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The word you're missing is "arithmetic". For most people "math(s)" and "arithmetic" are the same thing, when the latter is only the commonly encountered part of the former.
And for many of those people, algorithms are the only way they can use arithmetic to reach a goal because the intuition isn't there otherwise.
I say this in full knowledge that even though I'm pretty good at arithmetic, much of the intuition I have now took me years after leaving school, and sometimes for more advanced things it's still not there.
For example, on a good day I can complete the square, and I understand the geometric intuition, but most of the time I'm just going to plug and chug with the quadratic formula.