Well, I was all prepared to go on a "what's with those questions tonight" thing when I saw the headline, but as someone who has had to struggle with teachers who thought they were right just by the mere fact that they were teachers, you have all my compassion and pity.

You could link her to https://www.bundeswehr.de - I guess she probably does not know a word of German, but there are photos there of soldiers with German flags and eagles and everything on their shoulders.

The page of the German Ministry of Defence might be nice - the sidebar on the right has a three-part history of the Bundeswehr, in English and everything, if you switch the language to English. It explains how the German rearmament came about (spoiler alert: it's not exactly that Germans wanted it while the Allies wanted to prevent it, rather more the contrary). If she seriously thinks the Western Allies didn't "allow" West Germany to have an army during the Cold War...

Also, where did she think those supposed "NATO" troops come from? Some sort of Foreign Legion for militaristic Germans? It's scary that someone with that lack of common sense, quite apart from knowledge, is allowed to teach.

One caveat: In my experience, you don't "win" these kinds of conflicts with teachers of the sort you describe. At best, the teacher will ignore the whole thing when she has to recognise she was awfully, terribly wrong, at worst she's going to be resentful.

I was accused of stealing a book in elementary school - my parents believed me that I hadn't taken it (not all my peers' parents were equally trusting towards their children), but just to shut the teacher up, they finally said that they'd pay for it without acknowledging I had stolen it. Well, it turned out another child had "borrowed it without asking". No sanction for the other child, no apology to me from the teacher, she just dropped the whole matter. Can you tell I'm still bitter about it even though it's almost thirty years later and the woman is dead? (Sorry, that had to come out.)