December 7, 2001

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

On Friday, 7 December 2001, I reintroduced sequences to the Integrated Math 3 class; they had previously been given a worksheet on the topic, which many of them found confusing. Originally I had planned to cover sequences (and possible misunderstandings of their definition), infinite vs. finite sequences, recursive formulas vs. explicit formulas, and a lead-in to limits.

I am excited about this chapter because it leads to some important mathematics. Even though many of these students may never go on to formal analysis and, say, Cauchy sequences (which gave me nightmares for a week in college before they finally made sense), sequences arise in many places in "real life"; Fibonacci's sequence and the nautilus shell is but one well-known example. The ability to find and describe a pattern is important in many domains of knowledge, and limits are a keystone of modern calculus.

As it turned out, I had not anticipated how much the class would struggle with the notation for recursive formulas, since it had been a long time since it was new to me. As a result, we did not get to the rest, but I felt it was more important that students become familiar, if not yet comfortable, with subscript notation and ways to write recursive formulas. In general, mathematical notation often has the benefit of conciseness, but can often be a source of confusion and obfuscation. Fortunately, sequence notation is far from as bad as mathematical notation gets, but it still requires some acclimation.

(I should mention that one student--who ought to be getting an A in the class now that he's attending regularly again--did see the beginnings of an explicit formula, but I put that on hold. I did stop by his desk later to tell him we'd get to his idea the next class period.)

There are several things I would have liked to do better. First, students did seem comfortable producing the two elements of a recursive formula--the starting number(s) and rule for deriving one term from the previous one(s)--but, as mentioned above, writing these in the prescribed "math-speak" posed difficulty. While I attempted to give them some practice in "translating" words to notation, it might have helped to go even slower, especially with subscripts such as n or n+1. More practice with "simpler," obvious sequences (along the lines of 1, 4, 7, 10...) before moving into the Koch snowflake problem would be one possibility, with students checking each others' work.

Second, while I did attempt to "outline" the flow of the day's lesson on the whiteboard with questions and key terms (I never write fast enough on the whiteboard during class for my own satisfaction), I remain somewhat dissatisfied with my use of the whiteboard. The on-board notes started in an organized, hopefully readable fashion, but quickly became crowded and "complicated." I find myself drawing an unholy number of arrows in the course of a lecture or discussion, and from past experience I remember how hard it can be to reconstruct a thread of thought from an arrow-laden whiteboard if your attention wanders for a moment. It's sometimes difficult to plan ahead because I don't yet have enough pedagogical content knowledge to anticipate the issues that will come up and I tend to wing it a lot when students ask questions, but perhaps I can practice "organizing" the whiteboard content on the fly.

Third, as my supervisor pointed out, once students got working on finding patterns in the Koch snowflake, it would have been far more efficient (and perhaps helpful) to bring the class back together for discussion of recurrent issues or questions. That's something I need to remember to do more often. I think sometimes that my brain gets stuck in "tutor" mode because that was most of my experience "teaching" math in the past, so I have to remember to remind myself about whole-class needs and exigencies.

Fourth, one thing I observed in Megan's middle school classroom and would like to incorporate in the future is having students engage in inter-group discussion when they get "stuck." The more minds, the merrier; even asking students to share things they've tried in their groups might help stimulate new ideas and approaches.

I remain happy with the lesson's overall flow, but I'd definitely like to increase the interactivity of the class and ease students more gently into the often frustrating world of mathematical notation! (In some sense they're lucky that they may never see subscripts that themselves have superscripts and subscripts....)

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