When you get hung up on something, that means you are stuck on it in the sense that you are thinking too much about it or worrying about it. Getting hung up on grammar means to worrying so much about how you are saying something, that you can’t concentrate, or think about, what you are saying (or what another person is telling you) to have a fluent conversation. Paying too much attention to being grammatically correct becomes distracting to your brain, and then you cannot focus on the quick back and forth of a conversation in English.
This is not an unusual problem for learners. I have met many English language learners that score really well on exams and probably know more about English grammar than I do, but have a difficult time keeping a normal conversation going in English. Instead of approaching the complications of English grammar from an academic standpoint and memorizing the long lists of rules and exceptions to rules, it is better to learn it the way most native speakers learn it, by listening and practical usage. If you ask a native speaker why he uses grammar in a certain way, he will probably reply that “it sounds right”. That’s the way most native speakers figure out the correct answers on grammar exams too! (and the reason why so many native speakers have bad grammar habits)
Of course, some basic grammar rules must be learned, and even memorized. There is no way around that in any language. However, beyond the basics, listening to English conversations and learning the way phrases are used by native speakers goes much farther in helping a student become fluent. I personally hate teaching advanced grammar rules to students, mainly because it is boring and complicated both for them and me. Over the years, I have made an extensive study of the English language and all of its peculiarities, and I still am learning new things about grammar all the time. So don’t worry too much about having perfect grammar when you are speaking to someone, chances are that no one will even notice if you make a mistake or two.
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