Battling a Paradigm

The following exchange took place in the IMP listserv. I will follow in another post with additional reactions.

Nicole:
I am a first-year teacher at YWLCS. I have only taught IMP 1, and not
for very long, but in my observations and in conversations with the
other math teachers at my school, I have found a couple of things that
might be helpful.

(paraphrased)
1) POW's are too hard for my students, so I encourage them to work on simpler problems to practice math communication. "Having to focus on really dense math and writing up a report seems to be too much for them to do at once."

2) Homework completion is low, and I attribute this to the difficulty of the assignments. I give them modified assignments

My response:
I love your comments, Nicole, and I would like to respond to both of your points. I'll start with the second about challenging homework.

It is, of course, no surprise that students are more likely to complete and turn in work that they are able to do but I would caution anyone against making the work easier as the primary approach to address the problem of work completion. Too often in a concrete field such as high school mathematics we, in order to maintain the flow of the curriculum, push students so as to arrive at an answer and then move on, whether it be to the next problem in the set or the next topic in the book. I think instilling in students the persistence, patience, and understanding of what problem solving entails is getting short shrift in your post.

A very experienced teacher at my school made the following observation of IMP and traditional-background students in his BC Calculus class: While the traditional students were noticeably stronger in trigonometry fields and were more adept with the complex algebra of calculus, the IMP students were superior in their problem solving skills. This was most apparent when students were presented with a new problem, or an old problem in a new context. Traditional students (I'm about to generalize heavily here) would respond "hmm...This doesn't look familiar. We haven't been taught this yet. Teacher, how do I do this problem?" The IMP students start the same, "hmm...This doesn't look familiar," but finish quite differently, "I'll try this old method we learned and see what happens." The point is that through exposure to open-ended problems, some of which were unsolveable by some students based on their background and abilities, the IMP students were not afraid to try and fail. I assume that this is because they understand that failed attempts are as educational as successful efforts, sometimes more so.

Going back specifically to the issue of getting students to complete work, the emphasis should be on the process, not the solution and this needs to be instilled into a students disposition towards mathematics early and often once they begin engaging with the more complex ideas of high school mathematics. One of the more noticeable observations from following a cohort of IMP students beyond Year 1 is how their behaviour, comments and decisions shift away from the values learned in a traditional setting. In an environment of challenging work, where at times only a few students appear to follow the discussion, it becomes OK to not fully understand, and thus OK to commit an error, or fail to generalize completely. This isn't a lowering of expecatations as much as a removal of the pressure to be correct and an increased willingness for risktaking.

And this is the point where POW's become so important (here, I address the first point). The open-ended and challenging nature of the problems conveys two important messages to students: 1) A truly interesting problem is not something that can be solved in a few minutes or not solved at all. An answer is not a bar to be jumped over in an obstacle course of math, and it is not the case that any given student is either one who can jump over the bar or one who can't. Learning is a byproduct of multiple, persitent attempts, and the final resting point for a student is along a gradient of understanding and ability. 2) It is not the answer but the process that motivates these assignments. It may be the case that only a few students fully solve the problem or generalize their work completely, but that is not the point. Knowing how to find the weights of half a dozen bales of hay from limited information is not as important as learning and appreciating that unknown information can be obtained from limited facts through careful thinking and ingenuity (and the A students do not have a monopoly of these things).

I think I'm getting preachy here, and I want to empahsize that I don't disagree fundamentally with modifying assignments or practicing technical communication techniques with simpler problems. I think I'm particularly sensitive to how difficult or easy the IMP curriculum is perceived because I fear that we are going to lose the program in our school because of an ill-informed perception that IMP math is remedial math and doesn't serve advanced students.

Counseling

On Tuesday night, I got a call from a colleague via the math department phone chain. A student was found dead in his room - an apparent suicide - and faculty and administrators were being notified of a brief meeting the next morning. This is the second such incident this year, the first coming in the late fall - a junior just like this newest victim.

One thing that was particularly creepy about the next morning was how clear it was that the students, for once, didn't know as much as the faculty. Between the auditorium and my room, I saw many students' expressions changing as a friend passed on the tragic news - I was suddenly aware of how morose my expression must have been.

The day passed as expected, though. Early attendance was relatively high, and an occaisional class spent the period discussing issues related to the event, and by the afternoon nearly everyone had been dismissed by their parents, was convening to grieve collectively in the gym, or had simply gone AWOL. The remaining students were either the hardier ones, emotionally, or those with few connections to the events.

It's the days afterwards that I find the most difficult. There are more widely varied reactions from students as the intense emotions created by the turmoil rise and fall on their own whim. I myself am divided between my professional duty to maintain the classroom routines and my apathy-mixed-with-deep-concern for the students. It's not that I don't respect how intensely and unpredictably a teenage death effects these youngsters, it's that I find I have little patience for those students who are still drawing attention to themselves through melodramatic behavior or anger and envy at the deceased student for pulling so much publicity.

