Syllabus

[D]ifferential equations… represent the most powerful tool humanity has ever created for making sense of the material world. Sir Isaac Newton used them to solve the ancient mystery of planetary motion. In so doing, he unified the heavens and the earth, showing that the same laws of motion applied to both… [S]ince Newton, mankind has come to realize that the laws of physics are always expressed in the language of differential equations. This is true for the equations governing the flow of heat, air and water; for the laws of electricity and magnetism; even for the unfamiliar and often counterintuitive atomic realm where quantum mechanics reigns… [T]heoretical physics boils down to finding the right differential equations and solving them.
—Steven Strogatz

Overview

MA315 is an introduction to ordinary differential equations (ODEs). This is a very expansive subject with applications throughout the physical and social sciences. Part of the reason for the expansiveness of the subject is that there is no general theory telling us how to solve all possible ODEs; instead, we have to learn to apply idiosyncratic techniques for each of many different classes of ODEs. Broadly speaking, our focus in this class will be on (1) first order ODEs, and (2) first order linear systems of ODEs with constant coefficients.

Prerequisites

Linear algebra (at roughly the level of MA220) is an important prerequisite for this course. It won’t show up much during the first week, but by the second week we’ll be using it heavily. In case your linear algebra is rusty, I encourage you to use the beginning of the course to review the following concepts: vector spaces, subspaces, linear independence, dimension of a vector space, matrices, determinants, eigenvectors, eigenvalues, and diagonalization.

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 this, 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 can’t read a math textbook 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 the 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’s 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 towards the goal of becoming self-regulated learners of mathematics, 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. After class, you should be ready to tackle the problem set D. In E, you’ll find information about any afternoon events and deadlines that will occur on day A. More details follow.

Reading assignments

The reading assignments have two parts: reading some sections from of the book, and then doing a few exercises to help establish a basic understanding of the concepts introduced in those sections.

To show that you did the reading assignment, you’ll use an online form to submit a question that you have about the reading, and to indicate that you have completed the exercises associated with the reading assignment. For the day A reading assignment, you’ll submit this form by 11:59pm the night before day A, and then you’ll submit a hard copy of the exercises at the beginning of class on day A.

A few thoughts about these submissions:

• Remember that you don’t necessarily need to wait till the night before day A to work on and submit the reading assignment for day A. You’re strongly encouraged to get ahead. I’ve often found that letting things simmer in the back of my mind for a while helps me understand them.

• You might be asking, “What if I don’t have any questions about the reading?!” That’s fine. You can 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 understand but you think one of your peers might 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 as many of your questions as I can. 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 address your question in class or via email, please find me during office hours and ask again!

• I’ll check your solutions to the reading assignment exercises only for completeness. The exercises numbered in the 100s have solutions in the back of the textbook.

In-class structure

I’ll begin class with a very brief discussion 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 by working through an couple of examples.

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

• I’ll put a problem on the board.
• 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 solve the problem.

Problem sets

After class, you’ll hopefully be ready to tackle the problem set listed in the fourth column of the calendar. These problems won’t be collected, but some of the problems on the quizzes will heavily resemble problems from the problem sets, so you’re encouraged to spend time really figuring them out.

Assessment

Grades will be calculated as follows.

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

• Every reading assignment you submit 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. In other words, you don’t need to do submit both things every day to get a perfect score for this component of your grade.

• There will be a short writing assignment towards the beginning of the block. You’ll find more details on the webpage.

• There will be three quizzes during the block. They will take in class (see the calendar). There will be some free response questions, and some true-false questions. 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, at the usual class time. The format will be similar to that of the quizzes, except that the final will be 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 will be two projects during the course. The first will be an individual project that will be due at the beginning of week 3. The second will be a group project, about a topic of your choice related to class material, which you will present on the last day of class. You’ll find more details about both on the webpage.

• 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 are 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 is 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.