Syllabus

For an instant we hovered upon its threshold. But the impulse, the command that had carried us thus far was not to stop here. Into it and up it we were thrust, our feet barely touching the glimmering surface… —Abraham Merritt

Overview

This class is structured as a sustained meditation on a single concept: the derivative. We’ll start with a rigorous treatment of derivatives of single variable functions, before proceeding to derivatives of multivariable functions. Then we’ll define manifolds and tangent spaces, and finally we’ll discuss how pushing forward tangent vectors provides an even further generalization of derivatives.

Prerequisites

The two main prerequisites for this course are real analysis 1 (at the level of MA375), and linear algebra (at the level of MA220). We’ll make substantial use concepts discussed in both of these classes.

The linear algebra content will mostly pick up when we start discussing multivariable derivatives. If it’s been a while, I encourage you to use the first week or so to review some linear algebra.

Course Structure Philosophy

In the long run, more important than learning any particular piece of math is learning how to learn math independently (in technical parlance, that’s “how to be a self-regulated learner of math”). Improving yourself in this regard is my foremost goal for you for this course.

Research shows that the following three things are key aspects of becoming self-regular learners, and it will be good for you to keep them in the front of your mind as we go through the course.

1. Active reading. Reading math is very different from other kinds of reading. You cannot read math the same way you’d read a novel for pleasure if you want to get anything out of it. You have to stop constantly as you’re reading math. Try to work out examples yourself, instead of just reading through them. Doodle pictures to make sure you have some kind of a picture in your head of what’s going on. Formulate precise questions about things you don’t understand.

2. Peer communication. Talking to your peers about math is incredibly important. If you don’t understand a particular concept and ask your peers, you’re much more likely to get an explanation that you actually find helpful. If you think you do understand a particular concept and help a peer who’s struggling, you’ll almost certainly find that the process of explaining the concept to your peer will solidify your own understanding of it.

3. Self reflection. A key part of learning how to learn is reflecting on your learning and taking the time to ask yourself questions about your learning. What parts of your study habits are working for you? What parts aren’t working? How actively are you reading? Is there anything you could try changing?

All three of these are built into the way this course is structured. The first two are at the forefront of an evidence-based course structure known as peer instruction, which was pioneered by the physicist Eric Mazur at Harvard. It is predicated on the observation that information transfer (listening to lectures or reading books) is easier than information assimilation (solving problems and explaining concepts to others), so it makes sense to move information transfer out of the classroom and information assimilation into the classroom. There’s a growing body of data that suggests this format is quite effective: by a certain metric, it leads to a two-fold improvement in conceptual understanding over more traditional methods!

To round off our three-pronged attack, you’ll be asked to complete weekly self-reflection forms. I encourage you to take advantage of these and use them as an opportunity to tweak your learning habits as you find necessary.

Course Mechanics

On the course webpage, you’ll find a calendar that looks like this:

This means that, on day A, we’ll be discussing topic B in class. You’ll want to prepare for this by doing reading assignment C before class. In D, you’ll find information about problems we worked on in class on day A. In E, you’ll find information about any important events and deadlines that will occur on day A. More details follow.

Before class

Before class on day A, you’ll do the reading assignment C. The reading assignments have two parts: reading some sections from of the book, and then solving some exercise from that section.

Then you’ll use an online form to submit a question that you have about the reading, and to indicate that you have attempted the associated exercises. For the day A reading assignment, you’ll submit this form by 11:59pm the night before day A.

At the beginning of class on day A, you will also submit a hard copy of your solution to the exercises.

A few further thoughts about this:

• You don’t need to wait till the night before to do the reading assignment. I’ve often found that letting things simmer in the back of my mind for a while helps me understand them, so I’d encourage you to get ahead.

• What if you don’t have any questions about the reading? That’s fine; submit a non-question instead. For example, you might decide to send me a question that you had, but then you managed to figure out (either by yourself or by asking someone for help). Or you might send me something that you yourself understand but you think one of your peers will find confusing. I’m mostly looking for an indication that you read the assigned reading and made a sincere attempt to process it.

• I’ll do my best to tailor our in-class discussions to address the questions you submit. The earlier you send me your question, the more likely I am to be able to work it into our discussions. If I don’t adequately address your question in class, please ask me again during office hours!

