Learn Dutch: Mijn vader is aan het denken.

In Dutch, to say that someone is currently in the middle of doing something, you use a form of the verb zijn (to be) + aan het + the whole verb (the infinitive). Here, is + aan het + denken literally translates to 'is on the thinking', which is the standard Dutch way to express 'is thinking'.

What does aan het literally mean in this sentence?

Literally, aan het means 'at the' or 'on the'. This grammar structure treats the action almost like a physical place or an object you are currently busy with. So aan het denken is like saying your father is currently 'at the act of thinking'.

Could I just say Mijn vader denkt instead?

Yes, Mijn vader denkt is also grammatically correct and can mean both 'My father thinks' and 'My father is thinking'. However, using the is aan het denken structure strongly emphasizes that he is in the very process of thinking right at this exact moment.

Does the word Mijn ever change its spelling based on the noun that comes after it?

No, the possessive pronoun mijn (my) is very easy to use. It stays exactly the same whether the noun following it is a 'de' word (like the masculine word vader), a 'het' word, singular, or plural.