Some Thoughts on the Upcoming Recounts in Three Battleground States

Author’s Note: I took a momentary break from posting about math and education issues to discuss some threats facing the political system in the United States. I work hard to keep this blog apolitical, but no matter which party or ideology you subscribe to, the problems discussed below should concern you.


The upcoming recount in Wisconsin along with potential recounts in Michigan and Pennsylvania may or may not change anything, and while plenty of people on all sides are probably just ready to move past this horrible election, this is something that needs to be done. When one candidate–any candidate–wins the popular vote by a decisive margin of over two million votes and growing but still manages to lose the election, the situation is worth investigating–especially when exit polls in those battleground states suggest many people may have voted differently than what the results showed.

Of course, even if the results hold, the real issue here lies with the electoral college. Now, I know the electoral college was founded to prevent one small, population-heavy area choosing a president that all Americans, many in different areas with different needs, would then be forced to live with. (You probably have heard the horror of how California would choose every president if the electoral college were abolished.) The electoral college was supposed to give all states a proportional say in choosing the nation’s leader. But that hasn’t happened. In fact, we have ended up with exactly what we were trying to avoid–a few states picking the president. Our nation is so polarized that only a few states are ever toss-ups, and those so-called “battleground states” end up deciding every election.

For example, let’s say you are a Vermont voter like me. Hillary Clinton won Vermont by 61%. Of course, she only needed 51% to take all of Vermont’s electoral votes, so for 10% of voters, their vote for Hillary didn’t really make a difference. Of course, if you voted for Donald Trump or anyone else in Vermont, it was even worse; your vote essentially got ignored because Clinton took all the electoral votes. This same thing is happening in every state.

Of course, this is NOT an argument for not voting. If fewer people voted the problem would be even worse. But the fact that some people’s votes are not counting as much as those of other people in other places should concern everyone who loves democracy. So regardless of what happens in the recount, we should seize this opportunity to investigate how we can improve our electoral system to insure that every voter gets an equal say in choosing the nation’s leader (this also means examining our voting rights laws, which have taken a hit in recent years, leading to questionable voter purges). It should be obvious now that the very future of our society depends on solving these problems.


The Winter of Our Discontent–and Our Sweep Map Inversion

It’s been quite a winter. I was finally able to resume the math research that I had been neglecting over the semester’s frenetic final month. But even before I had become swamped with the work of writing tests, grading homework, and quelling student anxiety, my research progress had been slowing. Bogged down in endless iterations as I tested an algorithm and continuously revising computer code for that algorithm, I hadn’t made any new discoveries in quite some time. So this winter I decided to give the algorithm a rest and start playing-yes, just playing–with some different cases. Almost immediately, I stumbled upon an example that seemed to violate the conjectured bijectivity of the functions (called sweep maps) that I work with. It looked to good to be true–and it was. A silly misordering on my part had led to this erroneous “discovery,” but my mistake ultimately led me to a deeper understanding of the sweep map as well as to possible inversion strategies for the new cases I was looking at. Moreover, these insights invigorated my psyche, giving me the energy boost I needed so I could research on.

When I wasn’t researching, I was thinking–not about math, but about life. I have never been one for sappy end-of-year reflections, but last fall I began reflecting on just how I got to where I stand today. This was prompted by some conversations with an old friend, one whom I had met as a freshman undergrad and whom I had not spoken to in a long time. As I talked to him, the past six years flooded back into my mind, and I recalled a former self who was barely recognizable.

Searching for the reason for this change, I went digging through the paraphernalia of my undergraduate days. As I flipped through old issues of the campus paper for which I used to write, I was confronted with names belonging to people who had once been close friends but who now seemed like strangers, with words attributed to me but that I hardly remembered writing. Then I began digging through envelopes filled with everything from award certificates to old homework to mundane correspondence. I could not escape the realization that I had once been energetic and successful, blessed with opportunity and with a circle of supportive friends. I had been happy.

