A Problem Based Learning Starter Kit

You’ve seen the tasks. You’ve read the research. You’re basically bought in. But how do you begin? More importantly, how do you introduce students to inquiry driven learning?

Or maybe you’re not convinced. Perhaps you maintain that the teacher is the primary knowledge constructor. Perhaps you’ve been burned in the past by inquiry driven instruction. You tried it and didn’t see kids learning much and you feel like you wasted some amount of class time when you could have been actually teaching. I can speak from experience: if I wasn’t part of a cohesive team (all subjects, as part of an entire school effort) I quite possibly would have tossed inquiry, Problem Based Learning, groupwork and everything else in the trash after my first miserable experience with it.

Or maybe your students are burned out and beaten down on math. They’ve been labeled “remedial” and by golly, they’re living up to that stamp that your district has placed on them. To them, math is an arbitrary bunch of rules to follow and steps to regurgitate. Their test scores stink and they have difficulty applying math in new and novel situations. Applying math in new and novel situations is probably an entirely foreign concept. Up until now they’ve had example problems or math instructional software to guide them through their problem packet.

It’s always tough to be the first. In many cases, you might be the first teacher to actually ask students to solve complex math problems without pre-instruction. Students might look at you cross-eyed the first time you ask them to work in groups collaboratively on a problem that may not look like the stuff they see in their textbook. There isn’t an example problem for them to look at. Yes, you are the first line on the shores of Normandy.

Not all problems are created equally and some may be more easily acquired and delved-into by students. If you’re not careful with your first exposure of kids into a new way of mathematical task-posing, you and the students could easily frustrated with the process (if you even have one yet). As Dan states perfectly in one of my favorite posts this year on first-steps toward inquiry, “The Unengagables“, “you’ll be hearing from their attorney.” Dan poses three quick methods of introducing kids to mathematical inquisitiveness, be sure to check those out, and follow the comments. I’ll follow with a few tasks here that I think make for good first-foray’s into Problem Based Learning (PrBL).

I like these tasks as first-forays for a few reasons, pointing two directions.

For the teacher:

  • The problems kind of “implement themselves.” That is, there isn’t a whole lot to do to massage the task to make it implementable. While I don’t necessarily advocate a plug-n-play curriculum, it’s ready to toss in the oven.
  • It doesn’t take too long. Maybe a day, maybe two at most. I’m not sure any first-foray into PrBL should last more than a couple days.
  • The task includes facilitation notes and/or other supporting resources.
  • The task naturally fosters student and peer-to-peer dialogue. Obviously any good task should do just that, but these tasks especially do that with minimal teacher-prompting.

For the student:

  • It’s naturally engaging or intuitively interesting. Real-world is nice, mathematically perplexing is better. 
  • The problem allows for multiple ways of being mathematically smart. Hopefully some of these tasks will spur the conversation about being smart in math in multiple ways. Habits of a Mathematician type stuff.
  • The task at hand is clear. And gets to the point.

Here are a few problems that I’d consider starting with. Or, if you’ve been burned or you’re skeptical, problems to try and experiment with.


Why it’s a good starter problem:

It ties together a visual and number sense. There are several ways to prove or demonstrate a solution. It gets to the point.

Why it’s a good starter problem:

The task allows for guess and check. The task is intuitive and understandable. The scaffolding task involves analysis of samples of student work, a non-threatening way of fostering dialogue.

Why it’s a good starter problem:

The scaffolding involves manipulatives. The math naturally folds into multiple representations and modeling.

Why it’s a good starter problem:

The task prompts students to ask the question. There is an “either-or” possibility for initial guessing and estimating. The task allows for easy differentiated instruction (don’t know how to find the diagonal of a right triangle? how ’bout a workshop on Pythagorean’s Theorem?).

Why these are good starter problems:

You probably have a file cabinet in your room.You probably have a door through which students enter your room. Students have seen and interacted with post-it notes. Students have seen and interacted with styrofoam cups. And with a phone, you could recreate this exact Act 1 video. The task may incorporate multiple ways toward mathematical smartness. Kinesthetic learners might engage via experimentation with post-it-ing the file cabinet themselves.

Dialogue is an inevitability with Always/Sometimes/Never. It can be tailored to your specific classroom. The notion of finding counter-examples is one of the most mathematical ways of thinking I can come up with, and one that kids intuitively understand (it’s a shame we rarely bridge that). If you incorporate some Geometry-type Always/Sometimes/Never cards (like these), kids will be begging you for scratch paper.


