Happy Thanksgiving! I’m sure your wonderful family is the exception, but sometimes holiday dinner conversations can veer into unpleasant territory. If you don’t have Adele to bail you out, math blogs are here to help. (When your only tool is a math blog, every problem looks like…well, whatever a math blog can solve.) If you need a conversation topic for a group of turkey-bloated people with a diverse range of opinions about politics, religion, and morality, here are some suggestions from the math blogosphere.

What your part of America eats for Thanksgiving, courtesy of FiveThirtyEight. There’s nothing more satisfying than talking about food as you devour it. Green beans or squash? Canned cranberry sauce or whole berry? Are Thanksgiving cherry pie eaters totally depraved or just slightly misguided?

With a few diagrams scratched into the mashed potatoes, anyone can appreciate this elegant proof of the Pythagorean theorem attributed to Albert Einstein.

Tanya Khovanova’s math blog is rife with fun puzzles, but my favorite at the moment is this one: How many letters are there in the correct answer to this question?

Likewise, Math Munch has lots of fun topics and activities to look into. Hypernom seems especially appropriate for this particular holiday.

How many pentagonal tilings can you make with turkey bones and orphaned green beans after dinner?

Mathematicians have proved that English is trivial. In that article, Alex Bellos gives “aisle” and “isle” as an example. They are homophones, so we can do a little algebra to show aisle=isle, which means a=1. In fact, we can find homophones to reduce every letter of the alphabet to 1. Can you prove it around the dinner table? As a bonus, try to prove it for other languages represented around your table.

The Erdős discrepancy conjecture, one of the many (fairly) easy to state but hard to prove conjectures about sequences of integers, was recently proven. Everyone likes numbers, so you can definitely get people talking about these conjectures and maybe even make some progress at the Collatz conjecture.

Graph isomorphism! Those feisty computer scientists are at it again, whittling down the known complexity of NP-intermediate problems. You can get the whole family involved in this one: draw some graphs with leftover cranberries and green beans and see if they’re isomorphic. Were you able to do it in polynomial time?

What is the funniest number? Do you agree with *Catch-[insert funny number]* author Joseph Heller and his agent?

If you did an average of a 15:00 mile over the course of the race, is there a mile you traveled in exactly 15:00? Get thee to a turkey trot for some real-world experiments!

What’s up with Shinichi Mochizuki’s proof of the abc conjecture? And what does the saga say about the sociology of mathematics? For that one, you might need to brush up on what exactly the abc conjecture is. Luckily, Numberphile has you covered.

Can you wrap your mind around the function x^{x}?

If you want to stir the pot but prefer to avoid politics, Is 5×3 five threes or three fives, and does it matter?

If all else fails, you can always gather the family around the computer screen, watch Vi Hart’s mathematical Thanksgiving celebration, and lament the fact that you made a green bean casserole rather than a green bean matherole.

However you celebrate or don’t, I hope you have a peaceful holiday.

]]>Graphs, you’ll recall, are simply collections of nodes linked together by edges. If you look at the image below, you see three different graphs. Each has four nodes linked together with four edges. But wait, are they actually different? If you take the graph to the far left and twist the top and bottom, you get the graph in the middle. So we say these two graphs are the same, or in mathematical jargon: isomorphic. Any time you can twist and deform to get from one picture to another (without detaching any of the edges or nodes), you are dealing with isomorphic graphs. So what about the graph on the far right, can you twist and pull that one to make it look like the first two?

Sure, you can. This is the essence of the graph isomorphism problem: how do we decide when two graphs are isomorphic? And this example above is pretty easy, but as you increase the number of nodes, this quickly becomes a really hard problem. In the parlance of algorithms, it is even thought to be NP hard. Or rather, it *was* thought to be NP hard until University of Chicago’s Laszlo Babai totally pwned it last week. It was announced that Babai has discovered a quasipolynomial time algorithm for deciding isomorphism of graphs, which is a massive discovery in the face of classicaly super-duper hard problems in complexity theory.

When we measure the speed of this sort of algorithm, we measure it relative to the number of nodes *n*. A really fast algorithm would only increase in complexity at the rate of a polynomial in *n*. A really slow algorithm would increase exponentially relative to *n*. Babai’s algorithm seems to sit somewhere between the two. It’s faster than anything that’s ever been done before, bit it’s not quite polynomial fast either, so fittingly, it is called quasi-polynomial.

