Geoff Krall
This is a post in the ongoing Emergent Math mini-series: Routines, Lessons, Problems, and Projects.
Ah problems. I have to reveal my bias here: I love problems. Problematic problems. Problems are where I honestly cut my teeth as an educator. If you’re reading this blog, might have stumbled across my Problem-Based Learning (more on that specifically in a second) curriculum maps. I’ve blogged about Problem-Based Learning (PrBL) a bit. I’ve learned so much from teachers and math ed bloggers about what makes a good problem, how to facilitate a problem, what kinds of problems are out there. Some of that I’ll share here. Let’s just start with Problems.
The questions on voluminous review packets? Not problems. My first resource on problems, problem-based learning, and problem solving is NCTM’s research brief on problem-solving, Why is teaching with problem solving important to student learning?(2010). In it, it hints at the “what really is a true problem” question:
Story or word problems often come to mind in a discussion about problem solving. However, this conception of problem solving is limited. Some “story problems” are not problematic enough for students and hence should only be considered exercises for students to perform.
This brings us to my personal, current definition of Problems: Problems are complex tasks, not immediately solvable without further knowhow, research, or decoding of the prompt. Problems can take anywhere from one class period to three or four class periods.
So when I say “problems” I mean problems that are genuinely challenging to the problem solver. Even the difficult, toward-the-end-of-the-section questions may not be problematic enough for some students. Also, a problem ought not to be so obtuse or convoluted as to not be accessible for all students. Just because something is real hard doesn’t necessarily mean it’s a problem. If someone were to challenge me to make the U.S. gymnastics team, I wouldn’t consider that a problem; I’d consider it futility.
I like to think of good math problems like this: a good problem is accessible enough so students a couple grades lower can attempt it, yet challenging enough so students a couple grades above have to think about it. I actually think this of all mathematical tasks, but it’s particularly apropos of problems.
Here’s a good problem (from Illustrative Mathematics):
I like this problem for many reasons. One, it combines two not-often mathematical things: lines and quadratics. In most curricula you have your unit on linear functions and your unit on quadratics. Why aren’t these two things combined more often? I have no idea. Most textbooks presents lines in one unit and quadratics in another, as if they’re in a different universe. It’s like we’re reading a geography textbook about pre-Columbian South America and Europe. But back to our discussion of problems, it’s the confluence of these concepts that makes this such an interesting, challenging, and worthwhile problem.
There’s straight up problems – just give students a prompt and facilitate as you see fit.
There are countless other modes of problems, here are a few.
- Would You Rather? problems
I’m not sure of John Stevens is the first “would you rather” problem designer, but he certainly codified it with his stellar website. A Would You Rather (WYR) provides students two possible choices and students must decide which one makes more sense to choose: which one is cheaper? which one is better? what deal gives the greatest value? etc.
There are several things that make this format incredibly appealing: 1) Providing students an initial choice naturally facilitates guesses and estimates at the beginning of the problem. 2) Making it a choice makes CCSS.MP3, making arguments and critiquing the reasoning of others, a necessary part of the task. 3) In many cases, either answer may be correct, depending on how it’s interpreted, the desired outcome, or the input variables (in the WYR above, the answer may depend on how far away one is from the airport, how much airport parking is, etc.). And 4) there’s something delightful about the “would you rather” framing. Maybe because it reminds me of the “what’s worse?” scene from So I Married an Axe Murderer.
- 3-Act Tasks
Dan Meyer gave us this format years ago and countless of math teachers have built upon it sense. Following the narrative structure of movie, in act 1 the “conflict” is established and we’re drawn into the plot of the movie/problem. In act 2, our hero / students go questing for the solution. In act 3, we come to a resolution.
Most often these act 1’s kick off with a video or picture to pique the interest. What do you notice/wonder? What do you think will happen? In act 2, students will work through the scenario presented in act 1, sometimes provided with additional information or knowhow that might be useful to solve the problem. In act 3, students make their final answer and we come to some sort of resolution (often by playing the last part of the video).
Dan has the most comprehensive list of 3-Acts, but others have followed suit with their own libraries.
I’m sure I’m missing others. Please let me know in the comments who I’ve missed.
