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Dana Ernst
Works at Northern Arizona University
Attended George Mason University
Lives in Flagstaff, AZ
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Dana Ernst

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How SageMath, an open source mathematics software system, got its start.
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Dana Ernst

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+Hugh Denoncourt, Dustin Story, and I recently wrote up a 4-page paper that summarizes an interesting open problem involving the longest element of the symmetric group. The paper is available as a free download from "Open Problems in Mathematics." Here are some details.

Title: On the number of commutation classes of the longest element in the symmetric group

Original proposers of the open problem: Donald E. Knuth
The year when the open problem was proposed: 1992
Sponsor of the submission: Richard M. Green - University of Colorado Boulder
AMS Subject classification: 05
Status of the problem: Open


Abstract: Using the standard Coxeter presentation for the symmetric group S_n, two reduced expressions for the same group element are said to be commutation equivalent if we can obtain one expression from the other by applying a finite sequence of commutations. The resulting equivalence classes of reduced expressions are called commutation classes. How many commutation classes are there for the longest element in S_n?

Thanks to +Richard Green for being our sponsor. Potentially of interest to +Drew Armstrong and +Christopher Hanusa.
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On Friday last week, I gave a talk titled "The mathematics of Boggle logic puzzles" in our Friday Afternoon Mathematics Undergraduate Seminar (FAMUS). The talk was inspired by +Richard Green's post on Google+ about the topic:

https://plus.google.com/101584889282878921052/posts/gNmuwUfFDcU

In the second half of FAMUS, I discussed my path from hating mathematics as a child to falling in love with mathematics to eventually earning my PhD and becoming a professor of mathematics. I also shared a bit about what I love about mathematics, as well as the joys and struggles of teaching. The students seemed to really enjoy this.
Boggle is a popular word search game where players compete to find as many words as they can in a 4 by 4 grid of letters. On the other hand, a Boggle logic puzzle is the game of Boggle played in reverse. A list of words is given and you need to recreate the board. In this talk, we will discuss some of the mathematics behind Boggle logic puzzles. In particular, we will summarize some of the known results and highlight a few open problems. In th...
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It sounds like you gave a great talk, and I'm glad you found the post helpful.
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My undergraduate abstract algebra students had a great conversation about productive failure on the first day of the semester while we were doing my "Setting the Stage" activity. During our discussion, I mentioned that skateboarders likely attempt and fail at doing a kick flip hundreds of times before getting it. Well, I just had an awesome conversation with a group of skateboarders on campus about this. These skateboarders were pretty darn awesome by the way. I walked up and asked them how many times they tried a kick flip before getting it. Every one of them guessed it was in the thousands! One kid said he tried it roughly a hundred times everyday for six months. These kids had a lot of insight about productive failure and persistence. We can learn a thing or two from the skaters.
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+William Stein Shouldn't there be a rule "no skating while using your phone" in the same way it's "no driving while using your phone"? ;-)
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New at Math Ed Matters: +Angie Hodge discusses the use of a math autobiography assignment. 
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Here are the slides for the short talk I gave during the "Increasing Student Engagement & Understanding through Active Learning Strategies in Calculus I" minicourse at the 2016 Joint Mathematics Meetings.

Title: Student presentations in calculus: A vehicle for getting students to talk about mathematics
This talk was given on January 8, 2016 as part of the "Increasing Student Engagement & Understanding through Active Learning Strategies in Calculus I" minicourse at the 2016 Joint Mathematics Meeting in Seattle, WA.
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Those are fantastic, Dana. I hope it went well.
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The 2016 Nebraska Conference for Undergraduate Women in Mathematics (NCUWM) took place this weekend at the University of Nebraska-Lincoln. This was the 18th annual conference. Cool!

I wasn't there, but I had some students present stuff.

Joni Hazelman and Parker Montfort presented the following poster.

