How can I teach my illiterate mother to correct her mistakes in prayer without her thinking that I am showing off my knowledge?

You must correct your mother's mistakes in prayer because it is a form of cooperation in righteousness and advice, in fulfillment of Allah's Almighty saying: "And cooperate in righteousness and piety." Your teaching her should be done with wisdom and gentleness, and do not pay attention to the stratagems of Satan, which may deprive you of the reward for teaching her.