[henrikschroder]

To me, in Swedish, Ā and Ä and an A with two small dashes above would be the same letter, but with stylistic typeface differences. It doesn't change the letter itself. I have never heard of any German insisting their umlauts shouldn't be dots, so I think your German teacher was just a bit pedantic/insane.

Conceptually umlaut dots are different from trema dots, but who cares?