This. When I first started learning Russian, we immediately jumped into basic grammar rules. After two weeks of incredible frustration, I realized I did not have sufficient mastery of English grammar to be able to establish a framework for understanding Russian grammar. I often say that my first two months of learning Russian were spent learning English and it is not a joke.
Interesting. We had a lot of grammar parsing of Russian since the 2nd grade of school. Especially we analyzed parts of speech and constituents. For the latter, we'd underline words in sentences in different ways.
It's so widespread that today if you want to play word guessing with gestures, and you have several words, you just imitate that underline style, and everybody understands it. (Just remembered, we also did a lot of word analysis, marking up prefix, root, suffixes and ending, and everyone knows this markup too.)