I would...urge that people be introduced to [chaos] early in their mathematical education. [Chaos] can be studied phenomenologically by iterating it on a calculator, or even by hand. Its study does not involve as much conceptual sophistication as does elementary calculus. Such study would greatly enrich the student's intuition. Not only in research, but also in the everyday world of politics and economics, we would all be better off if more people realised that simple nonlinear systems do not necessarily possess simple dynamical properties.

Robert May, 1976

Basic Info

  • Instructor: Dave Feldman
  • Email:dfeldman@coa.edu
  • Pronouns: he/him/his
  • Problem Solving Sessions: TBA, via Zoom.
  • Teaching Assistants:Mafe Farias Briseņo, Nynke Ham, Will Valenzuela-Tyson

Course Goals

  1. Stay physically and mentally healthy and maintain intellectual and personal connection during a potentially difficult time.
  2. I want you to gain a solid understanding of the basic mathematical ideas behind chaos and fractals. As part of this, I hope you'll leave this course thinking somewhat differently about order and disorder, simplicity and complexity.
  3. I want to help you improve your basic facility with algebra and functions, your problem solving skills, your ability to create and interpret different types of graphs, and your overall mathematical confidence.
  4. I want you to gain a sense of how math and physics is done, and gain an awareness that these are not static, "dead" disciplines. I want you to gain a greater understanding of science as an institution and science as a culture, and how science intersects and influences other creative and scholarly areas.
  5. Have fun while growing and learning.

Texts

  1. Chaos: Making a New Science by James Gleick. This is a popular science book and is not a traditional text. I have around a dozen copies I can loan out.
  2. Chaos and Fractals: An Elementary Introduction, David Feldman, Oxford University Press, 2012. I'd suggest checking out the list of errata. Multiple copies will be on reserve on the library.
  3. We'll also read a few short papers, and quite likely a short story and some poetry. The exact assignments will depend on the interests of the class.

Evaluation

Your evaluation will be loosely based on the following:

  • Weekly Homework Assignments: 50 percent.
  • Final Project: 25 percent.
  • Class Participation/Engagement: 10 percent.
  • Short Writing and Responses: 15 percent.

In general, I recommend against grades; I believe they are more likely than not to interfere with genuine, reflective learning. This is especially true in this class, where I have found that many of the class activities really don't lend themselves to letter grading. I do not have any quota of A's, B's, etc.

Course Structure and Flow

This is the seventeenth(!) time I've taught this course, so on the one hand I have a very good feel for how this will go. But we are still in a pandemic, so nothing it quite normal.

  • This course is somewhat difficult to describe. I think the most important pre-requisite is to approach the material with an open mind and a sense of exploration and adventure. With a little intellectual initiative I think you will find many opportunities to make connections between the material of this course and other interests of yours.
  • There are some aspects of this class that are new this time around:
    • This class is larger than I've ever taught it before. I am trying a large class since I think there is strong student demand for the course and there is a large room in the new building that should work well.
    • There will be two smaller discussions sections/labs that will allow us to do some more interactive labs and investigations and also discuss some readings.
    • I'm also using electronic WeBWorK homework for the first time in this class. I've used WeBWorK before in many other classes, and I'm confident it will work well in this class, but there may be a few bumps in the road.
  • In many traditional math classes the instructor shows you how to do a bunch of problems, you go home and obediently practice the stuff your instructor showed you, and then you take a test. This class will not be like that. There will be some traditional-style homework, but there will be many other exercises too.
  • Some of the homework assignments are more like laboratory exercises. That is, rather than doing a quick problem and getting a simple answer, I'll ask you to explore some stuff, make observations, and look for patterns. Frequently these assignments will have open-ended questions. You will enjoy these assignments more if you approach them as you would a good laboratory exercise in a science class.
  • We will do a wide variety of things in the discussion sections, including some hands-on exercises and explorations.
  • This class is not a systematic review of algebra or trigonometry; it is not explicitly designed to prepare you for further math classes. Nevertheless, you will review and strengthen your understanding of algebra and functions by taking this course.
  • The material we'll be reading about in Gleick's book, and the work we'll do in my book and in class will shift in and out of phase with each other. The two books don't cover the same topics in the same order. In the past this hasn't been a problem; it's just something to be aware of.
  • In terms of the math stuff we'll be doing, it will take a few weeks to get to the chaos material. Before we do, what we're doing might seem odd. The initial exercises we do might seem boring or irrelevant, but this phase of the course is essential groundwork for what's to follow.
  • There will frequently be readings or exercises assigned for class. When this is the case, you should read carefully and be ready to discuss and share ideas and questions.

