"Why Can't You Just Tell Me?"
"Philosophy is a game with objectives and no rules.
Mathematics is a game with rules and no objectives."--unattributed tagline
A Lazy Teacher?
If you ever visited my classroom, these are some exchanges you'd have a fair chance of overhearing:
"Ms. Lee, I don't get any of this." (Usually from a student who has spent most of the class chattering about non-math-related topics. I stifle off-topic conversations that have overstayed their welcome, but you can't catch 'em all.)
Well, why don't you ask [Jane Doe].
"What do you mean, ask [Jane]? You're supposed to be the teacher!"
For one, [Jane] can use practice communicating math--mathematicians spend a lot of time communicating, and can answer your question perfectly well. For another, there are twenty of you and one of me...do the math yourself.
Or another common scenario:
"Ms. Lee, can you show me how to do #3?"
Sure can. *starts writing an example on the whiteboard*
"You're not doing #3!"
Nope. I'm doing a related example. (Usually the same situation with different numbers.)
"Why can't you just show me how to do this one?"
Because once you know how to do one, you ought to be able to do another. It won't kill you to exercise your brain a bit more.
(By now, you are probably getting the impression that 90% of my students are whiners. I assure you this isn't the case; you're seeing a distorted sample, after all, so that I can get my point across rhetorically.)
So does this make me a lazy teacher?
Selective Laziness
I'd say (though I'm biased): yes and no.
Yes, because much mathematics, like much computer science, is selectively applied "laziness." We take shortcuts. When you learn to count in kindergarten (or whenever it is), the teacher does not start with set-theoretic foundations. And I didn't even realize you needed a proof of the existence of real numbers until I took my first theoretical math course in college.
That isn't really the point, though. More important is the no, because it all depends on what your goals are as a teacher.
In the first scenario, my goal is for students to get a taste of classroom esprit--the classroom as a mathematical community--but also to counteract the years of "teacher as the arbiter of truth" conditioning they've had, especially in math. Something isn't true in math because I say so; it's true due to historical convention (in the sense that Euclidean and non-Euclidean geometries can both be "true") and the consensus of the mathematical community (as recently as De Morgan, negative numbers--familiar to students by middle school, if not earlier--were considered an absurdity). What makes the difference between a high school classroom and academia is that I have far more background in mathematics than my students do...but I can still be wrong, and that experience means I have more tools at my disposal for examining mathematical "truth" (itself a contentious problem in math philosophy).
There's a practical reason, too. I'm a fan of groupwork when it can be set up and taught to students (yes--you have to teach kids to work effectively in groups, because otherwise you end up with slacker syndrome) because it is far more efficient for several kids to figure things out and serve themselves as teachers of a particular topic than for me to run around and explain it to everyone. Side-benefits are that this improves students' communications and productive socialization skills--no bad thing--and exposes them to different styles of "teaching," increasing the probability that someone's explanation will make sense to a given student with a given learning style.
As for the second case: it's a truism that examples are helpful to many students, but I attempt to break away from the "show me how to do it" model as soon as the student's readiness makes it feasible. Why? Practically speaking, I will not be there to answer questions after the student leaves my class. Learned helplessness is not what I want to foster. The student must learn enough problem-solving skills to tackle the problem, however fumblingly, on his/her own. Doing a parallel problem will often suffice.
Even better, however, is teaching the student how to try different approaches. "Well, what have you tried?" If the answer is "nothing," some cajoling on my part is necessary. (Or in later stages, a mild rebuke to try something before asking me.) I'd much rather help him/her figure out his/her own approaches than feed them mine. If I had the opportunity to start a class (I came in just before Thanksgiving break), I'd start with a two-week unit solely on problem-solving, using logic puzzles and other brainteasers--because this is a skill that, if approached broadly, generalizes to fixing the car or calculus or whatever else comes up in life. If someone doesn't know how to calculate the length of a triangle midsegment ten years after s/he leaves my class and isn't in a situation where that information is critical, I really could care less. If someone leaves my classroom knowing how to brainstorm approaches and go through them methodically, or how to analyze possible solutions and check them for feasibility and/or correctness, I'm a happy teacher. (Not that I'm not happy right now.)
My end goal, which may or may not shock you, is simple: it's to make myself obsolete. The ideal (which is not practically realizable in the one year I have, in only one class) is for a student to leave his/her educational experience with the ability to direct his/her own inquiries into the world, for the teacher to fade into one resource among many. Mind you, I don't expect teaching as profession to disappear, for there are always more generations...but this is the scale by which I weigh my teaching, and I hope I don't fall too short.
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