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These both translate as "learn" or "study" in American English. When do I use which one?

Some examples of what I am confused about:

1. I study (will get a degree in) Physics at University

2. I am a student in δημοτικό / γυμνάσιο

3. I will study until midnight tonight

4. (To get a university degree in Physics, I need to take a course in Mathematics). I am taking a course in Mathematics.
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μαθαίνω κάτι means "I learn" something

μαθαίνω κάτι σε κάποιον means "I teach"something to someone.

Examples:

μαθαίνω Ελληνικά = I learn Greek.

μαθαίνω τον κανόνα = I learn the rule.

μαθαίνω κάτι απέξω = I learn something by heart.

μαθαίνω να οδηγώ = I learn how to drive.

μαθαίνω από τα λάθη μου =I learn from my mistakes.

 

μαθαίνω Αγγλικά στο φίλο μου  = I teach English to my friend.

μαθαίνω στο γιο μου να οδηγεί = I teach to my son how to drive.

 

σπουδάζω means "I am a student at university, or college". It refers to academic studies.

Examples:

Σπουδάζω ιατρική. = I study medicine.

Σπουδάζω μαθηματικά. = Ι study mathematics. (I am a student at university)

Σπουδάζω στο πανεπιστήμιο της Αθήνας.= I am a student at the university of Athens.

About your examples:

I am a student in δημοτικό / γυμνάσιο = είμαι μαθητής στο δημοτικό/ γυμνάσιο

I will study until midnight tonight = Θα μελετήσω (or θα διαβάσω) μέχρι τα μεσάνυχτα απόψε.

(μελετάω or διαβάζω  here mean "study my homework")

I am taking a course in Mathematics. = Παρακολουθώ μαθήματα μαθηματικών.

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