### Archives For Mathematics Posts

On July 27, 2013, I created a petition on Change.org to get math.ED – Mathematics Education added as a category on the arXiv. You can find the petition here. At present, there is no dedicated category on the arXiv for math ed. In addition, there is no culture of math ed folks utilizing pre-print servers like the arXiv. I’d like to change both of these facts. If you want to know how all this got started, check out this post.

As of August 1, there was just shy of 200 signatures. My initial goal was 50. The support has been quite impressive. Most of the signatures are from the United States, but there are others from around the world. As far as I can tell, support is coming from people with interests in math ed, physics ed, STEM ed, ed tech, math, stats, operations research, secondary education, and more. I even recognized at least one philosopher. Thankfully, it seems we have the support of a few prominent math ed researchers (e.g., Alan Schoenfeld), which I think is crucial for this to really work.

I’ve also had about a half dozen people contact me to say that they would be willing to serve as a moderator for a math ed category on the arXiv. This is one of the things the arXiv told me that we would need to move forward. The list of people I have is probably more than sufficient.

As exciting as all this has been, it hasn’t all been rainbows and unicorns. Without going into detail, I had one person post a response to a comment I left on a discussion board announcing the petition that essentially told me that mathematics education is worthless and that there is nothing worth placing on the arXiv. If you are interested, I’m sure you can find the discussion. I maintained control and didn’t respond.

Also, I used Change.org on someone else’s recommendation and so far it has seemed to work well. However, I received an email from someone that signed the petition that was upset that Change.org sent them follow-up emails. I sincerely apologize if anyone else was annoyed or offended. According to Change.org:

Every so often you can expect to hear from Change.org about campaigns we think you’ll be excited to join. If you’d prefer not to receive these emails, you can unsubscribe by clicking the link at the bottom of any message you receive from us.

I realize it is a hassle, but it appears that you can opt out of any future correspondence with Change.org.

There was also an interesting discussion on Twitter that Republic of Math (@republicofmath) and I (@danaernst) had. You can read more about that conversation here.

In addition, there have been a few people here and there that aren’t supportive for one reason or another of the endeavor to utilize the arXiv for math ed. I’m okay with that. Heck, maybe this is a bad idea and if someone has arguments about moving forward, I want to hear them. I’m not so dead set on this happening that I won’t listen to reason. In general, I’m in favor of sharing knowledge in ways that are open and easily accessible. This is my motivating principle.

OK, so where do we go from here? There have been a couple of developments with the folks at the arXix, which started with a comment that Greg Kuperberg left on my original blog post. Greg is the chair of the math advisory committee for the arXiv, which is the committee that would approve a new math category. According to Greg:

What I can tell you at this stage is that the “petition” that I would like to see is enough postings to math.HO to justify a separate math education category. Creating a separate category first just in the hope that it will attract interest hasn’t usually worked well in the past.

Another possibility is to change the name of the math.HO category to better reflect its purpose. That’s a more welcome option than multiplying the list of categories.

I would rather negotiate a change to the name of a category in private. However, I can say that the name “History and Overview” has never been all that great of a fit for the topics listed with it, so a name change of some kind could make sense. Of course those topics don’t just include math education, but also closely or loosely related topics such history of mathematics and recreational mathematics.

In any case, “a rose by any other name would smell as sweet”. Getting more math education submissions to the current math.HO is partly a separate matter. I’m certainly all for more encouragement of that. Again, we can discuss techniques in private.

Getting a moderator for math.HO is also a good idea; once again, we can discuss.

I’ve since followed up with Greg privately and it seems that the most likely scenario is a name change to the math.HO – History and Overview, which lists math education as one of the possible topics. A name change seems reasonable to me. However, in order to move forward, the arXiv would like to see math ed submissions to the math.HO category. Again, this seems reasonable. In addition, there is currently no moderator for the math.HO category, so we would still need to move forward with the list of folks I gathered.

At this point, what we need is for people to start uploading their math ed related manuscripts to the math.HO category on the arXiv. I think this will require some guidance, as well as a discussion about copyright and such. I think I’ll save that for a future post.