And then came grad school. It began jubilantly enough. I had been once again blessed with a great opportunity, this time the chance to earn a master’s degree in math, all funded by a teaching assistantship. I was eager to indulge my passion for math and prepare for my dream career. But my personal life was unraveling fast, and I was struggling to balance this fact with my new world of grad school, a world that I was surprised to find that I felt uneasy in. My life, my background, even my personality did not seem to fit that of the stereotypical grad student at all. I was reluctant to engage with the other students–and even more reluctant to seek help when coursework became challenging. I was ashamed–ashamed of what I was going through, ashamed to no longer be a straight-A student, ashamed to be different. I was isolated–isolated from the other students, from the faculty, and from my old friends, for I could not let them see how I had changed. Where I had once joyfully looked forward to the future, I was now operating in survival mode, just trying to make it through the next assignment, just trying to make it through whatever challenge was waiting at home. That is no recipe for success. Success requires enthusiasm, engagement, networking.  I knew that; I had been a master of it as an undergrad. But I was far too emotionally exhausted to do it this time. And so while I completed my master’s degree last May, it was not with the level of excellence I was used to. It was small wonder that I put together a half-hearted Ph.D. application proposing to do research in a field in which I had no background, even smaller wonder that that application was rejected.

So where to now? Well, first I must address some people, starting with my friends from undergrad days. I am sorry that I have been too caught up in my own pain these past few years to be the friend you remember, the friend you deserve. I would like to thank all of you for always standing by me, even if our relationships have been relegated to cyberspace recently. For those of you to whom I have not spoken in some time, I hope we can rekindle our relationships, for none of you have ever left my heart. For those of you who knew me only as a graduate student and saw me only as I was angry and overwhelmed, I am sorry that you never got to know the real me, and I want you to know that I was never angry with you personally. I hope that we can begin to develop the relationships that I know we can have.

As I write this, many new things are beginning. A new semester has started at Champlain College, and I am getting to know a fresh crop of calculus students. I can’t wait to see how they grow both as students and as people over the semester. Meanwhile, I nervously await admissions decisions from Ph.D. programs, hoping that at least one will say yes so I can get back on the path to my dreams. I can now say that I am ready to complete a Ph.D. My research has allowed me to prove to myself that I have something to contribute to the cannon of mathematical knowledge. I also now know that I-my entire person, with all my experiences–belong in grad school. My life has calmed, my personal growth continues, and my pain and shame is melting away. The winter of my discontent, to paraphrase Shakespeare, is thawing. My dream of becoming a math professor is regaining its strength, and despite the roadblocks it has faced, it will not be stopped. Today, I get back on the road to that dream.



Where Does Math Come From?

“Where does math come from?” is likely a far less exciting question than “Where do babies come from?”, but it is one that has been on my mind a lot lately–and for good reasons. As a mathematical researcher, it’s my job to create new math–though whether math is actually created or rather discovered is a question with no easy answer.  But as I contemplate my future (hopefully) in a PhD program and reflect on a semester of college teaching, this question becomes even more crucial. I will try to shed some light on it here.

I began thinking seriously about this question a month or so back when working with one of my calculus 1 students. Though a student of video game programming, she was eager to learn why certain math concepts worked the way they did; after all, she was becoming aware that calculus played an important role in computer graphics. Soon we were discussing not only the calculus 1 class material but also concepts from vector calculus and 3D mathematics. It was quite refreshing to have a student so eager not only to understand how to use the concepts but moreover to know where they came from and why they worked–especially a student who was not a pure mathematician. It forced me to think more heavily about how much of our lives hinges on pure mathematics: after all, pure math is the place where math is “born,” where it is brought into the world, tested, proved, and turned into useful tools that can then be applied to engineering and computer science and medicine and so many other of the things that our modern world depends on. And that’s when I decided that this importance could not be kept silent.

I was in the midst of wrangling with grad school applications, applying to several Ph.D. programs around the northeast while trying to figure out what to do about my application to UVM’s Ph.D. program, where pursuing research in pure math seems like a long shot. So I decided to write a letter to UVM’s president, first to thank the math department for the transformative experience I had in the master’s program and then to argue for more support for the pure math side of said department. I made the argument that UVM’s stellar engineering and complex systems program were all dependent on the math created by the pure math program. After a month, a figured I would never hear back, but last week I received a letter from the president saying he was glad to hear I had had such a positive experience in the master’s program and that he was forwarding my concerns to both the dean of the college of engineering and mathematical sciences and the dean of the graduate college. So while I still have no idea what will happen going forward with my Ph.D, I do hope that pure math will get a little more love, and maybe those who come after me will get to see how truly important this subject is.

