In 2008, I made a major life decision to shift my career path from software development to teaching. After leaving my teaching position in 2022, I’ve been actively searching for a new job that combines these two passions as an “instructional technologist” without much success. Part of the problem is the difficulty of communicating how I view my experiences as a teacher and programmer as two sides of the same coin – being a mathematician. As a potential method of resolving this problem, I thought I might be able to construct a “Mathematics Portfolio” that demonstrated how the skills I acquired as an educator are relevant to my qualifications as a software engineer.
There was only one problem with my plan: I’d never seen a “Mathematics Portfolio” before. I’d seen portfolios in other disciplines related to math, including both “Teaching Portfolios” and “Programming Portfolios”, but I’d never seen a portfolio I would specifically identify as being created by a mathematician. It’s common for mathematicians to provide a list of publications in a “Curricula Vitae”, but my working definition of “Portfolio” requires a purposeful act of curation on behalf of its creator. There’s a distinction to be made between a mathematician’s collected works and their self-evaluation of those works. A “Mathematics Portfolio” should necessarily include both the mathematics itself and a reflection on behalf of the mathematician producing it about why it is important to them.
If the word “mathematician” is defined as a “person who does mathematics”, it becomes a logical necessity to define what is meant by the word “mathematics”. This is not a trivial matter to unravel. Our definition of “mathematics” is based on a shared social understanding of what it means to “think mathematically”. Understanding the need to avoid circularities in our definitions of terms like this is one of the core components of mathematical thinking. However, it’s very rare to actually see mathematics as an active process. Usually this process takes place inside the mathematician’s mind and what we see is what they publish.
Perhaps it might be easier to define “mathematics” if we first define “physics”. Human beings coexist is a shared reality that operates by a set of rules we call “physics”. We don’t get to observe those rules directly, but discover them through the scientific process. In order for us to learn how our universe works, we must first develop a model and method for testing it experimentally. We have a shared world we all interact with so our internal models are inherently similar.
In contrast, I would define mathematics as a universe-agnostic model of truth. Whereas “physics” is the study of the real universe that we collectively live in, “mathematics” is the study of all possible universes. This distinction is often described as “a priori” and “a posteriori” forms of knowledge. However, where knowledge of physics is constructed through a separate process called science, the process of discovering mathematical knowledge is itself called mathematics. Our definition of mathematics is inherently circular because there’s no common external model for us to compare against. Truth in mathematics is established by convincing other mathematicians that something couldn’t possibly be false.
To resolve this dilemma, I thought it might be useful to decompose mathematical thinking into a set of characteristic thought processes. While this is a topic of much debate, allow me to propose the following definitions for discussion purposes:
Mathematical thinking is a formalized thought process for arriving at truth characterized by defining terms, stating assumptions, reasoning logically, creating useful abstractions, decomposing problems, and communicating information. A person who routinely engages in mathematical thinking is referred to as a mathematician, and the collective social behavior of mathematicians is referred to as mathematics.
I’ll address each of these components in turn and discuss why I think they are important. The goal here is simply to identify the skills that I would want my “mathematics portfolio” to demonstrate so that I can begin to organize a suitable collection of artifacts. I’ll conclude with a brief reflection about how this definition has impacted my portfolio development.
Defining Terms
I hope that the care taken in defining “mathematics” demonstrated above provides evidence of this skill. Attempting to define mathematics alone is not sufficient to make one a mathematician, but I’d argue that not including a definition of mathematics in a mathematics portfolio would arguably disqualify one as a mathematician. The effectiveness of mathematical thinking depends on a foundation of precise definitions. When I’m “doing math”, I mean what I say and I say what I mean.
Being a teacher has taught me that this type of formal communication is not the primary mode of thinking for most people. It’s far more practical for human beings to “code-switch” in and out of this state. After all, natural communication often involves words with double meanings that we must interpret based on the context. A word like “neighborhood” will have dramatically different meanings depending on whether I’m discussing “my house” or “a point”. Understanding where and when precise definitions are appropriate is a quality that makes for a good mathematician.
Stating Assumptions
One of the reasons I’ve defined mathematical thinking the way I have is that it allows me to distinguish between the thought process of an individual mathematician and the collective social efforts of many. What constitutes a good definition is how well it enables further communication between two parties and coming to an agreement on the definition of terms often includes implicit assumptions about what is true in the shared reality of the communicators. Once two people have agreed on a set of definitions it enables them to ask “what if?” questions using those terms.
Mathematics is built on a foundation of mutually agreed upon hypothetical statements within a given society. The presently accepted de facto standard in mathematics is a specific collection of assumptions colloquially known as “ZFC” – an abbreviation for “Zermelo–Fraenkel set theory with the axiom of Choice”. However, it’s important to note that mathematics as a whole is not limited to this specific set of rules and some interesting things happen when we choose different ones. What’s important for mathematical thinking is that we actively reflect on the assumptions we’re making and why we’re making them.
Reasoning Logically
Once we’ve established a set of well defined terms and agreed on a set of rules by which they are bound, we can start sorting out the true statements from the false ones Given our stated assumptions and definitions, what new information can we deduce from that knowledge? This logical reasoning is arguably one of my favorite parts of math because I tend to view it like a game. Like the classic puzzle game “Pipe Dream”, mathematics is often about building a path from a starting point to the goal with a limited collection of possible moves at your disposal.
