What makes it hard to add to math is knowing the math up to the point you will add. Something like this was said earlier in the 20th century, that if you learned the math in a field, then you would know or have ideas at least of what to do next. Often trivial seeming additions may become more important in retrospect.
Students in math or mathematical subjects should be educated to see that adding to math or their subject’s math is not a mysterious process of communing with the gods, but is a matter of learning an area well and adding what may seem trivial pieces bit by bit.
Like any act of writing, once they get started they may find they have added a great deal.
Priority disputes are often about what may have seemed trivial additions at the time. Simple formulas so obvious and so much expressing what was already known that to claim credit for them seemed of little matter at the time.
This is what students should be taught is what doing new math in the sense of research is. It is not so impossible.
The difference of passing math on and adding original math is not so different. A new expression of old math is this same type of reasoning. Rethinking old math, bit by bit will often turn out to be or to spin off new math.
There are many bits of new math to be discovered at even low levels of old math. These may suggest better structures for understanding math then the old ones.
Dedekind’s 1888 book on the numbers is an example of this process. Moreover, it is a mine for still doing this type of innovation. Understanding it leads to a better understanding of abstract algebra and topology. The structure choices of these subjects may not be set in stone as best.