A. The word (the reference is Deut. 6:7) derives from the verb shanah, to repeat, and therefore to learn by repetition. But some connect it with shen, a tooth; teaching diligently certainly implies biting deeply like teeth into a subject.
Rashi says it is an expression for chiddud, being sharply impressed. In the Talmud a certain sage with an incisive mind is called shin’na, a man with prominent teeth (Hag. 15b).
Rabbinic literature mentions teeth in a number of other metaphorical contexts. Teeth symbolise friendship: “He who smilingly shows his teeth to his fellow is better than one who (reluctantly) gives him milk to drink” (Ket. 111b, based on Gen. 49:12).
From teeth we learn the blessing of peace: teeth are dangerous like a sword, and better than turning your teeth or sword upon a person or nation is to beat your swords into ploughshares and your spears into pruning-hooks (Job. 16:9; 35:16, 58:7, 1245:6; Isa. 2:4; Mic. 4:6).
To provoke or irritate a person is to set their teeth on edge: this what we have to do (hak’heh et shinav) to the wicked son in the Haggadah (see Jer. 31:29).