Babai is unveiling his results in a series of three talks at the University of Chicago, the first two of which have already been graciously livetweeted by Gabriel Gaster — a sincere tip of the hat to Gaster for this exciting blow-by-blow.

Computer scientist and blogger at *Shtetl-Optimized*, Scott Aaronson, gives a nice breakdown of the striking features of this new result that make it such a dramatic step forward from its predecessors, including its relationship to string isomorphisms, finite simple groups, and Luks’ algorithm from 1982. For all the gory details on the mathematical content of Babai’s talk, Jeremy Kun posted a recap of the November 12th lecture, which he intends to update as further details emerge. In the meantime, there are some thoughtful discussions going on over in the comments section at *In Theory*, computer scientist Luca Trevisan’s blog.

As for the preprint, rumor has it that Babai said it will be available “soon, soon.” It’s all very exciting, but we have to remember to keep our proverbial hats on until everything is checked out. But it occurs to me that it suddenly became a lot more difficult to walk around glibly saying how *hard* it is to factor large numbers, and how *secure* our cryptosystems are because of that. Because who knows, graph isomorphism today…maybe factoring tomorrow?

People, especially sports fans, seem to believe that players can get “hot,” that they will hit more baskets (or succeed in whatever metric is of interest in their sport) more after a series of hits than after a series of misses. We statistics-savvy types are far too sophisticated to believe something like that, though, which is why we call it the fallacy of the hot hand. In 1985, Thomas Gilovich, Amos Tversky, and Robert Vallone wrote a paper suggesting that the “hot hand” doesn’t exist; it’s just a misinterpretation of the natural streakiness of independent events. But a few months ago, Joshua B. Miller and Adam Sanjurjo published an article that claims to find an important flaw in the 1985 paper. Perhaps there is a hot hand after all!

I’ve read several posts and articles about the fallacy of the fallacy of the hot hand recently. Jordan Ellenberg was on the case for Slate with a good explanation of the flaw (the moral of the story: averaging ratios is a dicey prospect), and George Johnson wrote about it for the New York Times. I also enjoyed the Gödel’s Lost Letter post about the problem, complete with a suggestion for fixing it.

Andrew Gelman has the most extensive coverage of the hot hand with four posts about it in the past few months and a nice one from 2014 as well.

One thing I really like about Gelman’s writing in general is thefact that he emphasizes effect size so much. He wrote, “A better framing is to start from the position that the effects are certainly not zero. Athletes are not machines, and anything that can affect their expectations (for example, success in previous tries) should affect their performance—one way or another.” To me, this puts the hot hand question into better perspective. (On the other hand, it also makes me want to give up—how can we possibly untangle all the contributing factors to any aspect of human behavior or performance? I suppose that’s why I’m in math and not psychology.)

The problem Miller and Sanjurjo found with the earlier research boils down to a fairly simple observation about probability, so posts about it eventually spill into questions about coin flips and boys and girls born in different families. Steven Landsburg writes about boys, girls and hot hands, comparing the problem with the Gilovich, Tversky, and Vallone paper to a probability question about the ratio of boys to girls in a village where families stop having children when they get a boy. I’m not sure I quite agree with him that the situations are analogous, but he has a good explanation of the hot hand issue, and there are some interesting threads in the comments.

It seems that these probability discussions tend to spill into the dreaded questions about boys born on Tuesdays. I have always bristled at these types of puzzles because they feel like semantics exercises rather than probability exercises. Once again, Jordan Ellenberg writes about them for Slate, leading off with this hilarious exchange:

Toby and Marla are playing a game with coins, because that’s what people in math problems do. Toby flips a fair coin three times, out of Marla’s view. “Did you get any heads?” Marla asks

“Yes,” Toby says. “For instance, the second coin came up heads.” (Because that’s how people in math problems talk.)

“I’ll bet you the next flip after that came up tails,” Marla says.

Is this a good bet?

Ellenberg’s point is that puzzles like this *are* psychological to some extent rather than mere exercises in probability—the right answer depends on our assumptions about how people communicate about coin flips or children and when they were born (mod 7). He also points to a great post by Tanya Khovanova about sons born on Tuesday which she followed up with an invented dialogue about the problem. I think I will refer to her posts the next time someone tries to trick me with a riddle from that genre.

