Despite an increased awareness of this thing called “Problem-Based Learning,” (PBL/PrBL) there’s some nebulousness in what that word “based” means. Does it mean that students learn content within a problem? Does it mean students are honing their problem solving skills?

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.

That wacky middle is difficult to capture and package in a PD session, a conference talk, or even a modeled lesson study. Consider this an attempt to unwind a loaded term.

There are three ways in which one can deliver a “problem based” lesson. At least as I’d define it.

**A problem in which students need to identify or find additional information in order to solve the problem.**

Consider Graham’s “Downsizing Ketchup” 3-Act lesson, and most 3-Act’ers for that matter. The problem is posed via Act 1 and the setup of the scenario (or “conflict” if we’re being true to the 3-Act narrative terminology). A student or teacher may ask about and will need to know the information contained in Act 2. Act 2 yield the information that students need. Ostensibly (and again I should caveat: generally) that should be enough to complete the problem, with possibly side workshops as needed.

**A problem in which students need to learn new knowhow in order to solve the problem.**

This is the model of lesson under which I tried to teach most often. Like in the previous problem, students are given a problem to solve via an initial event: a video, a letter, an image, or even a straightforward word problem. After some initial brainstorming and pulling apart of the problem, students begin working toward a solution. At some point throughout the student-working portion a *need* for new knowhow will emerge.

Consider a problem in which the need to solve a system of equations arises. Energy efficiency electronics and appliances work quite well. How about light bulbs? Upon developing a model for both the cheaper, but energy guzzling light bulb and the more expensive, but energy consuming bulb. Upon graphing these, the need arises to solve for this system of equations. When I facilitated this lesson in class, students had *not yet learned how to solve a system of equations*, graphically or otherwise. We would deconstruct the problem, create a couple models of energy usage and graph them. At this point in the problem-solving process, I’d deliver a quick class lesson on how solve a system of equations. Once I felt like students had the hang of it, I’d turn them back to their light bulb problem and allow them to apply that new knowhow.

The thinking is that students learn better when there’s an authentic need to understand, which is what the problem context can provide. I found this to be both highly effective and incredibly difficult. How do you design a problem that necessitates the knowhow? At what point do you take that problem “timeout” to deliver the lesson? I’ve written a bit on that before. But it’s certainly more of an art than a science.

**A problem in which students have everything they need and must demonstrate mathematical thinking in order to solve it.**

Of course, there are excellent problems that may be given when students generally may not need additional info or new knowhow. Perhaps there are multiple pathways or methods that yield a solution. Consider a “puzzle” type problem, such as Youcubed’s Four Fours or Leo the Rabbit task. These are interesting, rigorous problems that don’t require new methods per se, but rely on a more general notion of mathematical thinking, such as Bryan’s Habits of a Mathematician.

I’d also put Fawn’s Hotel Snap in this category. There isn’t any information students need or instruction from the facilitator in order to achieve a solution. But it does require creativity, persistence, and organization, all mathematical skills.

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Each of these types of (*extremely academic professor voice*) PROBLEM BASED LEARNING have their time and place, depending on the objective, the standards involved, the students, the problem itself, and teacher comfort level with Problem Based Learning. And even providing these three models perhaps draws unnecessary boundaries between Problem Based Learning and just generally good math teaching and even between each particular model mentioned here. Still, I hope it’s somewhat clarifying, if only to generate additional future conversations.

This is really interesting. Two points stick out in mind mind:

First, I don’t see the separation that you do between your first two types of problems. In general, all my problem-based lessons require people to acquire information. I also use them to introduce units regularly, and as such, I’d also classify them as needing to learn new knowhow as well. The lessons COULD be used independently and not require that, but it isn’t how I see them.

Second, where (if at all) would you see something like Open Middle problems fitting? My guess is that they are the third problem type, but you may disagree.

Dan, Graham, Andrew and I had been talking about something similar to this. I hope to post it in the coming months.

Thanks for pushing my thinking.

Yeah I’d definitely toss openmiddle into that third category.

That’s interesting you wouldn’t discern between the first two. I’d suggest there certainly could be overlap and one could be a “subset” of another. I guess the difference is for the second (“Need New Knowhow”) is that I’d make sure I have a lesson in my back pocket ready to address the content needed. If it’s an “Additional Info” problem, I might not have a content-oriented lesson ready to roll out.

Thanks Geoff. You’ve given me a really helpful way to understand problem-based learning, especially by providing examples of each of the three types. I know one thing I need to get better at is the debrief and the sense-making that comes at the end. But good to know I’m on the right track at least with 3-Acts, Open Middles, and Hotel Snap-like lessons. Would you see games fitting in here at all?

I’ve wondered about that, where games fit in. I almost never write about math games (secondary brain, here!), but I love reading about them when they’re good (see simplifyingradicals2.blogspot.com). I don’t know that I’d toss ’em in as Problem Based Learning, but I could probably be swayed. Actually, I’d be curious to hear what Robert has to say. He’s done a lot of task taxonomy-thinking.

The debrief is quite difficult. This is how I describe these “little moments” in my not-even-close-to-being-published book:

“…these small moments require a teacher to hear what a student is saying, process what the student is saying, develop a hypothesis as to what a student currently understands or doesn’t understand about the topic, followed by developing a question or response to what is said that will get the student to go one step further in their understanding, all often in the span of a few seconds.”

You can have a few scripted questions at the ready, but one needs to be quite vigilant and precise to make it really click. I’d submit I have at best about a 50% success rate with quality debriefs….

Thanks for this! I’ve been moving my 4th grade math teaching in this direction this year, so having a framework for thinking about it really helps.

How do you see this intersecting with unit design? Most of your examples are lessons so I’m curious.

How do you assure that problems are not isolated in the sense that students don’t see the connection with them? How do you assure that students learn “all things curriculum speaks about”?