Foundations of Math – Spring 2013

Welcome

Welcome to the course web page for the Spring 2013 manifestation of MAT 320: Foundations at Northern Arizona University.

Instructor Information

Instructor: Dr. Dana C. Ernst
Office: AMB 119
Office Phone: 928.523.6852
Email: dana.ernst@nau.edu
Office Hours: MWF at 10:00-11:30AM (or by appointment)
Webpage: http://danaernst.com

What is this course all about?

This course will likely be different than any other math class that you have taken before for two main reasons. First, you are used to being asked to do things like: “solve for $x$,” “take the derivative of this function,” “integrate this function,” etc. Accomplishing tasks like these usually amounts to mimicking examples that you have seen in class or in your textbook. The steps you take to “solve” problems like these are always justified by mathematical facts (theorems), but rarely are you paying explicit attention to when you are actually using these facts. Furthermore, justifying (i.e., proving) the mathematical facts you use may have been omitted by the instructor. And, even if the instructor did prove a given theorem, you may not have taken the time or have been able to digest the content of the proof.

Unlike previous courses, this course is all about “proof.” Mathematicians are in the business of proving theorems and this is exactly our endeavor. For the first time, you will be exposed to what “doing” mathematics is really all about. This will most likely be a shock to your system. Considering the number of math courses that you have taken before you arrived here, one would think that you have some idea what mathematics is all about. You must be prepared to modify your paradigm. The second reason why this course will be different for you is that the method by which the class will run and the expectations I have of you will be different. 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 absolutely love lecturing and I do believe there is value in it, 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 the Moore method (after R.L. Moore). Modifications of the Moore method are also referred to as inquiry-based learning (IBL) or discovery-based learning. If you want to learn more about IBL, go here.

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 on our course webpage that lives outside of Bb Learn. I suggest you bookmark this page. In addition, we will be utilizing an LMS called Canvas. We will use Canvas to manage grades and the submission of Weekly Homework assignments. Also, we will use the forums within Canvas to facilitate out of class discussion. I will send the class an invite to our Canvas page and spend time in class discussing its use.

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, but the first half of the notes are an adaptation of notes written by Stan Yoshinobu (Cal Poly) and Matthew Jones (California State University, Dominguez Hills). 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.

In addition to working the problems in the notes, I expect you to be reading them. 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 forum on Canvas. Here’s what [Paul Halmos}(http://en.wikipedia.org/wiki/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 forum, as I will post comments there for all to benefit from. Lastly, you can always contact me.