Three years and two months ago, I was back in college after a semester sabbatical. I was at that point still majoring in education, and consequently, most of my classes dealt with ... education.
One Thursday morning, I had a class with a woman (a classmate) who was obviously very bright and energetic, but she just did not understand math.
This was basically a class to teach math and science instruction, and she first did not understand the material, let alone how to teach it.
But seeing that someone can't work a problem is one thing. Teaching it without usurping the teacher is another things.
For the adults who do not understand because they did not have the right teacher.
Let's call her Jeannette. I don't know her name, and I'm not sure we were ever formally introduced, but anyone who knows me in person knows that if you're friendly, I don't need to be introduced to you to want to teach you.
Jeannette was in at least her late 30s and black. I didn't see or look for a wedding ring and didn't ask personal questions. Not my business. And unless I can use it as an instructional aid or to gain trust (some people plain old want to trust), it's not on my list of questions to ask.
Jeannette had a lot of confidence and a lot of energy. And one of the things she was energetic and confident about was her math illiteracy.
She expressed this not as a happily wounded soul ("I'm helpless, and I'm happy to admit it!") but as one of those rare creatures who just is not let down or dejected when she doesn't understand.
Jeannette is the kind of person who, if she were mentally ill, might well kill herself thinking she could fly if she put wings on. I mean no disrespect to the mentally ill (seeing as I grew up around 'em and count myself as one), but she just didn't quit. If she thought she could do something, she tried. And she kept trying.
She had sat near me in class, though most likely not because she hoped I'd teach her. (If it's early -- and it was -- and I don't want to deal with anyone -- and I didn't -- I look about as socially useful as wet sand.) She just saw an open seat and a new face -- I hadn't been in class on Tuesday, owing to not, er, being on the roster yet (long story) and sat.
I remember absolutely nothing of the lesson that day, which is probably just as well, as the teacher was a woman who thought the exo in exoskeleton meant not outside but lacking. (I later sent her a half-dozen science Web sites asserting otherwise, and the basic message of her reply was "I've picked my reality, thanks!" She later ambushed me in a meeting by saying that she couldn't recommend me as a teacher unless I bowed to her will. Have I mentioned lately how thoroughly I hate teachers who are threatened by energetic students who don't disrupt class?)
After the class, Jeannette and I managed to find each other (I'd since woken up), and between my desire to help people when I'm awake and her desire to be helped when her helper is awake, we made our way to a patch of sidewalk where deliveries were made.
Where deliveries are made, you have depressions in the pavement. They're shaped as sometimes trapezoids and sometimes extended triangles. The point of this particular area is that it had a lot of shapes.
Jeannette's biggest trouble in math was geometry. She was going to be extremely limited in the classes and ages she could teach if she couldn't guide students to at least a basic understanding of the math of shapes.
Now, I got a 67 in geometry in high school, but in my defense, that guy was so boring that even Jeannette would have struggled to stay awake. He clearly loved to teach, but he often didn't teach. He'd spend entire class periods (I once swore he'd spent consecutive periods doing this -- and we were all supposed to be in other classes after geometry) writing out long and painfully boring proofs while not explaining much of anything or asking students where they thought he should go.
So I didn't test well in geometry, but I can explain the shit just fine.
And Jeannette was at the stage where she didn't understand how two triangles become a square no matter what they look like.
Understandable. You have to imagine the triangles as basically just holding mass and being divisible into chunks of triangle, and that doesn't get taught in schools much because, let's face it, it would confuse the everloving crap out of students -- and it isn't on the SOLs.
But I could explain it to Jeannette that way. And fortunately for both of us, she understood. And as I frantically ran around showing her how two triangles of different size and shape combined to make a square that "clearly" had no relation to either triangle
I
got
the
weirdest
fucking
looks
from people walking past.
And really, if some long-haired kid was running around talking half to the pavement and half to an older woman staring at him as he delineated imaginary points with his shoes, you'd probably think "At least he isn't going on about the end of days or anything."
(The look is cultivated, by the way. Anyone willing to look past the long hair and not entirely kempt beard is willing to drop what lies they were taught were true and accept something else. It's a question of being willing to look past what you thought was acceptable/true. Also, I really just don't give a fuck about my hair unless it interferes with my eating or it feels bad.)
Then came the real test.
I told you above that there were lots of shapes in that sidewalk area. Really, it'd be a great place to review geometry in a real (as opposed to book) setting. Instead of dealing with units of triangles, which is fantastically boring and also not exactly realistic, you're dealing with centimeters. Or inches. Or feet. Or hundreds of feet. And you can use car parts as corners of a triangle. (The first trick to getting this to work is getting your students to accept math outside of books. The second is keeping them from staring at the bugs and detritus they'll encounter. Once you have that, in my experience, you have their attention unless the ice cream truck comes by.)
So of course, the big test is that shape that ... well, as Cookie Monster would say, one of these shapes isn't like the other shapes.
I am, of course, referring to the dreaded circle, which takes all your lessons about nice, crisp shapes with straight lines and corners and makes them
RUN FOR THEIR LIVES AAAAAAAAAAUGH A CIRCLE!
Seriously. People can work triangles like they're willing sex slaves. They'll graduate to squares, go on to extra super hard work in trapezoids, even the irregular ones. You can even get the advanced students to irregular shapes if you give them enough logical information that they can deduce stuff.
Circles, calculus and physics are probably the most psychologically daunting topics the average college graduate deals with when it comes to math. Calc and physics, you're warned about as a child. "Oh, he's smart," you hear. "He's a physicist."
It's put out there as some kind of genius job, like people are born to be physicists like they're born to be quarterbacks or something. In the Babysitters Club series, Claudia's older sister, Janine, is put forth as some kind of ridiculous genius. In almost every book of that series (of the 100+ I read), there's a line about how Janine was expected to be a physicist because she was just that smart. Like very smart people might go to Harvard or MIT, but only the top of the top of the top go on to be physicists.