During Susan Ruff’s talk in the IBL Best Practices Session that Angie Hodge, Stan Yoshinobu, and I organized at MathFest, she made reference to an article by Kirschner, Sweller, and Clark. The paper is titled “Why Minimal Guidance During Instruction Does Not Work: An Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential, and Inquiry-Based Teaching” (PDF) [1]. As a practitioner and serious proponent of inquiry-based learning (IBL), I am extremely interested in what this article has to say. Here is the abstract:

Evidence for the superiority of guided instruction is explained in the context of our knowledge of human cognitive architecture, expert–novice differences, and cognitive load. Although unguided or minimally guided instructional approaches are very popular and intuitively appealing, the point is made that these approaches ignore both the structures that constitute human cognitive architecture and evidence from empirical studies over the past half-century that consistently indicate that minimally guided instruction is less effective and less efficient than instructional approaches that place a strong emphasis on guidance of the student learning process. The advantage of guidance begins to recede only when learners have sufficiently high prior knowledge to provide “internal” guidance. Recent developments in instructional research and instructional design models that support guidance during instruction are briefly described.

I intend to read the whole article, but haven’t read much more than the abstract. Here are few thoughts before I dive in.

When discussing the advantages of an IBL approach with people, I’ll often cite academic work that supports the claim that it is beneficial for students. For example, see the work of Sandra Laursen et al. located here. However, to be honest, despite my interest in seeing data that validates my own opinions, the reality is that I don’t do IBL because the research told me to. I do it because I’ve seen it work! My students tell me it works. Alright, to be fair, my students told me that my lecturing worked, too. But the types of comments I get now from my IBL students make it clear to me that something really good is happening. For example, read this. IBL may not work for everyone in all situations and I’m okay with that. If it stops working for me, I’ll try something different.

The first thought I had when I saw the title and abstract was, “what does ‘minimal guidance’ mean?” I certainly provide a lot less direct guidance in my IBL classes than I do than when I lectured, but is it “minimal”? I do my best to provide scaffolded guidance to my students and to set up a support network in a safe learning environment. This is crucial in my opinion. I’ll have to digest the whole paper to see what their take is.

It appears that there are several reflections and discussions of this paper online already. For example, go here, here, and here. In addition, Kirschner, Sweller, and Clark have written a response to criticism that they have received in their “Why Minimally Guided Teaching Techniques Do Not Work: A Reply to Commentaries” (PDF) [2]. I’ll try to read this paper, as well.

### Bibliography

[1] P. A. Kirschner, J. Sweller, and R. E. Clark, “Why Minimal Guidance During Instruction Does Not Work: An Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential, and Inquiry-Based Teaching,” Educational Psychologist, vol. 41, no. 2, pp. 75–86, 2006.

[2] J. Sweller, P. A. Kirschner, and R. E. Clark, “Why Minimally Guided Teaching Techniques Do Not Work: A Reply to Commentaries,” Educational Psychologist, vol. 42, no. 2, pp. 115–121, Apr. 2007.

On July 27, 2013, I created a petition on Change.org to get math.ED – Mathematics Education added as a category on the arXiv. You can find the petition here. At present, there is no dedicated category on the arXiv for math ed and I’d like to change this. If you want to know how all this got started, check out this post.

The last time I checked, we were just shy of 200 signatures on the petition. My initial goal was 50. The support has been quite impressive. Most of the signatures are from the United States, but there are others from around the world. As far as I can tell, support is coming from people with interests in math ed, physics ed, STEM ed, ed tech, math, stats, operations research, secondary education, and more. I even recognized at least one philosopher. Thankfully, it seems we have the support of a few prominent math ed researchers (e.g., Alan Schoenfeld), which I think is crucial for this to really work.

There have been a few developments with the folks over at the arXiv and I’ll share the current state of affairs in another post. In the meantime, I thought you might enjoy a conversation that happened on Twitter between myself (@danaernst) and Republic of Math (@republicofmath). Matt Boelkins (@MattBoelkins) chimes in at the end, too. The conversation wasn’t linear, but I’ve done my best to list the tweets in an order that makes sense.

I’ve recently finished coauthoring two different math education papers. Both papers have been submitted for publication. Neither paper is likely to change the world, but I still want to openly share what we have. Like most mathematicians, I’ve posted my pure mathematics research articles on the arXiv. In case you don’t already know, the arXiv was started in 1991 as an electronic archive and distribution server for research articles. Covered areas include physics, mathematics, computer science, nonlinear sciences, quantitative biology, and statistics. Readers can retrieve papers off the arXiv via their web interface and authors can submit articles (and resubmit if they make changes). It is standard practice for mathematicians to post their articles on the arXiv prior to submitting them for publication. In fact, some articles only appear on the arXiv. When I want to find a particular article, I first look for it on the arXiv. In short, the arXiv is awesome.

