Objectivists advocate for the full privatization of public schools and education. However the deplorable state of public schools is limited only to some public school systems. Wouldn't a better solution be to to pool and spread school funding more evenly?
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Objectivists advocate for the full privatization of public schools and education.
Correct, but not because of "the deplorable state of public schools...." Objectivism advocates the complete "separation of state and economics, in the same way and for the same reasons as the separation of state and church." [Excerpted in The Ayn Rand Lexicon, topic "Capitalism," originally from VOS, Chap. 1 (TOE).] "State" means the power of retaliatory physical force; "economics" includes schools.
However the deplorable state of public schools is limited only to some public school systems.
In a free market, bad businesses go out of business; their "funding" is reduced, not increased. Education is no different in principle. Like all other forms of production and trade, education certainly is an important activity. Schools that perform well should be rewarded (by the judgments of parents willing to send their children to those schools and pay the tuition), and schools that perform poorly should be allowed to fail (for lack of customers).
Wouldn't a better solution be to to[sic] pool and spread school funding more evenly?
"Pool and spread" means penalizing good schools by reducing their funding so that bad schools can be rewarded with more funding. That is a perfect moral inversion -- penalizing the good for being good, and rewarding the bad for being bad. For more insight into what a proper approach to education should consist of, and thus what standard should be used to judge how good or bad a school is, refer to the entry on "Education" in The Ayn Rand Lexicon.
If there is an implicit claim here that bad schools are bad only because they receive less funding than good schools, that is a highly dubious premise that is routinely assumed and sometimes even expressed explicitly by defenders of government funded and government controlled schools. Schools need to function within their financial means. (If they are prevented from doing so by government requirements to admit more students, then the focus of political discussion should be on reforming those government mandates.)
"Pool and spread" also suggests the view that "society" is paramount, and individuals exist only to serve "society" (through the taxes they pay, for example). Objectivism rejects that view of "society" versus individuals.
There are numerous other references in the literature of Objectivism regarding education. The Lexicon "Education" topic mentioned above includes a very brief excerpt from a very long article by Ayn Rand ("The Comprachicos") that provides a penetrating critique of modern education. Ayn Rand's writings also include a proposal for tax credits for education (as an interim step toward reform), allowing parents the option to use their own taxes to help pay for the cost of private schools instead of having to send that money to the government for public school funding. See The Voice of Reason, Chap. 23.
Is it really best to privatize all government schools?
(Note: This is an edited version of the short form of the question. Refer to the editing history for the original formulation and my original response to it. The edited version still raises the issue of "best" for whom and by what standard. which has been covered by some good discussion in the comments. I have now edited my own original comments to match more closely the revised short form of the question.)
It is often suggested or claimed by defenders of government funded, government controlled schools that privatizing the schools is motivated primarily by a negative evaluation of them (which, in turn, allegedly isn't the schools' fault because they are "under funded"). The text of the question attempts to challenge the imputed claim that all schools are bad, suggesting that the ones that aren't should not be penalized by being privatized.
This view misrepresents the position of free-market advocates and the reality of what life under a free market would actually be like. A free market in education would greatly improve all schools that are able to meet the demands of their customers. Converting a school (or school system) to privately funded and privately operated is not a penalty for the schools at all, but a huge benefit. It would make the schools better, not worse, more responsive to their customers and, over time, more efficient at meeting the expectations of their customers. Schools funded by government, and controlled by government entities such as various elected or appointed school boards, severely disconnect the schools from the wishes of parents, despite any efforts to incorporate "parent input" into the deliberation and decision-making processes of the government authorities.
Even if one were to grant that there are some public school systems that have succeeded, the moral status of a policy is not determined by its success or failure. One could more easily go on and on about the many failures of the public school system, and its role in creating a fundamentally anti-conceptual population, as opposed to talking about its successes.
But let's not fall into the same anti-conceptual trap -- rather, the public school system is immoral because fundamentally education is a personal choice, either direct or delegated to parents in the case of children who are not yet capable of making that choice themselves.
First, and most fundamentally, education must be a personal choice because the standard of morality is one's life. If one is to live life to the best of one's ability, one must be free to make one's own decisions. There can be few fields in which this freedom to choose is more important than in education itself -- the field in which one learns how to live life by gaining knowledge about reality, and developing the ability to reason in order to prosper in that reality.
Secondly, the use of government force to extract value from some, in order to benefit others, is immoral regardless of the reason.
Therefore public education in any form is fundamentally incompatible with government. Ayn Rand's essay The Nature of Government, while not talking specifically about education, expands further on these ideas.