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On October 12th, I saw a post by Dan Christensen on Google+ about a list of five open problems posed by the mathematician John Conway that have monetary rewards associated with them. In particular, Conway is offering $\$1,000$ for solutions (either positive or negative) to any of the problems. Here are the five problems (as stated by Conway):

  • Problem 1. Sylver coinage game (named after Sylvester, who proved it terminates): The game in which the players alternately name positive integers that are not sums of previously named integers (with repetitions being allowed). The person who names 1 (so ending the game) is the loser. The question is: If player 1 names 16, and both players play optimally thereafter, then who wins?
  • Problem 2. 99-Graph: Is there a graph with 99 vertices in which every edge (i.e., pair of joined vertices) belongs to a unique triangle and every nonedge (pair of unjoined vertices) to a unique quadrilateral?
  • Problem 3. The Thrackle Problem: A doodle on a piece of paper is called a thrackle if it consists of certain distinguished points called spots and some differentiable (i.e., smooth) curves called paths ending at distinct spots and so that any two paths hit once and only once, where hit means having a common point at which they have distinct tangents and which is either an endpoint of both or an interior point of both. The right hand figure shows a thrackle with six spots and six paths. But can a thrackle have more paths than spots?
  • Problem 4. Dead Fly Problem: If a set of points in the plane contains one point in each convex region of area 1, then must it have pairs of points at arbitrarily small distances?
  • Problem 5. Climb to a Prime: Let $n$ be a positive integer. Write the prime factorization in the usual way, e.g., $60 = 22 \cdot 3 \cdot 5$, in which the primes are written in increasing order, and exponents of 1 are omitted. Then bring exponents down to the line and omit all multiplication signs, obtaining a number $f(n)$. Now repeat.So, for example, $f(60) = f(22 \cdot 3 \cdot 5) = 2235$. Next, because $24235 = 3 \cdot 5 \cdot 149$, it maps, under $f$, to 35149, and since 35149 is prime, it maps to itself. Thus, $60 \to 2235 \to 35149
    \to 35149$, so we have climbed to a prime, and we stop there forever. The conjecture, in which I (Conway) seem to be the only believer, is that every number eventually climbs to a prime. The number 20 has not been verified to do so. Observe that $20 \to 225 \to 3252 \to 223271 \to \cdots$, eventually getting to more than one hundred digits without yet reaching a prime.

If you solve one of these, you can reach Conway by sending snail mail (only) in care of the Department of Mathematics at Princeton University.

Around the same time that I stumbled onto these problems, I was brainstorming ideas for a couple of upcoming talks that I was slated to give (one for undergraduates and one for high school students). I decided that discussing open problems with monetary rewards with an emphasis on Conway’s problems would likely make for a nice talk. Here is the abstract that I settled on for both talks.

There is a history of individuals and organizations offering monetary rewards for solutions, either in the affirmative or negative, to difficult mathematically-oriented problems. For example, the Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. A correct solution to any of the problems results in a $\$1,000,000$ prize being awarded by the institute. To date, only one of the problems has been solved (the Poincaré Conjecture was solved by Grigori Perelman, but he declined the award in 2010). These are hard problems! The renowned mathematician John Conway (Princeton) maintains a list of open problems and for each problem on the list, he is offering $\$,1000$ to the first person that provides a correct solution. In this talk, we will explore a few of Conway’s problems, and in the unlikely event we come up with a solution, we’ll split the money.

On Friday, October 24, 2014, I gave a talk during the Friday Afternoon Mathematics Undergraduate Seminar (FAMUS) at NAU. Speaking at FAMUS is always fun and my talk seemed to be well-recieved.

After having a practice run during FAMUS, I was able to improve the slides I intended to use during my talk at the 2014 NAU High School Math Day, which took place a few days later on Tuesday, October 28, 2014. Here are my slides:

I had a blast presenting to the high school students. It cracked me up that there were a few students that immediately started obsessing over the Sylver coinage problem and likely didn’t hear a word I said after that. My goal was to give an engaging and high energy talk. I also slid in some humor and I was happy that everyone laughed when they were supposed to. Interestingly, the thing I said that the students thought was the funniest was something that I didn’t intend to be humorous. When I stated that “If you solve one of these, you can reach Conway by sending snail mail (only) in care of the Department of Mathematics at Princeton University,” the audience burst into laughter. Requiring snail mail seemed so ridiculous to them, they thought it was a joke.

