Learn Hardcore Greek: Η μαθήτρια πιστεύει ότι η δασκάλα μιλάει καλά. - The student female believes that the teacher female speaks well.

Greek marks definiteness and gender with articles. η is the feminine singular nominative article (the). You need it once for the main-clause subject η μαθήτρια and again for the subject of the subordinate clause η δασκάλα. Both nouns are feminine singular and both are subjects (in their respective clauses), so both take nominative with η.

Why is η δασκάλα also nominative? Isn’t it the object of the verb?

The direct object of πιστεύει is the whole clause introduced by ότι. Inside that subordinate clause, η δασκάλα is its subject, so it is nominative. Greek uses nominative for subjects even inside embedded clauses.

Can I replace ότι with πως?

Yes. ότι and πως both mean that (complementizer). πως is a bit more informal or conversational. Example: Η μαθήτρια πιστεύει πως η δασκάλα μιλάει καλά. Both are correct.

What’s the difference between ότι and ό,τι?

ότι = that (introduces a content clause). No comma.
ό,τι = whatever/anything that (indefinite relative). It always has a comma and accent. In this sentence you need ότι (that), not ό,τι.

Do I need a comma before ότι?

No. In standard Greek, you do not put a comma before ότι when it means that (complementizer).

Can I drop ότι, like English sometimes drops that?

Usually no. In careful Greek you keep ότι (or πως). Omitting it is colloquial and not always natural or acceptable in writing.

Why is it μιλάει? Can I use μιλά or μιλεί?

The verb is μιλάω/μιλώ (to speak). In 3rd singular present you commonly see:

μιλάει (very common, especially in speech)
μιλά (also standard, a bit leaner; frequent in writing) μιλεί is archaic/rare in modern usage. So both μιλάει and μιλά are fine here.

Is καλά an adjective or an adverb? Why not καλός/καλή or καλώς?

Here καλά is an adverb meaning well: μιλάει καλά = speaks well.

καλός/καλή/καλό are adjectives (good).
καλώς is a formal/literary adverb used mostly in set phrases (e.g., καλώς ήρθες) and is not used with verbs like μιλάει in everyday Greek.

How do I pronounce the tricky spellings (η, ει, ευ, and the -άει in μιλάει)?

η is pronounced like i (as in machine): η → /i/.
ει is also /i/: πιστεύει ends in /-vi/ because ει → /i/.
ευ is /ev/ or /ef/ depending on the next sound. In πιστεύει it’s /ev/: /piˈste.vi/.
-άει in μιλάει sounds like /ai/: /miˈlai/. (The shorter μιλά is /miˈla/.)

Can I change the word order inside the ότι-clause?

Yes, Greek allows some flexibility for emphasis:

Neutral: ότι η δασκάλα μιλάει καλά.
With focus on the subject: ότι μιλάει καλά η δασκάλα (common and natural). Fronting καλά (e.g., ότι καλά μιλάει η δασκάλα) is possible for strong emphasis but is stylistically marked.

Can I omit the second article and say ότι δασκάλα μιλάει καλά?

No. Bare singular count nouns are unusual in this position. You need the article: ότι η δασκάλα μιλάει καλά. Without it, it sounds ungrammatical or changes the meaning to a predicate-noun reading.

Why definite articles at all? Could I use the indefinite μια?

You can, but it changes the meaning:

Η μαθήτρια… = the student (specific/known).
Μια μαθήτρια… = a student (non-specific). Similarly, η δασκάλα vs μια δασκάλα inside the clause affects specificity.

Is μαθήτρια the only word for student? What about university students and masculine forms?

μαθήτρια = female pupil/student (especially school-level); masculine: μαθητής.
φοιτήτρια = female university student; masculine: φοιτητής.
δασκάλα = female teacher (usually primary); masculine: δάσκαλος.

Does Greek drop subject pronouns? What if I say Η μαθήτρια πιστεύει ότι μιλάει καλά?

Greek is pro‑drop, so that sentence is valid, but now μιλάει (she speaks) is ambiguous: it could refer to the student or the teacher. To disambiguate, specify:

ότι η δασκάλα μιλάει καλά (the teacher)
ότι αυτή/η ίδια μιλάει καλά (the student)

What tense/aspect are the verbs here?

Both πιστεύει and μιλάει are present indicative. Greek present covers both English simple and progressive readings (believes/is believing; speaks/is speaking). If you wanted a past belief, you’d use πίστεψε (aorist) or πίστευε (imperfect), depending on context.

What’s the nuance difference between πιστεύει, νομίζει, and θεωρεί?

πιστεύει = believes (often stronger conviction or trust).
νομίζει = thinks/assumes (more tentative or subjective).
θεωρεί = considers/deems (more formal, evaluative).

Why is the first Η capitalized, and what about accents on small words?

It’s capitalized because it’s the first word of the sentence. The article η is a monosyllable and never takes an accent. By contrast, the conjunction ή (or) is always accented to distinguish it from the article. Polysyllabic words like μαθήτρια, μιλάει, and καλά carry one accent mark each (monotonic system).