I'm not a very good teacher. I really am not. I'm scattered and disorganized and I work multiple jobs to make ends meet which means that I have never EVER been able to give 100% to a single class.
However, what I am, to MANY students, is the best math teacher they've ever had.
How's that work?
A couple points.
First, one common logical fail people have is to think that they can't have a favorite of things they hate. To put it another way, the adjective "best" is relative whereas the adjective "good" is absolute. "Best" just means better than all others—a super maximum, if you will. "Good" means past some certain standard of excellence. If none ever get past that standard, then the one that got closest to the standard is the "best".
Logic aside, most math teachers don't understand math as well as they need to in order to convey what's important or to teach it more than the way they learned.
To them, real analysis and abstract algebra were not the start of the amazing, they were the final hurdles before their finish line. They don't understand *real* mathematics, so they are stuck thinking that the "horror" problems from physics and any brief engineering applications they've been exposed to in their calc and diff. eq. classes are the end of mathematics.
The problem then is that, not understanding real mathematics, they continue to regurgitate the mathematics the way that they learned it in primary and secondary school. This has led to the promulgation of bullshit techniques like FOIL, PEMDAS, cross multiplication, learning to add integers by subtracting absolute values (WTF IS THAT? I mean, seriously?), and more.
Hung-Hsi Wu calls this "textbook school mathematics" and makes the point much more eloquently in his substantial publications (he's also a better source than I am). I encourage everyone to check him out if you haven't already: http://math.berkeley.edu/~wu/
A little more personally, I always have heard, "People that are too smart are not able to teach well," and if the person knows me at all but hasn't been one of my students, this has always been passive-aggressively aimed at me. Well, that's a ridiculous claim that any person with any amount of logic would be able to refute. The issue isn't how smart someone is, it's how well they understand the material. If someone has an intuitive understanding of the material, but not a cognizant understanding of it, then of course they're going to be shite at explaining it to others. On the other hand, if the person that has that intuition is able to be self-aware and understanding of what's going on in their brains, then that's the one you want to explain it, because chances are they are going to be able to share that intuition with you.
Everything that I am awesome at teaching—arithmetic, algebra, trigonometry, calculus, real algebra, analysis, number theory, operations research, differential equations, statistics, all of it—I fought tooth and nail for. I figured them the fuck out in ways I didn't think ANYONE was doing. The way I was able to figure out arithmetic and high school algebra is through my understanding of number theory, real analysis and abstract algebra along with my natural insights, intuitions, and understandings. This is how I'm able to teach students to the point that their emotions well up when they realize how shoddily they've been managed their entire lives (obviously not all of them, but it's not uncommon for at least one ADULT student to be overcome during the first class they have with me).
The question they ask me is always the same, "Why weren't we taught this way?"
Imagine my surprise that what I fought for, others did also. That there were entire groups of educators dedicated to revamping school mathematics (Wu on the front lines and for decades by the way) and teaching math the way I was teaching math. That not all math teachers sucked!
Just a majority of them do.
Now, now, I'm not trying to ruffle any feathers. And I don't necessarily blame the math teachers either. Or at least, only on a case-by-case basis. But in all seriousness, how could even teachers that WANT to be good or actually DO understand the material teach math the way it needs to be taught when we have curriculum that says pi is 22/7 or 3.14? That those bullshit techniques I mentioned above are wrote into the curriculum, and as a consequence, the textbooks???
And then there was the Common Core State Standards.
If teachers are serious about adopting these standards and adjusting, re-learning and re-teaching themselves material that they thought was sacrosanct for decades to teach it the correct way—the way that leads to further understanding, not cognitive dead-ends for ease-of-calculation sake—if those teachers start teaching the way they could teach, if understanding of the material holds at least equal weight with getting the right calculation, then I won't have so many people surprised when I come in and blow their understanding of mathematics away.
I won't have to apologize impotently when students lament their entire experience of math to date.
And I won't have dozens of students tell me every semester that I am the best math teacher that they've ever had.
But you know, if that is the case, if I can't effect a mathematical renaissance for those students because they've already been enlightened, well, it will have been worth it.