In November of 2011, I applied for a Category 2 Small Grant from the Academy of Inquiry-Based Learning (AIBL). I found out today (December 30, 2011) that I have been awarded the grant!

The grant provides summer salary that will fund the development of inquiry-based learning (IBL) course materials for an abstract algebra course that emphasizes visualization and incorporates technology. In case anyone is interested, below is a slightly edited version of the narrative for my grant proposal.

Update (July 7, 2012): I am postponing this work until the summer of 2013. This summer is far too busy with my family’s move from Plymouth, New Hampshire to Flagstaff, Arizona. By next summer, I should be settled in at my new gig at Northern Arizona University and have a lot more time for this project.

Abstract algebra with visualization and technology

Project Summary

I am requesting summer salary in the amount of $2500 via a Category 2 AIBL Small Grant to fund the development of course materials for a junior/senior level IBL abstract algebra course. Course materials will emphasize visualization of groups and incorporate technology.

IBL Experience

I have been teaching college-level mathematics, starting as a graduate student, since the spring of 1997. During this time, I have received several teaching awards, most recently being named the 2009 and the 2011 Plymouth State University Distinguished Mathematics Professor, an honor determined by the mathematics majors at PSU. My classes have always been interactive, but initially they were predominately lecture-based. I was aware of the Moore method and inquiry-based learning (IBL), but I had never had an experience with this paradigm of teaching as a student. Moreover, by most metrics, my approach in the classroom seemed to be working. It was not until I sat through a Project NExT workshop at the 2008 MathFest led by Carol Schumacher that I began to think that perhaps IBL was a way for me to get even more out of my teaching.

In the spring of 2009, despite having no prior experience or formal training, I decided to teach my very first IBL course. The course I chose is called Logic, Proof, and Axiomatic Systems, which is meant to be our introduction to proof course. Having heard positive reviews, and being unfamiliar with the various free IBL resources, I elected to use Schumacher’s book titled Chapter Zero: Fundamental Notions of Abstract Mathematics [7]. Perhaps surprisingly, the course was a huge success and I was immediately sold on the potential impact that IBL can have on a student’s learning and character development. I have loved teaching since the day I started, but nothing compared to the joy of watching students truly learn mathematics, and often completely independent of me. I had taught the same course two semesters in a row using a mostly lecture-based approach, and I had thought that the previous two iterations went very well. However, the IBL version was a vast improvement. I have since had students from all three variations in upper-level proof-based courses and the students from the IBL version are much more independent and, in general, better proof-writers.

Over the past few semesters, I have taught the following courses using IBL:

  1. Logic, Proof, and Axiomatic Systems (twice)
  2. Real Analysis
  3. Abstract Algebra (twice)
  4. Number Theory

My first attempt at using my own theorem-sequence was while teaching Logic, Proof, and Axiomatic Systems using IBL a second time. As a starting point, I was modifying notes written by Stan Yoshinobu (Cal Poly, San Luis Obispo) and Matthew Jones (University of California, Dominguez Hills). Midway through the semester, I was writing my own notes. My experience writing my own theorem-sequence was positive. Yet, I learned quickly that it is very time consuming to produce a quality output.

Beyond my experience in the classroom, I have attended the last two Legacy of R.L. Moore Conferences and was a participant at the IBL Workshop that took place prior to the 2010 conference in Austin, TX. In addition, I am currently an AIBL Mentor for a small cohort of mathematics instructors in the Northeast that are new to IBL. Lastly, I am in the midst of conducting a couple of small-scale studies with mathematics education specialist Angela Hodge (University of Nebraska at Omaha) about student perception of the effectiveness of IBL.

Project Description

My first experience teaching abstract algebra was during a workshop in the summer of 2009 that was designed for high school mathematics teachers that were pursuing their teaching license in the state of New Hampshire. The purpose of the course was to introduce the concepts of group theory to these working teachers. The course was not intended to be a proof-based course. In fact, with the exception of one student, none of them had ever had a proof-based course before. I decided to use Nathan Carter’s excellent book titled Visual Group Theory (VGT) [4]. The book assumes only a high school mathematics background and covers a typical undergraduate course in group theory from a thoroughly visual perspective. In addition, we made use of the free and open-source software Group Explorer (GE) [1], which is also written by Nathan Carter. In the words of the developer:

“GE is mathematical visualization software for the abstract algebra classroom. It helps the user visualize group theory, builds students’ intuition, and enables experimentation with groups.”