Adventures in Coding: A Research Love Story

As many of you know, computer programming has sometimes been the bane of my existence. Since coding is necessary to make any significant contributions to modern mathematics, I have learned to somewhat get along with it. I studied FORTRAN as an undergrad, at first simply as a program requirement, but I learned it well enough to write a program that measures how close a polygon is to being convex, which was an important part of an undergrad research experience involving gerrymandering. As a grad student, I was forced to learn MATLAB, again first as part of a class. That class didn’t go so well, but I later got a grasp on MATLAB and found it to be quite useful. Yet when I took on some combinatorics research this summer, my professor suggested learning Sage, which is based in Python. I had heard that Python was the simplest and most intuitive of the computer languages, but given my struggles with coding, I wasn’t sure how this would go.

See, my main issue with programming is the very different ways in which my brain and the computer “brain” approach solving a math problem. I like to compare programming a computer to conversing with a two-year-old. The computer has absolutely no intuition. It must be told EVERYTHING. Things that the human mind takes for granted and doesn’t even consciously consider when solving a problem must be spelled out in painful detail for the computer. And then of course, there are the inevitable errors, often as frustratingly simple–and even more frustratingly hard to find–as a missed colon. Of course, the computer never says “you missed a colon in blah blah line”; it gives some cryptic error message. Think of this as the two-year-old having a temper tantrum.

So I got a book and found some websites and began my adventures in Python coding: loops, if-else statements, print commands, and other simple fare. And it was going well. Python did seem to be a much easier language to understand than FORTRAN or MATLAB. Still, I was cautious, waiting for the inevitable problems to arise. And sure enough they did as soon as I tried to make some of my programs interactive. Sage quickly freaked out with cryptic error messages as soon as it encountered my “input” command. I could not understand this, since I was copying code directly from my book. I asked the masses on Twitter for help and got some, but I still had issues. I consulted various websites and even asked one of my students who knows Python for advice. And yet, it was only through some fooling around and following my nose that I finally stumbled on the proper command (Sage prefers “raw_input” rather than “input”). And suddenly my program was interacting flawlessly. And the feeling of joy was indescribable.

Suddenly, coding didn’t seem so scary anymore. I realized in that one instance that, while errors were inescapable, they didn’t mean I was incapable of coding; after all, I had just figured out how to code something entirely on my own. And the fact that I was doing it all in the context of a project I was very excited about made it that much better. And finally, yesterday I also saw a preprint of a paper based on that research I did as an undergrad, and I was reminded that I have indeed successfully coded important programs in the past and that I can do it again. I guess all the code stars were aligned yesterday.  Hopefully, things continue this way. With my new attitude, I feel that they will.

I Am Not Alone

As I continue to reflect on the things I did wrong or could have improved on as master’s student in mathematics, I realize that I was fearing failure and doing things that I felt would keep me from failing but would not necessarily help me learn the way I needed to. Today, I came across an article that made me realize that I am not alone in my experience. In turns out that fear of failure, especially in academics, is a regular American epidemic. I hope you will all read this article so you can recognize–and combat–fear of failure when you see it in yourself.

On Becoming a Mathematician

Today I have some very personal things to talk about. As I mentioned in my last post, I had some things I wanted to talk about regarding what I have been up to in the time I since have taken a break from posting. Today, I am ready to talk about those things.

Back in May, I completed my master’s degree in mathematics at the University of Vermont. YAY! However, for a while it did not seem like much to celebrate at all. I had applied to do my PhD in math here at UVM this past winter, and around the beginning of March I learned that that application was being rejected. I had proposed to study complex systems, an area of applied math in which I had little background. It was not my first choice of work–I had been studying pure math in general and abstract algebra in particular throughout my master’s–but I had been spooked by a reported dearth of funding and advisers in pure math, so, desperate to complete my studies at UVM, I tried to convince myself that I was truly passionate about complex systems. The grad committee wasn’t buying it, though, and that was, as they told me, why my application was rejected.

It was devastating to say the least, not because I was so eager to study complex systems for the next three or more years, but because I had planned my entire life around getting a PhD at UVM. I had a household to support and now was scrambling to find a decent job, and, more importantly, figure out what I was going to do next education-wise, since getting a PhD was still my ultimate goal. In the months since, I have taken care of some practical things: I found a good-paying part-time summer job working as a data analyst at UVM’s Center for Rural Studies, and at the end of this month I will be starting an adjunct teaching job at Champlain College, a small private college right here in Burlington, Vermont. Yet most of all, I have been blessed to have some experiences this summer that have brought my future into sharper focus.

