When I joined our faculty, the provost and president welcomed us with a free breakfast and a serious message: we value the teacher-scholar model at this institution. Research and teaching get priority here. Message received. However, it was 7 a.m. So, within one year of joining the faculty in my department, I had committed myself to two departmental committees, one university committee, two independent study courses, and advising one masters student. Yes. Yes. Ridiculous. I know. But I didn’t realize that as I was saying “Yes” to all these things, I was also saying “No” to other more important things. In that first year, I was also teaching three new-for-me courses with nearly 200 students in total, each semester. In that first year, I was also a mom for the very first time in a new marriage in a new town of a new state. I know. I had you at “two departmental committees” — this was way too much for any one person to handle, but did I change in year two? You betcha I didn’t. I added a two hour commute and took on advising undergraduate students…
As you can imagine, my first and second year reviews from the chair and tenured faculty were horrendous, but it fell on deaf ears. Afterall, I was doing so much for the department and the university — doesn’t that count for something? The answer: Not so much. At the start of my third year, my chair sat me down and we had The Talk. I needed to focus on research. My teaching could use some improving. Research and teaching should be my priority and nothing more. I realized then what I should have known all along: I said “Yes” to low-priorities when I should have been saying “No.” After The Talk and a box of chocolates to calm my nerves, I cleared my calendar and started all over again. After months of saying “No” and focusing on the high-priorities, my back-burner research projects turned into articles, the thesis projects of my masters students helped keep my research program alive while I worked fastidiously on an award for an undergradaute research program targeting Pacific Islanders. Whatever time I had left, I spent in crafting my lectures. Every so often, I felt left out of the activities I used to do, but I told myself that later I’ll have time to do them. I learned to say “No” in a most respectful way and since then, I have learned to be more discerning about the commitments I make.
If your chair, advisor, mentor, colleague, fellow student or friend requests your participation in something apart from what is highly valued in that particular chapter of your professional life, you should *take a moment* before you respond. Give yourself enough time to weigh the benefits and effects on your priorities before you commit that precious amount of time. Learn to say, “That sounds very interesting — can I get back to you in a couple of days?” This move not only gives you a chance to analyze the situation, but it also shows that you are thoughtful about the things you do. Remind yourself, when you’re feeling like you are letting go of an opportunity, that it is okay to *wait until you can confidently say that you are contributing successfully to what is highly valued.* There will always be plenty of “other things” to do and you will have plenty of opportunities to say “Yes,” but only if you learn in the beginning to focus on the things that matter most and just say “No, not right now” to all the rest.
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Today, I am still almost in the same place. My enthusiasm for matters mathematical has not waned (except for that one unbearable stretch of time toward the end of graduate school when I would have preferred to be doing anything else but mathematics …) I am now older but no wiser, in more ways than one … and yet the trip so far has been amazing!
Still there were many times I was anxious, scared, frustrated, overwhelmed. Some of these times were, now that I think about it, normal and natural. And many such instances and issues have been raised in this blog before: the typical expectations and realities of the first year in graduate school, the stressful times before the qualifying exam, dealing with illness and loss, looking for that first or second job after graduate school, the emotional challenges of being on the tenure track, and so on. Personal reflections and thoughtful recommendations are generously offered to the readers of this blog in these and many other posts. And I know that I would have benefited from them immensely if I had received these suggestions at the relevant junctures of my professional and personal life. Instead, being of an earlier generation, I chugged along at my own pace and within my own means. Some things naturally came to me easier than others. And at others I have utterly failed.
But through the years, I have read many books, ranging from self-help guides to creative writing manuals, attended workshops, frequented relevant websites, and reflected upon my own personal experiences. As a result, I have collected together a few ideas and tools on productivity and time management that have been working well for me. In this post I want to share with you the most basic of these principles, with the hope that it may assist you in your life, at whatever stage you are. Whether you are a graduate student, a postdoctoral researcher, a junior mathematician on the tenure track, or wherever else you may find yourself within the world of mathematics, I hope that this one basic idea will give you a sense of control in the crazy-making, soul-sucking times which will invariably come up.
The big deal about life is … that there are always a ton of things to do! For instance if you are in graduate school you have to:
So what do you do? How do you handle all of this stuff?
Everyone knows you just need to make a to-do list and then just follow through … Right? If only it were so easy … But the to-do list is indeed a good start! So the basic principle I want to offer you in this blog post is the following:
A list is a simple idea, but done right, its impact on your life might be phenomenal.
So how do you do it right? The main goal here is to get everything down. I mean everything. Include the cat food to be purchased, the homework to be completed or graded, the paper to be refereed, the rent check to be written, the summer school course schedule to be consulted, the fridge to be cleaned, the garage door to be repaired, the sister to be called, taxes to be filed, and … I hope you get the point. You write everything down. This is what the productivity gurus will call a mind sweep. You want to clear and clean your mind, get everything that needs to be done out of your mind onto your list.
