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?

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.

Something that I thought was extremely interesting in this chapter was the argument that teaching with the “big picture” of problem solving skills in mind is actually much more difficult for the teacher. This totally resonates with me and my colleagues who are making this shift this year. We are all feeling more tired, more spent at the end of our day. Now, this could be because of a new “block” schedule that we are using this year, but I also think some of it has to do with the way we are teaching.

Schoenfeld quotes another researcher, Burkhardt, who says that teaching problem solving has its unique challenges for classroom teachers. The first of which is that the teacher “must perceive the implications of the students’ different approaches, whether they may be fruitful and, if not, what might make them so.” This is clearly a challenge for teachers without experience with this type of pedagogy. Not only do you need to be free to pursue incorrect solutions, but you need to reserve judgement, find “kind” ways of telling students they are wrong, control the dynamic of discomfort that students have when the teacher is uncertain, to name a few challenges. During the summer, when we did professional development before adopting this new curriculum, we had days of discussions about all of these issues. However, discussion only gets teachers to a certain level. I firmly believe that the best education is the experience of doing it, while having some observation for another opinion.

Secondly, it’s difficult pedagogically. It’s a BIG change from what most teachers are doing currently. There are so many new decisions that the teacher has to make that he/she may not be trained for. The teacher must “decide when to intervene, and what suggestions will help the students while leaving the solution essentially in their hands, and carry this through for each student, or group of students, in the class”. How do you assess for each student individually when they are solving a problem in a group? How do you give them enough practice in order to allow them to be ready for an assessment? I think that our curriculum makes these goals pretty accessible.  I’ll try to assess that this year in some way.

Thirdly, it is difficult personally.  The implications of putting yourself out there and being vulnerable every day are many.  Burkhardt says “the teacher will often be in the position, unusual for mathematics teachers and uncomfortable for many, of not knowing: to work well without knowing all the answers requires experience, confidence and self-awareness.”  I believe that these are just the tip of the iceberg of the personal implications of teaching with PBL.  One of my colleagues and I got in a conversation recently about how we are so surprised at how much the students don’t know and are afraid to try new things.  My take on it is that not only might this be the first time that the students are being asked to think this way, but we are deliberately asking them to think this way.  In any other type of teaching, these issues might not arise because the students are not being asked to hunt down the prior knowledge of their past algebra skills or to be invested in the learning process, as we are asking them to do.  I feel that we are noticing these weaknesses mostly because we are forcing these issues with our students.  Instead of getting down on them and ourselves, we should feel proud that they are accessing and practicing these skills.

Does this make it any easier?  Not really.   Is this a reason to not do it?  Definitely not. In the article that I wrote called “Transition to a Problem-Solving Curriculum” I commented that teaching this way requires a great deal from the teacher including more patience, more insight, more flexibility and more mathematical knowledge.  This is a lot to ask of our already overworked and underpaid faculty.  But, as Schoenfeld says, true problem solving is “far more rewarding, when achieved, than the pale imitations of it in most of today’s curricula.”

I also “learned” that comprehension has to procede production..so does this mean that in order to produce a valid solution to a problem you have to understand how to do it first? Hmm, well, I have students every day coming up with ideas that I don’t necessary teach them and they have pretty good production, I think.

I just read in a chapter of a book by Collins, Brown and Newman (see below for citing) that the reason why teaching problem-solving is hard is because it “requires the externalization of processes that are usually carried out internally”. You can imagine how tough that is. It’s tough enough for a teacher to make adjustments to a skill like knowing how to write the equation of a line that they can see and observe. At the same time, students can have a hard time seeing an example of what they should be doing to problem-solve as opposed to seeing how to write the equation of a line. Collins, Brown and Newman say that “this externalization (can be) accomplished through discussion, alternation of teacher and learner roles and group problem solving” (p.458).

This makes me reflect on how I might be modeling problem-solving in class. In order for the kids to be able to follow a model, there has to be something there for them to watch and mimic, right? I think what I try to do sometimes is ask questions of the kids, and try to get them to answer me in moments when I really know what to do next, but want someone to help out. I know that I am not always right, so that at least models it. I also think that I do not hesitate to attack a difficult problem. I tell them, do like Ed Burger tells you to do. He came to our school last year and told the students “Do you know what to do when you get a math problem you can’t do? (Pause) Don’t do it!” (Cheers from the audience) “No, no, not what I meant, you do an easier one.” His message was that when presented with a problem that you can’t do, make it simpler by seeing what you CAN do. I thought this was a hugely important message for him to send to the students. It really speak to the message of empowerment and prior knowledge. This is something I’m still working on how to convey to students.

Collins, Brown & Newman, (1989). Cognitive apprenticeship: Teaching the crafts of reading,writing, and mathematics. In Resnick, L & GLaser, R (Eds). Knowing, learning, and instruction: Essays in honor of Robert Glaser. (pp/453-494), Hillsdale, NJ:Lawrence Erlbaum Associates, Inc.