You see, as I believe I have stated several times on this blog, I was the first in my family to go to college. Determined to make my dreams a reality, I poured myself into my studies. I earned a 2200 SAT score, which was no small aid in helping me secure the blessing of a free undergraduate education. As an undergrad, I earned straight A’s nearly every semester and graduated magna cum laude with a double major. I say these things not to brag–certainly many other people have achieved much more–but rather to show that I was accustomed to reaping the benefits of hard work. I began grad school at UVM with the same zeal, but I quickly discovered it to be far more challenging than I had imagined. I certainly wasn’t failing, but I definitely was not having the kind of success I was used to. Afraid to look like I didn’t know what I was doing or did not belong, I withdrew when I should have sought help. I also began to dabble in many different areas of math, hoping to find the ease of success I had always known. Sometimes I did find some success, but other times I found myself knee-deep in material in which I had no background–like complex systems.  To make matters worse, I imagined that the other grad students, who all seemed to me to be from vastly different backgrounds than I was, were not having these difficulties. This of course was not necessarily true, but that just goes to show the power of perception. I felt my passion waning.

After graduation, one of the math professors here at UVM gave me an opportunity to work with him on some of his research into the properties of sweep maps, a type of sorting function in combinatorics. I fell in love with the project. As soon as I began studying combinatorics, I instantly connected with it on a deep and intuitive level. The central question of combinatorics–to find how many ways something can be done–seems very concrete and natural, and yet it opens the door to limitless exploration. I quickly became immersed in my research, thinking about it day and night and delighting in the hours I was able to spend with it each day. It felt like working on a giant puzzle, and I gave myself the freedom to play and explore. As I followed my intuition, some of my ideas did not lead to results, but they did lead to new ideas. Soon I was getting results–and asking new questions. And that is when I realized what I had been doing wrong as a master’s student.

I realized that my lack of success–and my lack of enthusiasm–were due to the fact that I was focusing on the wrong things. I had been fixating on getting A’s and “doing it right”–after all, those were the things that had enabled me to get an education in the first place. I was worried about being self-sufficient and not looking incompetent. But all those things are the exact opposite of what being a mathematician is all about. Mathematics is frequently less about getting the “right” answer and more about asking the right questions. After all, you can’t possibly reach a valid conclusion if you don’t start from a valid premise. The work of expanding humankind’s knowledge of mathematics is messy, full of false starts and dead ends that–hopefully, eventually–lead you to the right questions and then the right answers. Collaboration is a must. Once I realized all this, I felt free. And more importantly, I rediscovered the passion that had first led me to mathematics.

Reinvigorated, I am now reapplying to UVM’s PhD program to start in the fall of 2016. I am also applying to several other schools in the Northeastern U.S. I hope to remain at UVM and continue to work on the project I have fallen in love with, but regardless of where I end up, I know that I now have the skills and attitude to successfully complete a PhD program. Most of all, I now have the perspective needed for a successful mathematics career. And I can’t wait to get started.

On Being a First-Gen College Student

Wow, it’s been a long time since I last posted (more on what I have been up to later). I recently found this article on The New York Times‘ website, and I instantly gravitated to it. I was the first in my family to go to college, and I know that many people believe that, if you get the grades and financial aid to pursue higher ed, then your worries are over. I myself believed this for a long time. And yet, the truth is that academia is permeated by socio-economic inequality and–let’s be honest–a culture of privilege that can often make the first gen student feel very out of place. I definitely felt out of place in grad school until I learned to embrace my background and use it as a strength instead of hiding from it. I encourage all students to read the article above. It will change how you look at your campus culture.

Musings on Algebra

What is algebra? To most of us, it’s that x and y stuff that we suffered through in high school. To the person who studies algebra on the college or grad school level–people like me–algebra is a much more abstract topic, a study of sets endowed with certain operations, entities that come with names like “group,” “ring,” and “field.” And to the most brilliant mathematicians, according to one of my professors, algebra means a set of overarching principles that govern the universe. So algebra obviously has different meanings to different people. And yet, the college and universal algebras would be nothing, says my professor above, if not for their foundation in that familiar old x and y high school algebra. I must say I agree with him whole-heartedly. I can’t count the number of times I have gotten a homework or test problem wrong not because I misunderstood the concept, but because I made some “silly” algebraic error. I’m sure other mathematicians can share this frustration; sometimes one becomes so focused on the “hard” concept that they carelessly mess up the “easy” algebra part. I often see this with my calculus students as well. Sometimes the problem is the carelessness mentioned above, but other times the problem is that these students never got comfortable with basic algebra to begin with. They may intuitively understand the concepts of calculus, but they don’t have the algebraic “toolkit” to solve problems. This hinders their advancement in mathematics.