Take some time to do this. It is not trivial. And even if you take an hour to create your first list, you may later on remember things that you did not include the first time around. That is fine. Just patiently go ahead and add those to your list as well.
It is important that initially you impose no constraints on yourself. No telling yourself “this is too silly / trivial to write down” or “that is not relevant to my work”. But eventually you will have this ginormous list. In fact, if you are like most people I know, by the end of this process, you will have a huge list, perhaps with over a hundred items. And then comes the natural question. What do I do with this unorganized mess of items?
There are several ways to organize a to-do list. There is the “one-minute to-do list”, there is the “world-class to-do list”, and once you start looking, you realize that all kinds of people want to help you with your to-do list (and most likely will want some money for it). There are several software packages for desktop / laptop computers, a wild range of apps for smart phones, and of course there are many attractive options if you want to keep things simple and on paper. You might wish to look at these various offerings to figure out what will work for you.
But here a confession (and a warning) is in order: I have fallen down into many a rabbit’s hole trying to find the best way to implement any one productivity / time management idea. The search for the optimal productivity / time management tool can ironically become a huge time suck on its own. So I am here to tell you, after all the hours I have wasted on all this stuff, that the ideal method is the one that will work for you. It is not what works best for so-and-so on the web, or what works best for your advisor or your fiancee, but what works for you. Indeed just jumping in and getting started is the best thing you can do in this realm. And for that, there is perhaps no better alternative to stacks of paper stapled together or a new folder on your computer that contains a handful of text-files. Or if you like, you can go out and splurge on a neat agenda or download some app that looks intriguing and has good online reviews and give it a go. You can always change your method, but you do want to get started. Soon.
The productivity gurus and time management experts alike wax poetic about how the mind sweep relieves anxiety and how it allows you the space to think creatively and plan ahead. I agree. The mind is not meant to store all that stuff. It is not meant to be your storage pod of tasks to be done. It is meant to be your creativity machine. It is where the math is supposed to be done. It is where the problems are supposed to be solved. So you need to get all the tasks out of your mind and onto some other storage space. So that your mind is free to do its thing.
There you have it. The Fundamental Principle of Time Management, the First Theorem of Productivity, if you will: Get things out of your mind and onto paper. Make a list. Nothing too ground-breaking, I know, and it probably sounds too simplistic, but please humor me, and give it a try. Set apart an hour this weekend and do a mind sweep. My guess is that you will find it makes a difference. And with the sense of serenity and control it offers, you can make progress in any aspect of your life.
I decided not to preach about this. Instead, I asked 4 first-year graduate students who were absolutely successful undergraduates and are now in their second semester of Ph.D. programs. I asked them 5 questions and their answers are enormously revealing. Some answers have been edited for length.
Was there a period of adjustment/transition when you started grad school? If so, how long did it last? What needed adjustment?
Student 1: “There definitely was. It probably lasted a little over 2 months. Specifically for me, I needed to adjust to living in an entirely new (big) city by myself, taking courses where I may not have had all the background (and having to catch up; this was and is still something I struggle with), and, actually, re-finding my motivation to be in grad school.”
Student 2: “There was most certainly a period of adjustment I went through when I started…and at times I still wonder if I am going through that adjustment? In the beginning I was very excited to just do work all the time since I knew that was what grad school was going to be like, but it didn’t turn out to be rainbows and sun shines like I expected. My first semester I never went home before dark and although I had an hour between each of my classes I didn’t make any use of that. As of my second semester I vowed to make use of the hours between classes and attempt to be able to go home at 5 o’clock by working incredibly hard throughout the day. That lasted a little while, but then tests came around and it wasn’t as plausible. But I found myself adjusting and even if I go home around 5:30 I make dinner and start to work again until 10:30.
Student 3: “There was an adjustment period, and to be honest it was quite brutal. A lot changes when you graduate college and move to a new city, even if you aren’t going to graduate school. I didn’t know anybody in [the new city], and felt very alone. Add to that the insane workload associated with beginning graduate school. So that was quite difficult at first. As an undergrad I would do what was necessary, however last minute. As a graduate student, every day was another adventure and there was no reason to stop once I finished an assignment. I poured myself into work and soon felt very comfortable and so happy with my studies. I began to realize I was doing all this for fun. This was (for me), the first step for gaining stability. Second was the realization that you can’t do everything. [At first] I worked all the time; it was time to start realizing that I needed to be a person too. So I would take a day off here and there, go to concerts, and having the realization that it didn’t end in disaster and stress was huge for me. I made this mistake my first term. [I was in] the office until midnight every day, including weekends. For me it was kind of an escape, but it became overwhelming. To free myself, I play music and go dancing. Finally, you WILL want to quit. A lot of times. For each day you find yourself overjoyed with the beauty of it all, you will have one where you KNOW that you aren’t cut out for it. This is good, I think. Each time you want to quit you convince yourself not to. This is valuable. The more you convince yourself not to quit, the more concrete reasons you have to keep struggling through.”