What is remarkable is that by the end of the workshop, the students had more intuition about many concrete groups than many graduate students, despite being exposed to very few proofs. The students had played with groups, like a child with a toy. They were familiar with groups and had a visual way of thinking about various concepts.

My second experience teaching abstract algebra was in the spring of 2010. This was also my second attempt at IBL. Despite the success of using VGT the summer before, I decided that I wanted to teach a more traditional proof-based abstract algebra course. So, I chose to use Tom Judson’s open-source book title Abstract Algebra: Theory and Applications (AATA) [6]. My intention was to incorporate the visualization approach used in VGT with the flow of the content in AATA. However, I found this very difficult to do, and moreover, it required me to lecture more often than I would have liked. One of the issues is that VGT starts off by defining groups in terms of generators and relations, which does not arise in AATA until about half way through the semester. In addition to the students proving theorems, I also assigned exploratory labs, which required the students to make use of Sage [3], which is an open-source mathematics software system meant to be an alternative to Magma, Maple, Mathematica, and Matlab. Using Sage allowed the students to explore large concrete groups and make calculations that I would never ask them to do by hand. This opened up a whole realm of problems for the students to play with. The use of Sage was a success, but making more frequent and regular use would make it go even more smoothly.

This semester I am teaching abstract algebra for a third time. My initial intention was to write my own theorem-sequence that incorporated all of the positive aspects of my previous experiences with teaching the course. However, due to time constraints, I decided to postpone the project and apply for a AIBL Category 2 Small Grant to fund this work over the summer of 2012. I am currently using David Clark’s Group Theory [5] notes from the Journal of Inquiry-Based Learning in Mathematics [2]. Using this theorem-sequence has been a valuable experience and has provided me with further perspective on what I would like my own theorem-sequence to be.

My plan is to write a theorem-sequence in the spirit of Nathan Carter’s VGT, but with the mathematical rigor of Judson’s AATA and Clark’s Group Theory. In particular, the goal is to develop intuition about each new concept by introducing it from a visual perspective, where students are encouraged to play with concrete groups via short exercises. As intuition develops, students will be asked to conjecture and then prove theorems related to the relevant concepts. In addition, exploratory exercises that require the use of GE and Sage will be included. This will require me to write short introductions to the use of both of these software programs. Once the project is completed, my intention is to submit the material to the Journal of Inquiry-Based Learning in Mathematics, so that others can freely benefit from them.


[1] “Group Explorer.” [Online]. Available:

[2] €œ”Journal of Inquiry-Based Learning in Mathematics.” [Online]. Available:

[3] “Sage.”€ [Online]. Available:

[4] N. Carter, Visual Group Theory. Mathematical Association of America, 2009.

[5] D. Clark, “Group Theory”, J. Inquiry-Based Learning in Mathematics, no. 3, 2007.

[6] T. Judson, Abstract Algebra: Theory and Applications, 2011st ed. Open-source textbook, 2011.

[7] C. Schumacher, Chapter Zero: Fundamental Notions of Abstract Mathematics, 2nd ed. Pearson, 2001.

Dana C. Ernst

Mathematics & Teaching

  Northern Arizona University
  Flagstaff, AZ
  Google Scholar
  Impact Story

Current Courses

  MAT 226: Discrete Math
  MAT 690: CGT

About This Site

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  Unless stated otherwise, content on this site is licensed under a Creative Commons Attribution-Share Alike 4.0 International License.

  The views expressed on this site are my own and are not necessarily shared by my employer Northern Arizona University.

  The source code is on GitHub.

Land Acknowledgement

  Flagstaff and NAU sit at the base of the San Francisco Peaks, on homelands sacred to Native Americans throughout the region. The Peaks, which includes Humphreys Peak (12,633 feet), the highest point in Arizona, have religious significance to several Native American tribes. In particular, the Peaks form the Diné (Navajo) sacred mountain of the west, called Dook'o'oosłííd, which means "the summit that never melts". The Hopi name for the Peaks is Nuva'tukya'ovi, which translates to "place-of-snow-on-the-very-top". The land in the area surrounding Flagstaff is the ancestral homeland of the Hopi, Ndee/Nnēē (Western Apache), Yavapai, A:shiwi (Zuni Pueblo), and Diné (Navajo). We honor their past, present, and future generations, who have lived here for millennia and will forever call this place home.