Over posting, but can't let it go. Here is a link to a very brief and compact description of topics in algebra, or different algebras.
http://www.csee.umbc.edu/help/theory/group_def.shtml
The point is these are the some of the topics and their relationship in a heirarchy of concepts that most school math teachers are never exposed to. For sets of course, but also groups and lattice have great little geometry-like toys to play with that open up the more abstract spatial ideas. On the other hand rings and ideals are somewhat familiar to anybody who has played around with natural numbers, thinking about multiples as cycles and so forth. These are structural features to numbers and/or spaces.
Speaking of which, just exactly why is there a profound and deep duality in the sense of correspondence between number and space? That's the kind of thing that I like dwell on...
CG .\