Student 4: “My first semester was definitely the one of the hardest experiences I ever had to endure. It was not about the content of the classes or the rigor but rather balancing homework assignments with TA responsibilities as well as non-academic stressors. Grad school is definitely a step up from any undergraduate class I have ever taken in which it demands more discipline and deep conceptual understanding of the subject. I took me a semester to realize that I was not supposed to know everything right away, that understanding takes continual practice and to always reach out for help. I needed to adjust my approach to taking courses where my goal was not only to pass the class with an excelling grade but also take in what is being taught and try to apply it to a project. Also my priorities in general had to change, I now had TA responsibilities mixed with challenging homework as well as building research project. It was not until the second semester where I started to utilize resources and ask basic questions that I had a better sense of how to manage grad school.”
Were there some aspects of your first year that were surprising or unexpected? What are they?
Student 1: “I was lacking in some background and needed some time to adjust to the schedule and find a sort of ‘weekly routine.’ Other than that, most of what I experienced is along the lines of what I expected for my first year.”
Student 2: “I can say that graduate school has taught me how students in Intermediate Algebra and College Algebra feel while they are in class. Some concepts you just don’t understand, specifically for me in our Modern Algebra class I feel like I am just spiraling downward into a never-ending pool of confusion. It’s difficult to pay attention in class and stay interested when you feel that far behind and it even causes you to have animosity towards your professor. I never expected to realize how students feel in a math class when math is not their forte.”
Student 3: “First off, the amount of work, it’s more than I thought possible. I thought I wouldn’t be able to do it all. But you’d be surprised at what you can do. Second, it doesn’t feel like work anymore. As an undergrad, even math homework feels like homework. [In grad school] every day in class, we’re given concepts, and then we’re told to go play with these concepts for a while until they stick. Then we are given new ones. But this is all we do during the first year. No research or pressure or stress. Just play time in math world. It’s incredible.”
Student 4: “I thought graduate school was going to be more classes with more work. However, grad school is much more than additional courses, it’s the ability to incorporate what you learned as an undergraduate and expand it to focus on a project or research where taking additional advance courses help you build. Being a graduate student has that underlying pressure of understanding everything that is in your ‘field’ and come up with an idea that can highly impact or advance the field. There is a lot more reading than I expected. I soon learned that self-motivation becomes the main component that could make you succeed as a graduate student. As a first year I was also intimidated by the other students who had established seniority, however, asking them for advice was not only helpful but [gives you a perspective] that you are not the only one that ever felt overwhelmed.”
Have you changed the way you study? How?
Student 1: “I study a good amount more on my own than in my undergrad, where I did work in groups all the time. I still work with other students here often, but I spend more time than before trying to learn the material by myself. I think this is because in my undergrad, I sometimes cared more about “getting the grade,” whereas in grad school I care more about understanding the material to help me in research.”
Student 2: “I CERTAINLY changed the way I studied! Even as I took tests throughout the year the way I approached every test was different. My first lesson: yes the other people in your cohort are smart and they CAN indeed help you better understand concepts. In undergrad I never relied on other students to better help my understanding, but now I seek help from others all the time!! We have days where we study together and go over problems we had troubles with and I always come out of those sessions knowing more than before I went in.”
Student 3: “Boy, have I. I used to be an all-nighter guy. Blowing off work until the last minute and then pulling together something awesome in an epic stressful night. I don’t do that any more. I’m in the office by 9 each day. Class 9:30-10:20. Then during the day I work through some commutative algebra or algebraic geometry reading and problems that I’m doing on my own. Then class 1:30-2:20, and then from 3:00 on I read and work on my coursework. And leave the office at 10 [pm] or so. Also, you work on more than just coursework. Currently I’m also doing all the problems [in a] commutative algebra book, and reading through Algebraic Geometry notes. No due dates, just curiosity. No need to cut out early. That being said, if one day it’s really nice out, I can totally cut out early and take a day off, just because I feel like it. This is the best thing about being a graduate student. You are NOT a slave to the clock.”
Student 4: “It’s hard to start studying and place yourself in a ‘study mode’ especially when you have many things in mind that have to be done before the end of the day. At first it seems there are always more things to do than what time allows and that you will never finish. To a certain extent that becomes true, you are always going to have something to do and the best way to deal with it is to prioritize; you have to make a list every day of what has to be done that day and what can wait. Grad school is going to test the amount of time you should spend in school rather than at home or with friends. The transition may be hard but it’s no impossible. Another suggestion that helped me is to start hanging out with the older students, notice their way of studying, mimic their methods and attain the same discipline.”
Once you started grad school, did you feel prepared to be in there?
