We can be pretty sure, because there are outliers. There is a math teacher in my city that thaught 170 kids a year for 20 years and none of them ever failed the

We can be pretty sure, because there are outliers. There is a math teacher in my city that thaught 170 kids a year for 20 years and none of them ever failed the ministerial exams (the average fail rate is 15 percent). The problem with these kinds of statistics is that when you are comparing performance relative to the population, low variance in teacher performance leads to the statistical artifact that they have low correlation. This isn't because teachers have little impact, it's because the variance in the quality of teachers is much lower than the variance of other factors. This shouldn't be surprising given the fact that the average student will go through 30-40 teachers, each having their impact. The variance in performance of any individual teacher will be severely blunted. This effect is actually fairly widespread in intelligence research, it's why adopted twin IQ studies greatly overstate the inheritance - by putting the cohorts of kids in standardized environments, you inadvertently control for everything but what is inherited, thus increasing the relative variance of the latter.