Right, speaking for myself, my parents were REALLY involved (they were both teachers themselves), so even though I attended a lackluster public high school in rural Pennsylvania, I ended up doing pretty well, going onto a top 20 university and becoming a software engineer in Silicon Valley. I also loved learning and computers though, so I learned a lot on my own just by tinkering in my spare time for fun.

Hm, I'm sure there are good proxies, like the number/length of parent-initiated phone calls/emails to the school/teachers.

These are good proxies for parental involvement with school, but as Mark Twain famously suggested, school != education.

Which is clever, but all that matters for validity of the metric here, is whether a parent with a child in school, would be willing to make all these calls to the school, but otherwise be an uninvolved parent. It doesn't seem likely IMHO.

Helicopter parents can be highly involved with school but not effective in supporting their child's education. I know far too many parents who do their children's homework for them, as an example.

There are also many parents actively engaged in their child's education who don't feel the need to be in constant contact with a teacher.

The issue is whether "being involved with the child's education" has predictive power over long-term outcomes that explains away the influence of the school itself. While it may weaken the correlation for the different involvement types to be more vs less helpful, I was only speaking to the issue of whether your could measure involvement at all.

And my statement was that involvement with school ignores other types of involvement in education.

The grandparent comment explains how it still doesn't tell whether the parental involvement matters or not.

The experiment results show only that on average the kids perform better in better classrooms. But it's still possible that kids with better parental involvement have no benefit of it, and the statistical difference happens due to better results of just the kids with worse involvement.

In other words. Yes, the article controls for the fact that better parents can send kids to better kindergartens. It doesn't control for the fact that kids of better parents benefit from the better kindergartens.

It's possible that adding an interaction term for classroom quality and involved parent would show that there's zero marginal benefit to a good classroom if you're an involved parent. That result seems orthogonal to their research goal, which was estimating the mean impact of a quality classroom for the "average" child.

The phrasing, "they should control for" has a specific statistical meaning, criticizing the model as having an omitted variable problem -- that they left out a regressor that is correlated with both the dependant variable and at least one of the other regressors. That is a different criticism than suggesting an interaction term.