In the first main section of this thesis, I investigate superirreducible polynomials over fields of positive characteristic and also over [set of rational numbers] and [set of integers]. An n-superirreducible polynomial f(x) is an irreducible polynomial that remains irreducible under substitutions f(g(x)) for g of degree at most n. I find asymptotics for the number of 2-superirreducible polynomials over finite fields. Over the integers, I give examples of both families of superirreducible polynomials and families of irreducible polynomials which have an obstruction to superirreducibility. The writing and results on finite fields in this section have come from a collaboration with Jonathan Bober, Dan Fretwell, Gene Kopp and Trevor Wooley. The results over [set of integers] and [set of rational numbers] are my own independent work. In the second section I determine the asymptotic growth of certain arithmetic functions A(n), B(n) and C(n), related to digit sum expansions. I consider these functions as sums over primes p up to n. I obtain unconditional results as well as results with better error terms conditional on the Riemann Hypothesis. The results over primes have come from collaboration with Jeff Lagarias. I also independently solved the analogous problem of summing over all positive integers b [less than or equivalent to] n. Finally in the third section, I discuss mathematical education via the lens of interviews and interactions. I consider my role as a teacher through multiple real-life anecdotes and what those stories have taught me. My interviews were conducted with young mathematicians from Bronx, NY that I got the opportunity to talk to as a result of my employment with Bridge to Enter Advanced Mathematics during the summer of 2019. The anecdotes I give are from working with teenaged students from a variety of different cultural, socio-economical and mathematical backgrounds. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com.bibliotheek.ehb.be/en-US/products/dissertations/individuals.shtml.]