Student 1: “Honestly, at the beginning, not so much. Now, however, I am doing fine in my academics and am talking to a few professors, but I am still a bit worried about my comprehensive exam coming up next year.”
Student 2: “I did feel prepared when I started, but now that we are going more in depth with certain concepts our professors are assuming quite a bit of our previous knowledge. For example, I didn’t have a strong linear algebra course in undergrad…it was primarily computational and not proof-based. Now, my professors assume many things that I should have learned as an undergraduate in linear algebra but was not taught (or perhaps don’t remember? it was my sophomore year). That can be quite frustrating, but I deal with it and make an effort to fill the gaps.”
Student 3: “Yes. Totally. It’s like I’ve been waiting for this all my life. I was prepared academically (though I wish I knew more math, but that’s going to be true for the rest of my life). I was prepared emotionally (my parent’s both have PhD’s so I knew what I had in store for me).”
Student 4: “When I started grad school I felt the complete opposite of prepared in the sense that I felt like I was not ready for the transition and that I did not have the knowledge other students had. The worst thing I did was to compare myself to the rest of the group. I felt like I did not belong and that this discipline was far beyond of what I could one day understand. Regardless, by talking to the other students and to faculty, I found that the path to understanding takes time and dedication.”
Are you enjoying it?
Student 1: “For the most part, yes. Some of the required coursework is not to my interest, but other parts of courses, as well as some of the discussions with my potential mentor are enjoyable. Living in a big city with lots of things to do outside of school helps too! “
Student 2: “I am somewhat enjoying it…I still see the light at the end of the tunnel. Some days I just want to cry from being frustrated and overwhelmed. I debate leaving with a masters so that I can go home to my family (this is also my first time living away from home). It was difficult for me to develop relationships with other students quickly since I was not in the “big office” of the first year graduate students, that office has 10 students and there are 3 students in my office. But luckily my boyfriend from home is with me as a graduate student in physics so having him there helped me to get through that phase. Now I feel as though I am becoming closer to the other students (or at least some of them). But if I wasn’t getting along with the people in my year I don’t think I would have continued in this program, having people you like to talk to is vital!!”
Student 3: “So much enjoyment. Sure, it sucks sometimes. [But] even when it sucks, it’s awesome. I don’t know how to explain it, but I’m quite happy to be honest.”
Student 4: “I enjoy the busyness of grad school and the things I learn every day. I know that I still have a far road ahead of me but I feel that I can endure more stress now than before.”
I thought to end this post with a quote from Student 3: “All in all. Adjustment is hard. It’s uncomfortable at first. But graduate school can be so rewarding. I’m not saying I’ve made the transition perfectly or completely. There are so many roadblocks ahead. But have faith, and don’t forget to have fun.”
]]>My suggestion: pretend like you’re the President of the U.S. and at the same time pretend that you were just granted five minutes to speak with the person with whom you would like to speak most to in this world (or equivalently to a lawyer who charges $1000 per hour but who is giving you 5 free minutes). The latter may seem strange but you must also realize that any good mentor likely has many other things to juggle and, while they are doing this to help you, it is important that you are aware of their time and grateful for it.
Why the President? He (and hopefully soon, she) surrounds himself with advisors from different aspects of life. His trusted advisors are not all lawyers, nor are they all economists, nor are they all friends to whom he owes favors. When a crucial decision is being made, he usually will get opinions from different viewpoints and will weigh them before deciding.
How does this apply to you? Whatever big decisions you need to make, it is important to hear different perspectives. This doesn’t necessarily mean to ask your math major friend, you high school friend who is now working in a fast food restaurant, and your favorite professor whether you should go to graduate school. Instead this might mean ask your professor, someone with a graduate math degree working outside of academia, and a down-to-earth graduate student that same question. Hopefully you see that of the first three, only one of them likely knows what you would be getting yourself into but that is only one perspective. In contrast, in the last three, all can give you concrete feedback and from likely different perspectives.
You can likely speak with a professor or graduate student during office hours and be sure to be grateful for their time afterward, but how about the non-academic post-graduate? At conferences such as SACNAS, SIAM, and others, one can find more and more Ph.D. mathematicians that are employed outside of academia. Some may work in government labs but others may work in another industry. Approach these people! Introduce yourself and tell them that you are at a place in your life (such as an undergraduate contemplating graduate school but not sure of the area, a graduate student contemplating academia versus industry, or a professor contemplating a new line of research) where you would like to know if they could give you a 2 minute answer about what they liked about their career path to date and what they would change if they could. And after the 2-minute answer be as gracious as possible and ask if they have a business card so that you may contact them if you have any additional brief questions. This helps build up a diverse network.