Perhaps the best description of algebra I ever heard was from an Air Force Veteran I had the pleasure of tutoring in the subject two summers ago: “Algebra is just finding an unknown quantity.” This idea, which underlies all of mathematics, is what we as math teachers must impart to our students if we want them to develop the skills and the confidence needed for success. I like to think I am doing my part. Hopefully others will join me, and together we can improve math education.


Overcoming Math Anxiety

Non-math folks think that those in the math biz never suffer from math anxiety. After all, since we’ve voluntarily decided to spend out lives studying the subject, we must love it unconditionally and have no fear, right? Wrong. I have a bachelor’s degree in math, am currently working a master’s, and hope to soon be working on a PhD, but one branch of mathematics has struck terror in my heart for nearly five years: differential equations.

The horror began in my first semester of college. I had taken Calc 1 during my senior year of high school and Calc 2 the summer before college. Calc 3 was the natural next course for me to take, but it happened to be offered at exactly the same day and time as a freshman seminar that I was required to take. Since this was a small school, there was only one section of Calc 3, and all sections of the seminar met at the same time, so I was in a bind. I needed another math class, so my adviser suggested Intro to Differential Equations, which he was teaching. The prerequisite was only Calc 2, but when I entered the class, I realized a lot more knowledge was expected. Without Calc 3 and–especially–Linear Algebra, I was lost. The material quickly surpassed my understanding, and I struggled to keep up. I barely passed with a C, and I was relieved just to have survived. Yet the experience was so traumatic that I changed advisers and never took my school’s second, more advanced course in differential equations. It was the only math class my college offered that I did not take.

Now fast forward nearly five years. This coming fall, I am taking a course in Fourier analysis, a subject that has significant overlap with “diffy” and requires some diffy knowledge. So reluctantly I pulled out my old diffy textbook, knowing I would need to finally get a handle on that subject if I hoped to have any success with Fourier. And what did I find as I began to do some problems? Not only did I finally understand the material, I actually enjoyed doing it. I am happy to say that I am finally beginning to banish the old diffy ghosts.

Having this experience gives me an understanding of and sympathy for math anxiety when I see it in my students. I hope I can now share my new success story as another tool to help my students overcome that anxiety, just as I finally have.

America’s STEM Shortage: Myth vs. Reality

Chances are you’ve heard about the shortage of STEM workers in the United States, as well as the many efforts–both public and private–to attract students to STEM careers. Yet this article by CBS News casts doubt on those shortage claims, asserting that merely one quarter of people with STEM degrees are working in a STEM job. So does this mean that the need for STEM workers is all just hype? Not necessarily. The article makes no claims as to WHY the STEM grads aren’t working in STEM fields: is it because they could not find such jobs, or did they simply decide to pursue other careers? The answer is not known. Also, the unemployment rate among those with STEM degrees is far lower than it is for the general population of workers, meaning that science and math skills are very much in demand outside of traditional STEM fields. All in all, the employment picture remains quite good for those with STEM degrees.

Yet the most powerful take-away for me came near the end of this article. While the value of STEM skills is no myth, the argument that the U.S. is falling behind other nations in math and science is in fact false. The U.S. remains dominant in these fields, and the unfounded fear of losing excellence has driven efforts to push more people into certain research careers, which has led to cycles of boom and bust in the research job market as well as gluts of certain workers. There is no doubt that we live in a tech and data driven society that depends on STEM skills, and the U.S. would do well to up its investment in scientific research, but fear-mongering is the wrong approach to take. We saw this 50 years ago during the height of the Cold War, when the U.S. poured funds into missile technology out of fear of falling behind the Russians. (In reality, the Russian technology was nowhere close to ours.) So let us continue our efforts to bolster math and science education and encourage our students to explore STEM careers, but let us do it out of an enthusiastic and optimistic desire to expand technology for our nation and the world, not out of a desire to best the “other.” Then our students will truly be able to focus on solving problems in a global community instead of focusing on beating some phantom foe.


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My name is Erin Milne, and I am a mathematics student and teacher. I earned my master's in math from the University of Vermont, and I received my undergraduate education from Lyndon State College. My goal for this blog is to make mathematics interesting, useful, and non-frightening, as well as to inspire other low-income and first-generation students to continue their education. I hope this blog will be helpful, inspiring, and thought-provoking for high school and college students facing the same challenges that I have faced.

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