As a side note, I used mtheme (available for free on GitHub) together with beamer/LaTeX to generate my slides. I’m really happy with the look of mtheme and thrilled to get away from the standard beamer themes.

Prior to this summer’s MathFest in Portland, I was a co-facilitator for a four-day workshop on inquiry-based learning. My co-facilitators were Stan Yoshinobu (Cal Poly, SLO), Matt Jones (CSU Dominguez Hills), and Angie Hodge (University of Nebraska at Omaha). I love being a part of these workshops. Even though I’m there to help others get started on implementing IBL, I benefit tremendously from the experience and always leave feeling energized and fired up to teach. If you are an aspiring practitioner or a newish user of IBL, I highly encourage you to look into attending a future IBL Workshop, which is run as an MAA PREP workshop.

On day three of the workshop, I gave a 30-minute plenary talk. Most of the sessions are designed to be highly interactive and this was one of the few times that we “talked at” the participants. At the end of day two, I had given the participants a choice of topics for the plenary and the request was to describe the general overview of my approach to IBL in proof-based classes versus a class like calculus. So, that’s what I set out to do. The slides I used for my talk can be found below.

I’d like to think that my talk was more than the content of the slides, however, the slides ought be useful on their own for someone that is curious about IBL. This talk was similar to others about IBL that I’ve given in the past.

The past two years, Angie Hodge, Stan Yoshinobu, and I have organized an Inquiry-Based Learning Best Practices special session at MathFest. We’ve had a fantastic turn out in terms of speakers and attendees both years. This year we thought we would try something new and decided to organize a poster session instead. Here’s the abstract for the session:

New and experienced instructors implementing inquiry-based learning methods are invited to share their experiences, resources, and insights in this poster session. The posters in this session will focus on IBL best practices. We seek both novel ideas and effective approaches to IBL. Claims made should be supported by data (student responses, sample work, test scores, survey results, etc.). This session will be of interest to instructors new to IBL, as well as experienced practitioners looking for new ideas. Presenters should have their materials prepared in advance and will be provided with a self-standing, trifold tabletop poster approximately 48 in wide by 36 in high.

One of our goals of the poster session is to increase interaction between presenters and attendees. We hope that someone can wander around and gather a lot of information about implementing IBL in a short period of time. I’m not usually a fan of poster sessions, but I’m looking forward to this one. The poster session takes place on Thursday, August 7 at 3:30-5:00PM in the Hilton Portland, Plaza Level, Plaza Foyer. If you are attending MathFest, please stop by the poster session. Also, if you think you have something interesting to share, we encourage you to submit an abstract. The deadline for submission is Friday, June 13, 2014.

Questions regarding this session should be sent to the organizers:

Angie Hodge, University of Nebraska at Omaha
Dana Ernst, Northern Arizona University
Stan Yoshinobu, Cal Poly San Luis Obispo

If you want to learn more about IBL, check my “What the Heck is IBL?” post over on the Math Ed Matters blog.

Undergraduate Student Poster Session

The Joint Mathematics Meetings took place last week in Baltimore, MD. The JMM is a joint venture between the American Mathematical Society and the Mathematical Association of America. Held each January, the JMM is the largest annual mathematics meeting in the world. Attendance in 2013 was an incredible 6600. I don’t know what the numbers were this year, but my impression was that attendance was down from last year.

As usual, the JMM was a whirlwind of talks, meetings, and socializing with my math family. My original plan was to keep my commitments to a minimum as I’ve been feeling stretched a bit thin lately. After turning down a few invitations to give talks, I agreed to speak as part of panel during a Project NExT session. The title of the panel was “Tried & True Practices for IBL & Active Learning”. Here, IBL is referring to inquiry-based learning. Last year I gave several talks and co-organized workshops about IBL, so I knew preparing for the panel wouldn’t be a huge burden.