So remember…pretend you’re the President and seek advice from calculated sources that will give you potentially different angles from which to view your decision. Even choosing two or three of each type of advisor (professors not all necessarily in math, etc.) is wise as it gives additional perspectives – just be sure that you are also openly grateful for their time so that you may come back again for this or other matters. With all the advice you receive, just remember that only you know what is best for you but choosing trusted mentors may help you realize more about you than you knew before. In the end, we are the one that must deal with the consequences of whatever good or bad decision that we make.
]]>1. Start early. – It’s never too early to start studying for your quals and I suggest beginning your very first year! (If you’re further along, no worries. Just keep reading!) By setting aside an hour a week to review the material as you go along, you will be ahead of the game when time comes to take the test!
2. Get copies of the old exams. – Most departments or more advanced graduate students will be happy to provide you with qualifying exams from previous years. You should make sure that you can solve every problem from beginning to end!
3. Create a solutions set of old exams. – One mistake I made was that I would look at old exam problems and write out a skeletal solution without filling in the details, because I thought I could solve the problem. But when I took my qualifying exams, I would lose credit on all those “details” that I didn’t write out. I suggest that you create a detailed solution for each problem that you work and show a few of them to your professor to verify that you would receive full credit.
4. Talk to your professors. – This is especially true if you have to take an oral exam like I did. By speaking to my professors individually, I gained insight into the type of question they would likely ask me. The conversation usually started like this: “Dr. Scott, I’m studying for my qualifying exams and I was wondering if you would give me some practice oral questions?” No one ever turned me down and I heard many of these same questions during my actual oral exam!
5. Relax. – Often easier said than done, relaxation can mean the difference between passing and not passing your exams. Your brain function and overall mental health improves when your body is in a relaxed state. So take a walk or go for a swim in the days leading up to your exam. And remember that no matter what happens, you’re still amazing!
]]>But why celebrate Pi Day? And how can we as educators use this as a learning moment in the classroom? I’d like to present ways you can discuss \( \pi \) in Religious Studies, Sociology, Computer Science, Mathematics, and Political Science. (Most of the post below comes from my web page here.)
The number \(\pi\) is defined as the ratio of the circumference of a circle to its diameter. This number has been around much much longer than one might expect. I was amazed to find that the Old Testament of the Bible even discusses the number \(\pi\)! For example, I Kings 7:23 reads: “[King Solomon] made the Sea of cast metal, circular in shape, measuring ten cubits from rim to rim and five cubits high. It took a line of thirty cubits to measure around it.”
Since a cubit is about 1.50 feet (about 0.46 meters), it’s easy to see that \(\pi\) is approximately 3. Here are some questions one might wish to ask for discussion in the classroom.
The African engineer Archimedes of Syracuse (287 BC – 212 BC) found an even better approximation of \(\pi\) via his “method of exhaustion”: by comparing the circumference of the circle with the perimeters of the inscribed and circumscribed 96-sided regular polygons, he found \(\pi\) is somewhere between 223/71 = 3.14085 and 22/7 = 3.14286. The Chinese astronomer Zu Chongzhi (429 – 501 A.D.) realized \(22/7 = 3.14286\) approximates \(\pi\) to 2 decimal places, whereas \(355/113 = 3.14159\) approximates \(\pi\) to 6 decimal places. (The fractions are known in Chinese as Yuelu and Milu, respectively.)
With the introduction of calculus, one can compute \(\pi\) to as many digits as desired. The German mathematician Gottfried Wilhelm Leibniz (1646 – 1716), Scottish astronomer James Gregory (November 1638 – October 1675), and Indian astronomer Madhava of Sangamagrama (c. 1350 – 1425) each discovered the series expansion \( \displaystyle \sum_{k=0}^{\infty} (-1)^k \, \dfrac {1}{2k+1} = \dfrac {\pi}{4}\). Of course, this follows from the Maclaurin series expansion \(\arctan(x) = \displaystyle \sum_{k=0}^{\infty} (-1)^k \, \dfrac {x^{2k+1}}{2k+1} \) where one substitutes \(x = 1\). An even better approximation comes from setting \(x = \dfrac {1}{\sqrt{3}} \): \[ \pi = 6 \, \arctan \dfrac {1}{\sqrt{3}} = \sum_{k=0}^{\infty} \dfrac {(-1)^k}{2k+1} \, \dfrac {2 \, \sqrt{3}}{3^n} = 3.141592653589793 \dots. \]
Gottfried Wilhelm Leibniz (1646 – 1716) and James Gregory (November 1638 – October 1675)
But is \(\pi\) a rational number? Can we find integers \(p\) and \(q\) such that \(\pi = p/q\)? In 1768, the Swiss mathematician Johann Heinrich Lambert (1728 – 1777) showed that the answer is no: \(\pi\) is an irrational number.
