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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?

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neurosynth is a useful tool for answering these kinds of questions. Go there and enter "arithmetic" in the search bar and you can find a number of studies through meta-analysis and their relevant brain regions on a brain map. Note it's in beta, so use the "studies" tab to confirm with the literature. –  Keegan Keplinger Aug 9 '13 at 0:20

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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:

Using functional MRI, we demonstrated a differential cortical representation of numbers between native Chinese and English speakers. Contrasting to native English speakers, who largely employ a language process that relies on the left perisylvian cortices for mental calculation such as a simple addition task, native Chinese speakers, instead, engage a visuo-premotor association network for the same task. Whereas in both groups the inferior parietal cortex was activated by a task for numerical quantity comparison.

(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).

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