Maybe that's fine? I mean I'm sure a lot of people who fix stuff up in their homes know very little of the physics behind their hammers, but they can still use

Maybe that's fine? I mean I'm sure a lot of people who fix stuff up in their homes know very little of the physics behind their hammers, but they can still use them just fine. Maybe a good tool does not need to be understood on a deeper level to be used. If we gave students more exposure to maths as a tool to be used, rather than arcane formulae and symbolic manipulations, they could build an intuition and appreciation for it that allow them to use it even if they cannot perform the mechanical transformations of equations on their own. I'm not saying x+1=7 is a good example of that -- my son, who is not even literate, could answer that in the blink of an eye. But recently I needed to get the implied marginal probability out of a partition of conditional probabilities, i.e. solve for p in something like e = ap + b(1-p). Could I have done it manually? Sure. Did I plug it into a symbolic solver? Absolutely. For me the insight lay in (a) knowing that was the shape of the equation I knew, and (b) that I needed to rearrange it to get p as a dependent variable. Actually going through the motions is not that important, in my mind.