The Hungarian mathematician Miklos Laczkovich (1948 — ) gave a simplified version of Lambert’s argument in 1997. The idea is to introduce Bessel functions, that is, certain series solutions to the second order differential equation \( x^2 \, y” + x \, y’ + (x^2 – n^2) \, y = 0\): \[ J_n(x) = \left( \dfrac {x}{2} \right)^n \left[ \sum_{k=0}^{\infty} \dfrac {(-1)^k}{k! \, (k+n)!} \left( \dfrac {x}{2} \right)^{2k} \right] \qquad \text{where} \qquad k! = \int_0^\infty t^k \, e^{-t} dt. \]
Laczkovich showed the ratio \( x \, J_{n+1}(2x) / J_n(2x) \) is never a rational number whenever \(x^2\) is a nonzero rational number. Since
\[ \left. \begin{aligned}
J_{+1/2}(x) & = \sqrt{ \dfrac {2}{\pi x}} \, \sin x \\[5pt]
J_{-1/2}(x) & = \sqrt{ \dfrac {2}{\pi x}} \, \cos x
\end{aligned} \right \} \qquad \implies \qquad
\dfrac {x \, J_{n+1}(2x)}{J_n(2x)} = x \, \tan 2x \]
we may conclude (1) if \(x\) is a nonzero rational number, then \(\tan(x)\) is irrational and (2) \(\pi^2\) is irrational.
Curiously, American mathematician Noam Elkies (1966 -) observed that one can use the latter fact to conclude there are infinitely many rational primes. Indeed, if there were finitely many, then the left-hand side of Euler’s identity \( \displaystyle \prod_{\text{$p$ prime}} \dfrac {1}{1 – p^{-2}} = \dfrac {\pi^2}{6} \) would be a finite product, and hence a rational number.
Johann Heinrich Lambert (1728 – 1777), Miklos Laczkovich (1948 — ), and Noam Elkies (1966 -)
More than a century after Lambert’s result, many began to wonder how “irrational” \(\pi\) really is. One question that arose was whether one can “square the circle”. That is, is it possible to construct a square, using only a finite number of steps with compass and straightedge, having the same area as a given circle? In 1667, James Gregory gave a proof (albeit incorrect) that this is not possible.
Allow me to put this into a larger context. A rational number \(x = p/q\) can be thought of as a root of a linear polynomial \(f(x) = q \, x – p\) having integer coefficients. More generally, a number \(x\) is said to be algebraic if it is the root of some polynomial \(f(x) = a_n \, x^n + \cdots + a_1 \, x + a_0\) having integer coefficients \(a_k\). Clearly, rational numbers are algebraic, and even some irrational numbers (such as \(i = \sqrt{-1}\)) are algebraic. In fact, numbers which can be constructed using a finite number of steps with a compass and straightedge are also algebraic; this is a fact shown in many advanced Abstract Algebra courses today. Numbers which are not algebraic are said to be transcendental. In 1768 – in the same paper where he proved the irrationality of \(\pi\) – Lambert conjectured that \(\pi\) is transcendental, and hence “squaring the circle” should be impossible.
In 1882, the German mathematician Ferdinand von Lindemann (1852 – 1939) showed \(\pi\) is indeed transcendental. This follows as a special case of the Hermite-Lindemann-Weierstrass theorem: \(e^x\) is never algebraic whenever \(x\) is a nonzero algebraic number. If \(\pi\) were algebraic, then \(x = \pi \, i\) would also be algebraic, contradicting the fact that \(e^x = -1\) is algebraic. Lindemann’s was the first definite proof that “squaring the circle” is indeed impossible.
Ferdinand von Lindemann (1852 – 1939)
While Lindemann’s result was celebrated by research mathematicians, it was not well-known to amateurs. In 1896, Indiana physician and amateur mathematician Edwin J. Goodwin believed he had discovered a way to square the circle. He had constructed a circle with circumference 32 and diameter 10, and inscribed a square having sides of length 7. As you can guess, there are several problems here: the ratio of the circumference to the diameter \( 32/10 \neq \pi \), and the square should have sides of length \( \sqrt{5^2 + 5^2} = \sqrt{50} = 7.07107\). It appears that Goodwin misunderstood the idea of “squaring the circle:” he thought it meant one should inscribe a square within the circle. (Which Archimedes had already done some 2,000 years before anyway.)
Goodwin’s Model Circle as Described in Section 2 of the Bill
Goodwin proposed a bill to his state representative Taylor I. Record stating that his result would appear free of charge in Indiana textbooks, and that the state would receive royalties for printing the result out of state. On January 18, 1897, Representative Record introduced the bill which read in part:
“[Goodwin's] solutions of the trisection of the angle, doubling the cube and quadrature of the circle having been already accepted as contributions to science by the American Mathematical Monthly … And be it remembered that these noted problems had been long since given up by scientific bodies as unsolvable mysteries and above man’s ability to comprehend.”
