What is algebraic about algebraic effects?

I started a thread last time Algebraic Datatypes were discussed that hit a similar vein. I assert most programmers know the middle school version of "algebra" and think of it in terms of what they can do with the values they are working with.

This is in contrast to doing operations on either the "types" or the "effects" that are happening during code. So, you have to show people what are the equivalent of + and * in types and effects to show them how these can be thought of algebraically.

I still think this would be easier if a type system was made that let you use + and * in defining something. It would give an obvious path to seeing how things relate to algebra. Would almost certainly make some things harder, of course. Maybe it would be a good worksheet asking questions about stuff?

The big underline, though, is getting people to realize what algebra there is is not on the values that your code represents. It is treating something else as the value for the algebra.