However, when I went to go look for a mathematics education category, I was surprised to see it was not among the list of mathematics categories. The math.HO – History and Overview category lists mathematics education as one of the possible topics, but it doesn’t appear to be commonly used for this purpose. In contrast, there is an active physics education category.

After exploring this a little further, it doesn’t appear that there is a culture among math ed folks to use pre-print servers like the arXiv. I think this is unfortunate and I’d like to help change this. If there is going to be cultural shift, I believe that there should be a dedicated place for math ed papers. Authors need to know where to submit papers and readers need to know where to look. A category called History and Overview doesn’t cut it in my view. A precedent has been set by the physics education crew and we should follow in their footsteps. I’d also like to point out that Mathematics Education is listed as one of the American Mathematical Society’s subject classification codes, namely number 97.

I have two proposals:

1. The category math.ED – Mathematics Education be added to the arXiv.
2. Math ed people start posting to the arXiv (when copyright allows it).

A couple days ago, I contacted the arXiv about the first item and here is their response:

The creation of a new subject class requires considerable support from the community that will use it. We do not want to create subject classes that will be useless because of under use.

More precisely, we require a commitment from a significant group of researchers to submit papers using the proposed subject class. This should include promises to submit a number of initial papers to get the subject class going (a solution to the chicken-and-egg problem).

If this issue is important to you then you must first start by canvassing support from your community. If you receive overwhelming support, and have a significant number of researchers who have agreed to use arXiv, please feel free to contact us again with more specific information.

I then followed up and asked for clarification about what “overwhelming support” and “significant number of researchers” means, to which they replied:

Overwhelming support would include quite a number (more than, say 50) prominent researchers who agree that such as category should be added, and who would agree to make use of it.

Okay, that sounds like a lot of people to try to get on board, but I say, let’s go for it. One obvious question: what’s the best way to gather interest and then record this interest to the arXiv? Would a petition be sufficient?

In the arXiv’s first response to me, they also said:

We also need a volunteer to moderate the class by reviewing daily submissions and flagging inappropriate submissions. This moderator should also review a significant number of already archived papers, looking for submissions that can be cross-listed to the new subject class and contact authors encouraging them to do so.

I don’t think I am the appropriate person for this and I’m not really willing to take it on anyway. Any volunteers?

Update, July 27, 2013: There is now a petition on Change.org. If you are in favor of the arXiv including math.ED – Mathematics Education as a category, please sign the petition. If you would also utilize this category by uploading articles related to mathematics education, please leave a comment (on the petition) indicating that this is the case. You can find the petition here.

Update, July 28, 2013: We’ve exceeded 100 signatures on the petition to get math.ED – Mathematics Education added to the arXiv. The next step is to round up 2–3 volunteers to help moderate category submissions. I don’t think this requires a tremendous amount of work. I’d like to have a list of potential moderators before I contact the arXiv again. Any takers? After all is said and done, there is no guarantee that the mathematics subject board at the arXiv will approve our request. However, they asked for support from at least 50 people and we have 100. Fingers crossed.

A couple of days ago, Peter Krautzberger sent me an email asking if I was interested in becoming an editor for Mathblogging.org. According to Mathblogging.org’s about page:

From research to recreational, from teaching to technology, from visual to virtual, hundreds of blogs and sites regularly write about mathematics in all its facets. For the longest time, there was no good way for readers to find the authors they enjoy and for authors to be found. We want to change that. We have collected over 700 blogs and other news sources in one place, and invite you to submit even more! Our goal is to be the best place to discover mathematical writing on the web.

Mathblogging.org is run by Samuel Coskey, Frederik von Heymann, and Peter. Felix Breuer also had a hand in the site’s creation. The current editors are Peter Honner, Fawn Nguyen, and Shecky Riemann.

Lately, I’ve been feeling stretched a bit thin, so I told Peter that I needed to think about it before deciding. I’ve been trying to be careful about the new projects I take on so that I don’t get in over my head. But…then I remembered the talk that Joe Gallian gave at the conclusion of my first Project NExT workshop in 2008. The theme of Joe’s talk (which he gives every year for Project NExT) is “just say yes.” His thesis is that by saying “yes” we open doors to new opportunities and by saying “no” we close ourselves off to what might have been. Okay, I’m sure Joe would admit that we shouldn’t say “yes” to everything, but I believe he would say that most of us say “no” too often.

