Welcome

Welcome to the course web page for the Spring 2021 manifestation of MAT 431: Introduction to Analysis at Northern Arizona University.

Course Info

Title: MAT 431: Introduction to Analysis
Semester: Spring 2021
Credits: 3
Section: 1
Time: 8:00-8:50AM MWF
Location: AMB 225

Instructor Info

  Dana C. Ernst, PhD
  AMB 176
  MWF 1:30-2:30PM, T 1:00-2:00PM via Zoom
  dana.ernst@nau.edu
  928.523.6852
  danaernst.com

What is This Course All About?

This course is an introduction to real analysis, which is a subject area of mathematics that deals with the set of real numbers. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real numbers, and continuity, and related properties of real-valued functions.

We will spend most of our time studying sequences and the continuity of functions. We will take an axiomatic approach (definition, theorem, and proof) to the subject, but along the way, you will develop intuition about the objects of real analysis and pick up more proof-writing skills. The emphasis of this course is on your ability to read, understand, and communicate mathematics in the context of real analysis.

The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.

An Inquiry-Based Approach

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).

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?

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 here.

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 at first glance.

As the semester progresses, it should become clear to you what the expectations are. For more details, see the syllabus.

Don’t fear failure. Not failure, but low aim, is the crime. In great attempts it is glorious even to fail.


Dana C. Ernst

Mathematics & Teaching

  Northern Arizona University
  Flagstaff, AZ
  Website
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Current Courses

  MAT 431: Intro to Analysis
  MAT 526: Combinatorics

About This Site

  This website was created using GitHub Pages and Jekyll together with Twitter Bootstrap.

  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 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.