>Anyone who actually can perform the "shortcut" you mention and successfully changes the operations is performing the algorithm correctly and solving the problem, regardless of whether they use the word "moving" or not.

Sure. The question is, if they move on to more advanced mathematics will they understand the underlying rules they're applying well enough to extend them into more advanced and finicky details, or will they have to "unlearn" their previous understanding first? Or if they hit a problem that doesn't fit the algorithmic pattern and needs something "tricky" to solve, will they be able to, or will they waste untold time trying to force an unsuitable algorithm to apply?

I tried to allude to that simply with my "+1" example - if they think in terms of "moving terms", then such an option will never occur to them, and that oversight can often be crippling.

Basically - if you're following an algorithm to solve a problem, your skills are extremely limited, and you'll only ever be able to solve problems that fit the algorithm. While if you correctly understand the underlying processes and how to use them, then you don't need an algorithm, you can approach every problem on it's own merits and work out how to solve it. Potentially less efficient for common patterns that fit the algorithm, but far more flexible.