You are right that relevant memories and analogous experiences come up and are used as building blocks in our evaluation/exploration of a problem, but it doesn't seem to me an algorithmic procedure at all.

You can trace out your journey in solving a problem, in retrospect, but could you encode it into a "solving-a-problem" algorithm?

I think you could extract some kind of generic template for problem solving: you come up with an idea, you evaluate whether it is the solution, you adjust the idea if not.

But this is a template, not an algorithm. Coming up with an idea has to do with filtering the old and new memories/perceptions that come to mind: does this one seem right? or this one? Evaluating whether it is right is also an active process of asking questions. It involves memory (of the problem to be solved), attention (to the potential solution), judgement (do they fit together?), etc.

None of these are a predetermined sequence of steps you apply mechanically, such as the child "solving an integral" above.*

*Of course, the child is problem-solving in the sense that it's trying its best to follow your instructions. "Did I cut it right?" "Are the scales even?" But this is not the problem of "solving an integral" to which it is completely oblivious to.