An algorithm that does something can in principle be ran by someone who doesn't know what the algorithm does. You could have a kid calculate an integral by giving it a sequence of directions whose purpose it doesn't understand (e.g. cut out some cardboard that matches the shape, put it on one side of the scale, place enough unit cardboard pieces on the other side until they are even, then tell me how many pieces you put).
Reasoning has more to do with how the problem came about. A person had to come against a certain problem, figure out a way in which they can solve it, then apply the (perhaps algorithmic) solution. The algorithmic part is only an artifact.
