"Do you remember we have a math test tomorrow?"
Certainly; However, if you pay close attention to the part of the sentence 明天有数学课时 (lit. "tomorrow has math test"), it refers to both the speaker and the listener that they will take the test tomorrow. This is the possible reason that "we" is implicitly emphasized in the translation.
The point of excluding characters is to shorten the amount of saying. In this course, sentences like those emphasize how a Chinese speaker would say in real life.
So, this is what I like to call a "frame" sentence. The real question is 你记得...吗? ("Do you remember...?") Any piece of information could be put inside the "frame" (in this case, it's 明天有数学考试.)
明天 is inside the frame, which means it's just part of the information that's being remembered (or not remembered). If you put 明天 at the front of the sentence, than it becomes part of the frame instead of the content. 明天你记得数学考试吗? ("Tomorrow, will you remember the math test?"). Honestly, that Chinese sentence sounds a bit awkward to me, and I'd probably add 会 or 要 like this: 明天你会记得数学考试吗?