Teach them the responsibility or teach them responsibility is correct?
Either one could be correct depending on context, but, if these are supposed to be independent sentences, the second one is the only right choice.
The full context is She feels it’s difficult to teach the children…responsibility.
I think the is necessary here.
'She feels it’s difficult to teach the children responsibility.
She finds it difficult teaching them to be responsible.
What Bee says. “Responsibility” is a general concept in that sense, there are not several responsibilities in this context.
- I will teach you responsibility (the general concept).
- I have many responsibilities (specific types of responsibility, tasks)
- She hates the responsibility of taking care of her rabbit (a specific instance of responsibility, the task of)
Indeed, I happened to meet many sentences ‘the’ is used carelessly, esspecially in communication.