I think that depends. If the subject is mentioned in a sentence tat's part of an article etc., it would be lower case (and it's "math" in AE; "maths" is BE). But if it's part of a listing in a class schedule, for instance, it would be capitalized. At least that's how I see it.

If it's a proper name, then capitalized. [On the schedule:] 10:00, AP Calculus. 
If a generic or casual reference, then not capitalized. What math class do you have this semester? I'm taking calculus.

In our school, trigonometry is taught in Advanced Algebra; in some schools, it's part of Pre-Calculus.


(You know, of course, that the question of capitalization isn't exactly a critical issue for us.)

As a rule of thumb, general subjects aren't capitalized unless they're proper names (e.g., languages).

She teaches math, science, and music in elementary school.
This year he's taking world history, geometry, biology, English, and first-year German. Next year he'll take American history, American literature, chemistry, computer science, and hopefully second-year German.
I have government second period, calculus third period, and choir fourth period.
He has gym last period and trig next-to-last period.
She wants to take biogenetics and drop astrophysics.
He teaches differential equations and engineering calculus.

An exception is P.E., which is another term for gym.

However, in American schools and colleges, the course name may be understood as a particular title (and thus capitalized) if it specfies a particular topic or year, not just a general subject.

She teaches German 1, German 2, and German 3-4.
He's signed up for Engineering 1215, Computer Science 2282, and Mechanical Drawing 1201.
For your science distribution class, are you taking Physics for Nonscientists or Biology of Local Ecosystems?

But the boundaries are fuzzy, and sometimes you could really go either way.

Our high school offers AP World History, AP American History, and AP Calculus A and B / AP world history, AP American history, and AP calculus A and B.
I really enjoyed Physics of Music / physics of music.


So I would say that in your examples, 'Twilight' is just wrong, and Harry Potter is understandable within the context of specific course titles as opposed to general school subjects.

Nur zum Verständnis: Bei Harry Potter ist "Divination" ein Schulfach, ebenso wie z.B. "Trig(onometry)" bei Twilight. Warum ist dann das eine falsch und das andere nicht?

In beiden Beispielen werden die Fächer auch großgeschrieben, wenn es um die Lehrkräfte geht:

Sybil Trelawney is our Divination teacher.
Mr Varner is my Trig(onometry) teacher.

++++

Aus den bisherigen Antworten schließe ich bereits, dass es keine feste Regel für alle Fälle gibt und dass es daher durchaus didaktisch sinnvoll ist, deutsche Kinder nicht mit unterschiedlichen Schreibungen zu verwirren.

At least for AE, I pretty much agree with #1 and #2 and the second half of #3 in particular.

As a general rule, I would consider the name of a particular class to be a proper name, which is consistent with the general "Eigenname" rule in the OP. (Another way of looking a the names of particular classes is that they are titles, which are then written in title/headline case in AE.)
Calculus 101, World History 204, Women in the Labor Market 210
Just speaking about the general subjects, though, would often be lowercase. ("I'm taking two world history courses next year.")

And then there's this bit of wisdom: (You know, of course, that the question of capitalization isn't exactly a critical issue for us.)

I have the prospectuses (?) of two very good schools in the south of England; both capitalize all subject titles within sentences.

#6 doesn't surprise me at all, but I didn't want to speak for BE.

dass es keine feste Regel für alle Fälle gibt und dass es daher durchaus didaktisch sinnvoll ist, deutsche Kinder nicht mit unterschiedlichen Schreibungen zu verwirren.

Does it make educational sense to teach them that the names of classes are written lower case even just because there is a usage pattern that distinguishes between caps and small? That would be like saying for German that "alles" is always written in lowercase just because it's not always easy for NNSs to differentiate when it should be in caps and when in lowercase.

Since it's apparently clear to the students that names are written in caps in English, why can't it be explained that something like "Geometry" is written with a cap when considered to be a name?

hbberlin, da wirfst du eine grundsätzliche Frage auf: Im Anfangsunterricht werden ja Regeln oft übergeneralisiert. Beispiel für Englisch ist die Wortstellung im Satz S-V(-O), von der es natürlich Abweichungen gibt (Inversion wie "out comes the snowman"). Dein Beispiel für Deutsch, "alles", passt auch: Großgeschrieben wird es nur bei "(mein) Ein und Alles", das rechtfertigt nicht, im Anfangsunterricht darauf einzugehen.

tomtom's example may indeed show that BE is more tolerant of what AE would often find needless capitalization. We've seen that in other contexts (the Queen vs. the queen, etc.).

On the other hand, in a prospectus or a course catalog there might indeed be more reason to treat everything as a title in AE as well, as dude mentioned in #1.

I'm still wondering, however, if BE students would really talk about Maths as opposed to just maths. A lot of Maths homework? A test tomorrow in History? Always gets chosen first for a team in Sport (or Games or whatever they call it)?

I don't really think I've seen that even in BE books, but maybe usage has changed more recently than the era of BE children's books I'm familiar with.

In any case, I have to disagree that there's no firm rule, or that it's useless for German students. In fact the rule that school subjects aren't proper nouns is a fairly logical one, so to me it is indeed useful to begin early distinguishing between languages, which are always capitalized, and other subjects, which aren't. I wouldn't necessarily expect young German speakers to understand that distinction at first, but it doesn't hurt to model correct usage for them -- especially if it makes them notice that languages are always capitalized, which is a good and absolutely firm rule that can be learned very early. (-:

I'm still wondering, however, if BE students would really talk about Maths as opposed to just maths. A lot of Maths homework? A test tomorrow in History? Always gets chosen first for a team in Sport (or Games or whatever they call it)?

No, Not a lot of Maths homework.
A lot of maths homework, a test tomorrow in history, ... a team in PE.

But in a list of subjects taken, I would write English, Maths, Physics, Chemistry, Biology, Geography, German.