And don't even get me started on calculus. (Actually, convenience of conveniences, I get started on it myself.) It's a language, and it's taught like a foreign language. If people taught foreign spoken languages like they teach math, global trade would refer not to international agreements but trade of globes.
Unless you have a mental handicap, failure to grasp calculus can generally be blamed on poor instruction (which is sometimes a matter of insufficient individualized attention) or not paying attention. I've tutored the stuff, and I have yet to fail to reach a student. The stuff is just not that hard, but most people who love calculus forget that it can be hard, and many people who teach math do a lot more math writing than they do math explaining. (Until I switched majors, I was going to be one of the ones who explained everything. Maybe one day.)
And there are two things about circles that throw everybody.
The first is that they have no straight lines. They subvert the dominant shape paradigm: straight lines.
Seriously. In terms of the closed, line-based drawings you see in the average grade school math book, circles and their cousin, the oval (whether flat-sided or not), are the only round shapes. Everything else is so straight-edge you know it's never done drugs.
And that's daunting.
And then you add this pi shit to the equation (literally), and you confuse the everloving fuck out of people who never did you harm, never made fun of your plaid tie or asked if there was a flood -- because your pants were an inch above where your shoes ended.
In my experience, here's what pi means:
"One of the circle's dimensions is the same as the squares: the distance across. The other circle dimension is basically just 3/4. Because if you look at a circle inside a square, it looks like it covers about 3/4 of the area. So we're going to multiply half the distance across by itself, which gives us a little bit more than the area of a quarter of the circle, and then multiply in a factor of 3/4 to account for the loss of area to the whole thing. Still sound confusing? Let's do a few, and then I'll explain." (Not perfect, but A: I am not a math teacher, and B: I haven't actually had to explain this since roughly this encounter three years ago.)
This, to me, makes a hell of a lot more sense than saying this:
"And then multiply it by pie, which is 3.14 blah blah blah I'm not going to explain this to you, but I expect you to memorize it."
That's what many people hear when you explain pie (not pi, pie) to them for the first time. And the spelling distinction is important; you are, again, subverting the dominant paradigm. And you need to explain it or your students are actually going to write things like 4*4*pie. I've seen it.
Here was my chance, then, to explain circles in a way that Jeannette would actually understand, as opposed to nod and smile through while not understanding a fucking thing because circles break the shape mold.
I pointed out to her a circle and a square. The circle's diameter was roughly equal to the square's side.
I then mapped out the four circle quadrants. She saw that it was imaginarily divided into quarters.
We set a value for the radius at something like six inches. So one of those quadrants was about 36 inches square, or a quarter of a square foot.
The whole thing was a whole square foot times 3/4, or about 3/4 of a square foot.
And the square was, rather conveniently, a square foot. (I put my shoe inside it to demonstrate.)
Now, while I was doing this, I was drawing on the circle with my left shoe, which of course left no marks on the concrete, and she was following along and not asking any questions, just absorbing. (She asked questions when she wanted more information. If your students don't do that, make them. Otherwise you'll have no idea what they don't get, and they'll accept not understanding. Epic fail.)
And a minute later, that was that.
I haven't seen Jeannette since.
I never saw her in class again, and I never saw her on campus again. I have no idea if she withdrew, transferred, took the PRAXIS subject tests (tests most teacher students must take) and passed and got licensed then, whatever. For all I know, she had shapes swimming around in her head on her drive back home that day and crashed when she was startled by a shape whose area she couldn't instantly roughly compute. (A little insensitive? Sure, but tell me you didn't at least chuckle.)
Maybe she lived just fine, and maybe she's teaching kids today (well, yesterday, and two days from now) based on what that anonymous long-haired hippy kid showed her with his work shoes on concrete between classes as oncoming students looked on and thought, "He's drunk -- no, he's moving too well. And he isn't stoned, either. Maybe he's on speed. Huh. I didn't know we had a speed problem."
There are a lot of maybes attached to this story, but there is one definite: She understood.
I know she wasn't trying to be nice because I know how she looked when she didn't understand. That was the face she had on for all of that class period.
If you've taught (formally or otherwise), you know even without having to think actively about it that life is full of random (or not so random) instructional aids.
Sidewalk geometry is a pretty easy one to use. Candy's also incredibly useful. I used red hots and green candies (conveniently in the same package) to teach a widow how to balance her checkbook. Took two months, but at the end, she understood how to manage her finances. Put some candy in a box, put a number on top and you have yourself a variable for use in manually working an algebra problem. (And for sure, eat the candy. Keep your blood sugar sufficiently high that hunger doesn't begin to be more important than learning -- see Maslow's Hierarchy of Needs.)
The key is to approach math, science, reading, whatever from perspectives your students will use. So often in school, as a student or teacher, I heard "But when will we ever use this?"
If you incorporate stuff like sidewalks and candy, or recipes (golden for teaching fractions, and if some manly man student who's a whole 6 years old says he's never going to have to cook, ask him how many sacks some football player will record in five games if he records a sack and a half a game), the math never has a chance to be anything but real. It is what those students are using. Take the path from your residence to your work and divide it into so many segments. Then have your students figure out the average length of those segments. (For more fun, have them figure out shorter routes, routes with fewer turns, a triangular route, etc.)
Do this enough and your students seek out their own examples, their own problems to solve, in math and elsewhere. They begin to be proactive. They do not wait to be in a classroom to learn, nor do they need a desk or paper in front of them for a situation to be an opportunity for education. They are instantly receptive to what they can extract from what is before them.
And at that point, you have taught them to teach themselves.