I am a private math tutor for high school and college students who are struggling with their math courses. I'm considering pursuing a masters degree in mathematics education to both improve my teaching abilities and deepen my understanding of the mathematical and physical science concepts that I teach on a regular basis. The courses that I usually tutor in are Algebra, Geometry, Pre-Calculus and High School Calculus and Physics. My own mathematics and physics education has gone formally up to second semester college calculus and second semester college physics, and informally into various areas of mathematics but without formal grounding and expansion. Having completed courses up to second semester college calculus, I'm a bit overwhelmed and lost in the various math courses that I could pursue in deepening my understanding of the subjects I teach, mostly because I don't know the order of progression that is needed to get a good grounding beyond the calculus level. I would like to come out of this process with a firm understanding of the following ideas and concepts, and I need to know which branches of mathematics and course names will usually cover this material: Rings, Groups, Fields, Defining Metrics and Spaces, Deeper looks at Symmetries and Transformations and their connection to Operations and Sets ; Continuity, Completeness, and Closure, Vectors, Vector Spaces, Components, and Operations on Vectors, Sets, Constructing and Defining Sets, Operations on Sets, Deeper Understanding of the Relationship between the Real and Complex Number Systems, finding out why the Trigonometric Functions and Complex Numbers have a relationship, Basic Non-Euclidean Geometries and converting the Theorems of Euclidean Spaces to their N-E Counterparts, Functions of Real and Complex Variables, and the usage of the above concepts to explain and model physical systems such as Kinematics, Forces, and Fields. I'd greatly appreciate the assistance of anyone who has an advanced understanding of mathematics at the college and graduate level who can help point me in the right direction. =) Many thanks!