The tips and notes section explains this. Here is the relevant part:
"You can use both «говорить по-английски» (adverb) and «говорить на английском» (на + Prep.)
«На английском» is specifically about content in the language or about linguistic ability. «По-английски» is about the way an action is done/ an object is made (for objects, it does not mean the language).
So a book can only be written «на английском». However, if you mean "English-style pizza", it can only be «пицца по-английски»!"
I think most of the user that use duolingo on smartphones can't access the Tips and notes section :-/
It's frankly astounding that they've not added this feature to the app version - the material already exists, you just have to let us access it!
In the Spanish course you can already access the Tips & Notes on mobile, so hopefully they will eventually add this feature to the other courses as well.
Uh? What particular по-английски culinar invention do you mean with english style pizza? Or ingredients? Brown sauce, pickles, gentleman's relish, mint sauce, A.1, marmite perhaps?
The kind of pizza commonly found in England, as opposed to the traditional kind found in Italy. They are very different.
It's an adjective in the prepositional case (the noun языке is implied), because of the preposition на before it .
So японском got declinated like an adjective because in real its японском языке but someone dropped the языке for a quicker/smoother sentence?
Because while "на японском" refers exclusively to the language, "по-японски" usually means "Japanese-style" or something like that. Рыба по-японски = Fish [cooked] in Japanese style. Though it's fine to use "по-японски" with a verb: говорить по-японски = to speak Japanese.
Super hard to decipher what the voice is saying. She kinda says "новостина и понском". I think the flow of speech is a little unfortunate. Shouldn't be used for listening exercises really.