I am teaching a class in the Spring for abstract algebra for high school teachers. The goals of this class is to give future high school teachers a deeper understanding of many of the patterns and concepts in geometry, introductory algebra, and trigonometry. What concepts would you consider necessary for students to learn? Also, what topics in high school/middle school math could be explained using abstract algebra? I have a general course outline in place listed below.
First couple of weeks- Review sets, functions, relations, binary operations, and common proof techniques
Next 6-8 weeks - Focus on groups. We will talk about the definition of groups, subgroups, cyclic groups, Cayley diagrams, multiplication tables, Direct Products, quotient groups, LaGrange's Theorem, cosets isomorphisms, homohorphisms, and plenty of examples of groups in geometry, trig, and algebra (such as rotations of a circle, plane isometries, symmetries of regular polygons, symmetry groups of the conics, modular arithmetic, even and odd functions, ect.) These concepts are not listed in the order they will be presented.
The last 4-6 weeks - Rings and fields. We will talk about the definitions of rings, fields, subrings, modules, ideals, Fermat's little theorem, and focus a fair bit on rings of polynomials. We will also talk about the differences between various number systems and compare them to less common number systems (naturals, integers, rationals, reals, complex, dual, and split numbers, and Gaussian integers). If we have time, we will wrap up with splitting fields and field extensions.
Are there any important concepts that are missing?