In the wake of mathematical enlightenment a profound understanding of basic notions bridges the gap between the conceptual and concrete. In many cases, problems that have an exterior of simplicity exploit the boundaries of comprehension and provide insight into extensive associations. From the mind-stretching inclinations of geometry and algebra emerges the intricate framework from which these connections form. Piece by piece, generalizations are built from the material of empirical understanding fabricated by the process of asking intrinsic questions.
Questions of this nature are entwined in reticent patterns found across the full spectrum of mathematics. Many of these inquiries encompass and ascertain the properties of special functions. Mappings such as Euler’s Totient function provide a strong basis for further investigation into characteristics of positive integers. Specifically, this function denoted by counts the number of positive integers less than or equal to a positive integer such that the positive integers counted and have only 1 as their common divisor (in other words they are relatively prime to , denoted such that ). In example, = because there are two positive integers less than or equal to 3 that are relatively prime to 3 (namely 1 and 2, given ). Euler’s Totient function is distinguished by several other properties as well. For instance, if is a prime number, then . It is also multiplicative, in the sense that if then = . By virtue of these attributes, several open problems in the field of number theory involve .
The mathematician Robert Carmichael proposed one such conundrum in 1907 that still remains unsolved. Basically, Carmichael conjectured that for every positive integer there exists a positive integer such that and . As a consequence, with the given properties, the conjecture is certainly true for odd numbers. This can be seen by letting be a positive odd integer and in the fact that = , which it follows
= = .
However, as easily proved as the conjecture is for the positive odd integers, the statement has not been shown true for the positive even integers. Maybe a clever argument will come from a thorough investigation of basic notions. Perhaps, rather, it will be stumbled upon in search of greater abstractions. Whatever the case of discovery may be, a resolution will certainly be achieved, if at all, by asking insightful questions.
]]>
1. Go watch Prof. Mahajan’s teaching course. Take notes. Seriously. I cannot stress enough the excellence of the advice he dispenses. I have returned to these videos many times in the past four years.
2. Be flexible. It’s easiest to manage a course or recitation when you maintain a mental model of the student’s learning. Prof. Mahajan discusses many techniques for building such a model which I have summarized in a previous post.
I almost derailed my vector calculus class in its first week. I assumed that because the students completed linear algebra, they would be comfortable reasoning about lines and planes in three dimensions. After receiving feedback from the students, I quickly realized that one of the goals of a vector calculus class is to develop spatial intuition. I corrected the mistake by devoting an extra lecture to the basic material, which brings me to my next point.
3. Less is more. If you are lecturing, you will cover one page of handwritten notes per ten minutes of class. Often less. Find your ratio and treat it as law.
4. Be clear and direct; doubly so when discussing course policies. Each character you write or figure you draw will be copied into 129 notebooks. Every quiz or exam will be read by 129 people. Your syllabus will be interpreted 129 different times. The exercises will be discussed in dozens of small study groups.A not insignificant fraction of students will attempt to bend a rule or ask for a more favorable interpretation of a grading policy. If you’re unsure how to respond, then don’t. Ask them to submit their case via email and sleep on it.
5. Work on your shtick. It sounds silly, but a gimmick can help break down the communication barriers which develop after twelve years of schooling. Some of the social norms students pick up are downright toxic to learning!
I learned about the importance of your shtick by observing a UCSB professor known for his elaborately choreographed, 850-student calculus lectures. But it didn’t really hit home until I went through my teaching evaluations with a fine tooth comb. My first time as an instructor, 88 students completed an evaluation. Eleven of them mentioned my beard or included a little picture of me, accentuating my beard. Ten of them talked about my effort to learn names. There were only a handful of comments with a greater “hit rate”.
I thought I was bad at learning names until I forced myself to try. Every time a student asks a question I either ask them their name or, if I think I know it, address them by name. The remainder of my system is based on two principles. First, I have a pretty good spatial memory. Second, students tend to sit in the same location every day. Give it a try!
6. When you begin lecturing, you will have to choose a side: slides versus chalk. I’ve written before about some of the issues with slides. For the record, out of those 88 evaluations, seven liked my use of the chalkboard and four suggested I use slides.
7. Spend a little bit of time reading research in mathematical education. At a minimum, this will give you a language to describe your existing habits in the classroom. This is the first step in improving or changing those behaviors.
8. Keep a journal. I’ve written before about my love of journaling; I view it as an integral part of the scientific method. Here’s a more literary take on keeping a diary. At a minimum, this record of your thoughts and experiments will provide great material for the teaching statement you will eventually have to write.