The bill was introduced to the Indiana House of Representatives, but it caused a lot of confusion; Goodwin’s mathematics had claimed that the previous formulas involving the area for the circle were incorrect, although the bill itself was contradictory about the correct formula. On February 5, about the time the debate about the bill concluded, Purdue University professor Clarence Abiathar Waldo (1852 – 1926) arrived in Indianapolis to secure the annual appropriation for the Indiana Academy of Sciences. (Waldo was the president of the Academy that year.) An assemblyman handed him the bill, and Waldo pointed out the obvious mistakes, that is, that it is impossible to square the circle. (Recall that this had only been proved some 15 years before by Lindemann!) Still, the bill was tabled by the Indiana Senate, when one senator observed that the General Assembly lacked the power to define mathematical truth.
Clarence Abiathar Waldo (1852 – 1926)
You can read more about this history at Mental Floss. We in Indiana in general and Purdue in particular are grateful for Waldo’s intervention. If this fortunate series of events had not transpired, Hoosiers everywhere would be the laughing stock of the country for being those who legislated \( \pi = 3.2 \) as a rational number. Hopefully this will not be forgotten on this year’s Pi Day!
]]>I can’t emphasize enough how transformative an experience this program is for the students, and how much I wished such a program existed where I grew up when I was a rising 8th grader. Teaching at SPMPS was a great joy, and I highly recommend it as an outreach initiative to get involved in!
The program is currently looking for Instructors and Residential Counselors for the 2014 summer program. For more information on how to become involved this summer, see “Jobs At SPMPS”
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The idea of mentorship is central here (the eMentoring Network!), so this is a good excuse to remind myself: I think all of us can act as mentors (and mentees). Undergraduates can act as peer-to-peer mentors, helping out classmates. Joining a local math club is also a good way for us to support each other. Graduate students can also help out in math clubs, and can reach out to mentor undergraduates. A nice word from a grad student can go a long way for an undergraduate. For folks that already have PhD’s, the opportunities for mentorship are limitless.
Two organizations that standout for fostering mentorship are SACNAS and the National Alliance. At SACNAS, one can join (or start!) a local chapter, or become involved in the national organization. And the National Alliance is always looking for new mentors. You could also start or joint a mentorship program at your home institution (there’s a cool and exciting program at my home institution).
Happy new year!!
]]>Because of the wealth of experience I had as an undergraduate researcher and as a teaching assistant for a highly successful summer undergraduate research program, I was ready, eager and willing to start directing undergraduate research. Unfortunately, however, very few of my colleagues had had these rewarding opportunities and some of them have expressed a desire to engage their students in a similar fashion, but are reluctant because they did not know how or where to begin. I now know that this is not uncommon.
Indeed, the number of undergraduates engaging mathematical sciences research has increased dramatically the past few years. Indicators of this growth are the size of the undergraduate poster session at the Joint Mathematics Meetings (e.g., over 300 posters at the 2013 meeting), the number of mathematics Research Experience for Undergraduates (at least 65), and the recent creation of journals devoted to mathematics undergraduate research (e.g., Involve at UC Berkeley). This success is in contradiction to the view held by some today and many in the past that “undergraduates cannot do mathematics research, because there is so much background needed to understand and successfully tackle a problem.”
Today, many mathematics faculty, some motivated by the success of colleagues with undergraduate research, want to begin their own undergraduate research program, but are hesitant to do so, because they are unsure how to get started. i.e., how to find/choose tractable problems, how to recruit students, how to get funding or release time for the endeavor, how to guide students towards a solution without solving the problem for them, etc.
For these reasons Herbert Medina, Professor of Mathematics at my undergraduate alma mater, Loyola Marymount University, and I decided to organize a panel on the subject for the upcoming 2014 Joint Mathematics Meetings in Baltimore, MD. The specifics of the panel are as follows:
Panel Title: Directing Undergraduate Research: How to Get Started
Time & Place: Thursday January 16, 2014, 2:35 p.m.-3:55 p.m. in Room 316 of the Baltimore Convention Center
Panelists: Michael Dorff, Brigham Young University; Angel Pineda, California State University Fullerton; Joyati Debnath, Winona State University; Sandy Ganzell, St. Mary’s College of Maryland
The panelists, all having enjoyed success in directing undergraduate research, will address these questions and provide concrete advice on how to get started with directing undergraduate research. Hope you can join us!
But before then, if you have suggestions or specific questions/comments/suggestions about undergraduate research, please enter them in the responses section below.
]]>Since we early-career mathematicians must be concerned about current and future career instability, illness and loss take on extra dimensions of complexity. In this post, I reflect on my own experiences and the lessons I’ve learned from those around me.
I have ulcerative colitis (UC). This is chronic inflammatory bowel disease, so I’ll have it for life. I was diagnosed back in 1999, so I’ve been dealing with it since my first days in graduate school. This is a disease that no one likes to talk about — the primary symptom is frequent, uncontrollable bloody diarrhea.
