εἰς τὸ τέλος τοῖς ἀλλοιωθησομένοις ἔτι εἰς στηλογραφίαν τῷ Δαυιδ εἰς διδαχήν Psalm 59:1 LXX
What meaning did the Septuagint translator want to convey to readers at the time by using the word teaching?
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Ps 59 in the LXX is the same as Ps 60 in the English Bibles.
The OP refers to the superscription of this psalm which has the following text in Hebrew and Greek:
לַ֭מְנַצֵּחַ עַל־שׁוּשַׁ֣ן עֵד֑וּת מִכְתָּ֖ם לְדָוִ֣ד לְלַמֵּֽד׃ בְּהַצֹּותֹ֨ו ׀ אֶ֥ת אֲרַ֣ם נַהֲרַיִם֮ וְאֶת־אֲרַ֪ם צֹ֫ובָ֥ה וַיָּ֤שָׁב יֹואָ֗ב וַיַּ֣ךְ אֶת־אֱדֹ֣ום בְּגֵיא־מֶ֑לַח שְׁנֵ֖ים עָשָׂ֣ר אָֽלֶף׃ = For the choir director; according to Shushan Eduth. A Mikhtam of David, to teach; when he struggled with Aram-naharaim and with Aram-zobah, and Joab returned, and smote twelve thousand of Edom in the Valley of Salt.
Εἰς τὸ τέλος· τοῖς ἀλλοιωθησομένοις, ἔτι εἰς στηλογραφίαν τῷ Δαυείδ, εἰς διδαχήν, ὁπότε ἐνεπύρισεν τὴν Μεσοποταμίαν Συρίας καὶ τὴν Συρίαν Σωβάλ, καὶ ἐπέστρεψεν Ἰωὰβ καὶ ἐπάταξεν τὴν φάραγγα τῶν ἁλῶν δώδεκα χιλιάδας. = For the end, for them that shall yet be changed; for an inscription by David for instruction, when he had burned Mesopotamia of Syria, and Syria Sobal, and Joab had returned and smitten in the valley of salt twelve thousand. (Brenton)
Thus, the LXX (in this respect at least) reflects the Hebrew well. In both cases, the intended use of the Psalm was for didactic purposes.
The first three verse recall the major defeat of the Israelites. We have this in the Pulpit commentary:
Verse 1. - O God, thou hast east us off, thou hast scattered us, thou hast been displeased (comp. Psalm 44:9-11). The expressions used imply a signal defeat, which, though not mentioned in the historical books, harmonizes with the account given in 1 Kings of the severe treatment of Edom by Joab. From the fact of the defeat the psalmist infers the ground of it - God's displeasure. O turn thyself to us again; rather, O restore to us (i.e. make restoration to us) again (see the Revised Version). Psalm 60:1
Therefore, the didactic purpose of Ps 59/60 was to recall Israel's dramatic defeat and failure in battle and their ultimate vindication by God when Israel finally gained the victory. That is, it teaches people to depend on God for victory and not their own prowess.