9. Be knowledgeable about your department. Know answers to common administrative questions and know where to direct students when you don’t know an answer. Inform students of the resources available to them. New students, especially, need to be reminded about office hours, drop-in tutoring opportunities and review sessions.
10. Use teaching to practice your public speaking. As a former software developer, I can tell you that this is an integral part of a professional career both in and outside of academia.
11. Go watch Prof. Mahajan’s videos. Seriously.
]]>
The paper describes an experiment where science faculty from research-intensive universities rated the application materials of a student who was applying for a laboratory manager position. The faculty evaluators were told that that the résumé was real and that the evaluation would be used to develop mentoring materials for science students. Each résumé was randomly assigned either a male or female name. The study found that faculty participants rated the male applicant as significantly more competent and hireable than his (identical) female counterpart. The faulty evaluators also made lower salary recommendations (by about 12 percent) for the female applicant and offered her less career mentoring. In addition, it was found that male and female faculty evaluators were equally likely to give the female candidate lower ratings.
It’s easy to identify blatant examples of sexism or harassment in the academic workplace. Telling a woman she is bad at math simply because she is female is an obvious example of sexism. However, it is the more subtle and, in many cases, unintentional examples of gender bias or discrimination (as in the experiment described above) that are harder to identify and even harder to address. If a female grad student is told by her advisor that she is not cut out for graduate school, it is less likely that she will question the objectivity and validity of this assessment compared to a more obvious putdown with a clear lack of factual basis. This is why having a forum to discuss sexism is so important.
In July 2014, a new website, Everyday Sexism in STEM, was launched with the aim of shedding light on the prevalence of gender bias in the STEM fields. The Everyday Sexism in STEM project was created to provide a place for women in STEM fields to share their personal experiences dealing with sexism on a daily basis. The site welcomes all types of stories – from the most outrageous displays of gender discrimination to the subtlest that are tolerated and even considered normal in the workplace. The goal of the Everyday Sexism in STEM site is to provide a sense of community for those who have faced gender discrimination in the workplace. Shared stories allow others to know they are not alone in their struggles and their experiences are not isolated. In addition, every story shared provides evidence that sexism is a real issue in academia that must be addressed. The Everyday Sexism in STEM website is a great resource for both men and women because it establishes a basis for communication and makes it clear that it is okay to talk about gender bias.
Sources:
Moss-Racusin, Corinne A., John F. Dovidio, Victoria L. Brescoll, Mark J. Graham, and Jo Handelsman. 2012. Science faculty’s subtle gender biases favor male students. Proceedings of the National Academy of Sciences of the United States of America 109, (41): 16474.
Everyday Sexism in STEM (http://stemfeminist.com/)
]]>It’s that time of year again – the summer is coming to an end, classes are getting started, and new grad students are arriving on campus. Graduate school can be an intimidating and challenging experience, especially in the first year. I asked some of my fellow grad students at Penn State what advice they would give to new grad students or what they wish they had known when they started grad school. Here is a list of advice that we came up with.
General grad school advice:
Academics advice:
Personal advice:
Good luck to everyone entering or returning to grad school this fall!
]]>Mirzakhani, who also received the 2014 Clay Research Award, was born in Iran and completed her PhD at Harvard University. As a child, she wanted to be a writer, but in high school she discovered a love for math and was a two-time gold medalist in the International Mathematical Olympiad. Her graduate thesis focused on geodesics on hyperbolic surfaces and moduli spaces, and she has continued to explore geodesics and moduli spaces in her recent work. She is currently a professor at Stanford University.
Quanta Magazine has an excellent piece on Mirzakhani, and you can read more about all the 2014 Fields Medalists and their work at the IMU homepage.
]]>
I like to use the Penultimate App by Evernote with the Jot Script. Penultimate has built-in support for the pen and will tell me when I need to replace the battery in my pen. A feature that I really like in Penultimate is the ability to search your notes. Suppose you are looking for something about measure in your notes. You can open the specific notebook and do a search for “measure.” The app will search through your handwritten notes for that word and put a yellow rectangle around it (as seen in the picture to the right). It saves a lot of time if you just need to look something up quickly in your notes. If you are an Evernote user, you can also have your Penultimate notebooks sync with your Evernote account. These can then be viewable and searchable from your computer.
What ways do you like to take notes electronically?
Good luck with the semester!
]]>The first time I entered the math library at Lebanon Valley College, I was struck by what I saw on top of the bookcases – a giant slide rule! Though I had never used one, I remembered my dad telling me about how he had to use a slide rule in his math classes in college. This iconic piece of mathematical technology owes its existence to the mathematical development that is celebrating its 400th birthday this year – the invention of logarithms.