Some Details and Logistics

  1. This class fulfills both the ES and the QR requirements. At the same time.
  2. Homework will usually be due Fridays at the end of the day. If you need extra time on the assignments, that's fine, but please let me know.
  3. You will do some short writing for this class in addition to problem sets.
  4. You will also do a final project. If you wish, you can collaborate with other students. Projects take a wide variety of forms and can be on almost any topic related to the course material.

Homework!

  1. There will be a homework assignment due almost every Friday. It is essential that you do these assignments, as this is how one learns math, and also this is most of what your evaluation will be based on.
  2. As discussed in this video there will be two parts to almost every homework assignment:
    1. Problems to be submitted on WeBWorK
    2. Problems to be submitted on "paper" (a scanned pdf) on google classroom
  3. WeBWorK is an online homework system. I use this system even when the world is not in a global pandemic. There are three reasons why I use WeBWorK:
    1. You get instant feedback while doing the work, so you can learn right away from your mistakes. You can submit solutions many times until you get everything correct.
    2. Some problems are randomized so that you will all get slightly different versions of the questions. This means that collaborating with other students will be maximally effective, since you'll have to share solution methods and not just the final answer.
    3. Since the problems are automatically marked, I can spend more time helping you and won't have to spend as much time grading.
  4. If you need extra time for one or two of the homework assignments, it's not a big deal. But be mindful to not fall farther behind every week.
  5. I expect the homework assignments to be productively challenging. The assignments will vary in length, difficulty, and style. I don't expect you to be able to complete the homework quickly in one sitting. Don't let yourself get frustrated. I strongly suggest working with others and seeking help if you need it.
  6. You are strongly encouraged to work together on homework. You can also consult me, class tutors, other faculty, friends, and family. However, the homework you hand in should represent your own understanding.
  7. As I plan on sending out homework assignments and other information via email classroom, it is important that you check your email regularly.
  8. You will want to have at your disposal a basic calculator or phone/tablet app. I strongly suggest an actual calculator. You do not need a graphing calculator for this course (or, in my opinion, ever).

Help Sessions

The Teaching Assistants and I will have a handful of help sessions every week. You are warmly invited and encouraged to attend these sessions. Help sessions are relaxed, informal, and hopefully fun. Things that happen at help sessions:

  1. The TAs and/or I am around to offer help on the homework.
  2. Some students do most of the homework while at a help session. They work through problems alone or with others, and find it comforting to know that help is immediately at hand if needed.
  3. Others do the problems at home and come to the help session with specific questions.
  4. Help sessions are also a chance to ask general questions about the course. Conversations also sometimes wander into other areas: politics, the state of the world, what's for dinner, what classes are offered next term, and so on.
  5. Help sessions are a great way to meet other students in the class.

Everyone is welcome at help sessions! Attending these sessions help students do well in the class and get as much out of it as possible.

You should also feel free to reach out to me and the Teaching Assistants with questions. If you're stuck on a problem, it might help to take a photo of what you've done so far and include it in an email to us. We might be able to help you with a short email, or we could set up a time to meet, perhaps on zoom.

Also, you can summon help via email from within WeBWorK. If you do so, I get an email and a link to your version of the WeBWorK problem, and I can see all the answers you've submitted. Usually I can see what's going on and quickly offer some help. Sometimes it'll be an issue that doesn't lend itself to email, in which case we find a time to meet and go over it.