• I won’t grade your solution to the exercises for correctness; I just want to see that you’ve put in an honest effort. If there’s something that you’d like for me to take a close look at, please ask during office hours!

In class

We’ll start class with a brief summary of the reading. This is not intended to be a substitute for having done the reading. Instead, the idea is to refresh your memory about what we’ll be discussing in class today (to “get in the zone”, so to speak).

We’ll spend most of class solving problems in the following format.

• I’ll put a problem up (usually, a true-false problem).
• You’ll think about the problem by yourself for a couple of minutes.
• We’ll vote on an answer to the problem.
• You’ll have a few minutes to talk to your classmates about the problem.
• We’ll vote again.
• I’ll tell you how I’d think about the problem.

Depending on how much time we have left after this, we may have some open-ended time at the end of class (eg, to work on other non-reading-assignment exercises).

After class

After class, you should be ready to tackle the other exercises from the reading (ie, the ones that weren’t already assigned for the reading assignment).

These exercises will not be collected (except for a couple of them of your choosing which will be submitted for the proof portfolio, see below). But, quiz problems may strongly resemble these exercises, so you’re encouraged to spend time figuring out as many of them as you can.

Assessment

Grades will be calculated as follows.

Here are some details about each of the components of your grade.

• Each reading assignment is worth up to 2 points: 1 for the question, 1 for the exercises. The reading assignments component of your grade will be the total number of points you accumulate in this way, out of a maximum of \(2(n-2)\) points, where \(n\) is the total number of reading assignments assigned (ie, you don’t need to do submit both things every day to get a perfect score for this component of the grade).

• There’ll be a short writing assignment towards the beginning of the block (see the webpage for details).

• The proof portfolio will consist of some proofs that’ll be graded for both mathematical correctness and quality of proof writing (see the webpage for details).

• There’ll be three quizzes during the block.

  • They’ll take place during regular class time (see the calendar for the dates).
  • There will be some true-false questions, and some free response questions (which could be calculations or proofs).
  • No books or electronic devices will be permitted, but you will be allowed one handwritten page of notes.
  • A day or two after each quiz, I’ll block off a couple of hours in the afternoon for “quiz revisions.” During this time, you can come by and meet with me one-on-one to discuss up to 2 of the true-false questions that you left blank. If you convince me that you now fully understand the solution, you’ll get points for that question.
  • If you won’t be able to to make it to a quiz, please reach out to me before the quiz to let me know and we’ll work something out.

• The final exam will be on the fourth Tuesday, during the usual class time.

  • The format will be similar to that of the quizzes, except that the final will be a bit longer than the quizzes, and there won’t be any “revisions.”
  • If your score on the final exam is higher than your lowest quiz score, I’ll use the final exam score to replace the lowest quiz score.

• There’ll be a project. The final product will be due on the last day of class. See the webpage for details.

• You’ll get a full score for the participation component of your grade as long as I am able to see that you are putting in a good faith effort to engage with the class. This includes completing the weekly self-reflection forms!

Accommodations

If you anticipate or experience any disability-related barriers to your learning in this course, please discuss your concerns with me as soon as possible and we’ll find a way to provide the accommodations that you need. Also, please contact the office of Accessibility Resources if you have not done so already.

Honor code

Please make sure that you’re familiar with the Honor Code at CC. Violations of the Honor Code will have to be reported to the Honor Council, which is really no fun for anyone.

Advice

• Mens sana in corpore sano.
  • It’s very challenging to get caught up after being ill, especially on the block plan. I encourage you to make sure that you’re getting enough sleep, that you’re eating well, and that you’re staying physically active. Take care of yourself!
  • If you do happen to get sick, please stay home to avoid getting your classmates sick, and let me know so that we can figure out how you’ll make up class work.
• It’s probably evident that the structure of this class will require you to be proactive about your learning. If you need help developing good study habits, please ask and I’ll be happy to help.
• Real analysis is hard. I remember really struggling with it. I also remember that the point when I started feeling more comfortable with it was the point when I started doodling pictures (instead of relying on symbol pushing to lead me down the right track). It can be challenging to develop good visual intuition, but I think visual intuition is really the key to succeeding in real analysis.