
2. Print, slice, and drop in Ziploc sandwich bags. (Hooray for Costco!)
3. Distribute one bag to each group (~4 students).
4. Give them about 5 minutes to sort. Then 3 minutes to describe in writing their thinking. (“How’d you match each graph with its integral expression?”) Then 2 minutes to debrief as a class.

Now, what to do with that extra half hour…? I’m still working on that bit. In the meantime, I’ll share some of the cards. Enjoy!

Wish Me Luck!

I’ll check back in tomorrow with a brief report on whether tomorrow is a success, or another borderline failure.

The Goods

All of the resources I created are available here.

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!”