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Have a read of the Wikipedia article on Catechesis: churches have long been using the terms "catechist" to refer to teachers and "catechumens" to refer to students, both based on the Greek word κατήχησις meaning "instruction". Then in the late middle ages "catechism" arose as a term for a specific tool for teaching. Apparently it was Luther who popularised the question and answer format of catechisms.
The Bible doesn't have anything to say about specific teaching methodologies, just that it's important to teach and instruct the next generation of Christians to know and trust God.
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I usually tell people a catechism is similar to the modern day FAQ although I don't think a catechism necessarily has to take the question and answer format.
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I would like to share some further research I did on this question. Here is a short extract from the Catholic Encyclopedia:
"Catechumen," in the early Church, was the name applied to one who had not yet been initiated into the sacred mysteries, but was undergoing a course of preparation for that purpose. The word occurs in Gal. vi, 6: "Let him that is instructed in the word, [ho katechoumenos, is qui catechizatur ] communicate to him that instructeth him [ to katechounti , ei qui catechizat ] in all good things." Other parts of the verb katicksein occur in I Cor., xiv, 19; Luke, i, 4; Acts, xviii, 24.”
The article goes on to explain that although “some sort of preliminary instruction must have been given” to the first century converts to Christians (Galatians 6:6), “we cannot infer from this that the full regulations were already in force. It was rather the danger of apostasy, or even betrayal in time of persecution which gave rise to special precautions as to admission into the Church.”
The article concedes that “by the end of the second century we find the catechumenate in force in all its main lines.” Quotes from the early Church Fathers follow (Justin, Tertullian and Augustine). Origen was in charge of the catechetical school (tou tes katecheseos didaskaleiou ) at Alexandria (Eusib., Hist., Eccl., VI, iii)." Source: https://www.catholic.org/encyclopedia/view.php?id=2661
The Synod of Dort approved the Heidelberg Catechism in 1619, and the catechism has become one of four “standards of unity” that define the beliefs of the Reformed tradition.
"The Heidelberg Catechism has 129 questions and answers and is divided into four basic parts: Introduction, Misery, Deliverance, and Gratitude. Within each category are subcategories dealing with more specific elements covering the basic doctrines of Christianity. The questions and answers of the Heidelberg Catechism are arranged into 52 “Lord’s Days” so that the whole catechism can be easily taught in one year. Each answer in written form also contains the Scripture references that support the answer." Source: https://www.gotquestions.org/Heidelberg-Catechism.html
During the first century new converts to Christianity were given instruction about what it meant to be a follower of Christ Jesus, but here was no formal catechism till the end of the second century. After the Reformation, the Protestant catechism came into use in the 16th century. Today, instruction is a necessary part of conversion, and no person should be baptised into the Church until they have a grasp of the fundamental doctrines.