British spelling: to instruct someone in religious doctrine through systematic questions and answers; to question closely about beliefs or knowledge.
From Greek 'katechizein' (to teach by word of mouth), ultimately from 'kata-' (down) and 'echein' (to sound). The British '-ise' spelling reflects Latin-derived verbs in British English.
To catechise someone is to really grill them with questions about what they believe—it's where we get the word 'catcall' (calling out) and why teachers use the 'Socratic method' which is basically sophisticated catechising.
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