Basic Info
Introduction to Chaos and Fractals: Official Course Description
This course presents an elementary introduction to chaos and
fractals. The main focus will be on using discrete dynamical
systems to illustrate many of the key phenomena of chaotic
dynamics: stable and unstable fixed and periodic points,
deterministic chaos, bifurcations, and universality. A central
result of this study will be the realization that very simple
non-linear equations can exhibit extremely complex behavior. In
particular, a simple deterministic system (i.e., physical system
governed by simple, exact mathematical rules) can behave in a
way that is unpredictable and random, (i.e., chaotic). This
result suggests that there are potentially far-reaching limits
on the ability of science to predict certain phenomena. Students
in this class will also learn about fractals---self-similar
geometric objects---including the Mandelbrot set and Julia
sets. We will also read about and discuss the development of the
field of chaos. In so doing, we will examine the nature of
scientific communities, with a particular eye toward how changes
in scientific outlooks occur. Throughout the course, students
will be encouraged to explore the relations between chaos,
fractals, and other areas of study such as literature, art, and
cultural studies. Students who successfully complete this class
should gain a quantitative and qualitative understanding of the
basic ideas of chaos and fractals, a greater understanding of
the cultural practice of science, and improved mathematical
skills. Evaluation will be based on class and lab participation,
weekly problem sets several short writing assignments and a
final project. Level: Introductory. Prerequisite: A high school
algebra course or signature of instructor. Lab fee: $10. Class
limit: 24. Meets the following degree requirements:
ES and QR
Who/when/where
- Instructor: Dave Feldman
- Pronouns: he/him/his
- Meeting Times: Monday, Thursday: 10:30 - noon; Tuesday
Labs 2:30 - 4:00 or 4:00 - 5:30
- Location: Room 103, Center for Human Ecology
- Help Sessions:
Help Session Schedule (updated frequently)
- Individual Meetings: By appointment
- Tutors: Helen Lloyd, Nonoka Toge
More Course Info and Advice
Course Goals
- Stay physically and mentally healthy and maintain intellectual
and personal connection during a potentially difficult
time.
- 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.
- 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.
- 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.
- Have fun while growing and learning.
Texts
- 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.
- 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 and I have copies to loan.
- We'll also read a few short papers, and maybe 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/Lab Participation/Engagement: 10 percent.
- Short Reflection Pieces: 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 nineteenth(!) time I've taught this course, so I have a
pretty good feel for how the class will go. That said, more than any
other course I teach, this course varies from offering to offering
depending on the interests and personalities of the students in the
class.
- 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.
- This class is large, by COA standards, at least. The class was
large the last few times I taught it, and I think it worked 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.
- 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
share ideas and questions.
Stuff about Homework
- There will be an 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.
- There will be two parts to almost every homework assignment:
- Problems to be submitted on WeBWorK
- Problems to be submitted on "paper" (a scanned pdf) on
google classroom
- WeBWorK is an online homework system. There are three
reasons why I use WeBWorK:
- 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.
- 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.
- Since the problems are automatically marked, I can spend
more time helping you and won't have to spend as much time
grading.
- 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.
- I do not expect all of the homework assignments to be easy; I
don't expect you to be able to sit down and do them easily the first
time. Don't let yourself get frustrated. I strongly suggest working
with others and seeking help if you need it.
- 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.
- As I plan on sending out homework assignments and other
information via email/google classroom, it is important that you
check your email/classroom regularly.
- You will want to have at your disposal a scientific 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).
Some Key Bits of Advice
- Come to class and lab. If you're sick, or have a
field trip for another class, or an unavoilable
appointment, that's ok. But you will get so much more out
of this class and it will be a better experience for you if
you come to class. This is especially the case if you are
feeling confused or lost.
- Submit the assignments. The homework for this
class is not intended to be tricky or arduous. It's very
important that you do it, since doing problems is how one
learns mathematical ideas and concepts. I do not expect
homework to be "perfect", and you can always resubmit an
assignment without penalty.
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:
- The TAs and/or I am around to offer help on the homework.
- 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.
- Others do the problems at home and come to the help session with
specific questions.
- Some students work in groups at help sessions, others work more
or less alone.
- 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.
- Help sessions are a great way to meet other students in the
class.
- Often there will be coffee/tea and/or snacks.
Individual Meetings
I am happy to meet with students one-on-one. The best way to set
up an appointment is to send an email. There are lots of reasons
why you might want to meet with me:
- You have some in-depth questions that there isn't time to
explore in a help session.
- You have a question that you think is too basic or you're
uncomfortable asking in a help session. (You shouldn't be
uncomfortable, since, as the saying goes, there are no dumb
questions! But I understand that you might be uncomfortable
nevertheless.)
- You want to explore possibilities for energy projects on campus
or in the community.
- You want to discuss some challenges you're facing in the class.
- You want to talk about other classes in energy or physics, or
internships, senior projects, etc.
Please don't hesitate to reach out if there's anything you want to
discuss. You should also feel free to reach out to the TAs.
Diversity, Inclusion, and Belonging
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. I aim 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 let me know. You can also reach out to:
Kourtney Collum, Provost;
Puranjot Kaur, Coordinator of the Office
of Title IX and Civil Rights Compliance;
or Maya Caines, Restorative Practices Team
Coordinator.
Gender-based discrimination: Title IX
COA is committed to fostering a safe, respectful, and
inclusive campus where all community members have equal access
to education and employment. We will not tolerate sexual
harassment or gender-based discrimination of any kind.
As a faculty member, I am considered a "Responsible
Employee." This means I am required to share any disclosures
of sexual or gender-based misconduct with the Title IX
Coordinator—even if the incident occurred before your time at
COA. The purpose of this requirement is to ensure that anyone
who has experienced sexual misconduct is offered support,
information about their rights, and access to available
resources. Please know that you are not required to respond to
outreach, and a disclosure does not automatically result in a
report to law enforcement.
For more information about Title IX, COA’s policies, and
available resources, please
visit: coa.edu/human-resources/title-ix.
If you would like to ask questions or explore support
options, you are encouraged to contact COA's Title IX
Coordinator, Puranjot Kaur, at pkaur@coa.edu. Speaking with
the Title IX Coordinator does not automatically initiate a
college resolution process. Much of her work focuses on
providing supportive measures to help you continue to fully
participate in COA's programs and activities.
Statements about Academic Honesty and Hours of Academic
Engagement
- I am required to remind you that: "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."
- I am also required to say that: You should expect to
spend 150 hours of academically engaged time on this
course, or 15 hours per week. In addition to 4.5 hours per
week in class or discussion section, in a typical week
you will spend 2 hours reading and preparing for class and
8.5 hours attending help sessions and completing
assignments.
- Can I use generative "AI" tools like chatGPT or
Gemini? But in brief:
- For writing and homework? Generally, no. Quoting
Martha
Donovan, "I expect all drafts of written work for this
course to be your own. I am interested in reading and
responding to your own original thinking and writing at
every phase of the writing process." But we'll talk
about the nuances of this.
- For helping with homework? Yes, use with
care, and cite your sources, as you would if doing a
"normal" internet search. Gen "AI" isn't bad at
explaining technical concepts sometimes. (I like to
think that one of the reasons it isn't bad at explaining
chaos and fractals is that both of my books were part of
its training set. So if you need something explained,
you could just check out those books. Or, you know, you
could talk to me. :)
- For coding? Yes. Definitely.