Welcome to the course web page for the Spring 2017 manifestation of MAT 320: Foundations of Mathematics at Northern Arizona University.

AMB 176

MWF 10:15-11:15AM, T 11:00-12:00PM

dana.ernst@nau.edu

928.523.6852

danaernst.com

The primary objective of this course is to develop skills necessary for effective proof writing. Students will improve their ability to read and write mathematics. Successful completion of MAT 320 provides students with the background necessary for upper division mathematics courses. Also, the purpose of any mathematics course is to challenge and train the mind. Learning mathematics enhances critical thinking and problem solving skills. The course description says:

The course trains students on methods and techniques of mathematical communication, focusing on proofs but also covering expository writing and problem-solving explanations.

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 inquiry-based learning (IBL).

This is not a lecture-oriented class or one in which mimicking prefabricated examples will lead you to success. You will be expected to work actively to construct your own understanding of the topics at hand, with the readily available help of me and your classmates. Many of the concepts you learn and problems you work will be new to you and ask you to stretch your thinking. You will experience *frustration* and *failure* before you experience *understanding*. This is part of the normal learning process. **If you are doing things well, you should be confused at different points in the semester. The material is too rich for a human being to completely understand it immediately.** Your viability as a professional in the modern workforce depends on your ability to embrace this learning process and make it work for you.

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, and 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). If you want to learn more about IBL, read my blog post titled What the Heck is IBL?

Much of the course will be devoted to students presenting their proposed solutions/proofs on the board and a significant portion of your grade will be determined by how much mathematics you produce. I use the word “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 solving genuine problems is difficult and takes time. You shouldn’t expect to complete each problem in 10 minutes or less. Sometimes, you might have to stare at the problem for an hour before even understanding how to get started. In fact, solving difficult problems can be a lot like the clip from the *Big Bang Theory* located below.

In this course, *everyone* will be required to

- read and interact with course notes on your own;
- write up quality solutions/proofs to assigned problems;
- present solutions/proofs on the board to the rest of the class;
- participate in discussions centered around a student’s presented solution/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.

Mathematics & Teaching

Northern Arizona University

Flagstaff, AZ

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MAT 441: Topology

MAT 526: Combinatorics

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