The other speakers on the panel were Angie Hodge (University of Nebraska, Omaha) and Anna Davis (Ohio Dominican University). Each of us were given 15 minutes to discuss our approach to IBL and active learning and then we had about 20 minutes for questions. For my portion, I summarized my implementation of IBL in my proof-based courses followed by an overview of my IBL-lite version of calculus. Angie discussed in detail the way in which UNO is successfully adopting IBL in some of their calculus sections. Anna spent some time telling us about an exciting project called the One-Room Schoolhouse, which attempts to solve the following problem: Small colleges and universities often struggle to offer a variety of upper-level courses due to low enrollment. According to the ORS webpage, the ORS setting allows schools to offer low-enrollment courses every semester at no additional cost to the institution. ORS is made possible by flipping the classroom.

If you are interested in my slides, you can find them below.

I’m really glad that I took the time to speak on the panel. The audience asked some great questions and I had several fantastic follow-up conversations with people the rest of the week.

As a side note, one thing I was reminded of is that speaking early during the conference (which is when the panel occurred) is much better than speaking later. The past couple JMMs, I spoke on the last day or two, and in this case, my talk is always taking up bandwidth in the days leading up to it. It’s nice to just get it out of the way and then be able to concentrate on enjoying the conference.

One highlight of the conference was having a pair of my current undergraduate research students (Michael Hastings and Sarah Salmon) present a poster during the Undergraduate Student Poster Session. Michael and Sarah have been working on an original research project involving diagrammatic representations of Temperley-Lieb algebras. I’ve been extremely impressed with the progress that they have made and it was nice to be able to see them show off their current results. The title of their poster was “A factorization of Temperley–Lieb diagrams” (although this was printed incorrectly in the program). Here is their abstract:

The Temperley-Lieb Algebra, invented by Temperley and Lieb in 1971, is a finite dimensional associative algebra that arose in the context of statistical mechanics. Later in 1971, R. Penrose showed that this algebra can be realized in terms of certain diagrams. Then in 1987, V. Jones showed that the Temperley–Lieb Algebra occurs naturally as a quotient of the Hecke algebra arising from a Coxeter group of type $A$ (whose underlying group is the symmetric group). This realization of the Temperley-Lieb Algebra as a Hecke algebra quotient was generalized to the case of an arbitrary Coxeter group. In the cases when diagrammatic representations are known to exist, it turns out that every diagram can be written as a product of “simple diagrams.” These factorizations correspond precisely to factorizations in the underlying group. Given a diagrammatic representation and a reduced factorization of a group element, it is easy to construct the corresponding diagram. However, given a diagram, it is generally difficult to reconstruct the factorization of the corresponding group element. In cases that include Temperley–Lieb algebras of types $A$ and $B$, we have devised an efficient algorithm for obtaining a reduced factorization for a given diagram.

Perhaps I’m biased, but I think their poster looks pretty darn good, too.

All in all, I would say that I had a successful JMM. I had a lot more meetings than I’ve had in the past, so I missed out on several talks I really would have liked to attend. There’s always next year.

On Thursday, October 17, I gave an hour long talk in our department seminar titled “An Iterated Prisoner’s Dilemma.” There were about 35 people in attendance, including undergraduates (mostly my calculus students), graduate students, and faculty from the Mathematics & Statistics Department at Northern Arizona University. I was pleased with the turnout since our seminars are usually on Tuesdays and I wasn’t sure how many people would come on a non-standard day. Here is the abstract for the talk:

The Prisoner’s Dilemma goes something like this. Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary confinement with no means of speaking to or exchanging messages with the other. The police admit they don’t have enough evidence to convict the pair on the principal charge. They plan to sentence both to a year in prison on a lesser charge. Simultaneously, the police offer each prisoner a bargain. If A and B both confess the crime, each of them serves 2 years in prison. If A confesses but B denies the crime, A will be set free whereas B will serve 3 years in prison (and vice versa). If A and B both deny the crime, both of them will only serve 1 year in prison. In this talk, we will first discuss this classic game theoretic problem and then introduce an iterative version that consists of a round robin tournament of teams, where the winner is the team that spends the least amount of time in prison.

I pretty much lifted this straight from the the Wikipedia page on the Prisoner’s Dilemma. So, thanks to the author(s) of that page!

