I really like that it allows me to compile documents on the go whether I am connected to the internet or not. It also connects with Dropbox, so I can include all of my packages in a Dropbox folder and it will allow me to edit TeX documents I have saved in Dropbox. One issue I am having is I have not figure out how to select text in it yet (I think this might just be a glitch with iOS 8, though).
What TeX editors do you use on your iPad or other tablet? What do you like about them? What do you dislike about them?
]]>The method of markers is a fair division method which is used when
The method (for N players and M discrete items) can be described by the following process:
Preliminaries – The items are lined up in a random order. For convenience, the items are labeled 1 through M, going from left to right.
Step 1 (Bidding) – Each player independently divides the array of items into N segments by placing N-1 markers along the array. These segments are assumed to represent the fair shares of the array in the opinion of that player. In order to keep the method fair, the players must all place their markers at the same time. One way of achieving this is if all players submit sealed envelopes with the positions of their markers. Everyone is guaranteed to get a fair share because the segments for a particular player never overlap.
Step 2 (Allocations) – Scan the line of items from left to right until the first first marker is located. The player owning that marker (let’s call him or her P1) goes first and receives the first segment in his or her bid. (If there is a tie, it is broken randomly.) Since P1 has received his/her fair share, the rest of his/her markers are removed. We continue scanning the line of items from left to right, looking for the first second marker. The player owning that marker (let’s call him or her P2) goes second and gets the second segment in his or her bid (which corresponds to the segment between his/her first and second markers). Continue this process, assigning one segment of his/her bid to each player. The last player gets the last segment in his/her bid, which is to the right of his/her last marker.
Step 3 (Dividing Leftovers) – If there are items left over after every player has received a fair share, divide the leftovers among the players by some form of lottery. For example, the players could take turns choosing leftover items or, if there are many more leftover items than players, the method of markers could be used again.
Consider the following example. Three children (Abby, Bryan, and Chloe) are dividing the array of nine candy pieces shown in the following figure using the method of markers. The players’ bids are indicated in the figure (with A for Abby, B for Bryan, and C for Chloe).
In this example, Abby would consider any one of these shares to be fair:
{1} (Abby’s 1^{st} segment)
{2, 3, 4, 5} (2^{nd} segment)
{6, 7, 8, 9} (3^{rd} segment)
We start at the left-hand side of the line of candy and look for the first first marker (in this case, it is A1). Abby will go first and will receive the first segment in her bid:
Notice that the rest of Abby’s markers have been removed since she has now received her fair share. We now continue scanning the line of items from left to right, looking for the first second marker. The next marker we come across is B1. However, since this is not a second marker we ignore it. The first second marker that we come across is B2. Bryan will go second and will receive the second segment in his bid:
Now we are down to the last player, Chloe. Chloe will get the last segment in her bid, which is to the right of her last marker (i.e. to the right of her C2 marker):
In this example, the final allocation of candy is: Abby gets piece 1, Bryan gets piece 3, Chloe gets pieces 8 and 9, and pieces 2, 4, 5, 6, and 7 are all leftovers (which can be divided among the children using some form of lottery).
Despite its elegance, the method of markers can be used only under some fairly restrictive conditions. In particular, the method works best if the items are roughly equivalent in value to each other and if the players’ preferences are fairly close. This is almost impossible to accomplish when there is a combination of expensive and inexpensive items, but is perfect for dividing items that are of small and equal value (like candy). So go out and enjoy, knowing you are now equipped with a method to fairly split the spoils from trick-or-treating. Happy Halloween!
Sources:
Tannenbaum, Peter. Excursions in Modern Mathematics (Sixth Edition). Upper Saddle River, N.J: Pearson Prentice Hall, 2007.
]]>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.
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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.
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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.
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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?
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