Just to clarify, do you mean between sentences or between paragraphs?
The reason I ask is that no spaces between sentences is just plain wrong. Any teacher who says that is not a good teacher and is probably making other mistakes. You and your son should find a different teacher.
If you mean between sentences: People who learned to type on a typewriter (older people, me included) tend to use two spaces. (That's what's needed for legibility on a typewriter). People who learned to type on a computer tend to use only one space. Either one should be fine for most uses. If someone wants a particular style, they'll specify. (Some programs even correct you, so you can't put too many spaces after a period). And again, no spaces is wrong. Wrong. Wrong. Wrong.
If you mean between paragraphs: There's no right or wrong answer, it is a matter of preference and your son should do whatever that particular teacher is telling him to do.