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This is the most familiar style to those of us who attended schools ourselves. Generally it involves four or more subjects a day, taught during specific time periods. Generally this style uses prepackaged purchased curricula, but certainly not always.
Classical (Trivium) -
Classical Education organizes education into three Biblical categories. These three categories are Grammar, Logic & Rhetoric, otherwise known as knowledge (learning the facts), understanding (organizes the facts into rational order), and wisdom (taking that knowledge and understanding and uses it in practical ways). This is the original liberal arts education. Memorization, dialogue, writing and languages are stressed.
Charlotte Mason -
A Christian-based philosophy of education that stresses good literature (rather than textbooks), copying of relevant materials, and dictation. Nature walks are stressed throughout. Structure is crucial and training of good habits begins in infancy.
is a non-Christian spiritually based program featuring delayed academics and a rich variety of music, arts and literature. The aim of Waldorf education is to educate the whole child -- head, heart and hands. The curriculum is geared to the child's stages of development and brings together all elements -- intellectual, artistic, spiritual and movement. The goal is to produce individuals who are able, in and of themselves, to impart meaning to their lives. Rituals of daily and seasonal life are strongly emphasized.
The original works of Maria Montessori have been gravely distorted here in America by a lack of copyrights on her name, but the original concept was to respect the child's inner desire to learn and allow him/her to make spontaneous and free choices within a carefully prepared environment (structure the environment, not the child). The role of the adult is to observe and use brief teachable moments to introduce new concepts (usually by doing the activity quietly herself and waiting for a child to ask a question about it).
Unit Study Approach -
Unit studies can be as flexible or structured as a family wants. They allow for a great deal of individual choice in both the choice of units to be done and in the materials used. It is usually an in-depth study of one specific topic (baseball, the planets, trees, puffins) that takes into account many areas of the topic, such as geography, science, history, art, etc. It is a complete immersion into the topic so that the student will see things as a "whole" instead as isolated subject areas.
Unschooling is not how something is done, but why. Unschooling is the belief that all people, no matter how old or young, have a built in desire to learn (unless that desire has been crushed by outside forces). It is a belief that if you allow a person of any age to pursue their own interests throughout life they will end up gaining the knowledge they will need in order to pursue the life they want. Unschoolers use textbooks, movies, classrooms and correspondence courses, museums and magazines, jobs and volunteer positions (and the rest of the world) to learn. Unschooling is not, however, never saying no and letting the wolves raise your children.
A unique to each family combination of several of the styles listed above. This is really what most homeschooling families are. Home school educators tend to, after the first year or so, pick and choose from a wide variety of philosophies and curriculum that meet the unique needs of their children.