Do you have any sons born on Tuesday? If so, what is the probability that you also have a daughter? Does she tend to make more baskets after making a few in a row, or does her basketball shooting record support the null hypothesis?

]]>This is quite easily the best low-budget Canadian horror movie to come out of the last millennium, truly thrilling. As a quick synopsis, a group of strangers wake up trapped in a cube. The cube has trapdoors on each face, on each trapdoor has a serial number consisting of a sequence of three-digit numbers. Through each trapdoor is another cube, and the film follows these strangers as they stumble from cube to cube. They quickly realize that some seemingly random cubes are booby-trapped with tripwires and acid. Luckily they have a math major in the cube with them (and hey, who doesn’t want a math major around during a tense situation?) and she realizes that as long as there are no prime numbers in the serial sequence, it’s safe to pass to the next cube. She starts to do rapid factorization in her head and deducing that the sequence 645, 372, 649 contains no primes, jumps through to a safe cube, and mouth agape, throws her head back in relief shouting, “PRIME NUMBERS! PRIME NUMBERS!” (much like I do most afternoons in the privacy of my study.)

The young math major at one point freaks out, saying that factoring large numbers is an “astronomical” task, one that she can’t possibly do in her head! While this is technically a true statement, in the world of primes, anything with less than 6 digits really doesn’t count as large. In fact, some of our most secure modern encryption is based purely on the fact that really big numbers are hard to factor. Nevertheless, it made me wonder. If I were trapped in a cube, and the responsibility fell onto my humble shoulders to find the primes, what are the smartest algorithms I could use to factor things?

Suppose we’re looking to factor a number *N*. First, there’s the classical check-all-numbers-and-see-if-they-divide-*N* method. Since you really only need to check up to *√N*, this method isn’t terrible for small values of *N*.

A similary efficient method is Fermat’s Factorization Method, which relies on the fact that any odd number can be written as a difference of squares. So if we can manage to write *N* as

N=a^{2}-b^{2}=(a+b)(a-b)

then we could reframe our quest for factors to the quest for suitable *a* and *b*. This method on its own isn’t much better than the old check-all-the-numbers method, but some combination of the two gives you a marginal improvement in efficiency.

Assuming *N* has at least one smallish prime factor, Pollard’s rho algorithm should give an improvement over the previous two methods. For larger numbers there are sieve methods. Carl Pomerance explains in the Notices of the AMS, “In 1990 my own quadratic sieve factoring algorithm had doubled the length of the numbers that could be factored, the record having 116 digits.”

The hardest numbers to factor are semiprimes — these are numbers which are products of two (not necessarily distinct) primes — which are the product of two really huge primes. To date, the largest one to be factored by computer was 232 digits long. A group did it by running a whole bunch of processors in parallel, but if you measured it in single CPU time, it would have taken 2000 years! Just so you can get a sense of what 232 digits looks like, here is the number:

12301866845301177551304949583849627207728535695953347921973224521517264005

07263657518745202199786469389956474942774063845925192557326303453731548268

50791702612214291346167042921431160222124047927473779408066535141959745985

6902143413

Spooky, huh?

]]>Last year, I had my math history students write a blog. The course counts as a writing credit, so blog posts seemed like good short writing assignments. But what about blogging in a math class that’s just a math class? My academic sibling Casey Douglas, who teaches at St. Mary’s College of Maryland, incorporates blogging into his Foundations of Math class, and I decided to ask him a few questions about it. If you want to jump down for links to some of his students’ posts, click here.

I hope other teachers interested in class blogging will get some ideas from him. If you’ve used blogs in your classes, feel free to share in the comments!

**EL**: First, can you say a little bit about the class that’s blogging: size, typical student majors, main topics covered, main learning goals?

**CD**: This is class called Foundations of Mathematics, a relatively fancy name for a standard proofs-writing course. The class size can vary, anywhere from 15 students to 30-ish students (this semester I have 21 students). Roughly half of the students are math majors (or will be.. this class tends to hook them), and all of them are at least math minors. (St. Mary’s offers an especially schedule-friendly math minor, one that requires Calc 1, Calc 2, Linear Algebra, Vector Calculus, and Foundations of Math (FOM); students who are already majoring in subjects that require some of these classes find it very tempting to complete the minor. Our department uses this as an opportunity to recruit more majors, encouraging students to take just one more course.) That said, the vast majority of students in this class are a bit “on the fence” when it comes to math, most being fairly nervous about the course as they’ve heard “it’s like no other math class before.” My primary learning goals are (1) to have students learn to critically assess logical arguments (especially proofs), (2) to have students learn to compose their own precise, logical arguments (in particular, mathematical proofs), (3) to have students understand the value and idea of a mathematical proof, and (4) to expose students to a wide variety of mathematics (including history and current research).

