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.