[tkosan]

I am in the process of writing a step-by-step elementary algebra equation solver that solves these problems like a human does. One thing I have learned from this experience is a good way for a person to discover how much they don't know about even the simpler parts of mathematics is to try teaching them to a computer. This leads me to suspect that Meai is correct about most mathematicians not having a complete understanding of the material they teach.

[jimhefferon]

> mathematicians not having a complete understanding

Oh, I'm happy to concede that there is always more to know. But that is a far cry from the poster's assertion that people teaching the class do not have any intuition about what they are teaching.

[oldandtired]

Please do share. Have you looked at the various CAS's that exist to see how these are implemented?

[tkosan]

I am the main developer of the MathPiper CAS (http://mathpiper.org), and I forked it from the Yacas CAS in 2008. MathPiper is a rewriting system that is similar in design to Mathematica. Most CASs don't solve elementary algebra equations the way humans solve them by hand, so these systems can't show the steps they took in a form that is easily understandable by a human.

However, a few CASs do solve elementary algebra equations the way humans do, and the most famous of these is probably PRESS (Prolog Equation Solving System). The story of PRESS's development is an interesting one.

In the 1970s and 1980s, a group of artificial intelligence researchers led by Dr. Alan Bundy at Edinburgh University conducted research on how computers can be "taught" to solve elementary algebra equations the way humans do. The first thing they did was to try and figure out exactly how mathematicians do mathematics. They were surprised to learn that a significant number of the techniques that mathematicians used to perform mathematics were not written down anywhere. They were not in any textbooks, nor were they in any journals or research papers. As the researchers dug deeper, they discovered that the techniques did not have names, and they were not taught explicitly. The researchers concluded that mathematicians were using these techniques unconsciously. (Alan Bundy. The Computer Modelling of Mathematical Reasoning. Academic Press, 1983, p.164.)

Why were these researchers the first people in history to discover this information? I think it’s because computers were the first "students" in history that absolutely refused to learn any mathematics that was not taught explicitly. The researchers then devoted years of effort to discovering and naming the unwritten techniques that mathematicians used to perform mathematics. When they "taught" PRESS these techniques, it was able to perform mathematics similar to the way humans typically would.

The step-by-step solver I am building is based on PRESS, and here is an example of what I have working so far:

https://www.youtube.com/watch?v=cy6bwNBkAK0

[oldandtired]

Thanks. I haven't come across PRESS before in my various investigations, nor have I come across MathPiper either. I'll have a close look at these to see what more I can learn.