When teachers begin to implement problem-based learning, several questions consistently come up and I have a slide to discuss them during my presentations. At the 2014 California Math Council (CMC) North conference in Asilomar, Michael Fenton mentioned that these questions would be worth sharing in a blog post.

Anyone else think @robertkaplinsky should write a blog post about his “Problem-Based Learning FAQ”? Drop him a line (or RT this). #CMCN14

— Michael Fenton (@mjfenton) December 7, 2014

So, here are my frequently asked questions…

– If you do two or more problem-based lessons per week you would be spending at least 2-4 days per week on these lessons. That would take you away from the balanced rigorous understanding we are striving for. Their procedural skill and fluency as well as conceptual understanding would suffer.

– On the other end of the spectrum, problem-based lessons are most certainly not something you do once a semester or “after standardized testing.” You definitely won’t have balance that way either.

– I believe that the sweet spot lies in doing about one to two of these lessons per unit. They are a great way to introduce a context that will provide a foundation you can use to build conceptual understanding and later procedural skill and fluency. By starting a unit with a lesson, you begin with this context you can keep referring back to (i.e., “When we did the In-N-Out burger lesson, do you remember how we kept adding $0.90?”). Having a second lesson at the end of the unit provides a nice bookend to the unit and acts as a culminating review activity. Using problem-based lessons in this manner leads well into my next frequently asked question.

Let’s assume you teach five periods and have three lower classes and two higher classes. I would begin any unit on surface area with my favorite problem-based lesson: Andrew’s file cabinet task.

<tangent>If you’ve done this task, you know how amazing it is. If you haven’t, please do it with your students. I don’t care if they are kindergarteners or college students. It is such an amazing lesson because even if you don’t know a single thing about surface area, you can do this task. It is easily scaffolded so if you are a young kid, you can count the squares on the front side . If you are an adult, you will definitely not find the problem beneath you. It also provides such abundant context for developing the idea of area as non-overlapping square units and of surface area being comprised of the net formed by the area of each of the object’s sides.</tangent>

So, you start the unit with this problem. Not surprisingly, the two higher classes are able to finish the file cabinet problem in one period on the first day. Also, not surprisingly, the three lower classes get stuck and even if you gave them a couple more days, you are not sure they would finish it. They do understand some of the foundational ideas such as having to cover the sides with stickies and that you would add all the sides together. Most likely they get stuck on the computations and converting between square inches and stickies as units. What they also have, very importantly, is the realization that there is something they want to figure out but don’t know how to figure it out. Those are two realities that are frequently missing when students learn mathematics.

Accordingly, after day one you have two classes who completed the task and three who haven’t. All classes now have the file cabinet as a context. Now continue on with however you would normally help students develop conceptual understanding and procedural skill and fluency. No one will be saying, “When will I ever use this?” because all new learning can be actively tied back to the context you’ve already created.

When talking about nets’ connection to surface area, students will be able to connect back to the file cabinet and more solidly grasp how the knowledge is interconnected. Eventually you will be done teaching the portion of the unit on conceptual understanding and procedural skill and fluency. To wrap up the unit, it is important to recall where the classes were at.

The two high classes have already finished the file cabinet lesson, so clearly you can’t do that again. Fortunately there are many other lessons on surface area including my aluminum foil office prank lesson. For the two high classes, have them give this a try and it will be another opportunity for them to demonstrate that they can apply their knowledge. As for the low classes, it is time for them to return to the file cabinet task.

Remember that this was something that they wanted to figure out (if you don’t believe me, look at all the positive feedback about it on Twitter) but weren’t able to. Now comes the litmus test. If they can figure it out now, it is a wonderful moment because this was a skill they needed and wanted yet didn’t have. They can now realize the growth they have made. If they cannot figure it out, then it is an honest reality check that the lessons on procedural skill and fluency were not enough and intervention needs to happen. The final result is that all students have a strong skill set that they can apply.

- Don’t grade it. Do you grade all your other classwork?
- Grade it using a rubric. I tried this and even integrated it into version 4 of my problem solving framework. What I learned was that it was so cumbersome and difficult to use that I couldn’t even figure it out… and I made the rubric!! Getting through five periods of students just seemed ridiculous. Ultimately I decided this was crazy to do and haven’t used it again.
- Use a scoring scheme similar to what the Smarter Balanced Assessment Consortium (SBAC) initially published for their constructed response problems. Essentially you get one point for a correct answer and one point for sufficient reasoning to support that answer. You can modify this to be out of ten and be five points for a correct answer and one to five points for sufficient reasoning.

SPOILER ALERT: In addition, the honest reality is that many of your first lessons will be awful. If you don’t believe me, check out my first lessons on the very bottom of this page. They are well intentioned but really not that good. In related news, if you are using one of my first lessons, I apologize.

If after all my discouraging, you still really do want to create these lessons, then I would say that the lessons average around 5 to 10 hours to make. I have done a few in a couple of hours and some have taken me over 40 hours by the time I had finished filming, editing, uploading, writing up the lesson, and posting them all over social media. Seriously, did you notice how many times Andrew changed clothes during the video of him covering the of the file cabinet? I think it was at least 5 times. That is a lot of time. I am so glad that was him and not me… and that he, and many others, are gracious enough to share them for free.

Please leave me any questions I missed in the comments.

Love this post, Robert!

Could you offer some guidance for implementing problem based learning in a college math classroom? (College Algebra, Trigonometry, Pre-Calculus)

We typically only have one class session per topic (1.5 – 2 hours), and I have found that the biggest obstacles in understanding occur because of rusty algebra skills and/or number sense.