Title: Explorations of Conway’s Sylver Coinage Game

Abstract: Sylver Coinage is a game in which two players, A and B, alternately name positive integers that are not the sum of nonnegative multiples of previously named integers. The person who names 1 is the loser! This seemingly innocent looking game is the subject of one of John Conway's open problems with monetary rewards. One such open problem is: If player A names 16 to start, and both players play optimally thereafter, then who wins? In this talk, we will discuss a simplified version of the game in which a fixed positive integer n (greater than 2) is agreed upon in advance. Then A and B alternately name positive integers from the set {1,2,...,n} that are not linear combinations with positive coefficients of previously named numbers. As in the original game, the person who is forced to name 1 is the loser. We will investigate who wins under optimal play for given values of n and determine the Nim-values for the simplified game under certain conditions. Joint work with Robert Voinescu and Ryan Wood.

Link: https://speakerdeck.com/dcernst/explorations-of-conways-sylver-coinage-game

+Hannah Paige Prawzinsky gave the following presentation.

Title: New coprime vertex labelings

Abstract: A coprime vertex labeling is an injective assignment of the labels {1, 2, . . . , n} to the vertices of an n-vertex simple connected graph such that adjacent vertices receive relatively prime labels. I will present new labelings for several infinite families of graphs. No prior knowledge of graph theory will be assumed. Joint work with Nathan Diefenderfer, Michael Hastings, Levi Heath, Briahna Preston, Emily White, and Alyssa Whittemore. This research was supported by the National Science Foundation grant #DMS-1148695 through the Center for Undergraduate Research (CURM).

Link: https://speakerdeck.com/dcernst/new-coprime-vertex-labelings
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I enjoyed this.
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This made me think that in the future, "scribe" might again become a career, just like it was in ancient times. I use computers so much these days that I have trouble with my handwriting -- I have to concentrate hard to even sign my own name sometimes!
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+Albert Schueller just posted about a new open-source abstract algebra text over on "Open Mathbook". I've also added it to my list of open-source math textbooks located at http://dcernst.github.io/resources/free-and-open-source-textbooks/.
Elementary Abstract Algebra: Examples and Applications is a new open-source abstract algebra text by a number of contributors: Justin Hill, Chris Thron (eds), Thomas Judson, Dave Witte Morris, Joy Morris, A. J. Hildebrand, Ho...
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I am teaching a course on Riemannian Geometry now and will endeavour to put my notes on github also. +Dana Ernst = Inspiration :)
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"A major obstacle for some students, in my opinion, is the misperception that mistakes are "bad," which is one of the unfortunate side effects afflicting many of our students, who have come up through a very rote memorization education. When mistakes are viewed as something to be avoided, then students holding this view are severely limited. Curiosity and the adventurous spirit of an explorer are sacrificed for the sake of not ever making a mistake. Productive failure is part of the set of topics within the general area of growth mindset."
 
IBL Calculus and "Learn by Doing" Assignments
This past fall I had the pleasure of teaching Calculus 1 to freshmen.  It was a blast!  I enjoyed every minute of it, and it's truly a privilege to be able to be one of the first professors that students see in college.  Not only is it a great experience fo...
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mistakes are bad in exam and test time for sure. while learning we all make mistakes. we become more perfect through our mistakes.
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Sweet!  Paper with +Sarah Salmon and Michael Hastings accepted to Involve. arXiv version updated soon.
Abstract: The Temperley--Lieb algebra is a finite dimensional associative algebra that arose in the context of statistical mechanics and occurs naturally as a quotient of the Hecke algebra arising from a Coxeter group of type $A$. It is often realized in terms of a certain diagram algebra, ...
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Congratulations! I hope the referee at least read it carefully.
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Quick Post: Re-Entering a New Term, Ken Robinson
Happy New Year!  Things have been relatively quiet on the IBL blog, but that's because we have some big things in the pipeline, such as a new NSF grant to expand IBL Workshop offerings. This is year 16 for me as a teacher (post PhD program).  Year 20 includ...
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Work
Occupation
Assistant Professor of Mathematics
Employment
  • Northern Arizona University
    Assistant Professor, 2012 - present
  • University of Colorado at Boulder
    Graduate Student/Teaching Assistant, 2003 - 2008
  • Front Range Community College
    Math Faculty, 2001 - 2003
  • Northern Arizona University
    Graduate Student/Instructor, 1997 - 2001
  • Plymouth State University
    Assistant Professor, 2008 - 2012
Places
Map of the places this user has livedMap of the places this user has livedMap of the places this user has lived
Currently
Flagstaff, AZ
Previously
Williamsville, NY - Fairfax, VA - Flagstaff, AZ - Boulder, CO - Plymouth, NH
Story
Tagline
Father of two boys, husband, mathematician, cyclist, trail runner, rock climber, and coffee drinker.
Introduction

