It's hard to imagine having only one instinctive visualization for integers.
Q: What's purple and commutes?
A: An abelian grape.
First number is 1, so its table of 1. Then x as multiplier sign. Then a count from 1 to 10. Then = sign. Then the result. We kids are supposed to write each line in left to right direction, then move to next line.
We use paper with square tables or graph on it. Most of the time, kids simply write 1, move to next line, again write 1, all the way till 10th line. Then we move to next column, write x, then move up, x all the way till 1st line. Then 1,2,3, in next column, = in next column coming up. Then the answers going down.
edit: But actually after that I used something called "Math-U-See", which used physical blocks to develop intuition. That was pretty cool.
For example, a string concat can be understood as an addition operation:
1 + 0 = 1 (identity)
1 + 1 = 2
1 + 2 = 3
2 + 1 = 3 (communitive)
"a" + "" = "a" (identity)
"a" + "a" = "aa"
"a" + "b" = "ab"
"b" + "a" = "ba" (non-communitive)
There's this whole intuition about addition itself that can be applied to something other than integers, and being able to reason about that is applicable to how you design software, particularly function interfaces.
Just as a note, my mother made me memorize the multiplication table when I was a kid, and I had ended up memorizing additions just through sufficient practice. I was able to intuit what additions and multiplications meant, but for the purpose of taking tests in school or doing homework, additions just pop out as answers because of the memorization. It wasn't until much later in life that I started encountering ideas such as, what if you were adding something other than numbers.