Time spent thinking about a problem is always time well spent. Even if you seem to make no progress at all.

Whenever I’m teaching via inquiry-based learning (IBL), it is important to get student buy-in. I often refer to this as “marketing IBL”. My typical approach to marketing involves having a dialogue with my students, where I ask them leading questions in the hope that at the end of our discussion the students will have told me that something like IBL is exactly what we should be doing.

In the past, I would just wing it on day one and it’s been different every time. However, I’ve had lots of people ask me to describe exactly what I do and I also thought it would be a good exercise for me to sit down and think carefully about the activity. So, in the fall of 2014, I created some slides to guide the activity, which I am now calling “Setting the Stage”. Since then I have shortened the activity and made some improvements. The current version of the activity is inspired by TJ Hitchman, Mike Starbird, and Brian Katz.

The main idea is that I want to get students thinking about why we there and what we should really be striving to get out of the course. In addition, it helps students understand why I take an IBL approach in my classes. Below is an outline of the the activity.

## Directions to the Students

- Get in groups of size 3–4.
- Group members should introduce themselves.
- For each of the questions that follow, I will ask you to:
- Think about a possible answer on your own.
- Discuss your answers with the rest of your group.
- Share a summary of each group’s discussion.

## Questions

- ￼What are the goals of a university education?
- How does a person learn something new?
- What do you reasonably expect to remember from your courses in 20 years?
- What is the value of making mistakes in the learning process?
- How do we create a safe environment where risk taking is encouraged and productive failure is valued?

Each time I’ve run the activity, the responses are slightly different. The responses to the first two questions are usually what you would expect. Question 3 always generates great discussions. The idea of “productive failure” naturally arises when discussing question 4 and I provide them with this language sometime while discussing this question. Listening to the students’ responses to question 4 is awesome. It’s really nice to get the students establishing the necessary culture of the class without me having to tell them what to do.

After we are done discussing the 5 questions, I elaborate on the importance of productive failure and inform that I will often tag things in class with the hashtag #pf in an attempt to emphasize its value. I also provide them with the following quote from Mike Starbird:

“Any creative endeavor is built on the ash heap of failure.”

I wrap up the activity by conveying some claims I make about education and stating some of my goals as a teacher.

## Claims

- An education must prepare a student to ask and explore questions in contexts that do not yet exist. That is, we need individuals capable of tackling problems they have never encountered and to ask questions no one has yet thought of.
- If we really want students to be independent, inquisitive, & persistent, then we need to provide them with the means to acquire these skills.

## Lofty Goals

- Transition students from consumers to producers!
- I want to provide the opportunity for a transformative experience.
- I want to change my students’ lives!

Below is the Spring 2015 version of the slides that accompany the activity.

You can always find the current version of the LaTeX source at my GitHub repo located here. Note that I’m using the beamer m theme for the slides, which require the Mozilla Fira fonts by default. Feel fee to steal, modify, and improve. And please let me know if do.

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.

My colleague Nandor Sieben and I recently submitted for publication a paper titled “Impartial achievement and avoidance games for generating finite groups.” The current arXiv version of the article is located here. My typical pure mathematics research interests are in the combinatorics of Coxeter groups and their associated algebras, so while I have a background in group theory and combinatorics, this was my first research experience in combinatorial game theory. In fact, prior to working on this project, I knew next to nothing about the subject. In the year and a half we worked on the project, I learned a tremendous amount of new material, which was a lot of fun. It was exciting to branch out and try something new.

Here is the abstract for the paper:

We study two impartial games introduced by Anderson and Harary and further developed by Barnes. Both games are played by two players who alternately select previously unselected elements of a finite group. The first player who builds a generating set from the jointly selected elements wins the first game. The first player who cannot select an element without building a generating set loses the second game. After the development of some general results, we determine the nim-numbers of these games for abelian and dihedral groups. We also present some conjectures based on computer calculations. Our main computational and theoretical tool is the structure diagram of a game, which is a type of identification digraph of the game digraph that is compatible with the nim-numbers of the positions. Structure diagrams also provide simple yet intuitive visualizations of these games that capture the complexity of the positions.

The fundamental problem in the theory of impartial combinatorial games is the determination of the nim-number of the game. This allows for the calculation of the nim-numbers of game sums and the determination of the outcome of the games. The major aim of this paper is the development of some theoretical tools that allow the calculation of the nim-numbers of the achievement and avoidance games for a variety of familiar groups. In the paper, we introduce the structure diagram of a game, which is an identification digraph of the game digraph that is compatible with the nim-numbers of the positions. Structure diagrams also provide simple but intuitive visualizations of these games that capture the complexity of the positions. By making further identifications, we obtain the simplified structure diagram of a game, which is our main computational and theoretical tool in the paper.

