February 13, 2002

"A mathematician is a machine for turning coffee into theorems."--Paul Erdös

On 13 February 2002, as a continuation of this unit on inequalities through algebra and graphing, my master teacher and I decided to introduce solutions to systems of inequalities through a patty-paper activity that would allow students to see the overlap between the graphs of two inequalities. (No wonder lesson-planning can be so unpredictable. The genesis of this activity came when she spotted a large box of forlorn, unused patty paper and thought, Why don't we use this?)

Class began with a warmup problem on the overhead in order to review solving a system of linear equations to lead into system of linear inequalities. Some of the students, I noticed, needed a lot of reminders in order to graph linear equations in the first place. It's occurred to me that note-taking skills are strong in some students but relegated to "make sure you copy this down for your reference" (whether it's understood or not). Unfortunately, as that was the extent of my note-taking strategy in math (as opposed to, say, history--my system of abbreviations determinatives, an idea I borrowed from Egyptian, was quite extensive), I'm not sure how to remedy that effectively. I've found that for concepts (as opposed to formulas), if I have to go to my notes, I'm unlikely to have the foundation to figure it out from them. I don't know how universal this is, though.

In any case, most students were able to find a solution graphically, but floundered when asked to do so algebraically. Though they had worked through several "canonical" examples during the previous class, not everyone had brought their notes or understood how to apply those examples to new problems. I remember working with one student who had difficulty picking out a parallel problem. This kind of problem-solving skill is something that we did not address near the beginning of the year and I really regret not thinking of it sooner. I wish I knew of a more (high-school) accessible counterpart of Imre Lakatos' Proofs and Refutations: The Logic of Mathematical Discovery, of which I've read tantalizing excerpts. (It's on my list of books to pick up, but that list has been ridiculously long for the last 15 years.

I went over the algebraic solution on the overhead, which is something I still am not comfortable using; overhead management could be a workshop in itself. In any case, I attempted to show the method of substitution, but ran into the misconception that unlike terms can be added. It made me realize all over again that variables/polynomials are tough. Polynomial rings, particularly the notion of having a polynomial modulo another polynomial, gave college juniors and seniors headaches in applied algebra; and the article we read in C&I on "What Are Variables, Really?" (I don't recall the author or citation information offhand, but I can look it up some weekend) made me wonder if I understand them in any depth, which I really doubt in the sense that I don't believe I could give a satisfactory definition that addresses their many uses.

Something that I didn't think of for this class, but have begun using, is to reserve a segment of the board as "tangent" space. This way students would have become less confused by the tangential nature of the discussion on like and unlike terms. It's sometimes hard for me to judge when such a topic should be acknowledged as a "tangent," since issues rarely come up without being deeply intertwined with the matter at hand, but the presence of such a reserved space at least reminds me to consider the possibility. I've begun starting off by using that space for a "math fact of the day" (e.g. the NSA hires more mathematicians than anyone else in the U.S. and probably the world--one might imagine the data would be sensitive!). On the other hand, I seem to run out of board space ridiculously quickly. (It's even worse on the overhead, which offers a far smaller area and feels far less open to student interaction.) I begin to appreciate why at Cornell, White Hall had blackboards in the corridors for passing mathematicians to scribble on. Too bad science-fictional holographic "screens" can't be pulled up anywhere in the classroom on demand. The worst part is, I don't know how to make whiteboard use more concise until after the discussion at hand has been wrapped up, by which point it's too late.

All this, and I still don't know whether that tangential discussion at all clarified the issue for students. I doubt it did, and it certainly confused an "outlier" student. Also, there must be a better way to address such "outlier" students. I've usually been an outlier student myself, and constant deference to the majority was never much consolation for me, either.

Finally, the most nagging issue: disengaged students (who show up regularly; we have a couple students that we're lucky to see once a month). By now one of them is probably so far "behind" that it would take a lot of individual tutoring and attention--which is what he seems to want, but working with him requires extraordinary patience, as he continually attempts to bypass questions and extract answers from the teacher (a process that I suspect he wouldn't be so persistent in if it hadn't given him results elsewhere).

The other student in this category is something of a mystery to me. He is extremely withdrawn during class unless confronted by my master teacher, at which point he becomes antagonistic (not entirely unreasonably), and is only animated when speaking with two of the other black students. I honestly have no idea what his math skills or abilities are; I'm not sure how to approach him about it in more depth when his interactions with my master teacher are so hostile on both sides; I don't know where to begin.

The entire class could probably benefit from cooperative/groupwork norms if they were set up, even now, but making group-worthy tasks is something that I've heard a lot of theory about and don't have a good model for implementing with regard to specific mathematical concepts. If it's difficult for Algebra II material, it's 10 times more difficult for Algebra I material. In my case, this is partly because I was taught Algebra I in a completely traditional, rote manner (and only by dint of my mom's persistent interventions did I ever bother doing my homework), but had an exceptional teacher in Algebra II who did foster discussion and conceptual thinking.

I also worry about consistency in setting up groupwork norms, which (in the minimal sense that they are implemented) meet resistance in my Integrated 3 class. (In particular, I think of a girl who has been continually adamant that "you don't act very much like a teacher, Ms. Lee.") I am quite aware that when my master teacher is in the room and offering help, particular students will bypass asking me ("You're not helpful!" or "Why can't you just tell us if it's right?" or "You can't leave! We don't get it yet" are common responses, though happily declining in frequency) and go to my master teacher, who will generally step them through a problem. If this consistency issue is a problem in Integrated 3, I suspect it would cause riots in Integrated 1, and I can't blame them. It must be awful to be a student "juggling" two teachers with differing expectations, and I haven't discovered a satisfactory way to resolve this.

Copyright © 2002 Yoon Ha Lee <requiescat@cityofveils.com>
Last updated on 21 May 2002