According to this Science Daily article several factors have caused this result including more certified math teachers and a better curriculum focusing more on geometry, measurement and algebra. What exactly are they doing in private schools?
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This is just anecdotal, but around here some of the private schools participate in Math League, and while they do respectably, there are two public schools that are always trading the top two spots.
On the other hand, I have a few friends teaching at privates, and they are doing very creative things. I wouldn’t characterize their teaching as “rote” at all.
At any rate, certainly not all publics are outperforming all privates. I don’t think the question is “what are privates doing wrong?” but rather “what are successful schools doing, independent of factors beyond their control, that can be replicated?”