**EL**: What are your goals for students’ blog posts? Or to put it another way, why did you choose blogging rather than more traditional writing assignments or in-class presentations?

**CD**: This class was first devised quite a while before I arrived at St. Mary’s, and it’s undergone a number of revisions since then. For much of the past decade or so, instructors for the course had students maintain a journal. They were expected to make regular entries where they could record their thoughts on/reactions to readings, assignments, and in class activities. One of the main points or benefits to this was having students regularly thinking and writing about this course; it encouraged (perhaps forced?) them to develop important habits—reading, re-reading, engaging with the text, working out examples—and kept the challenging topics on their mind. There was a significant drawback, though. At a few times during the semester, the instructor would collect these journals and assess them (how many entries, how in depth they were, etc.), and this left them with a daunting pile of books to sort through.

When I first had the opportunity to teach this course, I knew I wanted to avoid those piles, and requiring regular blog posts from my students seemed like an ideal way to do this. Like many others, I spend a great deal of time reading things online anyways, especially math blogs. I figured it would be easy to keep frequent tabs on my students’ blogs by incorporating them into my media diet. Its also easy to read through blog posts while, say, doing some other activity (like waiting for a child to doze off to sleep or washing dishes). I have them use WordPress, which supports (a version of) LaTeX, and so many of them also get the benefit of learning mathematical typesetting. Another benefit to the blogs is the creation of a community. Student bloggers often consult with others by reading through their old posts and commenting on them. (This usually takes some time, though, roughly about 2/3 the way into a given semester.) Finally, regularly-updated student blogs help me better assess students’ needs, struggles, and progress far better than homework assignments and nerve-wracking exams; I am able to customize class activities, lectures and discussions, I am able to write especially rewarding exams and I am able to create more helpful activities as a result. For all of these reasons I now require FOM students to maintain a blog whenever I have the pleasure of teaching the course.

**EL**: Is this the first time you’ve had students blog in class? If not, how did the earlier iterations go and have you made any changes as a result?

**CD**: This is my fourth time teaching FOM, and it is my fourth time requiring blogs. I have not implemented too many changes since the first time doing this (back in the Fall of 2012). After using this the first time, though, I realized one potentially serious issue: students may not wish to participate in such a public activity. I now make it very clear to students that, while they need to maintain a blog for this course, they do not have to use any identifying information, so they are not required to use their name, my name, the name of our school, etc. I think the next change I will implement for next semester’s FOM class is, in addition to regular blog posts, requiring students to comment on others’ blogs. Perhaps this will help establish a sense of community in a more timely manner.

**EL**: Overall, has blogging done what you hoped it would for your class? Have there been any unexpected difficulties or benefits to having the students write blogs?

**CD**: Overall, yes, I think having students maintain these blogs did achieve what I (and previous instructors for this course) wanted. (I mentioned several unexpected benefits in my answer to 2, so I won’t repeat them here.) There have not been any unexpected drawbacks, which is to say there have been some, but they were easy to anticipate. In particular, there are students who neglect their blogs and end up with a lower grade as a result. I am not especially bothered by this since it is usually easy to fix — a simple e-mail reminder to a student usually does the trick. Some teachers may not like the idea of students getting answers from others’ blogs, but I am unbothered by this; indeed, I view it as a net gain for the class.

**EL**: Were your students enthusiastic or reluctant about blogging? Have you had any privacy concerns or students who didn’t want to share their work publicly?

**CD**: Students are usually fairly reluctant to blog, mostly because it initially sounds like a lot of tedious work. However, the majority of them really get into it after two or three weeks. Many find great joy in developing their voice, their mathematical voice in particular, and quickly come to appreciate the benefits of regular blogging; they seem to agree that it helps them remember abstract ideas and definitions, and given the fact that FOM focuses so much on writing (especially proof writing), they also agree that their blogs provide them much needed opportunities to practice. While I have yet to have a student express privacy concerns, after my first semester doing this, I realized I needed to address this issue. I try to make it clear that blogs need not contain any identifying information, and I also provide them with the option of having their blog not listed on mine.

