Admonish – To urge, encourage, exhort, or give warning by reminding someone of something that they already know.
God used this word 9 times in Scripture: 3 times to Israel in their Masoretic Text Old Testament; once to the Body of Christ in the transition book of Acts; 4 times to the Body of Christ in the Gospel of Grace (Romans thru Philemon); and once to Israel in their tribulation book of Hebrews.
Teaching is the introduction of new information; admonishing is the reminding of something already learned.
See also Exhort, Rebuke, Revile