UC goes through periods of remission and flare up (it’s an autoimmune disease). In the first few years after my diagnosis, my UC was mild enough to keep private. I didn’t tell my fellow graduate students, my instructors or even my thesis advisor. Roughly once or twice per year, I would experience a few weeks or a month of bleeding, and then induce remission with increased or varied medication.
Things changed in 2007. I was a postdoc at Berkeley, and a flare-up started, as usual, with mild bleeding. This time, I was unable to control it as easily as before. My bleeding increased and eventually became constant, anemia set in, and I was unable to function. I developed an extra-intestinal manifestation of UC called pyoderma gangrenosum, wherein your immune system eats your skin, and creates open wounds. Even if you’re wearing adult diapers, it’s impossible to give a lecture while having bloody diarrhea! And you can’t function well at all with a bunch of open wounds!
For the first time in my life, I had to walk away from professional responsibilities. I couldn’t teach my classes. I couldn’t give seminar lectures. I couldn’t travel to talk or even participate in conferences. I was bedridden, and couldn’t even stay on top of my email. As an ambitious young mathematician, what was I to do? How was I to establish my reputation? Find and prove theorems? Develop outreach programs? Secure a tenure track job?
I have gone through severe flare ups several times now, the most recent of which is just ending. And I have known others forced cope with loss and tragedy as well. Two friends in particular come to mind; one lost a relative to suicide, another lost a friend to murder. Here are five lessons I’ve learned.
Career? Whatever. You can’t have a career from the grave. No job is worth risking serious injury or exacerbating illness. Think of your mother (or someone else you love dearly). Would you want her to risk her health for career advancement? Try to apply the same standard to yourself.
What’s more, you won’t be able to prove theorems from the sickbed. Your mental and physical health is correlated to your productivity. So, even from the perspective of career advancement, it’s important to make your health a priority.
Get yourself healthy first, and only then get back to work.
As a wise senior professor once said to a friend in need, your theorems will wait for you. Yes, of course, mathematicians are at times political, dishonest and capable of scooping each other. But the theorems that are truly yours are yours alone. They will be there for you.
Or, as a senior rock star once said, you can’t always get what you want, but sometimes, you’ll find, you get what you need. Perhaps, after your illness, you will find different theorems to prove. Evidently, those were the theorems the universe intended for you to prove.
You should allow yourself to focus on your healing. Deal with your loss. Rebuild your health. Focus on your recovery completely. Your theorems will be there for you when you get back on your feet.
This piece of advice is much easier for me to give than receive. My natural inclination is to turn inward. I want to find a solution to my problem without relying on the support of those around me. I want to keep my problems private. I don’t want to ask for help.
But to recover, I have needed large amounts of help and support from others. And help has come from unexpected places.
You might find staff members at your institution that have dealt with similar issues. Your colleagues may help in ways unimagined. People will surprise you with their generosity. This can be really affirming.
As a postdoc and a professor, I’ve been lucky to have tremendous support from my colleagues. (Not to mention my family and friends!) I wouldn’t have been able to make it without them.
And remember to let those you help you know that you are grateful for your help. This is important for your colleagues (and your marriage) — it’s not easy to take care of others!
The periods of child development are well known: the terrible two’s, the pre-teen years, etc. We expect children to go through these periods as part of a healthy progression of development.
Although it is less discussed in the general public, adults also have periods of development. Unfortunately these are usually characterized as identity crises, seen as negative, and expected to only happen as we enter our middle years.
These periods of adult development, or identity crises, are times in our adult lives when we grow, come to understand ourselves better or more deeply, or take time for serious self-reflection. They can be brought on through natural development, or through significant experiences. Often times crises occur when we face our own mortality or that of a loved one.
If you find yourself feeling in such a crisis as a result of illness or loss, don’t worry about it. Don’t criticize yourself. Forgive yourself. Allow yourself to be reflective. Allow yourself to grow. Allow yourself to change.
You may become a new or different person as a result of your experience, and this can be quite positive.
You will suffer as a result of your illness or loss. Learn from that suffering.
In addition to learning about yourself, as above, learn as much and as widely as possible. You are a mathematician, so you’re capable and inquisitive. You’re forced to focus on recovery. Learn about your disease or circumstance. Read up on suicide prevention, gun control, autoimmune disease or whatever your situation dictates. Study the impact of your illness on others including those close to you. Learn about the various ways that people deal with suffering. This is such a large subject, it’s difficult to even touch on. There is just so much to learn from illness and loss.
In the end, you may find yourself renewed, with a new sense of purpose, more informed, more able to help others, and more sympathetic to people experiencing various forms of suffering around you.
In short, I hope that the reader will remain infinitely happy and healthy. But should illness or loss come your way in your pre-tenure years, I hope these reflections will help. It is not the end of your career, only the beginning!!
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