Alternatives to Public School with accelerated academic standards, and smaller class sizes. There are three broad categories: parochial schools; boarding schools; and secular day schools. I attended the latter in my youth; a small school (Montessori through 8th grade) called Green Hedges, which is situated in Vienna, Virginia. The school was founded by Joyce Kilmer, and Joyce's descendants still lived in a house adjacent to the school while I attended. Many a day did we recite "I think that I shall never see a poem lovely as a tree..."

Private schools include Public Schools as a subset in the UK. Fees can vary widely, as can the educational ethos, although they tend to be moderately conservative and have one or more religious axes to grind.

See also: British Schools, State Schools.
In the UK, the term "Private School" has, as usual, a slightly different meaning. There are three layers of British schooling, the State Schools, the Public Schools and the very rare Private schools.

A state school provides free education to any children of the correct ages with its catchment area. It is owned and sponsored by the state and the local education authority.

A Public school is merely that - available to any member of the public who can pay its fees and pass its entrance exam. Catchment areas do not play a part in the role of public schools since without exception they offer some form of boarding.

A Private school is one which is not available to every member of the public. For example, some schools are available only to those of a certain religion, others cater specifically to children whose parents are involved in a particular profession, particularly the military.

The below is all UK-specific:

A private school is one which is not run by the state. It will still be checked by the state, but is far more free to run itself and use its own educational philosophy. Because they receive little to no money from the state, they generally cost money to go to.

A public school is a subset of the private school. It is, in theory, open to anyone. In reality it is open to members of the middle to upper classes who have a great deal of money. The most commonly-accepted current definition of a public school is one whose headmaster attends the Headmasters' Conference, a prestigious group which serves only to define public schools.

A state school is the opposite of the private; free for those in the area. It may or may not have an entrance exam; those which are academically selective are known as grammar schools; this originally meant that they taught Latin grammar.

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