Adam Clarke on Galatians 3:24

The law was our schoolmaster - Ὁ νομος παιδαγωγος ἡμων γεγονεν εις Χριστον· The law was our pedagogue unto Christ. The παιδαγωγος, pedagogue, is not the schoolmaster, but the servant who had the care of the children to lead them to and bring them back from school, and had the care of them out of school hours. Thus the law did not teach us the living, saving knowledge; but, by its rites and ceremonies, and especially by its sacrifices, it directed us to Christ, that we might be justified by faith. This is a beautiful metaphor, and highly illustrative of the apostle's doctrine. See the note on Rom 10:4, where this figure is farther explained.