Implied relations to children, students, teachers and others
I have been speaking Chinese for a while now and the Duolingo course seemed like a good opportunity to improve some of the imperfections in my grammar. One thing that is obvious by now is that I am generally using to many words in phrases like "I will see my children..." I would say something like "我会见我的孩子" but the course has made me aware of the fact that my relation to the children should be implied rather than stated since it expects me to write "我会见孩子." What I would like to know is whether there are circumstances where the 我的 is justified. If I, for example, would like to emphasize a contrast or to clarify what I refer to since we have been talking about so many different children for the last five minutes, would it be okay to say 我的孩子? Or should I always leave the 我的 out?
Don't worry too much as you will be understood inserting 我的 all the way. Certainly, you probably want to use the language more gracefully and are looking for the rules. Unfortunately I am native but I know no grammar about Chinese (haha...), so I can't tell. After all it must just be whether it is already implied in the context, or an emphasis is needed. You just said the rules in your question!
Just some examples:
I am going to Disneyland with my kids this weekend. (Who else's kids will you go with? No need 我的)
Your boss asks you to come to work on weekend. You reply:
I need to go to Disneyland with my kids. (Same logic).
My kids need to go to Disneyland! (It is preferred to have 我的 since obviously your boss is talking about work, not kids, and you raise this whole new topic of kids using kids as the subject.)
When you are talking to other parents about kids, certainly it would be more likely that you often would have to emphasis that you are talking about "your" kids, not the kids of the listener.
Finally, just a small suggestion... "我会见我的孩子" does not sound very natural to me for what you wanted to say. It is perfectly good by grammar, but hearing it I would have thought you were jailed or something... 见 means seeing or meeting but we use it mainly on more occasional or one-time events such as seeing friends or meeting a business partner. 看 (watch) would be a better choice because compared with 见 , it also implies taking care of and, yes, watching on the kids. Moreover, 会 puts an emphasis on the anticipation (remember that there is no tense in Chinese and we do not necessarily use 会 for all future tense sentences).
A better structure would depend on the context but something like this will be fine:
（小孩，孩子 are the same.)
Actually my Chinese is good enough for getting my meaning across, so yes, to me it is all about learning "to use the language more gracefully" as you put it.
Also, a big thanks for the clarification of the difference between 看 and 见. At first I intended to write a sentence with a 了 denoting that it had already happened, kind of like what my mother could have said about seeing me last month - since I am an adult, and we live in different cities, and we only had lunch together when she happened to be in town. (Would 见 be suitable in this context?) In the end I changed the example because I was afraid that I might mess up the grammar of 了. (Already during the first year of my Chinese class at the local university the test we had that included 了 was my worst, and I still haven't taken the time to correct this shortcoming. People still understand, especially if I throw in a word denoting the time frame, but I lack the ability to express nuances properly. I hope that duolingo will help me improve this.)