I took Joe’s talk pretty seriously my first few years post PhD and I think it has worked out pretty darn well for me. There have been numerous times I thought that I should say “no” but followed Joe’s advice instead. Most of the time it has worked out for the best. A good example is when Ivars Peterson asked Angie Hodge and I to start blogging for the MAA. Actually, let me back up a notch. First, Nathan Carter suggested that I apply for the editor position at Math Horizons. I implemented Joe’s philosophy and talked Angie into applying with me as co-editors. Alas, we were not chosen and instead the committee selected the most awesome Dave Richeson. However, as a result of our application, Ivars approached Angie and I about starting up Math Ed Matters. Around this time, I was beginning a new position at Northern Arizona University and I was concerned that my tenure committee wouldn’t value this sort of work. I dragged my feet for a couple months, but eventually Joe’s voice in my head won out. Angie and I have only been blogging for a few months, but we certainly made the right decision. Lots of new opportunities have presented themselves as a result of the blog. I could go on and on about similar choices.

Okay, by now you’ve already guessed that I agreed to Peter’s offer. So, what does being an editor entail? I already keep up with quite a few math-related blogs, but now I just need to “star” the ones on mathblogging.org that I find the most interesting/enjoyable/useful/compelling and leave a brief comment about them. Doesn’t sound too bad. Of course, to be fair I should start reading a few more of the blogs that pass through.

Yesterday was my first day on the job and I already selected two recent blog posts for Editors’ Picks:

I’m looking forward to reading more excellent blog posts and seeing if Joe is right again.

On April 23, 2013, I gave two talks at the University of Nebraska at Omaha. The first talk was part of the Cool Math Talk Series and was titled “Impartial games for generating groups.” Here is the abstract.

Loosely speaking, a group is a set together with an associative binary operation that satisfies a few modest conditions: the “product” of any two elements from the set is an element of the set (closure), there exists a “do nothing” element (identity), and for every element in the set, there exists another element in the set that “undoes” the original (inverses). Let $G$ be a finite group. Given a single element from $G$, we can create new elements of the group by raising the element to various powers. Given two elements, we have even more options for creating new elements by combining powers of the two elements. Since $G$ is finite, some finite number of elements will “generate” all of $G$. In the game DO GENERATE, two players alternately select elements from $G$. At each stage, a group is generated by the previously selected elements. The winner is the player that generates all of $G$. There is an alternate version of the game called DO NOT GENERATE in which the loser is the player that generates all of $G$. In this talk, we will explore both games and discuss winning strategies. Time permitting, we may also relay some current research related to both games.

The content of the talk falls into the category of combinatorial game theory, which is a topic that is fairly new to me. The idea of the talk is inspired by a research project that I recently started working on with Nandor Sieben who is a colleague of mine at NAU. In particular, Nandor are working on computing the nimbers for GENERATE and DO NOT GENERATE for various families of groups. If all goes according to plan, we’ll have all the details sorted out and a paper written by the end of the summer.

After a brief introduction to combinatorial game theory and impartial games, I discussed both normal and misère play for three impartial games: Nim, X-Only Tic-Tac-Toe, and GENERATE. You might think that X-Only Tic-Tac-Toe is rather boring, but the misère version, called Notakto (clever name, right?) is really interesting. In fact, there is a free iPad game that you can download if you want to try it out. Also, if you want to know more about the mathematics behind Notakto, check out The Secrets of Notakto: Winning at X-only Tic-Tac-Toe by Thane Plambeck and Greg Whitehead.

Here are the slides for my talk.

Immediately after giving the Cool Math Talk, I facilitated a 2-hour Math Teachers’ Circle as part of the Omaha Area MTC. The audience for the MTC mostly consisted of middle and high school mathematics teachers. The theme for the circle was the same as the Cool Math Talk, but instead of me talking the whole time, the teachers played the games and attempted to develop winning strategies.

This was my second time running a MTC at UNO. In February of 2012, I ran a circle whose general topic was permutation puzzles. You can find the slides from last year’s circle here.

Both talks were a lot of fun (especially the MTC). Thanks to Angie Hodge for inviting me out to give the talks.

