Math Tutor Looking to Deepen Understanding: A Journey Through Math & Physics

[ccb138]

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!

 

[Number Nine]

You can get an introduction to almost all of those things by taking courses in analysis, abstract algebra, and linear algebra.

 

[ccb138]

Many thanks, Number Nine! =) When you say almost all, do you happen to know which ones won't fit in those courses that I would need to supplement from outside?

 

[Number Nine]

Many thanks, Number Nine! =) When you say almost all, do you happen to know which ones won't fit in those courses that I would need to supplement from outside?

Non-euclidian geometry and complex variables are usually courses in their own right. You'll want some basic familiarity in analysis before studying complex variables, which is, itself, usually a prerequisite for non-euclidean geometry.

 

[micromass]

I've listed all the topics in an order which should be natural. I added the prereqs and listed the topics it would teach you.

Intro to proofs
Sets, Constructing and Defining Sets, Operations on Sets,

Linear Algebra (be acquainted with proofs before this):
Vectors, Vector Spaces, Components, and Operations on Vectors

Abstract Algebra (be acquainted to proofs before this):
Rings, Groups, Fields, Deeper looks at Symmetries and Transformations and their connection to Operations and Sets

Real Analysis (be very acquainted with proofs and calculus before this):
Defining Metrics and Spaces, Continuity, Completeness, and Closure, Functions of Real and Complex Variables,

Complex Analysis (requires real analysis):
Deeper Understanding of the Relationship between the Real and Complex Number Systems, finding out why the Trigonometric Functions and Complex Numbers have a relationship, Functions of Real and Complex Variables

Physics (requires multivariable calculus):
the usage of the above concepts to explain and model physical systems such as Kinematics, Forces, and Fields.

The only thing which does not fit any category neatly is:
Basic Non-Euclidean Geometries and converting the Theorems of Euclidean Spaces to their N-E Counterparts

Perhaps you can do this in differential geometry, but that depends on the class. However, there are many books which cover this (for example "the four pillars of geometry" by Stillwell) if you're ok with self-studying