**EL**: Have you gotten any feedback from students about whether writing the blog and reading other students’ posts were helpful for them?

**CD**: At the end of all of my previous FOM courses, all of the written comments I received about the blogs were positive, and for precisely the reasons you mention. Students found it convenient to read what others in the class were doing by simply skimming their blogs; they also found it helpful to practice expressing their ideas in written form. They also appreciated how easy it was for me to provide feedback (in the form of comments on particular posts). Although negative comments were never offered on a written evaluation, I have talked one-on-one with students who remained staunchly anti-blog throughout the course and after it concluded. These students did not see a point in writing their thoughts and posting them online, since they thought homework assignments were helpful enough. And, who knows? They may have been exactly right; perhaps their understanding of the foundations of mathematics was not helped by this activity. Having to do something that you don’t especially value or agree with, though, is also a good experience to have, too, especially if you hope to work in any job or career.

Also, one other noteworthy and unexpected (though obvious in hindsight) benefit: reaching introverts. Although I enjoy speaking in certain public forms (like this interview, in my classes, at conferences), I am a pretty standard introvert. Talking and interacting with others exhausts me, and there are lots of students for whom participating in class (be it discussions or activities) is similarly challenging. Blogging their thoughts and ideas helps them participate in meaningful and sustainable ways, I’ve found, and it helps them feel included. Their work is often the best in this arena, too, often resulting in more thoughtful and helpful blog posts.

**EL**: Those of us who are interested in blogging in the classroom are probably curious about some logistical details: how do blog posts figure into a student’s grade, and how do you grade them? How do students submit their posts to you?

**CD**: I have a relatively simple way to incorporate blog grades into a students’ overall grade. Here is a copy of my syllabus for this semester (https://mathematicalypse.files.wordpress.com/2015/05/fomsyll-21.pdf), where the Blog is said to count for 20 percent of the final grade. Roughly speaking, I expect students to post at least one entry on their blogs per class period. How long should each post be? How much math talk should they include? These questions can only be answered subjectively, but I provide some rough guidelines (both in class and in a blog post explaining some details, which can be found here: https://mathematicalypse.wordpress.com/2012/08/26/guide-to-fom-blogging/). If you have three posts per week, and they contain a rigorous and/or helpful discussion of topics we’ve discussed in class, then you’ll be awarded full credit. Alternatively, if you post only once per week, but the entry is especially long and thoughtful—so much so that it read like three “regular sized” posts jammed into one—you can also earn full credit. Similarly, you can post 6 times per week, publishing shorter thoughts and reactions and rack up a full amount of credit that way. I also encourage students to post about mathematics from other classes and that they’ve been thinking about or reading about on their own, so there are lots of ways to rack up “blog points.”

I check my students’ blogs multiple times a week, and blogs are graded on a weekly basis (on Fridays in this class). So every Friday I assess the number and quality of blog posts a student has done that week and assign a grade — usually its a nice round number, like 100, 90, 80, 70, 60, or 50.

By the way, I am not certain that 20% is the ideal number for this component, but it seems to work for my classes. People who are blog-negligent do not fail the class, but their grade does suffer, and no one is able to do well in the class by blogging alone.

**EL**: Any advice for math teachers who want to incorporate blogging into their classrooms?

**CD**: I think the most important advice I have is to maintain a blog yourself, and not just one about teaching. It can be a wonderful experience for teacher and student alike to share their thoughts on mathematics via their blogs, especially when the mathematics involves not just the topics discussed in class but the mathematics that happens to be on someone’s mind. For instance, I knew that I wanted to use my blog to keep track of some research interests I was pursuing the first time I taught FOM, and it was helpful for me to feel like a student again. I documented my progress, my setbacks, and my new ideas on my blog (never expecting or requiring students to read these entries, but never hiding them either), and it helped me understand better the progress, setbacks and ideas my own students were sharing on their blogs. It let me model for my students how to approach struggles and questions in mathematics, it also helped students view me as a member of their learning community.

