Author Archives: Trevor J Richards

About Trevor J Richards

I am an assistant professor in mathematics at Monmouth College in Monmouth, Illinois. I have diverse mathematical teaching and research interests. As a teacher of mathematics, I focus on drawing my students into the development of the mathematical theory they are learning, especially in the calculus sequence. Some of my favorite recent classes have been: Introduction to Proofs, Calculus II, and Actuarial Science. As a mathematical researcher, my interests primarily lie in the realms of real and complex analysis. In both these areas, I study the geometric, topological, and measure theoretic properties of the level sets of functions. I also greatly enjoy discrete math, especially logic and graph theory. In the fall of 2018, I will be teaching a course on the use of logic in private thought and public discourse. I have also recently developed an interest in financial mathematics. In the spring of 2018, I taught a prep course for the first actuary exam (SOA Exam P). I plan to make support of students seeking a career in actuary a significant part of my work going forward.


To share in the delight and the intellectual experience of mathematics — to fly where before we walked — that is the goal of a mathematical education.          William Thurston

Welcome to My Website

I am an assistant professor of mathematics at Monmouth College in Monmouth, Illinois.  See my curriculum vitae linked below for details about past education and experience with mathematics.


My Approach to Mathematics

The earliest experience with mathematics that I can remember is answering word problems posed by my father as a small child.  While we waited for the bus, he would ask me questions about trains leaving from a city at different times and at different speeds, and when would one overtake the other.  These problems were riddles, knots to untie, mysteries to solve.  This experience of math as a sort of detective game has shaped my approach to mathematics ever since.  Throughout my childhood and adolescence, and much more so now as a teacher and researcher, I have viewed mathematics in this way.  The metaphor of mathematics as a knot to untie is particularly apt.  When I approach a research problem, it is very much in the manner that I approach a complicated knot.  I first try to simplify it in any way I can, even if it is not clear how doing so will finally untie the knot.  I pull a bit, and try to loosen at one point.  When I am stumped at that point, I turn the knot over, and try to loosen at another point.  Even so with a mathematics problem.  If I am trying to establish some implication, say “Thing One implies Thing Two”, then the first thing to do is suppose that Thing One is true, and see what follows, even if it is not clear how what follows will lead to Thing Two.  Can I simplify at all?  Can I rephrase Thing One in other terms?  If I am stumped, then I turn the problem over, and try working on Thing Two.  Can I find some other, intermediate “Thing Three” which implies Thing Two?  In this way, working from both ends, I try to “meet in the middle” to form finally an unbroken chain of implication beginning with Thing One, and ending with Thing Two.

My early experience of mathematical problems as exploratory exercises also shapes my approach to teaching mathematics.  One of my first goals as a teacher of mathematics is to invite my students into the development of the mathematical theory they are learning.  If my students walk away from my class thinking that what mathematics is is a list of formulas which can be applied mechanistically to certain problems to “magically” obtain an answer, I have failed in this goal.  If on the other hand, my students walk away thinking that “mathematics is for them”, that mathematics is a way of exploring the world that is accessible to them and to which they can contribute in a meaningful way, then I have succeeded in this goal.  Mathematics is a house with many rooms, ever growing, and in my courses we explore together a small region of that house.