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The Trivium was the foundation of education in the classical world. The three legs of the Trivium are grammar, logic and rhetoric. The quadrivium includes arithmetic.
Essentially, the primary focus of classical education was to teach students the skills needed to develop and communicate skills.
Sister Mariam Joseph wrote an interesting book on what educators were trying to accomplish with the Trivium. It is far less oppressive than modern thinkers make it out to be.
Grammar is the study of the structure of the language. The classical world took great pride in the subtleties and versatility of their language. For example the tense structure can let people communicate extremely complex relations between objects in time.
Grammar is largely, but not completely, de-emphasized in modern education.
Logic is the study of the structure of ideas. With logic, one learns the structure of Rational Thinking.
Rhetoric is the art of forming a persuasive argument. This is where people learn to put together grammar and logic and communicate.
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