Does practicing mental math increase mathematical intuition?

Question

Has any experienced mathematician on this site found that practicing mental math was associated with the development of mathematical intuition and conceptual understanding? I was considering practicing multiplying large numbers in my head for about 30 minutes a day. It seems the greatest mathematicians were known for their mental arithmetic skills, such as von neumann.

Answer 1

I was typing a Comment which became too long.

Opinions & Suggestions are mine , though I am not a Mathematician, hence it should be taken with a Pinch of Salt.

Positives :
(1) Improve Concentration & Patience , similar to meditation , though Same can be achieved by Eg Playing Chess moves mentally with a given list of moves , Eg listing out all teams & team members according to your favourite games.
(2) Improve numerical calculation accuracy & speed , though calculators are indispensable & ubiquitous , when involving roots & trigonometry & Etc. It may have a minor usefulness Eg when trying to make sure your Supermarket Bill is accurate , Eg Sharing the Bill after sharing lunch & taxi.

Negatives :
(3) Blindly multiplying numbers & rote learning will not generate Intuition about Multiplication, unless effort goes into analysing the Computations to Identify Patterns.
What-ever Patterns we Identify via mental calculations can also be got through Paper & Electronic Calculations & there is no advantage to mental calculations.
(4) Disadvantage is that we may get wrong answers which we may ignore / may not be aware of / may have to verify with Calculators.
(5) Multiplying large numbers will not give Intuition about general numeric areas like : Prime Numbers , Polynomials , limits , rational numbers , irrational numbers , Differentiation , Integration , trigonometric functions , Etc.
(6) Multiplying numbers will not give Intuition about non-numeric areas like : Symmetry , geometry , groups , algorithms , Etc.

Why are some Mathematicians capable of Mental Math ? Why we can not emulate that ?

We are seeing the End Out-Come of Identifying Patterns , not the actions which made those Mathematicians great calculators.

Eg Gauss did not add all numbers from 1 to 100 & somehow saw that it was easier to try $55 \times 101$. Gauss had already (maybe a Days earlier or a few minutes earlier) figured out that $n(n+1)/2$ is easier than adding $n$ numbers & he just applied it.

He may have got the Insight quickly or he may made a lot of calculations & analysed those for Patterns (not just blindly calculating) & thus got the formula.

Eg , we may want to calculate $35256 \times 5$ , which we can try blindly multiplying mentally & yet get no Intuition.
Alternatively , we may analyse it to see that $35256 \times 5 = 35256 \times 10/2$ , hence we can try Dividing by $2$ with a $0$ suffix.

What are the Alternatives ?
Various mental activities may be more Instructive & useful.
(A) We might take a Definition with a list of Criteria & think about why those Criteria are necessary : try to generate Examples where a Criteria is not satisfied.
Eg Equivalence relation has 3 Criteria : try to figure out Examples where a Criteria is not satisfied. Eg Natural Number Definition has a Set of Criteria : try to figure out Cases where a Criteria is not satisfied.
(B) We might take a theorem & try to generalise it.
Eg Convex Polygon may have some Property. We might try with Concave Polygon & see where it goes.
(C) We might Prove a theorem.
Eg It is easy to mentally figure out Quadratic Equation formula. We can try Cubic Equation formula.
(D) We might analyse a Proof & try to simplify it / extend it / generalise it / apply it elsewhere.
(E) We can try to visualize a geometry theorem.
These activities will generate more intuition rather than Blindly multiplying large numbers without Patterns.