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It is true that the brain greatly outperforms the calculator, but it is apparent that humans make mistakes even with simple math problems, and take longer to answer, whilst the calculator just runs a bunch of bits through logic gates and pops out a correct result almost instantaneously.

Considering the above statement, it seems apparent that the human mind takes a completely different approach to arithmetic problems than the calculator.

Has there been any research that at least provides a general idea of the brain's approach?

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    $\begingroup$ If you could give the calculations to be made to a calculator telepathically then a calculator would probably beat any human brain with complex calculations. However when using a calculator it can only work as quickly as the person entering the calculation accurately. Also mathematical calculations can be done using quick tricks which are taught or just stumbled upon (for example see wisebread.com/…) $\endgroup$ – Chris Rogers Nov 15 '16 at 23:13
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    $\begingroup$ Just found a few papers of a cognitive architecture called ACT-R that tries to explain how people perform arithmetics: ra.adm.cs.cmu.edu/anon/home/ftp/anon/1998/CMU-CS-98-186.pdf , repository.cmu.edu/cgi/… and pact.cs.cmu.edu/pubs/… . Hope that I can find time to explain the lot. I suggest you first read the second article plus some questions about ACT-R here on CogSci. $\endgroup$ – Robin Kramer Nov 17 '16 at 14:50
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What I know is that there is evidence that the brain treats arithmetics very similarly as language. I also know that there is a cortical area more akin to encoding the sense of numerosity, which could be accessed while performing arithmetic calculations. However, I am no expert in the area and only know of some works done by some colleagues that I'll link here.

  1. Eye gaze reveals a fast, parallel extraction of the syntax of arithmetic formulas

  2. When order matters: Last-come first-served effect in sequential arithmetic operations

  3. The cortical representation of simple mathematical expressions

If you are interested, you should look at Stanislas Dehaene's work. He has many papers on how the brain does math and how it also is able to naturally encode numbers.

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