Angie Hodge and I are excited to announce that Math Ed Matters went live earlier today. Math Ed Matters is a (roughly) monthly column sponsored by the Mathematical Association of America and authored by me and Angie. The column will explore topics and current events related to undergraduate mathematics education. Posts will aim to inspire, provoke deep thought, and provide ideas for the mathematics—and mathematics education—classroom. Our interest in and engagement with inquiry-based learning (IBL) will color the column’s content.

Our first post is isn’t terribly exciting; it’s just an introduction to who we are. Here’s a sample of what we hope to discuss in future posts:

• How did Angie and I meet and how did we end up collaborating on this blog?
• History and impact of Project NExT
• Inquiry-Based Learning: What, Why, and How?
• How and why did Angie and Dana start implementing an IBL approach?
• What’s the Buzz? (Calculus Bee)
• A recap of the 16th Annual Legacy of R.L. Moore Conference (June 13-15, 2013 in Austin, TX)
• A recap of MathFest 2013 (July 31-August 3, 2013 in Hartford, CT)
• Pivotal Moments: How did Dana and Angie get to where they are now?
• Utilizing open-source technologies and text-books

We’d love for you to follow along and join in the conversation. What other topics would you like for us to discuss?

Thanks to the MAA for giving us the opportunity to share our musings with you!

On March 1-3, Jeff Rushall (he’s the Patrick Stewart-looking guy in the photo above) and I, together with a group of 8 Northern Arizona University mathematics majors, attended the 2013 Southwestern Undergraduate Mathematics Research Conference (SUnMaRC) at the University of New Mexico in Albuquerque. SUnMaRC is an annual conference that provides undergraduates in the Southwest an opportunity to give 20 minute talks in order to showcase the work they have been doing for their undergraduate research projects. Three of the 8 NAU students are currently conducting research projects under my guidance. That’s them in the picture above (taken during a break at the conference).

Two of my students, Dane Jacobson and Michael Woodward, have spent the spring semester studying the mathematics behind Spinpossible, which is a free game that is available for iOS and Android devices. Alternatively, you can just play the game in any modern web browser. The game is played on a 3 by 3 board of scrambled tiles numbered 1 to 9, each of which may be right-side-up or up-side-down. The objective of the game is to return the board to the standard configuration where tiles are arranged in numerical order and right-side-up. This is accomplished by a sequence of “spins”, each of which rotates a rectangular region of the board by 180 degrees. The goal is to minimize the number of spins used. It turns out that the group generated by the set of allowable spins is identical to the symmetry group of the 9 dimensional hyper-cube (equivalently, a Coxeter group of type $B_9$).

In a 2011 paper, Alex Sutherland and Andrew Sutherland (a father and son team) present a number of interesting results about Spinpossible and list a few open problems. You can find the paper here. As a side note, Alex is one of the developers of the game and his father, Andrew, is a mathematics professor at MIT. Using brute-force, the Sutherlands verified that every scrambled board can be solved in at most 9 moves. One of the goals of my students’ research project is to find a short proof of this fact. During their talk at SUnMaRC, Dane and Michael presented their current progress on this unexpectedly difficult problem. Here are their slides.

A week after SUnMaRC, Dane and Michael gave a similar and improved talk (which includes some random images of Samuel L. Jackson) during NAU’s Friday Afternoon Undergraduate Mathematics Seminar (FAMUS). As the name of the seminar suggests, the target audience for the FAMUS talks are undergraduates. You can find the slides for their FAMUS talk here.

My third student, Selina Gilbertson, has spent the spring semester working on extending the results of some of my previous students. During the 2011-2012 academic year, I mentored Ryan Cross, Katie Hills-Kimball, and Christie Quaranta at Plymouth State University on an original research project aimed at exploring the “T-avoiding” elements in Coxeter groups of type $F$. Coxeter groups can be thought of as generalized reflections groups. In particular, a Coxeter group is generated by a set of elements of order two. Every element of a Coxeter group can be written as an expression in the generators, and if the number of generators in an expression is minimal, we say that the expression is reduced. We say that an element $w$ of a Coxeter group is T-avoiding if $w$ does not have a reduced expression beginning or ending with a pair of non-commuting generators. My PSU students successfully classified the T-avoiding elements in the infinite Coxeter group of type $F_5$, as well as the finite Coxeter group of type $F_4$. At the time, we had conjectured that our classification holds more generally for arbitrary $F_n$. However, Selina has recently discovered that this is far from true. In Selina’s SUnMaRC talk, she relayed her current progress on classifying the T-avoiding elements in type $F_n$. Here are her slides.

