Abstract Algebra – Fall 2014

Welcome

Welcome to the course web page for the Fall 2014 manifestation of MAT 411: Introduction to Abstract Algebra at Northern Arizona University.

Instructor Information

Instructor: Dr. Dana C. Ernst
Office: AMB 176
Office Phone: 928.523.6852
Email: dana.ernst@nau.edu
Office Hours: 12:30-1:45PM on Mondays, Wednesdays, or Fridays, 2:15-3:15PM on Tuesdays (or by appointment)
Webpage: http://danaernst.com

What is This Course All About?

CayleyDiagramAbstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. This course is an introduction to abstract algebra. We will spend most of our time studying groups. Group theory is the study of symmetry, and is one of the most beautiful areas in all of mathematics. It arises in puzzles, visual arts, music, nature, the physical and life sciences, computer science, cryptography, and of course, throughout mathematics. This course will cover the basic concepts of group theory, and a special effort will be made to emphasize the intuition behind the concepts and motivate the subject matter. In the last few weeks of the semester, we will also introduce rings and fields.

An Inquiry-Based Approach

In a typical course, math or otherwise, you sit and listen to a lecture. (Hopefully) These lectures are polished and well-delivered. You may have often been lured into believing that the instructor has opened up your head and is pouring knowledge into it. I enjoy lecturing, but I also believe that in reality most students do not learn by simply listening. You must be active in the learning you are doing. I’m sure each of you have said to yourselves, “Hmmm, I understood this concept when the professor was going over it, but now that I am alone, I am lost.” In order to promote a more active participation in your learning, we will incorporate ideas from an educational philosophy called inquiry-based learning (IBL).

Loosely speaking, IBL is a student-centered method of teaching mathematics that engages students in sense-making activities. Students are given tasks requiring them to solve problems, conjecture, experiment, explore, create, communicate. Rather than showing facts or a clear, smooth path to a solution, the instructor guides and mentors students via well-crafted problems through an adventure in mathematical discovery. Effective IBL courses encourage deep engagement in rich mathematical activities and provide opportunities to collaborate with peers (either through class presentations or group-oriented work).

Perhaps this is sufficiently vague, but I believe that there are two essential elements to IBL. Students should as much as possible be responsible for:

  • Guiding the acquisition of knowledge, and
  • Validating the ideas presented. That is, students should not be looking to the instructor as the sole authority.

If you want to learn more about IBL, read my blog post titled What the heck is IBL?. Here is a great video about inquiry-based learning in mathematics. The video is part 1 of 3 (go watch part 2 and part 3).

Much of the course will be devoted to students proving theorems on the board and a significant portion of your grade will be determined by how much mathematics you produce. I use the work “produce” because I believe that the best way to learn mathematics is by doing mathematics. Someone cannot master a musical instrument or a martial art by simply watching, and in a similar fashion, you cannot master mathematics by simply watching; you must do mathematics!

Furthermore, it is important to understand that proving theorems is difficult and takes time. You shouldn’t expect to complete a single proof in 10 minutes. Sometimes, you might have to stare at the statement for an hour before even understanding how to get started. In fact, proving theorems can be a lot like the clip from the Big Bang Theory located here.

In this course, everyone will be required to

  • read and interact with course notes on your own;
  • write up quality proofs to assigned problems;
  • present proofs on the board to the rest of the class;
  • participate in discussions centered around a student’s presented proof;
  • call upon your own prodigious mental faculties to respond in flexible, thoughtful, and creative ways to problems that may seem unfamiliar on first glance.

As the semester progresses, it should become clear to you what the expectations are. This will be new to many of you and there may be some growing pains associated with it.

For more details, see the syllabus.

Learning Management System

We will be making limited use of Bb Learn this semester, which is Northern Arizona University’s default learning management system (LMS). Most course content (e.g., syllabus, course notes, homework, etc.) will be housed here on our course webpage (the page you are reading now) that lives outside of Bb Learn. I suggest you bookmark this page. In addition, we will be utilizing a Google Group to facilitate out of class discussion. I will send the class an invite to our Google Group and briefly discuss its use. The only thing I will use Bb Learn for is to communicate grades.

Course Notes

We will not be using a textbook this semester, but rather a task-sequence adopted for IBL. The task-sequence that we are using was written by me. Any errors in the notes are no one’s fault but my own. In this vein, if you think you see an error, please inform me, so that it can be remedied. The course notes are available here.

I will not be covering every detail of the notes and the only way to achieve a sufficient understanding of the material is to be digesting the reading in a meaningful way. You should be seeking clarification about the content of the notes whenever necessary by asking questions in class or posting questions to the course Google Group. Here’s what Paul Halmos has to say about reading mathematics.

Don’t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof
use the hypothesis?

Getting Help

There are many resources available to get help. First, I recommend that you work on homework in groups as much as possible and to come see me whenever necessary. Also, you are strongly encouraged to ask questions in the course Google Group, as I will post comments there for all to benefit from. Lastly, you can always contact me via email.