As someone who has used Problem-Based Learning for almost 20 years and sad to say has never been part of a full-fledged Project-Based Learning curriculum, what I know best is what I call PBL (Problem-Based Learning). I know there is a lot of confusion out there is the blogosphere about what is what, and with which acronyms people use for each type of curriculum. I did see that some people have been trying to use PrBL for one and PBL for the other, but I guess I don’t see how that clarifies – sorry.

So when I use the acronym PBL in my writing I mean Problem-Based Learning and my definition of Problem-Based Learning is very specific because it not only implies a type of curriculum but an intentional relational pedagogy that I believe is needed to support learning:

Problem-Based Learning (Schettino, 2011) – An approach to curriculum and pedagogy where student learning and content material are (co)-constructed by students and teachers through mostly contextually-based problems in a discussion-based classroom where student voice, experience, and prior knowledge are valued in a non-hierarchical environment utilizing a relational pedagogy.

Educational Psychologist and Cognitive Psychologists like Hmelo-Silver at Rutgers University have done a lot of research on how students learn through this type of scaffolded problem-based curriculum dependent on tapping into and accessing prior knowledge in order to move on and construct new knowledge. There was a great pair of articles back in 2006/2007 where Kirschner, Sweller & Clark spoke out against problem- and inquiry-based methods of instruction and Hmelo, Duncan and Chinn responded in favor. I highly recommend reading these research reports for anyone who is thinking of using PBL or any type of inquiry-based instruction (in math or any discipline). It really helps you to understand the pros and cons and parent and administrator concerns.

However, after you are prepared and know the score, teachers always go back to their gut and know what works for their intuitive feeling on student learning as well. For me, in PBL, I look at how their prior knowledge connects with how, why and what they are currently learning. One of the best examples of this for me is a sequence of problems in the curriculum that I use which is an adaption from the Phillips Exeter Academy Math 2 materials. I’ve added a few more scaffolding problems (see revised materials) in there in order to make some of the topics a bit fuller, but they did a wonderful job (which I was lucky enough to help with)and keep adding and editing every year. The sequence starts with a problem that could be any circumcenter problem in any textbook where students use their prior knowledge of how to find a circumcenter using perpendicular bisectors.

“Find the center of the circumscribed circle of the triangle with vertices (3,1), (1,3) and (-1,-3).”

Students can actually use any method they like – they can use the old reliable algebra by finding midpoints, opposite reciprocal slopes and write equations of lines and find the intersection points. However, I’ve had some students just plot the points on GeoGebra and use the circumcenter tool. The point of this problem is for them to just review the idea and recall what makes it the circumcenter. In the discussion of this problem at least one students (usually more than one) notices that the triangle is a right triangle and says something like “oh yeah, when we did this before we said that when it’s an acute triangle the circumcenter is inside and when it’s an obtuse triangle the circumcenter is outside. But when it’s a right triangle, the circumcenter is on the hypotenuse.”

Of course then the kid of did the problem on geogebra will say something like, “well it’s not just on the hypotenuse it’s at the midpoint.”

Dicussion will ensue about how we proved that the circumcenter of a right triangle has to be at the midpoint of the hypotenuse.

A day or so later, maybe on the next page there will be a problem that says something like

“Find the radius of the smallest circle that surrounds a 5 by 12 rectangle?”

Here the kids are puzzled because there is no mention of a circumcenter or triangle or coordinates, but many kids start by drawing a picture and thinking out loud about putting a circle around the rectangle and seeing they can find out how small a circle they can make and where the radius would be. When working together oftentimes a student see a right triangle in the rectangle and makes the connection with the circumcenter.

A further scaffolded problem then follows:

“The line y=x+2 intersects the circle in two points. Call the third quadrant point R and the first quadrant point E and find their coordinates. Let D be the point where the line through R and the center of the circle intersects the circle again. The chord DR is an example of a *diameter*. Show that RED is a right triangle.”

Inevitably students use their prior knowledge of opposite reciprocal slope or the Pythagorean theorem. However, there may be one or two students who remember the circumcenter concept and say, “Hey the center of the circle is on one of the sides of the triangle. Doesn’t that mean that it has to be a right triangle?” and the creates quite a stir (and an awesome “light bulb” affect if I may say so myself).

A few pages later, we discuss what I like to call the “Star Trek Theorem” a.k.a. the Inscribed angle theorem (I have a little extra affection for those kids who know right away why I call it the Star Trek Theorem…)

I will always attempt to revisit the “RED” triangle problem after we discuss this theorem. If I’m lucky a student will notice and say, “Hey that’s another reason it’s a right triangle – that angle opens up to a 180 degree arc, so it has to be 90.” and then some kid will say “whoa, there’s so many reasons why that triangle has to be a right triangle” and I will usually ask something like, “yeah, which one do you like the best?” and we’ll have a great debate about which of the justifications of why a triangle inscribed in a circle with a side that’s a diameter has to be right. So who are the bigger geeks, their teacher who names a theorem after Star Trek or them?

References:

Kirschner, P. A., Sweller, J., & Clark, R. E. (2006). Why minimal guidance during instruction does not work: An analysis of the failure of constructivist, discovery, problem-based, experiential and inquiry-based teaching. *Educational Psychologist, 41*(2), 75-86.

Hmelo, C. E., Duncan., R.G., & Chinn, C. A. (2007). Scaffolding and achievement in problem-based and inquiry learning: A response to Kirschner, Sweller and Clark (2006). *Educational Psychologist, 42*(2), 99-107.