Like WYR, there’s something inherently appealing about a narrative structure that we’re already used to. We’ve all seen movies, plays, TV shows, and read books. If you can provide a successful hook, we’ll want to see how the movie ends.
- Just straight up puzzles
While sometimes challenging to align directly to required content, give students mathematical puzzles. NRICH has a great library of puzzle-like maths, or perhaps maths-like puzzles.
And I don’t know if the authors (or you) would consider these puzzles, but I quite enjoy the tasks from Open Middle as puzzle-esque math.
Problem-Based Learning
Let’s take a slight birdwalk into the practice of Problem-Based Learning or PrBL. It uses problems as a means to teach new concepts or knowhow. The problem creates a need (and in the best cases, a desire) that requires the intended content knowledge, additional information, or mathematical dispositions.
I suppose in some ways this may not differ much from just giving students the problem and teaching as-needed, as you go. In PrBL there’s an intentionality (and even predictability) with how the problem is posed and how the learning is facilitated (for instance, you prepare the lesson beforehand, rather than just winging it).
Facilitating a problem
One of the biggest mistakes teachers make when using Problems for the first time is that they think that by posing a clever enough problem, students will intrinsically work their way through it dilligently, testing out different methods along the way. And to be sure, it’s understandable to think that when you watch a presentation on problem solving in math or participate in a conference session and the participants or audience dilligently work their way through a problem. But here’s the dirty little secret about conference sessions: the audience is entirely composed of adults who are excited about math and presenters are showcasing their absolute best problems. It’s easy to present engaging problems as a panacea when the audience is entirely bought in and the presenter gets to cherry pick which problem or lesson he or she gets to present. So it’s easy to walk away from these experiences thinking that – just like in that session – I’ll present this super-cool problem to my students and they’ll collaborate, problem-solve, and stick to it just like at that conference.
It’s never that smooth. Rather than – like Carrie Underwood – letting “Jesus Take the Wheel” – you need to keep your hands on the wheel and your foot on the pedal (and sometimes the brakes as well). Problems should be facilitated, not tossed in like a hand grenade. So how do we facilitate a problem?
Use routines. The biggest tip I can provide for facilitating problems is something we’ve already covered in this mini-series: provide routines. Routines to get started on the problem, routines to facilitate discussion in the middle of a problem, and routines when students are sharing their solutions.
Consider this sample Problem facilitation agenda:
- Introduce the problem
- Facilitate a Notice & Wonder routine
- Identify next steps and let students begin working
- 20 minutes later, take a quick problem time out and have groups do a gallery walk routine to see how and what other groups are doing
- Give a problem “time in” and have students continue working toward a solution
- After finishing the problem, have students show appreciations to one another via a routine.
One problem, three routines. And who knows? If students are struggling, you may want to hold a small workshop lesson in there as well. We’re starting to see our Routines, Lessons, Problems, and Projects framework become a set of nesting dolls.
Provide consistent group roles. Assuming students are working in groups, provide consistent, well-understood group roles.
And – like the problem itself – don’t just provide the group roles and hope for the best, check in with them and how they’re operating. Mix them up. Talk with them.
- “I’d like the Recorder/Reporter from each group to meet with me at the front of the class for five minutes to discuss your progress.”
- “Harmonizers – at this point give one of your teammates a compliment.”
- “I’d like all the Facilitators to swap groups for the next ten minutes.”
- “Resource Monitors – come up with a question as I’m going to go to each group and you can ask me one question.”
Use these roles, don’t just assign them.
Make Problems the cornerstone of your class
Quality problems won’t be the most often employed mode of teaching in your class, but make them the essential thing that students do in your class. Rich problems make for excellent assessment artifacts. They help teachers find the nooks and crannies of what students can do and know and what gaps in understanding still remain. They foster mathematical habits in a way that lessons and routines often can’t.
To be transparent, part of the reason I began thinking about this mini-series is because I was wrestling with the question: what’s the “right” number of problems to facilitate in a school year? And what are those problems? That’s when I began to think of the music mixing knobs analogy from my intro post.
There are endless ways to facilitate problems – use routines early, often and throughout a problem. Use Problems often and throughout a class. They are the bedrock of your class, and the discipline of mathematics more broadly.
For more information, go to Emergent Math.