If Concord schools had to take any number of first generation impoverished immigrants with poor English language skills, their test scores would implode. That doesn't happen, because working class immigrants don't buy the $800K homes in Concord.
And, what's the history? Were these schools neglected for years, then the state recently started pouring in money?
I mean, maybe there are some exceptions (perhaps partially gentrified cities where some tax base is moving back into cities at inflated prices?), but these would generally be outliers. Since wealthier people have fled most major cities for the burbs, the tax base and educational investments have generally followed them. You see this over and over again in communities throughout the country.
For example:
http://www.npr.org/2016/04/18/474256366/why-americas-schools...
Cambridge (expensive city) $27,569 per pupil
Concord (expensive suburb) $17,517 per pupil
http://profiles.doe.mass.edu/state_report/ppx.aspx
Rankings
Cambridge Ringe and Latin was 214 out of 340
Concord Carlisle High was 27 out of 340
https://www.schooldigger.com/go/MA/schoolrank.aspx?level=3
Not cherry-picked data.
A lot of the difference is that cities, even expensive ones, are cities. A lot more diverse student body. A lot more students with problems at home. More disruption, etc. You can spend more money but the average student in the city school almost certainly has less parental involvement and less of a home environment that's conducive to learning than in the tony suburb.
The linked article (and decades of debate) notwithstanding, it's also hard to really correlate spending per pupil with educational outcomes past a certain point--especially as measured on standardized tests. Poorer cities do spend less. Two Lawrence high schools have the worst test results in Massachusetts. But its spending, $14,475 per pupil, is still only a bit less than Concord.
Definitely does in my city. Teacher pay's also substantially higher in the inner city schools. Most teachers still don't want to work there because they're stuck in a seemingly-inescapable poverty-and-culture-driven "bad homes -> bad students -> burnt-out and bitter faculty" rut, which makes them a discouraging, abusive, and dangerous hell for faculty and students alike.
Source: my wife and probably half my circle of friends are teachers.
[EDIT] in fact, some of the better suburban schools pay worse than the so-so suburban schools, let alone the high-paying inner city ones, because they can. Good teachers will still work there because the environment is nicer and more rewarding (that is, the students are way easier to teach, and the community tends to be supportive)
Unfortunately, it's not that simple. The heart of the problem in many parts of the country is that we fund our schools with property taxes.
Hence, a cycle of better schools and increased demand in concentrated areas, which causes the supply-demand imbalance and price increases. Compounding this is the fact that people in such areas actively block "higher density" housing, which naturally means fewer houses per acre.
Building more houses would help to a limited extent, but the shortage would still exist and resistance is stout.
Hence, the shortage of housing is just a symptom of how we fund education, which is the real problem.
>Inequality is not the problem
Inequality absolutely enters here because if wealth/income were more evenly distributed, then funding for schools would be more evenly distributed, thus alleviating concentrated housing demand in small areas with better schools and driving up prices.