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.

This morning I spent an hour observing my boys’ classrooms at Haven Montessori. Wow. My wife and I sat in the corner and just watched for 30 minutes in each classroom. My younger son (age 5) is in a Primary Classroom, which is for children 3 years to 6 years (including Kindergarten). My older son (age 7) is in Lower Elementary, which is for 1st–3rd grades. Both of our boys started Montessori in January of this year and we have been thrilled with the outcome. We were hesitant to move them mid-school year, especially after having moved to Arizona less than a year ago, but we have no regrets about our decision.

This wasn’t my first observation of a Montessori classroom, but each time I do it, I am blown away. Here are a few quick observations:

• All of the students were working quietly and respectfully.
• Students were either working independently or collaboratively with another student or two.
• There were a variety of different things going on at the same time.
• Students were freely moving about the room, but always focused on their respective task.
• Students were smiling and enjoying themselves, but not goofing off.
• Students appeared to be working on stuff because they genuinely seemed interested.
• There were no incentives and no grades!

I could go on and on, but you get the idea. It seems like a Utopia. If you’ve never seen a Montessori classroom, go check it out. I’m sure the success of each classroom has a lot to do with the teacher, so we are tremendously grateful that our boys ended up with great teachers.

There are a lot of similarities between Montessori and inquiry-based learning (IBL), which is the approach that I try to take (to various degrees) in the classes that I teach. All of the wonderful things that I witnessed this morning are exactly the kinds of things that I strive for in my own classrooms. Sometimes it works and sometimes it doesn’t. But I keep trying.

While I was observing, I kept daydreaming; “What if students were provided with this type of experience throughout their entire education?” In particular, I spent quite a bit of time pondering my last observation above about grades and incentives. As a teacher, I try to de-emphasize grades as much as possible, but our educational system is so entrenched in their use. Is it possible to eliminate the need for incentives? Is there something that happens in our development that diminishes our curiosity flame?

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!

## Quote by Winston Churchill

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 the FAMUS talks are 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”.

Several weeks ago I was asked to take part in the Project NExT Alternative Assessment Techniques panel discussion at the 2013 Joint Mathematics Meetings, which recently took place in San Diego, CA. I was extremely honored to be considered for the panel, but at the time I was not planning on attending the JMM, so I declined the invitation. A couple weeks later, it turned out that I was going to make it to the JMM after all. At about 11PM the night before I was going to fly to San Diego, I received an email from the organizers of the panel discussion indicating that one of the panelists was unable to make it and that they heard was going to be there. They asked if I could fill in at the last minute and I accepted.

Here is the abstract for the panel.

Since classroom assessment is used to determine a student’s level of mastery, how can we vary our methods of assessment to accurately reflect the diversity of ways that students learn and understand the material? Traditional methods of assessment, such as exams, quizzes, and homework, may not accurately and robustly measure some students’ understanding. In this panel, we will propose alternative methods and discuss the following questions: – What assessments exist besides the traditional ones and how can I use them for my course? – How can I determine the validity of an alternative assessment? – How can I develop my own alternative assessments? – How can alternative assessments help me evaluate the effectiveness of a non-traditional classroom?

It is worth pointing out that I’m not an assessment expert by any stretch of the imagination. Also, given that I had less than 48 hours to prepare amidst a pretty full schedule, I didn’t have a lot of time to come up with something new and creative for my talk. Inquiry-based learning (IBL) is one of my passions and I’ve given quite a few IBL-related talks in the past few months, so I decided to “twist” the ideas from some of my recent talks into a talk about assessment. In my talk, I propose implementing IBL not only as a pedagogical approach but also as an assessment strategy. This isn’t really a stretch since in my view, an effective IBL class is all assessment, all the time.

My fellow panelists included Theron Hitchman (University of Northern Iowa), Bonnie Gold (Monmouth University), and Victor Odafe (Bowling Green State University). Theron gave a talk on using Standards Based Assessment (you can find his slides here), Bonnie spoke on a variety of summative assessment techniques, and Victor shared his experience with oral assessment. It turns out that the person that I was filling for is mathematics education superstar Jo Boaler. Me filling in for her is ridiculous.

Here are the slides for my portion of the panel.

Thanks to the organizers of the panel (Cassie Williams (James Madison University), Jane Butterfield (University of Minnesota), John Peter (Utica College), and Robert Campbell (College of Saint Benedict and Saint John’s University)) for providing me with the opportunity to speak on the panel.