"The woman is writing at the table."
Translation:Die Frau schreibt am Tisch.
I don't know how to explain this, but I'll try.. In this case, one can assume she was doing the writing on one spot the whole time, even though her hands were moving.. Using a scenario such as "Er springt auf dem Bett" vs "Er springt auf das Bett" to explain this.. The former would mean "He is jumping on the bed" (assuming the kid thought the bed was a trampoline. Lol).. Notice that despite the fact that he's actively jumping, it uses the dative case.
The latter would mean "He jumps onto the bed" (probably because he saw a spider crawling towards him wherever he was previously)
In summary, all that matters is the change in position.. Hope that helps
For much the same reason why it's not "The woman is writing on the table".
I think "auf" isn't actually wrong if you want to focus on the act of writing = the fact that the pen touches the paper on the table (not e.g. on the woman's lap), but it's much more elegant to use "am Tisch / at the table", focussing on the place where the woman is sitting while she does the writing.