When I was learning Hungarian grammar in Hungarian grades 1-8, the 'noun case' terminology was not widespread, although more and more academic sources began lifting it from works discussing Indo-European languages. Instead, what's taught is that Hungarian is a thoroughly agglutinative language, and there are different kinds of suffixes which play different roles. If they're present, the types of suffixes always occur in the same order.
The first kind are any word-creation suffixes that change the meaning of the root word, perhaps making it a different part of speech. Such a suffix is called a 'képző'.
The second kind are word-relationship markers that preserve the part of speech but signify a clarifying quality, like belonging to an adjacent noun, or plurality, comparative intensity, or an ordinal pointer -- the sorts of markers you'd use to answer questions like 'whose?', 'which?'. This kind of suffix is called 'jel'.
The third are role-in-sentence markers that signal the function of the word within the sentence. This suffix is called a 'rag'. For verbs, these are endings for grammatical person and mood. For nouns, the most common of these correspond to nominative case, accusative case, and dative case, but then there's one that expresses the notion of being used as an instrument, one that expresses a purpose, two that are used to signify a transformation, and a couple others. And, there are 10 more that show a location relationship; English would prefix such words with one of: at, into, in, out of, towards, by, away from, onto, on, from off of; in Hungarian, the word takes a corresponding suffix instead.
When I was learning English grammar in an American school, a structural examination at a level I was used to from Hungarian was largely absent -- it took until I was learning English as a foreign language, or learning German as a foreign language, that mechanics of English grammar became apparent to me. The dying usage of subjunctive mood and incorrect use of accusative and dative case is a natural simplification, but enabled by the lack of education about the origin of those forms and their meaning.