Yeah, that's how we were taught it anyway. Nodes are added to a priority queue as they're discovered while being popped off the queue in order. From the title I was expecting something like going from naive Dijkstra via A* to an optimized jump point search or something.

That said, the title aside, it's still a worthwhile post. It's sometimes easy to accidentally nuke your algorithm's performance with an innocent-looking function call or operation that's secretly turning it quadratic or worse.