I want to understand why mit seinem is used instead of just sein in the following example:
English: He is starting his German course.
German: Er fängt mit seinem Deutschkurs an.
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sein Deutschkurs (Nominativ)
seines Deutschkurses (Genitiv)
seinem Deutschkurs (Dativ)
seinen Deutschkurs (Akkusativ)
Anfangen mit wem oder was (Dativ)
Und hier sind Beispielsätze für alle Fälle:
Sein Deutschkurs hat gestern angefangen. (N - wer oder was?)
Die Lehrerin seines Deutschkurses macht ihre Sache gut. (G - wessen?)
Er hat gestern mit seinem Deutschkurs angefangen. (D - wem oder was?)
Er besucht seinen Deutschkurs regelmäßig. (A - wen oder was?)
A verbatim (and also correct) translation would be:
E: He is starting his German course.
G: Er beginnt (gerade) seinen Deutschkurs.
(There is no progressive tense in German, the best way to mimic it is to use the adjective gerade (just), but this is optional, that's why I've put it in brackets. I will omit it in my next examples). Without progressive form it would be even more verbatim:
E: He starts his German course.
G: Er beginnt seinen Deutschkurs.
You can replace the non-separable verb beginnen by the separable verb anfangen, which is a synonym of beginnen. But anfangen splits into an and fangen when used in present tense, where a declined form of fangen is nailed to position 2, while the former prefix an floats to the end of the sentence:
E: He starts / is starting his German course.
G: Er fängt seinen Deutschkurs an.
This construction is correct, but there is a second construction, that is more common in German. As far as I know, it also exists in English, but I don't know how common it is in English:
E: He starts / is starting with his German course.
G: Er fängt mit seinem Deutschkurs an.
In 1, 2 and 3 there is a subject (er), a predicate (beginnt, fängt ... an) and an accusative object (seinen Deutschkurs). This object has to be in the accusative case, because the verbs beginnen and anfangen need their object in this case.
But it can be replaced by a caseless prepositional object (mit seinem Deutschkurs). Prepositional objects themselves (the whole part of speech) have no grammatical case. But besides a preposition (here: mit) they contain an object that stands in one of the four cases, and it's not the verb that establishes this case. It is the preposition that defines this case, and mit always needs the dative case. This is the reason, why it is mit seinem Deutschkurs.
The key issue is that the two verbs used in English and German require different extensions for the sentence to become idiomatic. In English, if you are starting your course the course is just added as a simple direct object. Thus also, there is nothing between the verb and the personal pronoun.
He is starting his German course.
This sentence could be translated 1 : 1 into German. In that case, it would read:
Er fängt seinen Deutschkurs an.
This is a grammatically sound German sentence; however, it sounds weird to my ears. It is uncommon to use the verb anfangen with an accusative object directly if the thing you are starting is not a concrete thing. So you may say things such as:
Hast du deinen Bericht schon angefangen?
The report is something that is written on paper, which I can call finished when it is done and which I can hand on. A course cannot be handled in that way and that makes it less common to use etwas anfangen for things like courses.
Instead, in these cases the thing you are starting is typically attached with a prepositional object; in this case, mit etwas anfangen. The mit somewhat implies not only starting the actual thing but also starting the work around. So if you start a course, you’re going there, you’re preparing yourself for learning, you will be doing homework, etc. All these additions are implicitly covered by mit.
If you do use mit, it of course governs dative as it always does. Therefore, the final sentence will be:
Er fängt mit seinem Deutschkurs an.
Writing up this answer, I noticed these two similar yet distinct expressions:
Sie fangen das Spiel an.
Sie fangen mit dem Spiel an.
The first sounds like the kick-off to a football match or the first serve in a volleyball game. It could also be a board game and somebody has just begun there first move.
The second can also be used to express all of that, but furthermore it can also be used to mean something along the lines of ‘they’re starting their mind-games again!’.