We developed a software package that computes the simplified structure digraph of the achievement and avoidance games. We used GAP to get the maximal subgroups and the rest of the computation is implemented in C++. The software is efficient enough to allow us to compute the nim-numbers for the smallest 100,000 groups, which includes all groups up to size 511. The result is available on our companion web page.

In April of 2013, I gave two talks at the University of Nebraska at Omaha that introduce the two games that this paper is about, but did not elaborate on the nim-number aspect. I summarized those talks in this blog post.

Here’s a classic quote from RL Moore:

That student is taught the best who is told the least.

During his talk yesterday at the RL Moore Conference, David Clark provided a slight modification:

The student is taught the best who is told only enough to ensure that he or she will continue to work hard, stay engaged, and make progress.

I think David’s revision does an excellent job of capturing the experience I hope to provide the students in my inquiry-based learning (IBL) classes with.

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.

A few weeks ago, Stan Yoshinobu asked me to round up a few student quotes about their experience with inquiry-based learning (IBL). The intention is to use some of the material he gets for pamphlets and flyers for the Academy of Inquiry-Based Learning. I contacted a few of the students from the abstract algebra course that I taught in the fall and here is what they had to say.

“I’m a very shy person. Presenting math problems in front of an audience of math students was at first excruciating, but by the end of the course I realized I had gained an enormous amount of confidence. I truly feel that the IBL process has given me access to internal resources I didn’t realize I had available.”

“IBL created an environment for me where I felt comfortable enough to try proofs without the pressure of needing to be 100% right on the first try. So now in later upper division courses I am more comfortable with trying more complex problems, which ultimately lead me to do undergraduate research. And in all honesty, the classroom culture created by the IBL setup is what sold me on pure mathematics and has made me a better independent learner.”

“IBL helps prepare the student for the real world by teaching them how to create intuition. When you get to the real world or higher level mathematics courses, you will not always have someone there to tell you how to solve the problem.”

“By far, and without a doubt, inquiry-based learning is the best way to learn mathematics. Most methods for teaching math involve an instructor showing how to “do” various problems often involving computations and formulas, and then the students mimic the process for similar problems. IBL, however, asks the students to use what they know (or assume) to be true in order to create their own ways to solve problems or form logical arguments to validate other ideas. And logical arguments, not computations, not formulas, are the basis of all mathematics. Being able to form logical arguments is not something that can be mimicked, it must be discovered on one’s own, which is exactly how IBL works. Hence, when it comes to math, real math, and not just computations, IBL is the way to go.”

It would be a crime if I didn’t mention my all-time favorite student quote about IBL that was written on a course evaluation at the end of my introduction to proof course from the spring 2013 semester.

“Try, fail, understand, win.”

I believe that this last quote perfectly captures the essence of an effective IBL experience for a student. If you want to know about IBL, check out my post, What the Heck is IBL?, over on Math Ed Matters.

A couple months ago, my colleague Jeff Rushall and I co-applied for a Center for Undergraduate Research in Mathematics (CURM) mini-grant to fund a group of undergraduate students to work on an academic-year research project. Jeff and I had both individually applied in the past, but neither of us were successful in our proposals. If you are interested, you can take a look at my previous proposal by going here. Jeff and I are both passionate about undergraduate research and work well together. We decided that a joint application would likely be stronger than two individual proposals. I’m happy to report that we recently found out that our proposal was funded. We’re thrilled!

Here are a few more details. For the upcoming project, we recruited a diverse group of 7 talented undergraduates: Michael Hastings (one of my current research students), Emily White, Hanna Prawzinsky, Alyssa Whittemore, Levi Heath, Brianha Preston, and Nathan Diefenderfer. Students are expected to spend ten hours per week during the academic year working on the research project. In return, each student will earn a $\$3000$ stipend. Money from the grant will also be used to buyout a single course for both me and Jeff. In addition, Jeff and I will team-teach a topics course each semester that will include our research students but will also be open to other interested students. CURM will cover most of the travel expenses for Jeff and I to attend the Faculty Summer Workshop at Brigham Young University (BYU). CURM will also cover most of the travel expenses for the nine of us to attend the Student Research Conference at BYU.

