like playing 和 enjoyed teaching

Many students like playing basketball after school.

结构

Subject(主语) Many students
Verb(动词) like
Object playing basketball(动名词短语)
Adverbial(副词) after school(介词短语)

这里不能使用 Many students like play basketball after school，因为 play 在动词 like 之后，有两个动词了。在英语中，当一个动词跟在"like"、"love"、"enjoy"、"hate"等动词后面时，第二个动词（后面那个动词）需要使用 gerund 形式（动词+"ing"）或不定式（to+动词），具体取决于上下文。

也就是说，Many students like to play basketball after school 在语法上也正确，但 "like playing" 在很多情况下听起来更自然，尤其是在谈论一般喜好时。

I really enjoyed teaching English to school children at night.

结构

Subject I
Adverb(副词) really
Verb enjoyed
Object teaching English
Prepositional Phrase to school children
Prepositional Phrase at night

这里使用了 enjoyed 是因为它指的是过去的行为或过去的经历。说话的人说的是他们过去做过的事情，并从中体验到快乐或满足。

例如，I enjoy teaching English to school children. 现在时态，这意味着说话者目前喜欢这项活动，而且经常做。I enjoyed teaching English to school children. 过去式，这表明说话人在过去的某个时候曾在这项活动中找到过乐趣。I will enjoy teaching English to school children. 将来时，这意味着说话人希望将来能享受这项活动。

综上，Many students liked playing basketball after school. 在语法上也是正确的，使用"liked"表示你在谈论学生过去喜欢的事情。