Hiebert et al made many references to Dewey's views on problem-based learning (PBL). For this, I was very grateful. Like Dewey, I found the reading easy and insightful. Overall, Hiebert et al were discussing the importance of making the subject problematic. This is opposed to learning a skill so that you can find the answer as quickly as possible. The difficulty lies in attempting to foster a desire to solve a problem in the students. Numerous times the idea of making the problem a "real-world problem" came up. This idea in itself can be difficult to wrap one's head around. Is a real-world problem one that only occurs outside of school typically? Does it have to be related to an occupation, that isn't a mathematician or mathematics teacher? It was decided that perhaps we don't need to focus on the real-world problem, and instead focus on the process. By establishing a culture in a classroom where reflective inquiry, and PBL are fostered and have become second nature, any type of problem can produce effective "residue". What is residue you ask? Residue is a term that has been defined in this paper as the "understandings that remain after an activity is over" (p.17). If there is no residue, then what was the point?
I find that it can be very difficult to produce PBL without the necessary basic skills. I believe that it is a fine balance between being a teacher who stands at the front of the classroom and provides knowledge to their students, and being the teacher who provides a classroom with an embedded culture that promotes reflective inquiry and PBL. I feel that this is perhaps one of those things that appears scary to try out, knowing that it could very easily crash and burn, but at the same time produce interesting results, whether it goes the way you think it will or not.
The question I have is:
Do you think that this style of teaching is more appropriate at certain age levels than others? On a spectrum does PBL lead towards understanding of relationships, or does teaching of basic facts/relationships lead to PBL which leads to deeper understanding of said relationships?
I find your question about appropriateness at different age levels quite interesting. I know that for the adults I teach, PBL works very well - but there are always challenges. The most obvious being they typically have never seen math done in this context and are hugely uncomfortable to begin; later I think this works to PBL's advantage. And there are always one or two rather vocal students who persist in their displeasure, wishing instead we went over the basics for the full two hours. But again, in such instances I remind them that is where homework comes in. Perhaps that is why PBL works so well in my instance: we have the luxury of asserting students should complete several hours of homework and study for every hour spent in class.
ReplyDeleteWhat is best about PBL though, which ideally resonates at all age levels, is that when students get stuck, when they really need to learn something new in order to solve a problem, that is when they are so receptive to instruction - be it on the basics or otherwise.
I love the idea of problem-based learning at all age levels. Unfortunately, I think that the notion of PBL is much more common in the elementary grades than in secondary, but I'm unsure as to why. I think part of the reason might revolve around the fact that since most problems (exercises) at the secondary level take more time, that some instructors consider these "problems." True. there are numerous steps involved to finish the exercise, but numerous steps does not imply that it is a problem that involves reflective inquiry. Personally, I find it helpful to distinguish between exercises and problems, but I think that these are often muddled to be the same at the secondary level.
ReplyDeleteWith respect to your comment on the spectrum, I'm unsure. My gut tells me it's valuable to work in both directions. Sometimes, working through exercises can lead to reflection and developing an understanding of relationships. Indeed, one could make a problem of relating numerous exercises and trying to find patterns and relationships among them. I did this recently with my class when we were working through integration by parts. They worked through some exercises and I asked if they could generalize the method and remark on why one way would be better than another. Conversely, PBL can certainly lead to developing a connected understanding, particularly with mathematics outside the context of the current topic. I think this is where PBL really shines; it can bring mathematics into a connected web, rather than a series of disconnected ones.