My name is Dana Ernst and I am an assistant professor in the Department of Mathematics and Statistics at Northern Arizona University in Flagstaff, AZ

My primary research interests are in the interplay between combinatorics and algebraic structures. More specifically, I study the combinatorics of Coxeter groups and their associated Hecke algebrasKazhdan-Lusztig theory, generalized Temperley-Lieb algebrasdiagram algebras, and heaps of pieces. By employing combinatorial tools such as diagram algebras and heaps of pieces, one can gain insight into algebraic structures associated to Coxeter groups, and, conversely, the corresponding structure theory can often lead to surprising combinatorial results. The combinatorial nature of my research naturally lends itself to collaborations with undergraduate students, and my goal is to incorporate undergraduates in my research as much as possible. See my scholarship page for more information.

Furthermore, I am passionate about mathematics education. In particular, I am interested in inquiry-based learning (IBL) and the Moore method for teaching mathematics. This educational paradigm has transformed my teaching. I am currently a Special Projects Coordinator for the Academy of Inquiry-Based Learning and a mentor for several new IBL practitioners. Moreover, I actively give talks and organize workshops on the benefits of IBL as well as the nuts and bolts of how to implement this approach in the mathematics classroom.

I am also interested in utilizing technology to enhance the teaching and learning of mathematics. Specifically, I choose free and open-source software and technologies when appropriate. For example, I have been incorporating Sage and GeoGebra into my teaching. Sage is a free open-source mathematics software system licensed under the GPL. It combines the power of many existing open-source packages into a common Python-based interface.  For examples of a few of the cool things you can do with Sage, check this page. According to their webpage, GeoGebra is free and multi-platform dynamic mathematics software for all levels of education that joins geometry, algebra, tables, graphing, statistics and calculus in one easy-to-use package. There are tons of awesome GeoGebra examples located here.

In addition to using free and open-source software, I am inspired by the recent open-source textbookmovement and I strongly believe that educators should choose free, open-source, or low cost textbooks when a viable alternative exists. For a list of open-source textbooks, go here and here.

Angie Hodge and I are coauthors for Math Ed Matters, which is a (roughly) monthly column sponsored by the Mathematical Association of America. The column explores topics and current events related to undergraduate mathematics education. Posts aim to inspire, provoke deep thought, and provide ideas for the mathematics—and mathematics education—classroom. Our interest in and engagement with IBL color the column's content.

I also maintain a personal blog, which is part of the Booles' Rings network of academic home pages/blogs.  On my blog, I typically write about topics related to mathematics, education, and technology.  In addition, I occasionally post about my cycling, trailing running, and rock climbing adventures on my Elevation Gain blog

Lastly, I am a husband and a father of two incredible sons. Oh, I enjoy drinking copious amounts of coffee, too.

Education
  • George Mason University
    BS, Mathematics, 1993 - 1997
  • Northern Arizona University
    MS, Mathematics, 1997 - 2000
  • University of Colorado at Boulder
    PhD, Mathematics, 2003 - 2008
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