I recently came across a link to a Science News article by Tom Siegfried entitled Logarithms celebrate their 400th birthday. The article discusses how John Napier revolutionized mathematics in 1614 with his invention of logarithms. Though many considered the logarithm a godsend, not everyone was convinced.
Napier’s mathematical wizardry wasn’t universally appreciated, though, as rumors swirled that he was actually a dark wizard, à la Lord Voldemort. For one thing, Napier’s grass seemed to be greener than other landowners. And he allegedly trained a magical black rooster to identify thieves among his workers.
The article also discusses how the invention of logarithms led to the development of the slide rule.
It wasn’t long until others figured out how to put the logarithms to use in mechanical calculations using sticks. Inscribing numbers on the sticks at intervals proportional to their logarithms made it possible to multiply numbers by proper positioning of the sticks. Edmund Gunter, a London clergyman and friend of Briggs, had the germ of the idea in 1620. But the honor of first to slide the sticks is usually accorded to William Oughtred, an Episcopal minister, who also devised a circular version of the slide rule in the 1620s.
The invention of logarithms introduced a new way for quickly performing complicated calculations, with one mathematician claiming that “logarithms effectively doubled a mathematician’s useful lifetime.” Without the tools that logarithms offer the sciences today, our understanding of the world would be greatly limited.
]]>Is Mathematics an art or a science? Calvin has a different perspective. Hmm…Calvin & I might need to converse. #ijs pic.twitter.com/99zx8nB6op
— Karen Morgan Ivy (@Afrikanbeat) March 7, 2014
(Transcription below by http://blog.onbeing.org/post/250746172/calvin-and-hobbes-math-is-a-religion)
First frame
Calvin: You know, I don’t think math is a science. I think it’s a religion.
Hobbes: A religion?
Second frame
Calvin: Yeah. All these equations are like miracles. You take two numbers and when you add them, they magically become one new number! No one can say how it happens. You either believe it or you don’t.
Third frame
Calvin: This whole book is full of things that have to be accepted on faith! It’s a religion!
Fourth frame
Hobbes: And in the public schools no less. Call a lawyer.
Calvin: As a math atheist, I should be excused from this.
Related to this, I recently saw a link on my Facebook page to an article on NYTimes.com by Elizabeth Green entitled Why Do Americans Stink at Math? The article discusses a Japanese teacher who has tried revolutionizing mathematics pedagogy.
Instead of having students memorize and then practice endless lists of equations — which Takahashi remembered from his own days in school — Matsuyama taught his college students to encourage passionate discussions among children so they would come to uncover math’s procedures, properties and proofs for themselves.
The article also talks about how the American mathematics teaching practices have seen several failed reform attempts in the past.
It wasn’t the first time that Americans had dreamed up a better way to teach math and then failed to implement it. The same pattern played out in the 1960s, when schools gripped by a post-Sputnik inferiority complex unveiled an ambitious “new math,” only to find, a few years later, that nothing actually changed. In fact, efforts to introduce a better way of teaching math stretch back to the 1800s. The story is the same every time: a big, excited push, followed by mass confusion and then a return to conventional practices.
How can we, as mathematicians, work to ensure new reform (we are in the midst of the Common Core) actually works?
]]>This weekend, I was helping paint flats for a play when an interesting problem arose – we wanted to use three colors of paint to create rectangles of different sizes on a rectangular flat, with the stipulation that no two adjacent rectangles were the same color. Prior to painting, we labeled the rectangles with their colors just to make sure everything worked out. We didn’t run into any problems, but one person did ask the question – was it possible to come up with a configuration of rectangles that was impossible to paint with our condition? Since we were only using three colors of paint, the answer, of course, was yes! In fact, when making up sample designs the night before, our director ran into the issue of three colors not being enough.
In mathematics, the Four Color Theorem (or Map Coloring Problem) states that, given any separation of a plane into contiguous regions (producing a figure we will call a map), no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color. For this problem, two regions are considered adjacent if they share a boundary that is not just a corner. It seems like a simple problem – given a map, how many colors do you need so that no two adjacent regions are the same color? However, this problem, so simply stated in 1852, was tantalizingly difficult to prove, leading to many false proofs and false counterexamples. It was not until 1976 that, with the aid of a computer, a proof was obtained!
