Not sure whether including 1000 dummy variables is the best method or whether there is a better one I'm trying to measure the impact of teachers' performance on student outcomes. I'm not entirely sure how to do this however. As my dependent variables, I have the grades of the students, mapped onto a 1-10 scale. 
I have a number of independent variables, including aspects about the students. 
My concern is in how to incorporate the teachers into the regression. It would seem like the simplest way would be to introduce them as dummy variables, with a 1 representing they taught the student, and a 0 representing that they did not teach the student. 
This seems problematic though, as the dataset has around 1000 teachers, and there are 600,000 students in the dataset. It seems that this would be far too many variables to regress on at once. 
Do you have any advice in how I can incorporate the teachers' performance into my model?
 A: Multilevel models, which are a special case of mixed-effects models, were created to address this exact challenge.  The intuition is that there is a random "teacher effect" associated with each teacher.  Instead of using 1000 dummy variables, you assume a normal distribution of teacher effects and only need to estimate the variance of those 1000 effects.  This becomes an added random variable in your linear model, in addition to the usual 600,000 student level random variables.
It makes sense to do this if you aren't interested in the individual teacher effects per se, but just want to identify the overall relationship between teachers and performance.  If you particularly want to measure Mr Smith's performance, you probably want fixed effects ie 1000 dummy measures.  With 600,000 observations you may have enough data anyway.
One nice feature of a multilevel model is that you can decompose the overall variance into individual randomness, teacher randomness, individual structural effects (eg student gender, parent's occupation) and teacher structural effects (eg qualification, years of experience, gender).
It probably also makes sense to include a "school" effect as I'm presuming your 1,000 teachers are across several schools!  And there would be complications from subject too perhaps, but that's a question for what sort of data you have.
There's a big literature on applying multilevel models to education data.
