Do neuroscientists already know what areas of the brain are involved in mental calculations? Has any fruitful research been done on that or not yet? Do you think that if we understand what areas of our brain are responsible for doing arithmetical tasks we may one day have the ability to improve someone's calculation skills by non-invasive methods?
This is a part answer to your question.
According to "Arithmetic processing in the brain shaped by cultures" (Tang, 2006), the parts of the brain that are activated while doing arithmetic is to an extent, governed by culture, in the article, the authors assert:
(The article also includes fMRI imagery examples).
It also seems that recognition of simple arithmetic starts at an early age, according to "Infant brains detect arithmetic errors" (Berger et al. 2006).