Translation:Please don't look at the notebook.
To help clear up some of the ambiguity I'm seeing here: Duolingo lacks contextual clues, and therefore refers to the most basic and simple meeting of the vocabulary, regardless of what can be said colloquially. To that end メモ is the word I'd expect them to use for a handwritten note (regardless of length), while ノート most directly refers to a notebook.
Because of Japanese linguistic magic, one can refer to the stuff inside a ノート as ノート as well, but it's a more indirect sort of reference. This same sort of indirect referencing happens a lot with colloquial Japanese, and works just fine because of context clues.
Say it with me everyone: "because... context!" ノート in Japanese can refer to the notebook itself, the stuff written inside it (notes), or small pieces of paper for writing messages like sticky notes.
In this particular question, I assume the most common interpretation of the situation is that it is an instruction given during an test or the like, and the speaker is asking everyone to do it from memory. In this case, "Please don't look at your notes" is a perfectly valid translation. Presumably, looking at the notebook itself, i.e. its cover, is fine. At the same time, one can imagine an unruly student holding their notebook open during the test for everyone to see. This time, "Please don't look at the notebook" is a more correct translation. Note (pun intended) that in both scenarios, 「ノートを見ないでください」could be used, though arguably 「そのノート」is probably more likely in the latter case.