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In the sentence, 'Honor your old teacher', should the noun be definite or indefinite? I'm tempted to say definite.
τίμᾱ τὸν παλαίον διδάσκαλον σου.
But the one doubt I have is that there is no definite article 'the' in the English. I think this reveals that I have a less-than-perfect understanding of definite and indefinite nouns. Would 'your old teacher' be definite, because the noun is referring to a specific person? Is it more a matter of specificity, than whether the article 'the' is used?
Greek uses definite articles somewhat differently than English does. In English, the definite article is used before a definite noun, unless the noun is proper or modified by a pronoun.
In Greek, the definite article is used before definite nouns, even if the noun is proper or modified by a pronoun, or any nouns which don't change their form (to show their case).
In your situation, it sounds like there's one particular teacher who deserves to be honored: you're not just honoring every teacher you happen across. This is a perfect place to use the definite article.
The Greek definite article can also function as a non-emphatic possessive. So ἡ σκηνή could also mean his tent, or your tent, or their tent, and so on. So in your example you could get away without using a possessive pronoun: ὁ παλαιὸς διδάσκαλος in this context would clearly refer to your teacher, rather than someone else's. But adding σός puts considerably more emphasis on that fact.
If you're interested, this page provides more information on the semantics of the definite article. It's a very interesting and complicated topic; many papers have been written about the connotations of θεός versus ὁ θεός in John 1:1.
(*) With proper names, the definite article isn't mandatory. It's used generally for people who would be recognized immediately such as Socrates (if I mention "Socrates" you probably won't ask "which one?"), or people who have been mentioned earlier in the conversation. The aforementioned link has more information on this as well.
