2. Teaching
3. MAT431

Welcome

Welcome to the course web page for the Fall 2023 manifestation of MAT 431: Introduction to Analysis at Northern Arizona University.

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. According to Laursen and Rasmussen (PDF), the Four Pillars of IBL are:

• Students engage deeply with coherent \& meaningful mathematical tasks.
• Students collaboratively process mathematical ideas.
• Instructors inquire into student thinking.
• Instructors foster equity in their design and facilitation choices.

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

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

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!

In any act of creation, there must be room for experimentation, and thus allowance for mistakes, even failure. A key goal of our community is that we support each other—sharpening each other’s thinking but also bolstering each other’s confidence—so that we can make failure a productive experience. Mistakes are inevitable, and they should not be an obstacle to further progress. It’s normal to struggle and be confused as you work through new material. Accepting that means you can keep working even while feeling stuck, until you overcome and reach even greater accomplishments.

You will become clever through your mistakes.

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 and textbook 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.

Tell me and I forget, teach me and I may remember, involve me and I learn.