Quick note:
When we add covariates to the model, we expect variance to reduce, but we can get negative R-squares from between-school variance. This is because when we adjust for predictors the school averages/intercepts may change for both directions. The between school variance may reduce or may enlarge as the meaning of school averages/intercept changes. Can I simulate the case where between-school variance in fact enlarges?? Maybe this happens when pretest is very different by the two groups.