Are Mathematical Understanding and Algorithmic Performance Related?

An algorithm is first defined by an example of making pancakes and then through discussion of how computers operate. The understanding that human beings bring to a task is contrasted with this algorithmic processing. In the second section, the question of understanding is related to learning algorithmic performance, with counting used as the vehicle. It is noted that perhaps subskills must be mastered algorithmically before understanding can develop; perhaps the separation between algorithmic performance and understanding is impossible at any stage of the learning. In the third section, two studies dealing with the relation between algorithmic performance and understanding are described. Both fail to demonstrate that understanding improves algorithmic performance. The fourth section addresses the question of teaching for understanding and leaving algorithms to computers. Deciding what kind of division of labor we wish to establish between ourselves and our computers is the focal point. (MNS)