The key concept in elementary math is successor. That is after function. Functions we already have at least two notations. The ordered pair (x,y) and a function symbol y=f(x). We also have operator notations like +.
x+y = S(x,y)
Where S(x,y) means start at x and take successor y times.
S(x,0) = x
S(x,1) = S(x) = x’
The successor function is the base function of numbers.
(n,n’) is the link ordered pair.
We need these notations to teach concepts at lower age levels.
If lower age levels have a harder time learning not to leave steps out of proofs, i.e. explanations, then we need props to help them learn what the steps or structure is.
Manipulatives are at least partly using props to help teach concepts this way.
The successor function notations are a type of manipulative.
So are truth tables.
The more that young minds have a harder time with abstract critical thinking, the more we need symbolic structure to show them explicitly what rests on top of what.