Electronic Distractions

As much as possible, I'd like to minimize the use of laptops, tablets, and phones during our class meetings. There is good evidence that these wonderful devices can harm the learning environment, even for those who are not using their own devices and are merely exposed to the devices of others. An excellent, reasoned discussion of this is Why I Just Asked My Students To Put Their Laptops Away by Clay Shirky. I highly recommend giving his essay a read.

There is no need to ban laptops entirely—that just seems silly to me. There will be times in class when we will do exercises that require computers, and there will also be moments when we might want to grab a device and look up the meaning of a word or a historical tidbit or something. So let's use common sense and keep devices holstered except when they're needed, at which point we should use them proudly and without apology.

Diversity and Inclusion

It is my intent that students from all backgrounds and perspectives be well served by this course, that students' learning needs be addressed both in and out of class, and that the diversity that students bring to this class be viewed as a resource, strength, and benefit. It is my intent to present materials and activities that are respectful of diversity: gender, sexuality, disability, age, religion, socioeconomic status, ethnicity, race, and culture.

Learning about diverse perspectives and identities is an ongoing process. I am always looking to learn more about power and privilege and the harmful effects of racism, sexism, homophobia, classism, and other forms of discrimination and oppression. Your suggestions are encouraged and appreciated. Please let me know ways to improve the effectiveness of the course for you personally, or for other students or student groups. If something was said or done in class (by anyone, including me) that made you feel uncomfortable, please talk to me about it. You can also reach out to Provost Ken Hill, or Associate Deans Bonnie Tai or Jamie McKown.

Discussions

We will do a lot of different things during class meetings, including discussing readings. These discussions are chance for us to explore some of the philosophical, conceptual, historical, and social aspects of chaos and fractals. The readings we do will range widely in content, style, and difficulty. There are several goals and motivations I have for discussions:

  • The discussions are a chance to figure out some things together. We will be doing some readings that are sometimes difficult and subtle and, I hope, interesting. Working together we can solidify and deepen our understanding.
  • In some of the discussions it will be particularly valuable to hear a diversity of opinions. I expect there will be a range of reactions and responses to some of what we read, and I think that there is tremendous value to listening to and engaging with a diversity of thought.

Expectations for class discussions:

  • Prepare. Do the reading and think about it. Expect to spend an hour or two preparing, although obviously this will vary a lot from reading to reading. Take notes, write down questions, and prepare responses to any questions I might have posed.
  • If you can work with a hard copy of the reading. I think it is easier to engage with a text via a physical copy, and as much as possible I don't want us to have laptops open when we meet.
  • Be present. Listen to others, respond, and contribute your own ideas and questions.

We should all work to create an inviting atmosphere and ensure that there is opportunity for all to contribute. At the same time, there is no need for everybody to contribute equally. It is natural for some people to talk more than others, and I think this is normal and good. Also, I expect that students will engage and contribute at different levels, depending on prior coursework. I see this diversity of backgrounds as a strength and not a weakness; there are roles for everyone to play. Asking good questions is as important as providing answers.

Finally, it goes without saying, but I'll say it anyway: the point of discussions is not to figure out who is right and who is wrong, but to help all of us get to a deeper, and not necessarily uniform, understanding.

Standard Disclaimers

  • You should expect to spend a minimum of 150 academically engaged hours associated with this one-credit course. These 150 hours will be spent roughly as follows: 4.5 hr/wk "in" class, 2 hr/wk reading, 8.5 hr/wk on homework.
  • By enrolling in an academic institution, a student is subscribing to common standards of academic honesty. Any cheating, plagiarism, falsifying or fabricating of data is a breach of such standards. A student must make it his or her responsibility to not use words or works of others without proper acknowledgment. Plagiarism is unacceptable and evidence of such activity is reported to the academic dean or his/her designee. Two violations of academic integrity are grounds for dismissal from the college. Students should request in-class discussions of such questions when complex issues of ethical scholarship arise.