I remember once hearing a (probably apocryphal) story about a university math professor that tried to teach a gradeschool class about algebra by telling them a few simple axioms and definitions and then making deductions. As the story goes, he thought he would be making things as easy as possible by minimizing the number of things the students would have to learn, but that turned out to be a bad idea because the way you get kids to do good on the SAT is by minimizing how much they have to think.
I am curious about what that lecture series would actually look like. I would like to find a presentation of elementary algebra that treats it from an abstract standpoint, but that requires no prior knowledge. Essentially, I am looking for algebra explained in the "professional" style that the above story depicts.
I have no idea where to find it. Textbooks that present the abstract algebra view of elementary algebra tend to assume you already know elementary algebra. University textbooks about elementary algebra written in the 1700s (when elementary algebra was a dominant research topic) come close, but abstract algebra was not around back then. Does such a book or series of notes exist? Can it exist?