The trivium refers to a three part educational grid. In this proven and effective program the child is first taught the grammar of learning. This means the basic foundation that supports any given subject. The rules of phonics are the grammar of reading and spelling. Memorization of addition and subtraction facts and the times-tables, the grammar of mathematics, prepares the way for learning the calculus.
Second place involves a period of questioning and challenging the facts.
This often begins sometime during the 5th or 6th grades. Basics are still being learned, but now the child may initiate some thinking on his own. Proofs are needed. Trial and examination may be called for to spark further learning. Eventually, a third stage is reached.