**EL**: Can you share some of your favorite posts from students?

**CD**: There are so many to share. I actually asked some of my current students if they would have any objections to their blogs being mentioned in your post on this, and all were excited at the possibility. So feel free to explore other posts from the ones I mention here. Here is one by a current student, https://smcmfom.wordpress.com/2015/10/03/can-money-buy-happiness/ —in it she works out some examples of if-then statements (and their contrapositives), all on her own. The benefit of creating her own examples and sharing them with others is far more meaningful than that obtained by reading examples I come up with or our textbook uses. Here is another interesting post, https://scbergen.wordpress.com/2015/09/25/proof-of-math/, where a student wonders aloud about, essentially, metamathematics. When we spend some time talking about Gödel later this semester, I will be able to turn to this post to generate relevant discussion.

Here is one about a Problem of the Week (POW): https://tripletintegral.wordpress.com/2015/10/02/the-one-with-the-pow-oct-1/. What’s so great about this post is how helpful it could be/was to other students who struggled with this question. The student’s thought process is so clearly laid out that it will no doubt ring true for others in the class who also attempted this question.

This blog was from 2013, and the last two entries (on paradoxes and on blogging) are especially interesting, I think: https://cdmachlin.wordpress.com

In general feel free to pick through any of my old or current student FOM blogs for interesting posts. Warning: there are a fair number of expletives. Also, this 2013 student was obsessed with working in lots of cat pictures: https://myhypotamoose.wordpress.com/page/2/

[Editor’s note: Casey shared a couple more posts with me: https://fomdamentals.wordpress.com/2013/04/25/the-scavenger-hunt-solutions/, https://fompwomp.wordpress.com/2013/05/09/im-a-big-deal/. You can find the list of FOM blogs here: https://mathematicalypse.wordpress.com/2012/08/26/guide-to-fom-blogging/]

**EL**: Are there any questions you wanted to answer that I didn’t ask? Please feel free to add them!

**CD**: Maybe! So I’ve blabbed long enough about all of this, but I also do one other “unusual” thing for this FOM course. Their final exam is self-referential; the first question on the exam is to find the final exam. Students have to complete an elaborate, mathematical scavenger hunt to find the final exam. They are expected to blog about their successes and failures with this hunt and collaborate in completing it. This seemed to be one of the best uses of the blogs; students responded very positively to it.

Last week The New York Times ran an article by Tim Chartier of Davidson College — and math miming fame — and Sharon Jones of Central Piedmont Community College, all about using sports analytics in the classroom. For those interested in bringing sports or data analytics into the classroom, this article is a fantastic primer. It explains how to gather large data sets and then visualize, analyze, and interpret them.

This is of course a valuable skill in general for anybody looking to pursue a career in data science or data journalism. However, I would argue that sports analytics as a particular teaching tool for this skill can be a bit fraught. In my personal experience, the deeply gendered aspects of professional sport (the players, the spectators, the money handlers) can make it difficult for female students to, quite frankly, care at all. And I’m not suggesting that women don’t like (or play) sports, but one doesn’t need to look too far to find data sets that aren’t as *blatantly* gendered as NFL, MLB, and NBA sports stats.

But as an immediate counterpoint to what I just said, the ever-awesome Laura McLay of the blog *Punk Rock Operations Research* just posted about using game theoretic strategies or linear programming to decide if a football team should pass a ball.

So, I guess I should conclude by saying that nothing drives home a mathematical concept like an applied approach, but let’s just remember that WHIP, WAR, and VORP aren’t necessarily meaningful to everyone. Also, if you know the answer to my batting average questions, tweet it at me @extremefriday.

]]>Math3ma was originally created as a tool to help me transition from undergraduate to graduate level mathematics. Quite often, I find that the ideas of math are hidden behind a dense fog of formalities and technical jargon. Much of my transition process has been (and still is!) learning how to fight through this fog in order to clearly see the ideas, concepts, and notions which lie beneath. Throughout this process, I’ve found that writing helps immensely.

Math3ma would probably be a helpful study aid to a first-year graduate student or a nice preview for someone getting ready for graduate school. The posts include topics from first-year analysis, algebra, and topology, from Lebesgue and Borel Measurable sets to simple and non-simple groups to connectedness. I’ve been thinking about open sets recently, so I especially enjoyed her post about why open sets are everything. (They really are!) Her introduction to Galois theory is a nice overview and includes a suggestion for an abstract algebra TV commercial tagline. “Groups: it’s what they do.”

