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.

24 thoughts on “A Problem Based Learning Starter Kit

  1. What a great thing to do for others! Tried and tested activities and support are a wonderful way to enter into this type of teaching. I am trying one tomorrow!

  2. Reblogged this on lttmaths and commented:
    Geoff at Emergent Math leads you into inquiry-based learning – this post is packed to the rafters with good starting points for investigations (whether you have much experience with delivering this kind of thing or not) – follow the links for a wealth of other sources of rich tasks.

  3. Great tasks. And I appreciate the thoughtful analyses of why these are good “starter problems.” While I have been doing these types of tasks–though not these specific tasks–for many years, it’s always nice to have a resource like this to point to for teachers who are new(er) to the process of more open classwork.

  4. I usually start with the 8 Queens Puzzle, which does not match any standard as far as I am aware but I’ve had even the most remedial of classes manage with minimal setup. For the starter problem, it definitely isn’t about coverage, but about the process.

    However, I think the coverage problem is a pretty genuine one. I also try to be careful with any argument that runs along the lines of “you already lose time on X it’s ok to lose time on Y too”. A teacher looking at a three-week deficit (routine in remedial classes) needs every second they can get. (I also saw an argument once along the lines of “surely you have fluff activities you can cut out” which is also less than convincing.)

    1. The 8 Queens Puzzle is a good one and surely just more puzzles in general is a net benefit for a math class, especially if students have been labeled “remedial.” Speaking of which, while I don’t mean to belittle the coverage issue, I’m pretty sure if students are traditionally struggling in math (either intrinsically or explicitly labeled as such) that “covering more content” is going to be the salve that gets them to reengage with the subject. There’s much more foundational work to do there that I think spending some time away from linear content coverage would be better in the long run.

  5. You are wonderful!! You have posted exactly what I teach and who I teach! I, too, am struggling with how to introduce problem-solving techniques.
    I cannot thank you enough.
    You have made me excited, giving me a guide to begin with.
    Thank you from the bottom of this OLD teacher’s heart!!

  6. I am trying not to be a negative Nellie but I was a victim of this approach when I was in elementary school. I recovered in 4th grade all that I had lost in 3rd with project based learning and here it come again?! Now my youngest son has fallen victim to a math teacher trying to be all fancy with projects while the curriculum is slipping by. I got a progress report saying he can’t do the math required via the county’s pacing guide. Apparently, his group’s business plan for a mock Shark Tank they did for FOUR weeks was fine though. Little did she know, my child did nothing, nada, zip in math for four weeks. He’s too low functioning and weak in basic skills so he was nudged out of any math the group did as they designed a product to sell to another grade level for money given for free. So many holes in the implementation and I don’t know what to do as the suffering parent. The county and state evaluation process remains the same yet this group mess has masked the fact that no real math is going on. I’ve been to see the teacher who is getting married in May and she said this is the wave of the future. The community has asked the school system to graduate students who can collaborate and work in groups. The issue I have is that my children aren’t going to graduate from high school without learning basic skills. Nor will my boys get into any college since the SAT and/or ACT are still required for admission. Strong teacher with appropriate implementation and support, yes. This? NO.

    1. Laura, did you, like, look at any of the tasks here? Or read the words in this post? None of these should take longer than a couple class periods.

  7. Very new to PrBL and trying to rework my Geometry curriculum. Thank you for having a nice starting point. All help is appreciated!

  8. I am VERY new to PrBL. Thank you for having this starter information. All help is appreciated.

  9. I love PBL as I think it will help to answer the question “when am I going to use this in my adult life”. I agree that the foundational math facts and principles need to be understood and these projects build upon that.

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