Hi Erin. One problem based lesson could probably be finished during a single period. So what if you started a unit on graphing trig functions with this problem (http://www.101qs.com/2450-ferris-wheel). They would probably get to a place where they could plot the points and notice that the graph seems to repeat itself like a wave but they likely wouldn’t have the formal notation or the ease of using an explicit function to determine a location after a specific amount of time.

Next with a context set and the basis for conceptual understanding of sine and cosine waves begun, you can take them through the procedures as well as the other trig functions. Then, as a culminating activity you could come back to this problem and have students apply their new knowledge.

Overall you will probably have to skim off 10-30 minutes per topic in each unit but it seems like that should work. Am I missing something?

Once again, you have shared wonderful advice. I laughed when you advised not to score it with a rubric, I couldn’t agree more. Scoring a problem based task (in my opinion) detracts from the inspiring nature that they bring. I often contemplate creating one, but I am happy to leave it up to you, Andrew, Dan, etc. and use with glee. Thank you for so generously sharing your ideas with the great mathematical community. You all have no idea how grateful I am for our network. Sidebar, one of my students mentioned the Zoolander activity yesterday-I had this student last year.

Thanks for the kind words Merryl and continuing to be such an enthusiastic supporter. Doesn’t it feel great when students remember activities we worked hard on? Also, you might enjoy this blog post which builds upon the ideas I shared around assessment: http://robertkaplinsky.com/4-ways-assess-3-act-tasks/

Thanks a lot for sharing this post with us. Regarding assessment my plan is to make a test with envelopes and cards:

1) First envelope: a card with a link. Student take the tablet, go to the link and see the video. In this envelope, one blank card is present. Student has to write “what she notice” and what she wonder.

2) Go to teacher table and pick up second envelope where aiming question is exposed anb all possible variables and data for response it. Students has to write in blank card which information will she use

3) Go to teacher table and pick up 3rd envelope with the all available data. Students must reponse with a resoned answer.

End of the test.

I have NOT tested this test. So I no warrants. But my plan is to implement in mid february.

I am trying your How much were those pennies? problem next week as an intro to Decimals. I feel better after reading this. Wish me luck.

Good luck. It’s rarely easy but often worthwhile. Let us know how it goes.

My co-taught kiddos did the best work. Creative. On task. More counting and estimating. I had the pictures on the wall so that they could get out of their seats to estimate. My co-teacher was impressed with their problem solving skills. Granted it took a day, but so worth while. We will definitely include more into our year.

Yay! Glad you had a success you can build upon. It’s so nice to see them shine as problem solvers.

I found that I was hearing from the same group of students when doing these whole class (5th grade). I’ve started putting them into hyper-docs for my more eager math nuts and added an extension to keep the challenge going.

https://goo.gl/LxyBmd

Would love to hear your thoughts.

Have you read the book the 5 Practices for Orchestrating Mathematical Discussions (http://amzn.to/2ylVE04)? It’s the bible for how to facilitate these types of conversations. I would recommend checking that out as it contains pretty much all my thoughts on the matter.

Do you have any recommendations for lessons that are good for those of us starting out? You should rank them like a recipe. 🙂 I teach high school. If you have any suggestions on where to begin, it would be appreciated!

Yeah, that is certainly a question I’ve pondered over the years. My recommendation for an intro to problem-based lessons would be to use Andrew Stadel’s file cabinet task (http://www.estimation180.com/filecabinet.html). It is great for all secondary students and allows for a variety of strategies to be compared. That’s my top recommendation.

I just saw the file cabinet task on OpenUpResources’ new middle school curriculum. I’m wondering if you have any thoughts about OpenUp (Illustrative Mathematics) curriculum? It’s received a lot of good press recently, but I have mixed feelings on it based on the units I’ve used thus far.

Hi Eunice. This is not the answer you’re looking for, but I am really not qualified to give feedback on it. While I’ve observed teachers using the lessons, I have not actually taught using one myself. So, I really don’t have enough experience to make an informed statement.

What I will say is that I know and respect many of the people on the team. I know them to be brilliant and thoughtful people. It’s challenging to please everyone. For example, some people will love a curriculum that spends more pages developing conceptual understanding instead of procedural skill practice. Others will hate the curriculum for the exact same reason.

So, I think it is definitely worth investigating it and seeing if it is a good fit for your situation.

Thank you so much for sharing all of this helpful information!! I tried to click on your Five Practices link, but it says it no longer exists! Can you post a new one??

Thank you!!

Thanks for the heads up. I just updated it with a working link.

Great blog post. I’ve only recently started to use these kinds of problems in class and trying to find the balance between randomised group investigations (think P Liljedahl), and the rest.

Question: how long is a typical unit? I’m sure they are all different, but let’s say it was three weeks for the unit, would you do one problem per week?

Thanks James. I think this blog post might answer your question well: https://robertkaplinsky.com/4-ways-assess-3-act-tasks/

Thank you, but unless I’m missing something, this post doesn’t address my question.

I won’t grade any of the tasks. As a PYP teacher, we don’t grade anything in that way. I’m more interested in how you balance problems and the other activities to help with conceptual and procedural understanding etc. and how this looks over a unit.

Sorry, wrong link. Is this good: https://robertkaplinsky.com/two-ways-integrate-problem-based-learning-unit-another-avoid/?

Thaaaaat’s more like it! Excellent blog and I really like the idea of losing something that will definitely get them stuck, and then coming back to it with new skills etc. A slight issue I have is that I have a combined grade 3 and 4 class (including a grade 2 and grade 5 student). Recently, I’ve been using quite a few 3-acts within each unit so that they get loads of opportunities to apply but also practice on the job and learn from others.

Is it possible to overdo using 3-act tasks? I think I might be at the moment.

Final question about the unit length.- how long (ish) did you spend on each shape in the volume unit?