The Four Color Problem was first formulated by Francis Guthrie in 1852 when he was coloring a map of the counties of England and noticed that four colors sufficed. He asked his brother Frederick if it was possible to color any map using only four colors while also requiring that adjacent regions (i.e. those sharing a boarder) be of different colors. Frederick, who at this time was a student of Augustus De Morgan, posed the problem to his advisor. De Morgan, in turn, communicated the proposition to his friend William Rowan Hamilton, stating:
“A student of mine asked me to day to give him a reason for a fact which I did not know was a fact — and do not yet. He says that if a figure be any how divided and the compartments differently colored so that figures with any portion of common boundary line are differently colored — four colors may be wanted but not more — the following is his case in which four colors are wanted. Query cannot a necessity for five or more be invented… “
There were several early attempts to prove this theorem, though all of them failed. The first widely accepted proof was given by Alfred Kempe in 1879, with another proof given independently by Peter Guthrie Tait in 1880. It wasn’t until 11 years later that Kempe’s “proof” was shown to be incorrect by Percy Heawood. In 1890, in addition to exposing the flaw in Kempe’s proof, Heawood proved the Five Color Theorem and generalized the Four Color Conjecture to surfaces of arbitrary genus. In 1891, Tait’s proof was shown to be incorrect by Julius Petersen. Though both early proofs turned out to be fallacious, they were not without value. Kempe discovered what became known as Kempe chains, while Tait found an equivalent formulation of the Four Color Theorem in terms of 3-edge colorings.
During the 1960s and 1970s, German mathematician Heinrich Heesch developed methods using computers to search for a proof. Unfortunately, he was unable to procure the necessary supercomputer time to continue his work. However, others adapted his methods and computer-assisted approach.
The Four Color Theorem was finally proven in 1976 by Kenneth Appel and Wolfgang Haken, with some assistance from John A. Koch on the algorithmic work. This was the first time that a computer was used to aid in the proof of a major theorem. The Appel-Haken proof began as a proof by contradiction. If the Four Color Theorem was false, there would have to be at least one map with the smallest possible number of regions that requires five colors. Two technical concepts were used in the proof:
1. An unavoidable set is a set of configurations such that every map that satisfies some necessary conditions for being a minimal non-4-colorable triangulation (such as having minimum degree 5) must have at least one configuration from this set.
2. A reducible configuration is an arrangement of countries that cannot occur in a minimal counterexample. That is, if a map contains a reducible configuration, then the map can be reduced to a smaller map (using fewer countries). If this smaller map can be colored using only four colors, then the original (larger) map can also be colored using only four colors. This implies that if the original (larger) map cannot be colored with only four colors, then the smaller map cannot be colored with only four colors either. Thus, the original map could not be a minimal counterexample.
Using mathematical rules and procedures based on the properties of reducible configurations, Appel and Haken found an unavoidable set of reducible configurations, thus proving that a minimal counterexample to the Four Color Theorem could not exist. Instead of examining all possible map configurations (an impossible task), the Appel-Haken proof was able to reduce the problem to looking at a particular set of 1,936 maps. Each of these maps, it was shown, could not be part of a smallest-sized counterexample to the Four Color Theorem. Since checking all of these maps by hand would be tedious and time consuming, Appel and Haken used a special-purpose computer program to confirm that each of the maps had the desired properties. Checking the 1,936 maps one by one took over one thousand computer hours. The reducibility part of this work was independently checked by different programs and different computers. However, the unavoidability portion of the proof needed to be verified by hand, leading to hundreds of pages of analysis. The Appel-Haken proof concluded that no smallest counterexample exists because it must contain, yet cannot contain, one of the specially chosen 1,936 maps. This contradiction shows that there cannot be any counterexamples and therefore the Four Color Theorem must be true.
Initially, this long-awaited proof of the Four Color Theorem was met with trepidation and concern. It was the first major theorem to be proven with extensive computer assistance. In addition, even the human-verifiable portions of the proof were highly complex and prone to error. These two facts caused many mathematicians to reject the proof and led to considerable controversy. In the years that followed the original publication of the proof, it has become more widely accepted, although doubts still remain.
To this day, many mathematicians and philosophers debate whether or not the Appel-Haken proof is legitimate. Some claim that “proofs” should only be accepted if they are proved by people, not machines, while others question the reliability of both the algorithms used and the ability of machines to carry them out without error. Others argue that even proofs written by people can be found to be faulty and so the reliability argument is meaningless. No matter what your opinion on the matter is, the Appel-Haken proof of the Four Color Theorem has led to a serious discussion about the nature of proof which continues today and will continue into the future as we make use of computers that are growing ever more powerful.
Sources:
Appel, Kenneth; Haken, Wolfgang (1989), Every Planar Map is Four-Colorable, Providence, RI: American Mathematical Society, ISBN 0-8218-5103-9
Wilson, Robin (2002), Four Colors Suffice, London: Penguin Books, ISBN 0-691-11533-8
]]>
To apply and see all qualifications, visit https://www.mathprograms.org/db/programs/280. The deadline for applying is July 18, 2014 at 11:59 p.m. Eastern Time.
]]>