In mathematical terms, I tend to associate logical reasoning with the properties attributable to a generative grammar. Much like natural language has rules of grammar that determine what may or may not constitute a sentence, logic has an alphabet and rules of grammar that determine what may or may not constitute a proof. What makes this game so fun, is that you may or may not be able to prove a given statement from within the system! Mathematics can sometimes require you to step out of a system and look at the bigger picture.
Creating Useful Abstractions
One of the most powerful aspects of mathematics lies in the way its tools effectively transfer between seemingly disconnected domains of study. As a teacher, I often heard students often ask “when am I ever going to use this?”. The truth of the matter is “I don’t know” and this is precisely why learning math is so exciting to me. I never know where a mathematical concept will show up, but I also wouldn’t recognize them in their natural environment if I hadn’t played with them in a “toy problem” first.
A good example of this in action is The Matrix. In mathematics, one is generally introduced to the notion through a study of linear systems of equations. However, the applications of matrices extend so far beyond this initial example that they are almost unrecognizable. From modeling rigid transformations in physics to modeling key changes in music, matrices provide a useful model for “connecting things” in general. In order to answer the question “What is the matrix?” one needs to both understand the properties of the mathematical object and connect them with the properties of the situation in which it is applied. Doing so essentially defines a matrix, thus giving us an abstraction for the very process by which we “step out” of a system.
Decomposing Problems
I’d like to believe that one of my strengths as a mathematician is my ability to take a seemingly intractable problem and decompose it into a collection of individually solvable subproblems. As a kid, deciding that I wanted to be a mathematician was accompanied by the simultaneous acknowledgement that I could potentially devote my entire life to working on a particular math problem and still not solve it. Learning to accept that some problems in mathematics would be forever outside my grasp allowed me to focus on learning how to break down these problems into components which I might reasonably be able to solve one at a time. Rather than immediately classifying something as impossible, I reframed the problem probabilistically over time. Given some arbitrary problem in mathematics, what is the chance I will solve it in my lifetime?
Mathematics itself is a version of the “Many-Armed Bandit” problem. Each branch of mathematics is its own little slot-machine full of interesting problems and you have to pay-to-play this game through hours of effort. A mathematician goes around from discipline to discipline looking for that one “jackpot theorem” – a problem that is both important and within reach. Recognizing this situation as a probability problem allows you to start to separate the problem into known strategies of “playing the machine you’ve had the best luck with” and “exploring new machines”. This, in turn, gives you new information to work with and helps you to develop more efficient methods of addressing the overall problem.
Communicating Information
One of the ways my philosophy towards mathematics has been changed by teaching is through recognizing the importance of the communication process in the construction of mathematical knowledge. There’s a distinction to be drawn between “knowing something is true” and “being able to convince people something is true”. Mathematics needs to do both. I think this distinction often gets lost while doing mathematics because the vast majority of the time the person I’m trying to convince of the truth is myself. We all think mathematics is objective because we all want to think we’re objective self-critics. It’s effectively a self-fulfilling prophecy.
The nature of mathematical communication is probably most salient in Analytic Geometry, where there are clearly defined and connected “symbolic” and “graphical” representations. We can correlate equations like “y=ax+b” with the shape given by “a line” as determined by plotting the points which satisfy it. They say a picture is worth a thousand words, so it should be no surprise that a good data visualization can explain a phenomena more effectively than the numbers or narrative alone. Mathematics provides me with a vocabulary to connect the representations of thoughts between “words” and “pictures” inside my head, which is a necessary prerequisite for me to explain them to another person.
Self-Reflection
Having established a set of skills I want to showcase, I’m starting to see the difficulty in using this approach to organizing my work as intended. More often than not, these skills have such considerable overlap that the differences between them are almost invisible. Even this document itself, which began with explicit focus on “Defining Terms”, could equally be viewed as an act of “Decomposing Problems” when looked at in the larger context of my portfolio development. This isn’t necessarily a problem per se, but rather indicates that the terms were chosen exceedingly well.
My original idea was to pick out portfolio artifacts that “show off” each of these skills, but I’m now beginning to wonder if I should focus on selecting artifacts that “demonstrate growth” in each skill instead. In my attempted definition of “mathematician”, the phrase “routinely engages in mathematical thinking” is doing a lot of heavy lifting. People don’t develop routines without repetition, and it’s normal to stumble the first few times when learning something new. In contrast, when we see “published mathematics” most of the failures have been omitted in favor of presenting the final results in the most concise form possible.
I realized that not only have I never seen a “mathematics portfolio”, it was exceptionally rare to see a professional mathematician actually “doing mathematics”. I had a guitar instructor once tell me that when people see you play on stage they don’t see the thousands of hours you spent practicing, they only see the end result of those efforts. Likewise, a mathematician needs to project an air of confidence in their work and showing the mistakes one made along the way might be interpreted as a sign of weakness. However, I don’t think people learn much of anything without making mistakes and mathematics is no exception.
I concluded that in order to create a mathematics portfolio I would need to start by allowing myself space to make mistakes. It’s for this reason that I resolved to publicly work my way through a text on category theory. This may seem counter-intuitive for a portfolio, but I think that categorizing my works by the areas where I had difficulties will more effectively communicate how much I’ve grown in each of these areas. The first step to making a successful portfolio is for me to make an unsuccessful one and learn where it went wrong. It’s so logical it just might work!