A collaborative research program has many advantages over operating several disconnected projects, as Jeff and I have done in the past. One of our goals is to build a self-sustaining research group. Ideally, this group will consist of students at different stages of their education, each participating for multiple years. The opportunities afforded by a CURM mini-grant will provide a catalyst for our endeavor in several ways. First, the visibility of a student research group with a CURM mini-grant will help our department recruit mathematically inclined students. NAU has many such students, but some are enticed by existing research groups and grants in our science programs. Second, we would like to take advantage of the mentoring and training the CURM program provides for faculty. One weakness in my past projects is getting students to finish writing up their results for publication. I am hopeful that my involvement in CURM will help remedy this. Third, we believe that the stipend money will enable our students to forgo some of their part-time work and instead devote their time to mathematics. Lastly, we want to use the experience as a stepping stone towards obtaining an externally funded REU program.

Next year’s research project involves “prime labelings of graphs,” which is outside my typical research interests. Jeff and I believe that we have found a project that is accessible to undergraduates yet rich enough that we won’t even come close to running out of stuff to do. I’m really looking forward to branching out and exploring something new.

If you are interested, here is the project description that we submitted.

## Project Description

This research project is motivated by a conjecture in graph theory, first stated in a 1999 paper by Seoud and Youssef [1], namely:

All unicyclic graphs have prime labelings.

This is a viable choice as a research problem for undergraduates because it is interesting yet accessible, in large part due to the minimal amount of background information required. To wit, a **unicyclic graph** is a simple graph containing exactly one cycle. An $n$-vertex simple graph $G$ with vertex set $V(G)$ is said to have a **prime labeling** if there exists a bijection $f: V(G) \to \{1, 2, 3, \ldots, n\}$ such that the labels assigned to adjacent vertices of $G$ are relatively prime.

As discussed in Gallian’s “A Dynamic Survey of Graph Labeling” [2], many families of graphs have prime labelings; the “simpler” types of unicyclic graphs that are known to have prime labelings include **cycles**, **helms**, **crowns**, and **tadpoles**. The goal of our project will be to discover additional classes of unicyclic graphs with prime labelings, in hopes of bringing the aforementioned conjecture on all unicyclic graphs within reach. The families of graphs we will investigate include, but are not limited to:

- double-tailed tadpoles, triple-tailed tadpoles, etc.;
- irregular crowns (crowns with paths of different lengths attached to each cycle vertex);
- unicyclic graphs with one or more trivalent trees attached to cycle vertices;
- unicyclic graphs with one or more complete ternary trees attached to cycle vertices; and
- unicyclic graphs with a specified number of non-cycle cubic vertices.

Seoud and Youssef have established necessary and sufficient conditions for some graphs to have prime labelings, but they are somewhat limited in scope. Seoud has also published an upper bound on the chromatic numbers of prime graphs. These and other results may be beneficial to our students as their research project progresses.

Ernst and Rushall have already made some progress on these specific cases. We will use these initial results as a starting point with our team of students. More precisely, the 7 students involved in this project will attend a 3-credit research seminar during both semesters of the 2014-2015 academic year. Ernst and Rushall will team-teach the seminar, but eventually the students will play an equal role in leading discussions, presenting research results, etc.

Our recent experience with Seoud’s Conjecture has indicated that this problem is ripe with potential and highly appropriate as an undergraduate research project. The students can begin productive work in a single afternoon, and yet we anticipate the students producing original results worthy of publication in refereed journals by the end of the academic year. Moreover, there appear to be a virtually unlimited number of families of graphs to investigate, which will hopefully lead to a sustainable research program for undergraduates in future years.

The 7 students we have recruited to work on this research project are mathematics majors in our department. In addition, they all have very good academic records, and have proven themselves to be hard-working, reliable and creative students in previous courses that we have taught, including vector calculus, linear algebra, abstract algebra, foundations, discrete mathematics and number theory. These students regularly attend our weekly departmental undergraduate seminar, so they are familiar with the rigors associated with research and are motivated to investigate deeper problems in mathematics.

It should be noted that several presentation venues (departmental, university-wide, as well as regional conferences) will be exploited to allow our students an opportunity to showcase their efforts during the 2014-2015 academic year.

## References

[1] M.A. Seoud and M.Z. Youssef, “On Prime Labeling of Graphs,” Congressus Numerantium, Vol. 141, 1999, pp. 203-215.

[2] J.A. Gallian, “A Dynamic Survey of Graph Labeling,” The Electronic Journal of Combinatorics, Vol. 18, 2011. http://www.combinatorics.org/Surveys/ds6.pdf

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.