I was eager to read this week article, as I've always been particularly intrigued by Schoenfeld's work. This piece took a more expository stance, as he used his teaching experience in a problem solving classroom and related it to his work within his research cohort. In his research cohort, although he is the "leader" of the group, he is often unsure of what his data means or where it will take him. In contrast, he states that in his problem solving class, where he is also an authoritative figure, not only does he "know all the answers", but he also has a particular "lesson image" for how the class will go. His "lesson image" is so detailed that he can predict the questions he and his students will ask, as well as the responses that he will provide. My immediate reaction was "well, that must be kind of boring." For me, I enjoy the unique questions and inquiries that come from my students when we are solving problems, but I can certainly relate to his statement (particularly when I reflect on group based problem solving sessions I've assisted in). He goes on to state that his students are probably unaware of his ability to provide such detailed descriptions of what they will do, but that what ultimately matters is that he is "reacting directly and honestly to what the produce."
His ultimate goal in the course is for student's not to see him as the mathematical authority, but the mathematics as the authority. For me, this took awhile to develop. I don't think it was until much later in my mathematical career that I realized mathematics could "speak." I knew whether my argument was flawed or not, since the mathematics I used told me so. I didn't need a professor to tell me if it was "the right answer," but rather their reaction to my argument and advice on perhaps making it a bit more elegant. By the end of the course, this is the type of "community of inquiry" that exists in Schoenfeld's classroom. Students no longer request his "certification" at the end, as they have realized on their own, through the mathematics, that they understand the problem and it's solution. If at the start of the course a student would ask him "is this right?", he would react with " don't convince me; convince the class. Do you folks buy it?". He admits that students in his class have occasionally discovered results that were new to him and he claims that it didn't really matter to the students that it was new to him - that they were "simply doing mathematics."
I found his story compelling, but it has brought many questions to mind. As an instructor in a University mathematics course, I often struggle with the idea of not being able to answer my students questions. Especially as a graduate student, the amount of "authority" and "experience" I have, I suspect, is significantly lower than that of a tenured professor. Through my classes, I make a strong attempt to make the mathematics a joint effort between myself and my students, so that I am not perceived as the knower of all. Even still, I get emails referring to me as "Professor Radzimski"! A question I would ask Schoenfeld is, as a graduate student instructor not much older than most of my students, would you think that it might be easier to foster a community of inquiry within the classroom? Particularly in a non-problem solving based class, filled with theorems and definitions, what action can we take to foster inquiry among students in general?
I'm surprised to read that a tenured professor would make claims that they understand material so well that they can predict the solutions given to them by their students. While I may predict some common errors that students will make, and although I may have a certain lesson image (I assume Schoenfield meant a lesson plan?), there have been questions that have baffled me while I was teaching me - these weren't questions I couldn't answer, so far - but they were very broad questions which needed to be carefully answered.
ReplyDeleteI think that this kind of inquiry learning occurs well in an honours classroom, but I don't know if the same can be said for a remedial mathematics classroom where students are reaching for their calculators for any given problem without knowing exactly what they'll enter once they get there. I can only imagine the deer-in-headlights look that my students would have the first time they were asked to defend their solutions. To be honest, I can remember the gut-wrenching feeling it would produce in undergrad. It's a great thing to do, though; most of the time crunching away at a problem, it can become a challenge to defend a solution. I would suggest (and this is something I do) putting up a set of solutions on the board (all the solutions students give), then walk through the problem slowly - perhaps identifying where the error occurred - to no-one in particular, to maintain some degree of anonymity.
I certainly agree with the concern about being seen as the "authority" of mathematics - I tend to worry in a new teaching setting that my mathematics "authority" will be far below that of the teachers' authority there, and that I will not be taken seriously as a teacher if I'm not "nerdy" or "math-y" enough.
I really enjoyed reading about how mathematics can "speak" to students (whether it contradicts or confirms the teacher). It's definitely important in a classroom, of any age level, to establish exactly what your role as teacher is. Is it to simply teach a concept, give the students practice with said concept, and then assess whether the concept was learned? Or is it to introduce a concept, explore why it is useful and how it can be used to solve a host of problems, and ultimately develop an inquiry-based learner?
ReplyDeleteWhile I would wish the math to speak to all of my students, I understand that sometimes this simply isn't so. In this respect, I like to think of the voice of mathematics as something that can be spoken through the teacher, the textbook, the peer, or the self. Depending on the make-up of a classroom, sometimes the teacher's voice is your best option. With a willing class, that knows the rules of civilized conversation, many voices may be heard, agreed upon, or challenged. Then again, now that I think about it, maybe it's the problematic children who always disrupt class, who actually need a voice, and need to be heard.
One final thought. . .if we only ever use the teacher to act as a voice for mathematics, eventually the children will learn to distrust their peers, their textbooks, and even worse, themselves.