december: *“I’m well-qualified to teach arithmetic, because I had so much trouble learning it” *
Just a side note: you seem to be quoting this remark (by, presumably, your daughter’s first-grade teacher) as patently ridiculous, but it may make more sense than you think.
Naturally, anybody who’s teaching math of any sort should have arrived at the stage of understanding the subject well, no matter how difficult they may have found it when they first encountered it. However, it is in fact true—it’s certainly been true in my twenty-nine years of studying and teaching mathematics and working with mathematicians—that people who initially struggled with math concepts often turn out to be more effective math teachers than the naturally mathematically gifted types to whom those concepts seem intuitively apparent and simple.
The difficulties most students have with learning math are based not so much in simple stupidity or inattention as in crucial mistaken assumptions about the concepts. As you know, the logical structure of mathematics is very hierarchical: if you make one mistake in your fundamental notion of what some mathematical definition means, none of the subsequent statements based on that definition are going to make any sense at all. Unfortunately, many mathematically gifted people are often completely mystified how somebody else can not immediately grasp the idea that seems so obvious to them, and so they have a very hard time figuring out what it is that their students aren’t “getting.”
Whereas somebody who has a clear recollection of going through similar struggles themselves will often respond immediately with something like “Oh! I think I see the confusion. You’re assuming that (this triangle has to be equilateral, the result of the subtraction has to be positive, whatever wrong conclusion the student has jumped to). I can see how you might think that, but in fact we specified earlier that…” And so the student’s confusion is cleared up.
Now, many people who are gifted at mathematics and to whom math thinking comes very naturally are capable of the necessary imagination to understand where the confused student is coming from, but alas, many others are simply not very good at “seeing how somebody else might think” anything that is so blatantly incorrect. So their math teaching is often much more frustrating, both for them and the students, than learning math from a less mathematically brilliant person would be. That’s why I don’t necessarily agree that it’s absurd for someone to think that having struggled with learning arithmetic made them better at teaching it—assuming, of course, that they actually did learn it successfully and now understand it well.
End of side note. As for the point of the OP (which actually seems mostly unrelated to the thread title), I agree with wring that the people who are advocating testing only math teachers, only in underperforming schools, should explain more clearly why they think that is a good use of resources in improving the students’ math education.