Why it’s a good starter problem:

This task, like all of Yummymath’s, include well thought out worksheets with questions that allow for deep conceptual understanding. If you’re not comfortable with driving the car, let the questions that Bryan provides steer for a while.

Why it’s a good starter problem:

For teachers, I think this models nicely how to modify a textbook problem to something more interesting. For students, they have specific math-like things to do. It gives them exposure to modeling from an authentic scenario.


Why they’re good starter problems:

It’s got the presentation – usually with video – ready to roll. The Mathalicious team is adept at both humor and conceptual understanding. Like Yummymath, they can steer the ship for a while until you’re more comfortable with less lesson plan structure and organizing groupwork. The lessons are all aligned to CCSS.


So, there are a few problems to experiment with.

“But they don’t address my particular standards.” you say (For the record, all of these tasks do address specific content standards, see?, but possibly not yours). To that I’d say don’t worry about “coverage” for a day or two. Teachers lose teaching days all the time due to pep-rally schedules, fire-drills, or whatever. And (this is a whole other post waiting to happen but) coverage is overrated. If you can get kids buying into math – possibly for the first time ever – that’ll go a lot farther than coverage.

What are some of your favorite “starter problems”? What other advice would you give teachers that are starting out? Or maybe, better yet: what was your first experience with any kind of inquiry-based instruction like? What was it like for your students? Feel free to share in the comments.

Update 11/22/2013: Andrew, as always, doing great work. I feel like weekly POPs might be another way to dip you and your students’ toes into non-routine problem solving.

A Critical Ingredient Missing From My Math Blogging

Recently I came to a convicting realization recently with the help of a friend. If I’m painting with a broad brush, I’d suggest that effective math classrooms have three things in place:

  • Quality mathematical tasks
  • Effective facilitation
  • Social and emotional safety

I’m not sure if there’s a rank-order of the importance of these ingredients to a successful math classroom, but let’s just say that they have equal weight: a third, a third, a third.


As I look at my own posts, others’ posts that I’ve bookmarked and favorited, and where I spend most of my time, it looks roughly like this.


I’d give it a 60%-30%-10% split, and that 10% might even be generous.

In other words, social and emotional safety is something I rarely think about, and even more rarely blog about.  In Jo Boaler’s “How To Learn Math” course, she shares interviews with former math students who exhibit signs of trauma. That’s the word she and they use: trauma. While I think most math teachers that I read and follow probably understand the need for developing a safe place emotionally and socially for the classroom, it’s something I don’t come across that often. Much of that is my own fault. I spend most of my time blogging and talking about cool, engaging tasks and nifty facilitation moves. I don’t spend much time at all trying to flatten the spoken and unspoken social structures that crop up in nearly all classrooms. Part of this is that it’s actually much harder work to do that than designing a cool task based around an article or something. It’s also something that really can’t be modeled in a few-hour single PD session.

It’s also probably in part because I’m a dude. This post on the prevalence of males and math tasks has been lodged in my brain like a tumor. I can forget about it for a few days at a time, but every now and then it’ll pop up and I’m reduced to a stammering mess. I hope I’m not gender-norming when I suggest males tend to be more in to designing tasks and “troubleshooting” student engagement and less adherent to the social and emotional component accompanied with mathematics classrooms, which is just as if not more important than the other ingredients. I certainly have spent 10-to-1 time or more on designing, finding, and thinking about mathematical tasks versus flattening the social status of mathematics learners. Again, the ratio of what makes for a successful math learning experience is off, either in what I’m reading or what’s actually out there. Probably both.

Consider this post part confession, part imploration. My lack of writing on the social and emotional structures is something that has shaken and convicted me. It’s also something that damn hard to find posts on throughout the Math ed blogosphere. Maybe I’m looking in the wrong places, but I am looking and not finding much. Consider this not only an imploration for math bloggers to blog about developing social safety in math classes, but also to share these more often (starting in the comments here, pretty please?).

I can’t promise that I’ll start blogging about how to develop a flatter math classroom as I can’t be sure I’ll acquire any unique insight other than copying and pasting what others have said. But I do plan on making this a huge point of learning for myself this year. I want to read more about developing an emotionally and socially safe math classroom that flattens structures and gives students an in to mathematics. I want to write more about it too.


Update 2/5/2014: Glad I’m not the only one struggling with this. Erica Synder finds a nice analog of the social and emotional safety structures with Digital Learning.