Algebras we love
I really dislike the standard definition of a group - that it's a set S together with a binary operation * such that a bunch of properties hold. The definition doesn't build intuition, and doesn't motivate the introduction of the concept of "group".For me, a group is the set of isomorphisms of a graph. If you expand the definition of "graph" a little to include continuous spaces, that is sufficient to define all groups. And yet, this nice, intuitive definition of a group might show up at the end of a course in group theory - if you're lucky.It really is a shame how much intuition is stripped from mathematics teaching in the name of formalism.[0] https://en.wikipedia.org/wiki/Frucht%27s_theorem