Mean, variance, heritability confusions

This passage is typical (Mary Wakefield in the Spectator, interview with Robert Plomin, July 27, 2013) of a certain confusion:

Crucially, I suppose, what educationalists of a leftish bent must consider is this: if IQ is measurable (it is) and highly heritable (that, too), then the diversity we see now in exam results isn’t going to melt away. In fact, in the best school, with excellent teachers and rigorous exams, a normal, randomly selected bunch of kids will see a greater spread of results, reflecting their inherited abilities. The little Plomins, rich and poor, will pull away. The other kids’ results will get better too, but the gap will grow.

If we could give everyone a good school environment that is identical in all the relevant factors, then the heritability of school achievement will be 100%.   That is, all the variation in the trait will have to be the result of genetic variation, because the variation due to environment, by construction, is zero.   Moreover, having eliminated the variance caused by environment, the variability in this population would be less than the variability if the same population was placed in a diversity of environments. Moreover, while reducing the variance, we would also have increased the average (which is vaguely noted in the above as "the other kids' results will get better too".   We will see a lesser spread of results (and not "the gap will grow").  As a side effect, we will see higher heritability, which just serves to remind that heritability is not the same as inheritability.