– Paulo Freire

Any “philosophy” intended to guide our activity should be predicated by its ends. With respect to education, and particularly to high school, my personal goal is not so much to produce informed or functional young adults (which is addressed earlier in their lives, perhaps to a fault) as to encourage thinking questioners, or ‘lifelong learners’. Encouraging the development of skills is more important than subject matter, because facts are more easily forgotten and the right skills can regenerate content. In the same way, dispositions are more seminal than skills. School should encourage an inclination to combine content and skills into something beyond both, whether this means looking deeply into an issue and eventually owning it, or creating something new. These faculties are some of the greatest pleasures of being human; without them we cannot live full lives. The function of high school should be to promote learning and the desire to learn; functionality and “knowledge” will follow.

Pressed to describe learning itself I would use two metaphors. The initial stages of learning anything involve a lot of relatively accessible skills or information, but not much depth or breadth. As we become more expert our standards go up and our gains, while smaller, become more significant. It’s like an hyperbola: our knowledge gets asymptotically closer to a truth that’s always moving away, without ever catching up. Learning is also like a perverted version of the Bardo system in the Book of the Dead. The third bardo is normal life, the second is twisted, and the first is like the third but elevated, somehow. My kids are too often trapped in the third bardo: at best they want the right method; most only want the right answer. I want them to risk confusion and question the questions or ask their own. I’m not excited about the first bardo, just as I’m uninterested in them crossing the asymptote: these imply an answer, an answer implies stasis, and stasis is the opposite of wonder.

Essential to this proposition is the idea of teacher-as-fellow-learner. Modeling is generally the best course in any case, but it would be ridiculous to expect children to learn they can always look deeper into a question if the teacher acts as though he or she has the only answer. Likewise, it would not suggest continuing learning if the teacher betrays complaisance. My personal experience demonstrates that (for me) there is a clear inverse relationship between my effectiveness “teaching” a subject and my expertise. Also, when I am occasionally tempted to impress students with my erudition, I have to remember that the class is for their benefit, not mine. (On the other hand, when I take risks of my own in class it’s good modeling, and I allow myself to crow a bit if I’m successful.) Student-centered classrooms are a necessary fallout. My students have (still limited but) ever-increasing control over what we study, how we study it, and how much emphasis we give each focus area.

One way modeling can serve us poorly is in approaches to problem-solving. I like and stress a lot of pictures and geometric proof. This doesn’t work for everybody. I once met a guy in Honduras who didn’t recognize Spanish as distinct from English. His approach to making himself understood was to speak louder. It was funny there, but we do the same thing in high school math. Algebra II is only a louder version of Algebra I: the same material presented the same way. When students still don’t understand, we force Precalculus on them, which is a louder version of Algebra II. We need new languages. It’s an extreme example, but last week in one trigonometry class we used regular math, a board drawing, role play, and a demonstration with a banjo; in between, we replicated the work on graphing calculators and two kids presented their own versions of the material. Variety or adaptability is an effective classroom strategy; more to the point, it is a valuable lesson in itself.

Despite coming at complex trig functions from so many angles, they’re still hard to master. Even “visual” kids find graphing approaches obscure, and purely theoretical presentations are certain death. In contrast, it’s easy for students to put these concepts to work to solve real problems, and usually the process clarifies difficulties produced by the theory. Graphing the arc tangent of x is a hard sell, but using it to find the angle between a door and an upper window is not only easy but enlightening. Experiential, “hands-on” education can be incredibly effective. When used in small groups, together with challenging, open-ended and moral questions, an effective presentation plan, and integrated debriefing, it becomes almost magical. An irresistible need-to-know situation arises, kids learn thoroughly, and they take what they’ve learned further. It produces theory, as opposed to merely helping kids integrate it. Best of all, students tend to learn to enjoy learning.

Even so, I’ve never found the experiential approach wholly satisfying. It appears to encourage teachers to aim low, which in my opinion is a high crime. Algebra, for example, rarely lends itself to application, so we resort to elementary school projects. Moreover, these activities not only provide subtle extrinsic motivation (power: otherwise one of its chief benefits), but immediate gratification, which undermines my chief message. As a math teacher I’m constantly reminded that non-Euclidean geometry and complex numbers started out as pure math, unencumbered by any practical application. Now, two hundred years later, we know they describe our universe better than any previous models. But I’m not even interested in the idea that obscure pursuits may someday become “useful”. If the pursuit of deeper understanding or new perspectives is the ends of learning, they need no justification. In fact, justification in terms of a purpose or answer obscures the point, obscures the lesson.

Finally, as teachers, I’m afraid we tend to praise original thinking in our students only when it coincides with our preconceived ideas (perhaps this is especially true in math). We also personalize learning. As a result, by the time they get to high school, students have developed a fundamental confusion about who benefits from their work. Kids who do poorly often apologize, as if it’s the teacher who is hurt. The more specific our questions, assignments, and expectations, the more we encourage this confusion. Clearly there have to be parameters, but I’m finding they can be pretty wide. Students can also generate their own standards, or at least contribute to the process. These days I’m even slowly moving away from deadlines, because I’d much rather encourage quality work than compliance to a timetable. Even GM and Ford want self-directed, problem-solving workers these days; it’s time we did less to produce industrial revolution worker-bees in our schools.

There are many other issues I’d like to address, like deductive vs. inductive approaches, coverage vs. depth, and (its corollary) assigning a few essential questions vs. rote learning through numerous practice problems. These are, however, other papers. In the end, the model I would promote, ideally, would involve a lot of variety, because my students are diverse. I’d try to make sure we look beyond the classroom and get our hands get dirty, since it’s so much fun and so rewarding. My approach would stress questions for the sake of questioning, because while practicality is excellent and necessary, in the end it must defer to simple, undirected curiosity. It would be as open-ended and student-centered as possible, and try never to reward conformity, though it would be grounded in reality (two plus two would still be four, homework has to be done). In Freire’s terms, I want my students to be subjects, not objects, to direct their own learning and lead the most examined, most fully human life they can.