Learn Hardcore German: Auf dem Gymnasium hat die Schülerin viele Hausaufgaben in Mathe. - At high school the pupil female has a lot of homework in math.

German has a strict verb‑second (V2) rule in main clauses: the finite verb (here hat) must be in second position, but “second” means second element, not second word.

Elements can be:

a subject (e.g. Die Schülerin)
a time phrase (e.g. Heute)
a place phrase (e.g. Auf dem Gymnasium)
an object, etc.

In your sentence:

First element: Auf dem Gymnasium (a prepositional phrase of place)
Second element: hat (the verb, as required)
Then the rest: die Schülerin viele Hausaufgaben in Mathe

You could also say:

Die Schülerin hat auf dem Gymnasium viele Hausaufgaben in Mathe.

Same meaning, just a different element in the first position. In both cases, the verb stays in second position.

Why is it Auf dem Gymnasium with dem, not something like auf das Gymnasium?

The preposition auf can take dative or accusative, depending on whether you mean:

location (where?) → dative
direction (where to?) → accusative

Here you are talking about a location (where the student has homework), so you use the dative:

das Gymnasium (nominative)
dem Gymnasium (dative singular, neuter)

So:

Auf dem Gymnasium = at the gymnasium / secondary school (location → dative)
If it were direction: auf das Gymnasium gehen = to go to the gymnasium (direction → accusative)

Could I also say Im Gymnasium instead of Auf dem Gymnasium? What’s the difference?

You can say im Gymnasium, but there’s a nuance:

im Gymnasium (in + dem) literally means inside the building.
auf dem Gymnasium is the usual idiomatic way to mean “at that type of school” (as an institution), not focusing on the physical interior.

Compare:

Sie ist auf dem Gymnasium. = She attends a Gymnasium (type of school).
Sie ist im Gymnasium. = She is inside the building of the Gymnasium right now.

For “attends that school”, auf dem Gymnasium is more natural.

What exactly is a Gymnasium in German? Is it a sports hall like in English?

No. In German, das Gymnasium is a type of academic secondary school that prepares students for university, typically ending with the Abitur (university entrance qualification).

It is not:

a sports hall
a place with exercise machines

For a sports gym in German, you’d say for example:

die Turnhalle (school gym / sports hall)
das Fitnessstudio (commercial gym / fitness center)

So in the sentence, Auf dem Gymnasium means “At the (academic) secondary school …”

Why is it die Schülerin and not der Schüler?

German marks grammatical gender and often also natural gender:

der Schüler = male student / pupil
die Schülerin = female student / pupil

The ending ‑in is a common way to form the feminine version of a profession or role:

der Lehrer / die Lehrerin – male / female teacher
der Student / die Studentin – male / female university student

In your sentence, the subject is explicitly female, so die Schülerin is used.

Why is it die Schülerin and not der Schülerin or der Schülerin with some other ending?

Case matters here:

In the sentence:

die Schülerin is the subject → nominative case
viele Hausaufgaben is the direct object → accusative case

For die Schülerin (feminine, singular), the forms are:

Nominative: die Schülerin (subject)
Accusative: die Schülerin
Dative: der Schülerin
Genitive: der Schülerin

Because she is the one who has the homework (subject), you use nominative → die Schülerin.

Why is it viele Hausaufgaben and not viel Hausaufgaben?

German distinguishes between:

viel = much, a lot of (for uncountable or singular mass nouns)
viele = many (for countable plural nouns)

Hausaufgaben is grammatically plural and countable in German:

eine Hausaufgabe = one piece of homework
zwei Hausaufgaben = two homework assignments
viele Hausaufgaben = many homework assignments

So you must use viele with the plural:

✅ viele Hausaufgaben
❌ viel Hausaufgaben (incorrect)

Why is Hausaufgaben plural in German when English usually says “homework” (uncountable)?

The concepts don’t line up perfectly between the languages:

In German:

die Hausaufgabe (singular) = one specific task set as homework
die Hausaufgaben (plural) = several tasks, “homework assignments”

In English, homework is usually uncountable, so you say:

I have a lot of homework. (not homeworks)
I have three homework assignments. (you make it countable with assignments, tasks, etc.)

So:

viele Hausaufgaben = “a lot of homework / many homework assignments”

Why are words like Gymnasium, Schülerin, Hausaufgaben, and Mathe all capitalized?

In German, all nouns are capitalized, regardless of their position in the sentence.

In your example:

das Gymnasium – noun (type of school)
die Schülerin – noun (person)
die Hausaufgaben – noun (things)
(die) Mathe – noun (subject, short for Mathematik)

Adjectives, verbs, and most other word types are not capitalized (unless they’re at the beginning of a sentence or part of a proper name).

Why is it in Mathe without an article, and not in der Mathe?

For school subjects, German normally omits the article:

in Mathe – in math
in Deutsch – in German (class)
in Geschichte – in history

So:

✅ in Mathe
❌ in der Mathe (would sound wrong in this school‑subject meaning)

You do use articles with some other kinds of nouns after in:

in der Schule – at school / in the school
im Kino – in the cinema

What is the difference between Mathe and Mathematik?

Mathe is the colloquial, shortened form of Mathematik:

Mathe – informal, very common in everyday speech, especially among students:
- Ich habe viele Hausaufgaben in Mathe.
Mathematik – more formal, used in written language, official names of subjects, etc.:
- Sie ist gut in Mathematik.

Both mean the same school subject, just with a different level of formality.

Can I change the word order to Die Schülerin hat auf dem Gymnasium viele Hausaufgaben in Mathe? Is that still correct?

Yes, that sentence is fully correct and natural.

Two common variants:

Auf dem Gymnasium hat die Schülerin viele Hausaufgaben in Mathe.
– Place phrase first (emphasis on where), verb second.
Die Schülerin hat auf dem Gymnasium viele Hausaufgaben in Mathe.
– Subject first (more neutral / default), verb second.

German allows some flexibility in the order of subject, time, place, objects, etc., as long as:

the finite verb stays in second position in main clauses, and
you don’t break up fixed expressions unnaturally.