PBL for Engagement and Empowerment in the Mathematics Classroom

While I was holding a workshop this summer I got into a discussion with some teachers about the term “leading questions”. In most constructivist classrooms the term “leading” questions is actually looked upon favorably – as a good thing that allows students to often discover ideas on their own while the teacher leads them to the correct ideas. This is a very positive thing when you think about it, as opposed to the more traditional classroom where most main concepts are merely directly stated as fact and told to the student without any higher-order thinking involved or practiced. Although some students need that every once in a while (and that’s another blog entry altogether..), I believe it is the goal of mathematics education for students to learn those higher-order thinking and self-regulated learning skills. So in general, if we make the assumption that those are skills that a teacher wants to encourage in the classroom, how does one go about doing that without simply spoon-feeding students information?

The idea of handing out leading questions is a very good one and guiding students through either partner or group discussion, exploration and practice, with teacher or student summary of findings in the classroom is an excellent model. This happens in many classrooms around our country to varying degrees of success in student learning. Many teachers who use a problem-based approach tend to use the “guiding” or “leading” question model a great deal in order for students to gain more ownership or the coursework and material, as this is a major part of the pedagogical practice and goals. However, I’ve never been a fan of the word “leading” question. It somehow brings to mind walking a dog. I picture taking a student by the neck on a “lead” and dragging them to the correct answer or point that I, as the teacher, am trying to make. This goes directly against the focus of PBL in general, where the students are guiding the discussion and their ideas are taking the lead. Maybe my analogy is a little overboard, but I’m trying to make a point here – sorry.

So in this workshop, as I often do when I’m public speaking, I struggled to find the right word to describe what I was trying to say. I wanted to describe pushing the students in the right direction. We all know the goal of the problem or where it’s going, where we hope the conversation will lead to the next problem or topic in the curriculum. However, we don’t want to lead them there. Hopefully, if the curriculum is well-written the students have the tools that they need to move forward and with the cooperation of each other, they should be able to follow the path laid out for them. I came up with the visual analogy of those water games where you push the little button and the water is pushed up and little hoops have to get whooshed around onto hooks or into buckets and you win by getting all of the balls or hoops in. Do you know what I mean? In those games the player pushing the button is the force behind the water which is the impetus for the motion the causes you to win. In a PBL classroom, the teacher is the force behind the students who are the impetus for the learning, so it is really the teamwork between the two that causes the success.

After thinking this through, I thought of the word “thrust” instead of “lead”. I guess if you think of a question as possibly thrusting the motion of learning out there and then letting the students go with the information and background knowledge they have and see where it goes. Of course, you may have to continue to thrust those questions at them, just like you would have to in that water game in order to win, but that’s half the fun of the game isn’t it?

I was asked recently by a wise young teacher about the idea of relegating so much of the practice of problem solving in pbl to “homework” time that is outside of class. In many teachers’ mind this takes the observation of the practice of problem solving out of the classroom. I’d love to discuss this a bit here, as I think it might clarify a lot about the procedures of learning problem solving through pbl that I believe have not bee articulated (at least that I haven’t really found articulated in any meaningful way in research articles).

When a student is assigned a problem that they have never seen before and take it home and grapple with it, what do we expect to see the next day? What were they supposed to do that evening? These expectations need to be clear to the student in order for pbl to be successful in terms of learning on many levels. First, a student must be able to dig into their “toolbox” or “toolkit” as many math teachers call it and find remnants and memories that strike a chord with that problem. Others might call this “recall of prior knowledge” or “making connections” or “transfer of skills”, whatever you want to call it, it is something that we want to make happen where a student can find a piece of a problem or solution that have seen before and connect it with a new problem that may not know the solution to in the present. Not only do we want them to connect the idea of a solution, but we want them to connect the idea of the conept as well. Now, do we expect this all to happen in one night of grappling with a new problem for a ninth grade geometry student? Most likely not, that would be practically impossible for a student of average caliber to come up with some mathematically significant new material on their own in one evening. However, if the foundational pieces are laid for them in a way with problems that are stepping stones and arrows pointing in the right direction, it does seem that with the right motivation, they might be able to put down enough recalled information to have a substantial amount of ideas to have productive discussion. This allows the classroom the next day to be the microcosm of the problem-solving real-world of the(substitute appropriate career path here…engineer, business manager, etc.) So in reality, although some of the independent problem solving happens during out-of-class time, a great deal of both the independent and group problem solving practice happens right there in class during the discussion and resolution of the problem the next day.