My personal reaction is also angry, though. I'm not so much angry at the student for their selfish act, or the parents and counselors for not intervening more effectively, or the student body for contributing to an environment toxic enough to drive one to the ultimate end, but angry at the fact that someone was so convinced that there was no other solution that they made such a dramatic and irrevocable decision. I am also mildly angered by the encouragement from contemporary psychiatry to avoid serious discussion with the students of suicide and related issues. Not only are many of the students craving some kind of discourse or forum, if only to spectate, but they are often offended by the perception that the school is ignoring the importance of the event and their own feelings. For the most part I have mildly encouraged a discussion, prompting open discourse by sharing my own personal reactions - my frustration and undirected anger. I am most certainly not a trained counseling professional, but I play one at work.

My Homework Dilemma

I'm doing it again. Every year (except my first), by the first of the year, I have pretty much stopped checking to see if students have done their homework. Keeping tabs on this just isn't a priority for me, although pretty much everything else goes smoother when everyone has done their homework.

The question of how to deal with homework has been one on my mind since my first week in my MiT program. I certainly didn't want to check it everyday (maintaining daily routines isn't in my skillset) and I was reluctant to include homework in my grading scheme at all. After all, we were being rightly advised not to grade first practice efforts. And yet if they didn't have any external incentive, students were unlikely to even look at the assignment much less earnestly engage with the material (which is what we teachers *really* want).

I haven't come much further since then - I have conceived of a dozen different systems for dealing with the issue of homework, and all of them have failed in certain, fundamental aspects. My next big idea is the Problem Set, much like a university course.

My wife, in her graduate math courses, is doing Problem Sets nonstop these days. They're great - challenging, thought provoking questions which must be addressed and completed by a due date at which point they are collected, closely graded and returned as feedback. The collection/grading frequency is relatively low (which suits my tastes) and there is flexibility for the student to prioritize their work in terms of timing - as long as they get it in by the due date. It doesn't quite address the "first practice" issue, though. I might assign additional problems (from the book?) to be their first practices. Then the Problem Set would be a concise and robust set of questions of my own design - these are really the best problem set questions anyway. Then I've just transfered the homework dilemma to the book problems - we'd still have to talk about them in class.

I toyed around with a lesson schedule which completely ignored time in class to discuss the homework assignments until the end of a section - right before a summary assessment. There were other problems with the idea (trying to fit different topics into the same 5-day time frame) so I've abandoned the idea, but maybe the model of delayed and concentrated homework discussion can be adapted to other lesson schedules.

The trick is that besides the lessons, in which the students are exposed to the ideas and/or develop the important concepts and skill themselves, a good learning model should have some independent, reflective practice built in with time scheduled to debrief these experiences. So what exactly is it that we, as teachers, would like to get out of homework? Better put, what does the learning classroom need from homework? Before designing a system, what will be the systems purpose?

Like a 15-lesson CD for learning a new language (which has other important lessons for the classroom teacher - another post), the students need instructional/discovery experiences, in which new material is presented or old material developed further, and practice experiences in which no new material is presented so as not to distract the student from securing the old information as knowledge. The interactive environment of the classroom is best utilized for the former, leaving independent time outside of the classroom for the latter.

Again, I'm thinking of the university model in which there are lecture meetings (direct instruction of new material), section meetings (more interactive concept development) and office hours (for reflection/questioning of the material). It's no wonder I have difficulty with homework discussions - I don't have any office hours. Or do I? Can I dedicate a class period to homework questions with some sort of optional participation? Then I'm addressing student concerns and not forcing every student to sit through a response to a question they don't need answered. Maybe it could be an open structure day of work - I can be helping and answering questions from those who have been working on the problem set (I'm already sold on this idea) and the rest can be catching up on the work in groups or independently. Such a day should probably not be the day before any kind of assessment - that would set up the less motivated for failure. Somewhere in the middle, then, or maybe to impose a delay between the question session and the assessment I can spend the next day or two working on the next topic, although I don't like the sound of that idea - too disruptive of the lesson sequence.

I suppose there is a third type of experience - reviewing and reflecting on the practice time. Since this is another interactive experience, it would work well in the classroom, although I have considered moving the discussion online via a chat room, blog, or threaded discussion. I haven't found any user-friendly and free services that would adapt immediately to classroom use, but the online forum has been successfully implemented at the university level for years now. Hell, the kids are already doing it with instant messaging. I should probably explore IM services more since free threaded discussions are clunky and full of high-school inappropriate adds.

Of course, I haven't even mentioned the problem of constructing the Problem Sets.