In addition to the main posts, keeps a few tidbits in her back pocket: an unspoken rule of algebra (Stuck? Try the first isomorphism theorem!), two ways to be small, and other helpful morsels.

All of Bradley’s posts can help students get from the “OK, I understand the words in the statement and proof” stage to the “how does this fit into the big picture” stage. For me at least, that is one of the biggest challenges when learning new math. In my first algebraic topology course, it took me a couple months to realize that the fundamental group wasn’t just something we defined in passing but a major focus of the subject. (In retrospect, the word “fundamental” should have tipped me off.)

Reading Math3ma, I wonder whether my first year of grad school and qualifying exams would have been easier if I had started blogging a few years earlier! A blog like this definitely would have made a good resource when I was studying for quals. I’m looking forward to reading more from Bradley as she advances in grad school and her career in math.

]]>Suppose you have some arbitrary sequence of 1 and -1, something like this

1, 1, -1, -1, -1, 1, 1, 1, 1, 1, -1, -1, ….

And suppose you start plucking entries from fixed intervals and adding them together. For example, if I just pick every third entry from the sequence, and add them all together, I would get

-1 + 1 + 1 + -1 + …

If I carry on doing that for some finite amount of time, will that sum get as big (positive or negative) as I want? Very simply, this is the idea behind the Erdős Discrepancy Problem, which claims that for any arbitrary sequence *{x _{1}, x_{2}, x_{3}, …}* where the

x

_{k}+ x_{2k}+ x_{3k}+ … + x_{nk}> C.

Sounds easy enough, right?

Not quite. And indeed this problem perplexed the great Paul Erdős to such an extent that he offered a $500 reward for a solution in the 1950s. And let’s hope that the late Erdős left a stack of cash lying around, because as of last week the problem has been solved by Terence Tao and the contributors to Polymath5.

In his blog, Tao gives a series of technical explanations of how he and the people of Polymath5 solved the famous problem. He explains how the key to cracking open the Discrepancy Problem was actually solving a totally different problem called the Elliot conjecture. This is such a common occurrence in math (and all fields, I suppose) that solving a really hard problem turns out the be *just* solving another really hard (but maybe a wee bit easier) problem.

The Polymath project — whose most well-known result to date is the tightening of the prime gap bounds of Maynard and Zhang — is a crowd-sourcing effort to drive mathematical breakthroughs. Mathematician Timothy Gowers, over on his webblog, gives a great commentary on the collaborative nature of this result. Although Tao was admittedly the one to put bring the whole thing to a culminating result, the terrain of the foundation would have looked much different without the crowd-sourced contributions. Gowers comments,

“My own experience of polymath projects is that they often provoke me to have thoughts I wouldn’t have had otherwise, even if the relationship between those thoughts and what other people have written is very hard to pin down — it can be a bit like those moments where someone says A, and then you think of B, which appears to have nothing to do with A, but then you manage to reconstruct your daydreamy thought processes to see that A made you think of C, which made you think of D, which made you think of B.”

This result is a marked victory for polymath, speaking to the impressive power of collaboration in mathematics, and an exciting success for Tao. And Tao will not be spending his well-deserved $500 on high-priced journal fees, since he will be publishing his paper in the new Arxiv overlay journal Discrete Analysis.

]]>Although I’m new to teaching using IBL, I’m not new to IBL. The class that made me want to be a mathematician was taught using an IBL model where students presented all theorems and proofs. The class structure was perfect for me, but it turns out my experience as a cocky college student in the class isn’t very helpful for me now that I’m a less cocky teacher trying to run the class. Luckily, between my facebook friends and math blogs, I do have a lot of places to go for guidance.

There are many different ways to implement IBL. I decided to emphasize student presentations of material. I give them notes with definitions, theorems, and exercises, and they supply the proofs in class. The notes I’m using are by Richard Spindler, and I found them in the Journal of Inquiry-Based Learning in Mathematics. I am of course making some modifications, but it’s been incredibly helpful to have a base to work from. I follow more IBL-ers than I can mention here, but for course structure, I am probably most indebted to Dana Ernst. I borrowed heavily from his past syllabuses when I was setting mine up. I have also benefitted from reading Dave Richeson’s posts, especially the one about his inquiry-based topology class. While I was preparing for the first few weeks of class, I found Carol Schumacher’s Instructor’s Resource Manual (pdf) invaluable. It is designed to be a guide to use with her abstract mathematics textbook *Chapter Zero*, but the first 30 pages are about course organization and strategies for various types of active learning in the classroom, and her advice in those pages applies to many different math subjects.

