How much of mental math is subconscious?

For instance lets say I asked you:

What is 7+7?

Odds are you wouldn’t need to think about this at all and would immediately respond with “14”

In your mind you wouldn’t have to think: “(1+1+1+1+1+1+1) + (1+1+1+1+1+1+1)”
You would just know that the answer is 14

So this leads me to my main question:

For people who have been doing mental math for awhile how far does this go?

Do you need to consciously think about a problem like 347+823 or do you just know the answer by looking at it?

What about a problem like 27x36?

How much of this are you doing consciously and how much of it is automatic?

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Interesting question, but I’m not sure “subconscious” accurately describes what I think you mean.

All mental math is combining mathematical facts together to solve problems. I do some work in coaching professionals, and the first thing I do is usually to assess whether their greatest priority is getting better at solving problems, or in knowning mathematical facts.

E.g. if you don’t know your times tables, then solving 8 × 37 is going to be difficult because you have to somehow calculate e.g. 8 × 3 as part of your solution. But if you know them, you just do 240 + 56 = 96. But if you had memorized your 37-times table, then you would know the answer instantaneously, without any working.

You ask how far this extends for experts. I have the UK record for calendar date calculation, and for speed, I memorized 483 codes (okay I don’t know them perfectly) and simply perform two additions modulo 7. So it’s officially mental math, but practically it’s more memory recall.

I’m also advanced with inexact square roots, e.g. square root of 123456. For this I memorized the squares up to 99² = 9801, but otherwise the real difficulty is doing lots of multiplications of 2-digit numbers, and lots of subtractions of 2- and 3-digit numbers. That has extensive calculation; the memory is just the easy part.

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