Most math instructional software that I have seen tends to fall in two categories, either they want you to follow procedure and check your answers (occasionally checking the steps along the way) or they play some games and try to allow you to discover math concepts along the way.
This coincides with the two dominate teaching philosophies, teacher led examples or some form of constructivist/discovery/inquiry teaching. (yes I hate lumping them together but they aren’t the subject of the post)
There is a third way that most good teachers have used naturally, but few have considered it a separate teaching method. That is teaching from misconceptions. When your math teacher asked you to show your work, they wanted the opportunity to follow your thinking and then either explain in writing or during a conference the exact part of the problem where you made your mistake. That is, they were looking for your misconception.
It turns out that this is a very powerful method of teaching, or more correctly reteaching a concept. Now to the point of this post. I haven’t seen a math software that teaches this way. I’ve seen a science dissertation on teaching through misconceptions (this explains it quickly), and now I’ve seen an English theory on teaching through misconceptions. When are we going to see a math program that teaches from misconceptions?