Integer Multiplication in Terms of Counting

Visual intuition for integer multiplication

$5\times 3$ is the number of dots in a grid of dots with $5$ rows and $3$ columns (so each row has $3$ dots, and each column has $5$ dots).

By symmetry ("flipping" a grid swaps rows and columns), this is the same as the number of dots in a grid of dots with $3$ rows and and $5$ columns (so each row has $5$ dots, and each column has $3$ dots).

So the fact that $5\times 3 = 3\times 5$ is quite down-to-earth, no matter what others tell you!