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.
— 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.