In the past I consistently struggled with making the turn from the excitement toward problem-based learning (PrBL) to the actual design of complex, engaging problems. Typically I would spend the morning building the buy-in (the “why”), another part of the morning conducting some sort of problem simulation to showcase PrBL (the “how”). Then my instructions were along the lines of “OK gang, after lunch you’ll start designing your own tasks!” If you’re like me, you find it difficult to be creative on demand*. (I mean, if you’ve been keeping up with the infrequency of my blog posts in the past year you probably know that already).
Don’t get me wrong, I have little patience for math teachers who say “they’re not the creative type.” And I do think creativity is an under heralded attribute teachers need to have. It’s difficult to be creative at gunpoint.
I’ve started codifying what I believe is a more agreeable framework. Many of the successful implementation of inquiry-based, complex tasks has followed this progression (often over the course of multiple coaching sessions):
First: Find
Then: Adapt
Finally: Create
We start by finding (and often trying out) a task; then, at a later date, we try adapting a task (which we then implement); finally – and this is a tall ask – we try out creating a task more-or-less from scratch. This final step is probably more of a slow-walk from adapt than a full on design sprint.
Find
There are countless websites with open accessible tasks of ever-increasing quality and navigability. You know ’em, you love ’em. You can find a bunch on the “Math-like Blogs” list on the right side of this page. You can also find well organized tasks at IllustrativeMathematics, Shell Centre, Teacher.desmos.com, openmiddle.com and NCTM’s Illuminations.
An afternoon of PD spend simply clicking through your favorite, say, three of these resources is an afternoon well spent. That’s how the ol’ curriculum maps came to be.
Find something compelling and pretty soon you’ll find a ton of stuff you find compelling.
Adapt
Once you’ve found some good stuff, try to see if you can take something that’s pretty good and make it better. I’ve presented about that before: [NCTM] Adaptation.
This requires a bit more discussion and contemplation. You start to turn from “I like this task” to “What do you like about it?” Once we start adapting we are developing an implicit or explicit criteria for what makes a quality problem.
Maybe you adapt a problem by removing the sub-steps. That would suggest you like problems that allow for a lot of “open middleness.” Maybe your colleague adapts a problem to a hands on activity a la Fawn. That speaks to how much you value tactile experiences and students actually doing stuff.
Now – and only now – ought we turn to the ever challenging work of creation.
Create
Most of the time, creation of a task comes from either inspiration and/or sheer luck. I’ll see an advertisement or watching a movie and see something that’s kinda mathematical. Like I said, really tough to do on-demand, and also really tough to do in any kind of standards-aligned way.
But it’s also absolutely crucial! Not only does it work out your creative muscles, it generates tasks for the rest of us to find! It’s a give-a-penny / take-a-penny situation. Even if you’re not teaching, say, geometric constructions in your Algebra 2 class, maybe you get struck by a divine lightning bolt of inspiration that the rest of us can draw on. In that respect the Find –> Adapt –> Create framework could be seen as a cycle.
Find –> Adapt –> Create –> Other people find your creation
But yeah, it’s difficult to do on a good day, it’s much more difficult to achieve when I’m hovering over individuals harping on them: “got anything yet?”
And this isn’t just true for tasks. Consider other instructional tools.
| Find | Adapt | Create | |
| Rubrics | NTN Learning Outcome Rubrics (Math) | Pull from a few of the rubric indicators | Design your own, based on your grade level, school context and content area |
| Lesson Plan Template | Problem Planning Form | Modify based on your class time | Design a lesson plan template that works for an entire department |
| Math attitudes survey | Here’s one I developed | Steal a bit from it, but identify a few of the specific things you’re trying to deduce | On your next iteration, make it totally your own! |
I do believe that the best instructional experiences students have are by-and-large teacher-designed. Getting to that point is challenging so start with the stuff we have and slow-walk yourself into creation mode.
Y’know, unless inspiration strikes you like a lightning bolt while you’re sitting on the couch. In that case, disregard everything I said and go nuts, Creator.
* – Note: to contradict myself, this was not true of PBL Chopped! That was absolutely a fantastic experience of solely creation with incredible project ideas.
If one were to ask me “what makes a lesson problem-based?” I honestly don’t have a great, specific definition at hand. To me, I think of a problem based lesson a thing where students are given a complex problem and they have to solve it. In the middle though, all kinds of wacky things happen: new learning is acquired, old learning is readdressed, information is researched, attempts are made at a solution.
ra. For others still, college Calculus is the first stomach punch. For many it’s all of the above. For my daughter, it was when she got that schedule and received the message that she was no longer one of the “smart kids.” Of course, the Original Sin was the message that there exist smart kids and not-quite-as-smart kids in math to begin with.
mmer curriculum 
Is bad context worse than no context?
In elementary classes we consider it a good thing to be able to move from the abstract to the concrete. We ask students to count and perform arithmetic on objects, even contrived ones. We ask students to group socks, slice pizzas, and describe snowballs. A critical person might suggest these are all examples of pseudo-context, and they’d be right! These are more-or-less contrived scenarios that don’t really require the context to get at the math involved. Why do we provide such seemingly inessential context? I’m venturing a little far away from my area of expertise here, but I’d guess it’s because it helps kids understand the math concept to have a concrete model of that concept in their heads.
My question is this: in secondary classrooms, is there inherent value in linking an abstract concept to an actualized context? Even if the context is contrived?
I mean, yeah that’s bad. Comically and tragically bad. It doesn’t do anything to enhance understanding. I’d say the context actually hinders understanding. The thickness of an ice sculpture dragon’s wing? That’s about three bridges too far.
But what about a slightly less convoluted, but also-contrived, example. Say:
This problem is still most certainly contrived: dimensions of tanks aren’t often given in terms of x. I’m not even suggesting this problem will engender immediate, massive engagement, but it might help students create a mental model of what’s going on with a third degree polynomial. Or at least the context allows students to affix understanding of the x- and y-axes once they create the graph of the volume of this tank.
We provide similarly pseudo-contextual in elementary classrooms in order to enhance understanding of arithmetic and geometry.
From Burns’ “About Teaching Mathematics”
Of course, we also compliment such problems with manipulatives, games, play, and discourse, which secondary math classrooms often lack. In the best elementary classrooms we don’t just provide students with that single task. We provide others, in addition to the pure abstract tasks such as puzzles or number talks.
Perhaps the true sin of pseudo-context is that it can be the prevailing task model, rather than one tool in a teacher’s task toolbox. In secondary math classrooms pseudo-contextual problems are offered as the motivation for the math, instead of exercises to create models and nothing more.
(See also: Michael’s blog post on Context and Modeling)