When a student is presenting the work they have come up with from the previous night’s attempts at a problem, she is presenting the problem solving that happened outside of class. However, even within that presentation she is problem solving. Perhaps she realized her error and because of a comment a classmate made, or an insight shared, or even a prodding by the instructor, she has self-corrected and changed her presentation entirely. This risk-taking and on-her-feet thinking is practice in problem-solving in and of itself. The rest of the class is practicing problem solving because they are learning what they did incorrectly in their recall, how to find the right ways to tell their classmate what she did incorrectly, and also how to be fair and reasonable in their comments to others. The group dynamic in this situation is very important.

So, do I have a problem relegating so much problem solving to students’ work alone for “homework” – no, not really. It does take a lot of coaxing and talking to them about the expectations of the teacher and changing their idea of success on homework. Being explicit about what you value on their nightly efforts is very important and having a grading rubric that does that for them helps – as I do. However, I do believe that having a pbl curriculum sends the message to students that the major goal of the mathematics classroom is that they should be learning to be problem solvers, and that that should be happening everywhere, all the time.

Hi everyone! Just returned from a week of discussing PBL at a conference with two wonderful groups of people, and other individuals along the way. I met lots of great math teachers who all had me thinking and justifying what I do almost continuously 24/7 for the whole week. It was exhausting, but exciting to discuss teaching so intensely for so long with so many. The best part about talking about PBL with people who are new to it is the fresh vision that I get; the new perspective on what I take for granted in so many ways. I want to thank everyone so much for all of the great input and the questions that you all asked during the week.

One of the most important discussions I had with my group was after we had just watched a video of my class for about 20 or so minutes. I had put forth the caveat that it was by no means the perfect class and there was one student that had struggled for a while with one problem about similar triangles. I could tell that it was making my participants very uncomfortable that I had let the student go on, and perhaps it was making other students confused. This is an excellent opportunity for us to discuss classroom management – or maybe a better term for it is “learning mangement”. In other words, when do the “needs of the many outweigh the needs of the few, or the one” (does anyone get that Star Trek reference)? When does the individual student’s learning become more (or less) important than that clarity for the group or vice versa? When is there a point of diminishing returns for the whole group?

Part of the philosophy of PBL is based in social constructivist learning theory where the class will co-construct the knowledge together, even with students at different learning abilities and levels all discussing problems. It is feasible that with consistency and the values of such a learning environment repeated to them that students become acclimated to this environment over the school year and begin to value it as well. This has been my experience at least. However, at certain times when a student is taking “too much” time and others might not jump in to help co-construct, it does leave a moment up to the teacher to decide to either help with the construction, give direct instruction (i.e. perhaps go against the pedagogical ideology of the classroom), redirect the construction to others in the room, or even be patient and allow the organic process even more time.

This was a very difficult thing for people who had not yet seen this type of learning in practice. Although all teachers have heard the adage to what ten seconds before you react to students comments or wait for an answer, the fact is we are all uncomfortable with silence or discomfort in the classroom. We are all uncomfortable with disorganization and potential chaos. However, it is clear that there is great value in allowing a student or a group of student to grapple with a problem, as they practice problem-solving skills. I think it will be important for me in the future to have better sets of skills to give teachers in order for them to aptly deal with, and be ready for, this discomfort.

However, the question of time committed to individual students needs as opposed to the groups needs, I feel is something that you can ask in any instructional approach – even direct instruction where students are allowed to ask questions (probably the only instructional approach where that question doesn’t apply is a lecture where student don’t respond at all). In any classroom, the instructor needs to think about organizing the time committed to dealing with students on an individual basis vis-a-vis dealing with the classroom as a whole – this is no different in PBL. The biggest difference may be that a teacher following this instructional approach may feel more comfortable allowing the student the freedom to grapple with the problem on their own for longer, which honestly is probably a good thing, if you have the time in your class period, of course.