There were several motivating factors in choosing this topic. First, every once in a while, I like to give a talk about something that I don’t know much about. Doing this forces me to learn something new. Also, I’ve found that some of my best talks are on things that I am not an expert on. Certainly, one of the reasons why this is true is that I’m likely to pitch a talk at a lower level if I’m talking about something unfamiliar. I don’t know about you, but I much prefer sitting through a talk when I understand most of what’s going on. Culturally, it seems acceptable to give talks where most of the audience doesn’t understand most of the talk. I’m trying to give talks where this doesn’t happen.

It’s expected that our graduate students (we have a masters program at NAU) attend our weekly seminars, but lately their attendance has been poor. I wanted to pick a topic that might entice them to start coming.

I ended up choosing the Prisoner’s Dilemma as a topic because I wanted to learn more about game theory and I figured the topic would be accessible. Moreover, I was inspired by Google+ and blog posts by Vincent Knight and Paul Harper (both from Cardiff University). There was also an excellent Radiolab episode about the Prisoner’s Dilemma back in 2010 that planted a seed for me. I’d like to thank Vince and Paul for helping out while I was preparing my talk. In particular, my slides are a modification of Vince’s slides, which he discusses here.

Without further ado, here are the slides for my talk.

As you can see, the talk began with an activity involving the Two Thirds of Average Game. During the activity, the audience made two different guesses. While I was giving the rest of the talk, I had a volunteer enter all the guesses into a csv file on the Sagemath Cloud. At the end of the talk, I ran Vince’s python script on the csv file in the Sagemath cloud. The output told me who the winners were for both rounds of guessing and provided a dandy looking graph, seen below.

Results for 2/3 of Average

I provided the winners with some chocolate.

Around slide 18, the plan was to conduct an Iterated Prisoner’s Dilemma tournament involving 4 teams, but I was a little worried about running out of time. So, I decided to wait until the end of the talk and do it if I had time. I ended up squeezing in a 3-team tournament that we probably flew through too quickly to get much out of, but it was fun nonetheless. The three team names were the United States, North Korea, and Russia. North Korea ended up winning, but only by a small margin.

Talk: Proofs without Words

September 30, 2013 — 2 Comments

On Friday, September 20, I gave a 30-minute talk titled “Proofs without Words” during NAU’s Friday Afternoon Undergraduate Mathematics Seminar (FAMUS). As the name of the seminar suggests, the target audience for FAMUS is undergraduates. I usually give a couple talks at FAMUS each semester and this was my first of the semester. Here is the abstract with words for my talk.

In this FAMUS talk, we’ll explore several cool mathematical theorems from a visual perspective.

The talk basically went like this. I displayed a figure or drawing and then the goal was for the audience to come up with the corresponding theorem. I had a ton of fun and the audience seemed to enjoy it. The initial idea for the talk came from the book Charming Proofs: A Journey into Elegant Mathematics by Claudi Alsina and Roger B. Nelsen. This is a wonderful book that incorporates lots of visual proofs. If you don’t have a copy, I highly recommend it. I borrowed lots of ideas from it when I taught a class titled “Introduction to Formal Mathematics” while I was at Plymouth State University.

My original plan was to recreate a lot of the figures I had in mind using TikZ, but I should have known that I wouldn’t have time for that. When I was brainstorming the talk a couple days before, I decided to do a Google search in the hope that I could find some figures to borrow that others had already made. In my search, I stumbled on several references to “proof without words”, which is what I ultimately named my talk. In fact, there is a Wikipedia entry and Roger B. Nelsen also wrote a book called Proofs without Words: Exercises in Visual Thinking. Moreover, I was thrilled to find lots of cool figures on the Internet. For my talk, I borrowed images and content from the following sources:

In the end, my slides ended up being a sample each of these three sources. If you are ready to see some cool figures, check out the slides below. I’ve left the pauses in so that you can ponder the theorem before you see it.

As anticipated, I had more figures than I had time to discuss, but we did get through most of them.

The custom at FAMUS is to interview a faculty member after the 30 minute talk. The usual FAMUS host, Jeff Rushall, was out of town, so I was filling in for him. I had the honor of interviewing Michael Falk. This was fun for me because Mike was my masters thesis advisor.

On Thursday, August 22, I was one of four speakers that gave a 20 minute talk during the Department of Mathematics and Statistics Teaching Showcase at Northern Arizona University. My talk was titled “An Introduction to Inquiry-Based Learning” and was intended to be a “high altitude” view of IBL and to inspire dialogue. I was impressed with the turn out. I think there were roughly 40 people in attendance, from graduate students to tenured faculty and even some administrators. Here are the slides for my talk.