Both sets of students gave wonderful talks. In fact, all 8 NAU students gave fantastic talks. I was impressed. Below are some photos from our trip.

On Tuesday, February 5 (my birthday!), I gave a talk titled “A diagrammatic representation of the Temperley-Lieb algebra” in the NAU Department of Mathematics and Statistics Colloquium. Here is the abstract.

One aspect of my research involves trying to prove that certain associative algebras can be faithfully represented using “diagrams.” These diagrammatic representations are not only nice to look at, but they also help us recognize things about the original algebra that we may not otherwise have noticed. In this talk, we will introduce the diagram calculus for the Temperley-Lieb algebra of type $A$. This algebra, invented by Temperley and Lieb in 1971, is a certain finite dimensional associative algebra that arose in the context of statistical mechanics in physics. We will show that this algebra has dimension equal to the nth Catalan number and discuss its relationship to the symmetric group. If time permits, we will also briefly discuss the diagrammatic representation of the Temperley-Lieb algebra of type affine $C$.

And here are the slides.

Despite the fact that this was a 50-minute talk, it was intended to be an overview of one aspect of a long and complex story. The subject matter is intimately related to my PhD thesis, as well as a series of papers that I have written.

• Ernst, D. C. (2010). Non-cancellable elements in type affine $C$ Coxeter groups. Int. Electron. J. Algebra, 8, 191–218. [arXiv]
• Ernst, D. C. (2012). Diagram calculus for a type affine $C$ Temperley-Lieb algebra, I. J. Pure Appl. Alg. (to appear). [arXiv]
• Ernst, D. C. (2012). Diagram calculus for a type affine $C$ Temperley–Lieb algebra, II. [arXiv]

In addition, there is (at least) a part III that goes with the last two papers that is in progress.

On Friday, February 1, I gave a 30-minute talk titled “The Stargate Switch” during NAU’s Friday Afternoon Undergraduate Mathematics Seminar (FAMUS). As the name of the seminar suggests, the target audience for FAMUS is undergraduates. Here is the abstract for my talk.

An episode of Stargate SG-1 features a two-body mind-switching machine which will not work more than once on the same pair of bodies. The plot centers around two disjoint pairs of individuals who swap minds but subsequently wish they could reverse the process. We will discuss the mathematics behind the solution to this problem, as well as some generalizations.

This was my third FAMUS talk of the academic year, but my first of the semester. My first FAMUS talk was titled On an open problem of the symmetric group and my second talk was titled Euler’s characteristic, soccer balls, and golf balls.

In the 1999 episode “Holiday” from season 2 of Stargate SG-1, the character Ma’chello tricks Daniel into swapping minds with him. In an attempt to save Daniel, Jack and Teal’c accidentally swap minds, after which they then discover a limitation: the machine will not work more than once on the same pair of bodies. Physicist Samantha Carter saves the day by improvising a sequence of 4 switches that brings everyone back to normal.

In two recent papers, R. Evans and L. Huang from the University of California, San Diego, study the mathematics behind the Stargate Switch Problem, and generalize the solution to permutations involving $m$ pairs of bodies.

• Evans, R., & Huang, L. (2012). The Stargate Switch. [arXiv]
• Evans, R., & Huang, L. (2012). Mind switches in Futurama and Stargate. [arXiv]

The limitation of the machine in Stargate SG-1 is the same as the one suffered by the mind-switching machine in Futurama’s 2010 episode “The Prisoner of Benda”. However, the Stargate Switch, and Evans and Huang’s generalization, is a special case of the problem posed in Futurama. In the general case, we need to introduce two outsiders to solve the problem. But the Stargate Switch can be solved without the addition of outsiders (as long as we have at least 2 pairs of bodies).

The solution to the dilemma in Futurama is known as the Futurama Theorem and was proved by Ken Keeler, who is one of the show’s writers and has a PhD in applied mathematics. I’ve given a few talks about the Futurama Theorem and if you want to know more, check out my recent blog post located here.

As with my previous talks about the Futurama Theorem, I utilized deck.js to create HTML slides and I used MathJax to typeset all of mathematical notation. One advantage of this approach is that it allows you to view the slides directly in your web browser. You can view the slides by going here. To advance the slides, just use your arrow keys. Also, you can get an overview of the slides by typing “m” or go to a specific slide by typing “g”.