California State Standards

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

Standard 1

Engage and support all students in learning.

Fundamentally, schools exist that people may learn; and if people are not learning, something needs to be done. This is an area where I have succeeded. It is also an area where I have failed spectacularly.

Two classrooms, two situations, two outcomes.

Successes in Integrated Mathematics 3: Despite earlier protests, students have stopped asking, "Why are we in groups now?" and "Why don't you just teach us?" when I ask them to ask someone else. One who'd failed math for a year got his first B on a test on data analysis. The quiet Latina began speaking up in group discussions. A junior with extremely spotty attendance started coming to class regularly. After grading each other's work, a student noted that "Grading is a lot more difficult than I expected. You have to make accurate and excellent decisions on what you grade." The girl in special education became an "expert" on polar-Cartesian coordinate conversions. Students speak up--politely--and challenge my fiat; I am no longer the arbiter of knowledge, and this is as it should be. Teacher as resource, not ruler.

Failures in Integrated Mathematics 1: In my reflection on February 13, 2002's lesson, I note some equity dilemmas. They have not changed through the course of the year. When half to a third of the class--and not always the same half to a third--is missing on any given day, and a dozen students have either dropped out, been transferred elsewhere, or been expelled, something is wrong. What have I done? I've smiled, spent individual time hashing through problems, and talked to students after class. In the end it doesn't matter. I could not figure out how to help reach them. I don't know what to try next. I don't know what to adjust. Toward the end, the Accelerated Math program made possible more individualized instruction even as it shoehorned students into a last-ditch series of multiple-choice questions.

In some sense, this standard is irrelevant. What counts, in the end, is what the students--any of them--walk away with, or don't walk away with. Of course, that is what this standard attempts to address. Its external imposition is superfluous. I know what I've done, and what I've failed to accomplish.

With the number of factors involved, I'm at a loss to analyze the why. One class allowed considerable freedom in pacing and coverage, while the other was constrained by district-mandated testing. One class consisted of students who had already succeeded in math to some extent, while the other consisted almost entirely of students who had failed the class one or more times. The students and the content were different, of course. How does one address the myriad factors that go into a class when all the factors operate in conjunction, not isolation? I haven't found any answers.

Unfortunately, it's not as simple as plucking activities and wholesale from one class and transplanting them onto the second, which would result in a Frankenstein's monster model of instruction anyway. For one thing, the activities don't apply. I have an idiosyncratic approach to the topics at hand and as a result I end up creating, customizing, improvising, and/or modifying everything I use.

It haunts me, but I have no miracles, and I never will.

Standard 2

Create and maintain an effective environment for student learning.

The short answer: see Standard 1, above.

The long answer: By some miracle, my Classroom Management Plan seems on the right track. The mantras that my primary class know by heart are:

• Be polite.
  
• Be constructive.
  

and the implicit

• Be relevant.
  

Though this fell apart on a couple occasions (see Journal: April 16, 2002 and Journal: April 18, 2002), I think the students involved learned something from the experience. These rules are meant to foster a space where no student feels silenced or "stupid," and where mistakes are an illuminating part of the learning process, not a searchlight thrown onto their vulnerabilities.

The important question is, of course: does this work?

The answer this time is a resounding "maybe." I think I can safely say that most students are willing to give math a try most of the time. Like the rest of us, students, too, have their off days. Is it effective? "Progressive" teaching methods such as collaborative learning and student autonomy must, like any other teaching methods, be evaluated against what they do for the students. Having never taught an Algebra II-level class before, I don't feel that I have an accurate baseline on which to judge the effectiveness of the classroom environment; failing that, I'm left with the judgement I do have, and even there I see flaws. I accept that I am not going to single-handedly "save" every student I ever meet, but neither am I resigned.

A seemingly banal stumbling block is organization and lack of time-sense. I have had the experience of standing in front of the classroom with notes in my hand covering details such as collecting homework, handing out activity sheets, posing brainteaseres, and so on, then looking down half an hour later realizing that the notes are still there, but I forgot their existence. Until I internalize such classroom routines (and a year is not nearly long enough), I will have to rely on my improvisational skills and student "administrative assistants." While this year I have been fortunate enough to have an experienced teacher remind me of such things as "The bell will ring in 2 minutes," I will not always have this crutch available.

Standard 3

Understand and organize subject matter for student learning.

My understanding of mathematics, though highly eccentric, is my one great strength as a teacher. My exposure to mathematics has ranged from tutoring calculus to teaching my sister why there are an infinite number of primes, from teaching myself computer programming to college-level coursework, from reading assorted books on math to a high-school attempt to derive transformations in hyperspace. Practically speaking, this allows me to concentrate on figuring out what's going on in the students' heads instead of figuring out my own understanding. By no means do I consider my knowledge solidified, in the sense that there are always new connections to be made, especially cross-disciplinary ones.

