Undergraduate Math Student Worried About Graduate School

I warn that this is a long read and it has been a culmination of the past year or two of thoughts and worries.

Background

• 
  

  I am currently in my fourth year of undergraduate school at a small public university (liberal arts). I transferred in during my sophomore year from community college. I plan on spending another year here in order to graduate with the degrees I want. I came in as a biology major, taking mathematics courses for fun. I decided if I could take linear algebra, multivariable calculus, and proof/discrete math, and do well while taking my regular biology and chemistry lecture and labs, I would pursue a double major in mathematics.

  
  

  I ended up getting an A in each math course and was hooked. Unfortunately, due to losing credits (unknowingly) I had to drop my double major into a major in mathematics with a minor in chemistry and a minor in biology, in order to graduate in less than five years and still not let the biology and organic chemistry courses to go to waste.

  
  

That being said, I've taken the following classes:

• Pre Calc - Calc 1 - Calc 2 - Calc 3 (multivariable)


• Proof/discrete math


• Linear Algebra


• Real Analysis


• ODE


• Advanced ODE - (ordinary not partial)


• Abstract Algebra


• Number Theory


• Special Topics - Excursion in Classical Analysis (Special class offered by a professor who wrote the book, etc)


• Math Methods Operation Research


• Independent Learning Studies (fractals, Cantor set, some real analysis, measure theory, etc)

Currently in:

• Complex Variables


• Real Analysis - 2


• Mathematical Modeling

Planning on taking:

• Abstract Algebra - 2


• Independent learning study on primality with a professor


• Advanced ODE - Partial (if time allows)

GPA

• Math GPA 4.0


• Overall 3.92

I am not required to take many of these courses to graduate, but I have taken them because I enjoy math and to broaden what I know. I have pass failed only 2 classes (mathematical modeling and operations research) because I did not have enough time to devote to get the grade I wanted, due to pledging into a fraternity and being ill. I attained a pass of course in these two classes. Plus, these are more applied courses which I did not care as much as the purer courses.

Ultimate Goal

• My plan is to gain some undergraduate experience over this summer and the next year, before applying to graduate schools. As of right now, my plan is to go into number theory, mainly focusing on pure mathematics and primality.

The Questions

• My main problem is that over the years, I have taken more math courses than the average undergraduate would at my university and plan to do more. Although I've maintained an A average in all my math courses, I have managed to forget a lot of information I have learned. This mainly has occurred due to the fact that I've focused so hard to maintain this GPA that I often rather than not work very hard to get that A and then the knowledge is never used again for the most part. I am very, very afraid that when I attend graduate school, I will not know as much as I should. I am afraid that I will be subpar, much far behind other applicants who have more mathematics memorized. Granted I have decent grades, it almost feels as if I don't deserve it; due mainly to the fact that I can not recall a lot of the information I knew so well at one point or the other. Surely I am not the only one who has gone through this? I want to pursue a career in mathematics as I love it. I am just worried.


• Aside from one very gifted student in my small class of math majors, I do not know of any other students with such grades currently in the mathematics department here. Most are females (no offense) who are pursuing teaching masters and not worried about the pure math as much as I am. They do not care much for what builds math, which is fine, but it is not as important for them to recall the information I am worried about. I feel as if I am not truly gifted like most applicants who want to go to graduate school and have the mind that can recall information from years back. Many of my classmates believe I am intelligent or gifted of some sort, but truly I do not feel like it. I feel like I should know everything if I attained an A in every course. Of course this is not reasonable but I am very worried that I won't recall many things that I should have down. The problem this is late in the game and I'm not sure what to do. Having a course load so difficult and challenging for the past few years, with taking organic chemistry/biology lectures and labs, with 2-3 math courses each semester, I barely have time to sit there and recall information from earlier courses to jog my memory.


• I don't believe I have the mind like some select few who just know how to do things or apply things. Sure I can work on something for a few hours and eventually get it but it does not just spark into my head to do certain things.

Is there anything y'all could recommend me? Maybe help put my fears at rest? Maybe share some experience that is relevant to my situation? I honestly do not believe that my GPA is a good representation of what I know, but more how hard I can work. I know the simple answer is to just take the time and review things, learn it again, and use it so it isn't forgotten; but surely I can not do this for all the course I've taken.

I truly appreciate anyone who takes the time to read this very long post, and I apologize if this is in the wrong section or too personal or broad.