(Sorry, this is a long one! and caveat: I am not claiming that Wikipedia is the be all and end of definitions!)

So according to Wikipedia, PBL means two things (well, three if you count Premier Basketball League, but that’s neither here nor there). If you look up PBL on Wikipedia, the first hit is, in fact, Problem-Based Learning. Why, you may ask? I believe that this is because Problem-Based Learning has been around in various forms longer than Project-Based Learning, but the term itself was coined in the 1960s by Howard Barrows at McMaster University in Ontario, Canada ( http://en.m.wikipedia.org/wiki/Problem-based_learning). You can read more about Barrows’ specific definition of Problem-Based Learning at this site and in my tab at the top of my website that says “Problem-Based Learning.” However, what I feel is one of the biggest parts of Barrows’ definition is the fact that “the problem is a vehicle for the development of problem solving skills” – that is it, that it is the problem – hopefully well developed and set in a context that is interesting, challenging and meaningful to students – through which the students will development and learn the problem solving skills.

Wikipedia names Project-Based Learning (http://en.m.wikipedia.org/wiki/Project-Based Learning) almost the same thing, however, they connect it more to Greeno’s theory of situated learning – “learning by doing” and “teaching by engaging students in investigation.” However, all of these theorists ideas range from about 1991-2006, so it would seem that a PBL by any other name…is not really the same?

One of the most important distinctions in Project-Based Learning (which I will write as PjBL, because you know, it came 2nd, for the record) is the authenticity of the task that is motivated by a larger “driving question” – students learn by creating a project and investigating what they need to do in order to organize or structure their presentation for the project.

So what does this mean in mathematics? A few weeks ago, there was quite a discussion going down on Twitter about what constituted Problem-Based Learning.

Dan Meyer seemed to be criticizing Problem-Based Learning saying that it was discussed by others as “too much curriculum and too little time for PrBL” and he says that he saw “a lot of fluff in PrBL” – to which I would say, there’s much more to Problem-Based Learning than just doing problems, Mr. Meyer. Nat Banting also asked for a clarification of what the difference was between a project and a problem in math education.

In response to this discussion and Mr. Banting’s question, I posted an image of a table I created listing differences that I saw between project- and problem-based learning in mathematics education and had hoped for some feedback.

Then I thought that maybe my description on the image of the differences wasn’t clear enough, so I thought I’d try one more time to make the distinction between the authenticity clear.

If we revisit the idea that Problem-Based Learning has at its core, problems as the vehicle for learning and constructing knowledge, I think this is at the heart of the difference of the learning/teaching experience. When I look at purposefully scaffolded and written curriculum for problem-based learning, yes there are outcomes that need to be met, there are topics that are discussed that are set by the teacher (or curriculum writer). Mathematics has within it many very interesting abstract concepts that are worthwhile to teach even though the “real-life” applications don’t have much context for students (will a student appreciate the logic and problem solving skills that are learned in factoring a polynomial, for example, or do they just accept and take it as a skill they need, or use technology and not use that part of their brain – who knows?) The point I’m trying to make is that in problem-based learning the problems would lead students to figuring out (through discussion and with prior knowledge and experience) the skills needed to create a process of factoring and perhaps “unfactoring” an algebraic expression and hopefully what that expression might represent. This is what’s known as “preauthentication” – when the curriculum writers or teachers try to come up with some kind of experience for the students that simulates the mathematicians authentic experience or “ah-ha” moment of understanding or realization. Then there are other problems that allow the students to dig deeper and apply those ideas to other areas of mathematics, and yes, real-life problems too.

In Project-Based Learning, “emergent authenticity” allows the mathematics to emerge from the ideas of the project (or driving question at hand) which is up to the students to then find out what they need to know. This is where Project-Based Learning can fall short in the area of secondary mathematics – in a world of standards. Where does the learning take place and how does it happen? Once students understand a concept, having gone through rather traditional instruction and some type of formative assessment, the project can then be given. Yes, Mr. Meyer, I believe that there is a hard balance to make there between traditional instruction and when to do the projects – quite a dilemma of time (although schools do it with very nice interdisciplinary time schedules).

However, I do believe that in problem-based learning the rigor, content, problem-solving and all the other “4C’s” skills that project-based learning also promote end up happening in the discussion and presentation of problem ideas and solutions. So I would have to argue with those who say “it doesn’t matter what you call it” and “they’re all the same thing” because the learning process in mathematics is so very different in these two methods. Hopefully, I can shed some light on the differences between PBL and what I hope will be called PjBL soon!