Ah, variation. The spread of skill in each class from the average. Since I teach in a private school, some of my students are literally the brightest math students in town. If they're bored and not working, they set an example of behavior for everyone, and no one thinks they'll have to do any work. So the class has to be at least a little bit challenging for them. But then there are the students in the very same classes who, against recommendation, went with the faster-paced track instead of a perfectly good slower-paced one. And they look at me with this expression that says, "You seem so nice...how could you betray me with a test that hard?" Of course, did they come in for any extra help? Only one or two of them; it's the decent students who think (probably correctly) they can earn better grades that come in for help most often.
Even in my slower-track course there's significant variation. I gave a series of speed tests (I called them "challenges" because I didn't want them to think that it would affect their grade if they didn't do well.) on basic skills. The most telling one was: 25 minutes to give the prime factorizations of all numbers from 2 through 100. They had plenty of warning and I gave them practice tests (identical except for order). The best scores were 96 and 97 (out of 99, of course). The worst was 57. That kid missed a significant portion of the ones he tried, and didn't get to the rest. And they all put forth a decent effort.
I'm sure that teachers in public schools get even greater variation than I do. And with our president brilliantly requiring that No Child be Left Behind, they are forced to teach to the bottom of the class, not the top. I'm pretty sure that the president's educational philosphy is Leaving our Best and Brightest Behind. But even in a place like I teach, I have to deal with variation, and with so many kids in a panic about getting in to a good college, or at least in a panic because they're at the mercy of parents who are in a panic about it, emotions can run pretty high when achievement is uneven.
I wish we could just accept the simple fact of variation. Some people are just going to get better grades than others. Some will work harder than others. Some are (*gulp*) more naturally talented than others. And some will be luckier than others. Of course, the ones at the top have more of a responsibility to help those at the bottom, and those at the bottom have a responsibility to give a solid effort. But this expectation (coming mostly from parents and students, not from teachers) of "everyone can get into Harvard or be president if they just work hard enough" is unrealistic and demoralizing. Kids need some degree of affirmation at any level if they're working hard, and they don't need to be made to feel like being the best is the standard.
Heh...see how I did that? How I made that paragraph apply to the kids at my school, but then also general enough to apply to all kids, and even to society at large? That was on purpose. We need to be more supportive of our lower achievers and make them believe that hard work will pay off. The standard they hear is "you can be president!" or "you can make millions in the NFL" or "you could live in this mansion". And when it becomes clear that only a very small fraction of them will actually be president, they have no more reason to work hard. It's demoralizing. How about just setting realistic standards like "you can have a comfortable life as a teacher, electrician, cook, bus driver, mechanic, military officer, etc., etc., etc."? This relentless drive for the top doesn't really do anyone any favors.
And for those of you waiting for cool math stuff, I'll have to put that off for a little while; it's crunch time at school, and I have to write evaluations of all of my students this week. I promise, though, soon! I have a special property of cubics in mind...or maybe Morley's theorem...hmmmmmm.....