In particular, as someone who defected to math from a prospective major in history during my undergraduate years, I am especially interested in the connections and applications math has in diverse applications and diversions. In my Curriculum Unit on coordinate systems and vectors, I drew upon cartography (TREASURE HUNT Part 1), board games (NINE MEN's MORRIS and CHESS), and even a water-gun shooting example to illustrate various concepts and applications.

My background in math is also my Achilles' heel, as shown in a lesson on logistic growth functions. It is appalling to turn into the math teachers I never understood because their knowledge had been internalized and integrated so deeply that it became "intuitive," which is rarely the case for a student encountering the material for the first time. I need to relearn that "first time" experience and accumulate an understanding of common misconceptions as time goes by. It's not a fast solution, but as I never took notes during my high school years on my learning processes, it's what I need to do.

Standard 4

Plan instruction and design learning experiences for all students.

The short answer: again, see Standard 1. There are many students I do not know how to engage or reach, predominantly in the Integrated Math 1 class where the school's curriculum is far less flexible in comparison with the students' needs and prior knowledge. The downside of my idiosyncratic understand of mathematics and creative streak is that I am often tempted to create everything from scratch instead of working with existing resources. When there are definite constraints to what I may to pitch in favor of something more offbeat, this is a handicap. I am frustrated by my inability to work within a system to create learning experiences that will benefit all students. "They're not motivated" may or may not be an accurate diagnosis, but it doesn't in itself address the question of what to do about it in a given situation.

The first long answer: see Coordinate Systems & Vectors. By no means was this unit "perfect," but the principles behind its design, no matter how flawed in execution, are ones I hope to refine in future years.

My guiding idea is that I will inflict nothing on my students that I wouldn't find interesting, unless it has been demonstrated that they do like it despite my own preferences. To that end, I strove to bring anecdotes and lesser-known examples into lectures, foster a sense of mathematical community so students could learn from each other, and pace instruction so that as many students as possible had the opportunity to participate in doing math. This is one area where the formidable reputation mathematical ability has garnered can actually be beneficial, if students can get past the initial "I can't do this": once they see they can, the ego boost sometimes seems to spur them onward.

The second long answer:

Standard 5

Assess student learning.

While Wiggins & McTighe's "backwards design" is an illuminating key principle, it doesn't address the issue of truly unanticipated answers. This, by the way, is why I don't use rubrics. I have come across some fascinating examples of misconceptions (students believing that reflections across y=x could not actually touch or cross that line) and lateral thinking (a group assignment on expected value where students designed a deliberately "rigged" carnival game, complete with shills to work the crowd).

I have known for a while now that I will need far more extensive pedagogical content knowledge to design more effective assessments to address the issue of how I can identify "learning" when I see it. "Novel" assignments, such as the math book report that students wrote last semester, will no doubt require several years of "field trials" before I am able to identify strands of thought with any certitude. When in doubt, I find, ask the students. I may agree or disagree with them, but the salient point is that their impressions will force me to think about the messages they receive, and that I send, about what's really important to know, think, and do.

In the meantime, I've wrestled with how to weight effort vs. accuracy, thought vs. expression, and in both cases I'm tending to come down more on the side of the first. Given encouragement of effort, and appropriate feedback from peers or myself, accuracy will follow; and this encouragement can have the handy side-benefit of developing metacognition as students learn to check their own thinking processes and answers. Given encouragement of thoughts, I can then work with students on expression in standard mathematical notation as well as more creative media.

Standard 6

Developing as a professional educator.

I am just beginning to learn how to sift through forgotten textbooks and libraries for resources and materials. My strength is in seeking out written materials that are not normally thought of as "textbooks," such as various math-related books. I've consulted with others, whether my master teacher or fellow STEP students (particularly in numerous carpool discussions); during spring quarter I also audited Math Directed Readings so I could get a sense of the issues other math STEP students were grappling with.

What I've learned from this year is that I need to figure out how to become a better human being before I can address how to become a better educator. For teaching deals intimately with people, with their hopes, fears, and frustrations; with the products of their hands and the utterances of their mouths, the movements of their thoughts; with the problems and dreams they trail after them like caravans, whether the teacher is prepared or not. Teaching deals with the interplay between people and the families, cultures, and societies that shaped and continue to shape them; it cannot exist separately from that network, but must find a way to entwine itself, vinelike, and bear fruit from the soil it is given.

Copyright © 2002 Yoon Ha Lee <requiescat@cityofveils.com>
Last updated on 8 June 2002