In a programming language, you have syntax and semantics. Syntax tells you what you can write and how; for example what is a valid program. Semantics tells you what a program evaluates to, and maps your programs to another ___domain.
Here you have the same thing. In math you have syntax, and semantics. The other ___domain from semantics might be abstract, or it can be purely mechanical, but it can also be connected to reality. If your operations map to nothing that makes sense to you, the operations are just mechanical; you solve equations in some way because you know it's right but you don't understand why. If your operations map to other domains that you understand (also if they are abstract domains in your imagination), you can understand why the operations work like they do, and you know why you have to solve them the way you do it.
Maybe the equations don't have a specific meaning per se; but if they don't have any meaning for you, there is no way you understand what you are doing when you solve them.
A programming language semantics is a set of mutually recursive equations describing how a well formed program manipulates values. The equations themselves are as mechanical as the they can possibly be.
The recursive equations are the means by which you obtain a mapping from one ___domain to the other (eg. from the programming symbols to the program values). For equations there are many ways in which you can give meaning to each equation in the same way, such that the mechanical process makes sense.
For example:
a x = b (text) --> x is unknown, it is the right one if f(x) = a x equals f'(x) = b; both functions are programs you can compute and play with
from there you go mechanically to:
x = b/a (text) --> x is unknown, it is the right one if f(x) = x equals f'(x) = b/a
while in the first step it was hard to tell much about x, now we can see that it is trivial to guess which is the right x; x must be b/a
This is the first mapping from the ___domain of symbols to another ___domain that I could think of. There must be more natural mappings that can be used like this.
Check out abstract interpretation and Galois connections. These require a different kind of mechanical manipulations than simple algebra. I wonder if there is a gamification to be found in this direction.
Seriously: there is something to be said for the claim that mathematics is the search for beautiful tautologies.
Like javelin throwing vs hunting, running vs outrunning a predator, or painting vs making a portrait using paint because that is the only way to do it, there is a difference between being doing math and using math to reach a goal.
I think it would be very nice if one managed to give kids, even those with little mathematical talents, a glimpse of that difference.