If you take a look at the slides, you’ll see that I mention some recent research about the effectiveness of IBL by Sandra Laursen, et. al. During my talk, I provided a two-page summary of this research, which you can grab here (PDF).

After about 15 minutes, I transitioned into an exercise whose purpose was to get the audience thinking about appropriate ways to engage in dialogue with students in an IBL class. I provided the participants with the handout located here that contains a dialogue between three students that are working together on exploring the notions of convergence and divergence of series. After the dialogue, five possible responses for the instructor are provided. I invited the participants to discuss the advantages and disadvantages of each possible response. It is clear that some responses are better than others, but all of the responses listed intentionally have some weaknesses. We were able to spend a couple of minutes having audience members share their thoughts. It would have been better to spend more time on this exercise. I wish I could take credit for the exercise, but I borrowed it from the folks over at Discovering the Art of Mathematics.

If you want to know more about IBL, check out my What the Heck is IBL? blog post over on Math Ed Matters.

On Friday, June 14 I gave a 15 minute talk in one of the parallel session at the Legacy of R.L. Moore Conference in Austin, TX. The Legacy Conference is the inquiry-based learning (IBL) conference. In fact, it’s the only conference that is completely devoted to the discussion and dissemination of IBL. It’s also my favorite conference of the year. It’s amazing to be around so many people who are passionate about student-centered learning.
This was my fourth time attending the conference and I plan on attending for years to come.

Here is the abstract for the talk that I gave.

In this talk, the speaker will relay his approach to inquiry-based learning (IBL) in an introduction to proof course. In particular, we will discuss various nuts and bolts aspects of the course including general structure, content, theorem sequence, marketing to students, grading/assessment, and student presentations. Despite the theme being centered around an introduction to proof course, this talk will be relevant to any proof-based course.

The target audience was new IBL users. I often get questions about the nuts and bolts of running an IBL class and my talk was intended to address some of the concerns that new users have. I could talk for days and days about this, but being limited to 15 minutes meant that I could only provide the “movie trailer” version.

Below are the slides from my talk.

One of my goals was to get people thinking about the structure they need to put in place for their own classes. When I wrote my slides, I had a feeling that I couldn’t get through everything. I ended up skipping the slide on marketing, but in hindsight, I wish I would have skipped something else instead. Two necessary components of a successful IBL class are student buy-in and having a safe environment where students are willing to take risks. Both of these require good marketing and I never had a chance to make this point. Maybe next year, I will just give a talk about marketing IBL to students.

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On Saturday, May 5, 2013, I was joined by TJ Hitchman (University of Northern Iowa) for the Michigan Project NExT Panel Discussion on Teaching Strategies for Improving Student Learning, which was part of the 2013 Spring MAA Michigan Section Meeting at Lake Superior State University. The title of the session was “Teaching Strategies for Improving Student Learning” and was organized by Robert Talbert (Grand Valley State University). The dynamic looking guy in the photo above is TJ.

Here is the abstract for the session.

Are you interested in helping your students learn mathematics more effectively? Are you thinking about branching out in the way you teach your courses? If so, you should attend this panel discussion featuring short talks from leaders in higher education in employing innovative and effective instructional strategies in their mathematics classes. After speaking, our panelists will lead breakout discussions in small groups to answer questions and share advice about effective instructional strategies for college mathematics. Panelists will include Dana Ernst (Northern Arizona University) and Theron Hitchman (University of Northern Iowa), both noted for their effective use of the flipped classroom and inquiry-based learning.

Sweet, I guess running my mouth often enough about inquiry-based learning (IBL) gets me “noted.”

Each of TJ and I took about 10-15 minutes to discuss our respective topics and then we took the remaining time to chat and brainstorm as a group. The focus of my portion of the panel was on “Inquiry-Based Learning: What, Why, and How?” My talk was a variation on several similar talks that I’ve given over the past year. For TJ’s portion, he discussed his Big “Unteaching” Experiment that he implemented in his Spring 2013 differential geometry course.

Here are the slides for my portion of the panel.

Despite low attendance at the panel, I think it went well. Thanks to Robert for inviting TJ and me!

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.