There’s a new IBL blog on the block that has also been interesting for me: the Novice IBL Blog. It’s a joint blog by David Failing of Quincy University, Liza Cope of Delta State University, and Nick Long of Stephen F. Austin State University that’s been running for about a month, the same length of time as my class. It’s nice to feel like there are a few other people with some of the same questions I have, and reading about their experiences. For example, I am currently not entirely satisfied with the quality of course presentations and of my assessment of them, so I am grateful that Cope has written about the way she evaluates presentations. I also keep an eye on Stan Yoshinobu’s IBL blog. I’m trying to incorporate more of his tips for positive coaching into the way I give feedback.

I’m not going to pretend things have gone entirely smoothly for me so far. I had more students drop the class at the beginning of the semester than I expected, and the drop rate for women was higher than it was for men. One of the reasons I decided to go for IBL was that many people believe that IBL and other active learning techniques are more fair to people who aren’t well-off white men. (See for example this recent New York Times article about whether lectures are unfair.) Of course I could not do exit interviews of students who dropped (they tend to ghost), so I don’t know why they left, but I am disappointed that my class has not retained women as well as men. If I set up a similar course in the future, I need to think carefully about how to get better student buy-in and assuage students’ fears that they are not prepared for the class.

So far, my semester has been a bit of a roller coaster, but I’m glad I’m trying something new. And I’m extra glad to be teaching complex analysis. For never was a math topic of more *whoa* than this of complex differentiation and its lovely Cau…chy-Riemann equations. (With apologies to Shakespeare and all people of good taste.)

To quickly recap, academic journals are ridiculously expensive to administer, subscribe to, and even publish in. This is all so crazy when you imagine that I, as a junior faculty looking towards tenure, am required by my institution to publish in the leading journals. So I submit a paper, the fruits of months of my own labor and hand it over to the Journal of Whatever and Such-and-Such. If it is accepted, I’m then given the option: publish it open-access for a cost of upwards of $1,500 (typically paid by the university, I guess, to be honest I’ve never picked this option) or publish it normally, in which case the only people who can read it are those who pay the — some huge amount of money — per year subscription fee to the journal. And in the age of the internet, what is it that you’re really paying for? Official journal typesetting, a meager 200 KB of server space to host the paper, and a whole bunch of pointless overhead.

But as long as you can use LaTex, the official typesetting really isn’t all that important. Reach some minimum baseline aesthetic and your paper is more than readable. And I can think of a place where you can upload all the KB of paper you want: the ArXiv. So really, all you’re paying for is the pointless overhead part. Enter Tim Gowers, host of the great *Gowers’s Weblog*, and the editorial team of the new journal *Discrete Analysis*.

What they are launching is called an ArXiv overlay journal. What this means is, from the editorial standpoint it looks exactly like a traditional journal. The editorial board deals with submissions, sending them out to be peer-reviewed by appropriate reviewers, and managing the quality of content of the journal. Accepted papers are edited and the typesetting is cleaned up, per the referee’s suggestions, and uploaded to the ArXiv. Then *Discrete Analysis* hosts a page of ArXiv links with abstracts. Your paper has gone through all of the rigors of the traditional peer-review process, and it will even get a special *Discrete Analysis* stylized bibtex entry, but you’ve saved everyone a lot of time and money.

I think it’s an interesting model. I mean, we all want to do math, we want to publish math, and we want to read math. It makes sense to keep the process as close to the mathematicians as possible. Right now the electronic end of the journal is being run on a relatively cheap platform that costs $10 per submission. That’s currently covered by a grant, but when that runs up, Gowers points out, “the absolute worst that could happen is that in a few years’ time, we will have to ask people to pay an amount roughly equal to the cost of a couple of beers to submit a paper, but it is unlikely that we will ever have to charge anything.”

Gowers, who won the Fields Medal in 1998, gives a great point-by-point explanation of every facet of the publishing process, and whether you agree with the feasibility of the model or not, I think it’s a worthwhile read.

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