You may wish to bookmark the
course website: http://math.hws.edu/~mitchell/Math135F05/index.html where
I will post most of the course documents.
About Math 135
This course is likely to be very different from any other math course you may have taken.
For this reason the course may be
unsettling to you.
I assume that you are taking this course because you have a very strong interest in mathematics. Math 135
is intended for those who are seriously considering a major or minor in mathematics. It lays the foundation
for the methodology of most of the later courses in mathematics.
One of my colleagues in the Mathematics Department calls
First Steps into Advanced Mathematics a language course.
Indeed, mathematics does have its own precise language.
It is this precise language that permits rigorous proofs and
reveals the beauty and elegance of mathematics.
So this course is also about aesthetics.
Obviously some of the material in the course will be new to you. But the emphasis in
Math 135 is on doing mathematics: It is more about process than content.
The major goal in the course is for you to produce, create, and discover mathematics.
Your previous mathematics courses likely emphasized solving variations on
problems that were illustrated in class or in your text.
Think about all the derivatives, max-min problems,
and volumes of solids of revolution you have calculated.
While some problems might have been harder or trickier than others, almost all
were similar to problems you had seen earlier.
In fact, some beginning mathematics students
really enjoy this rote aspect of elementary mathematics:
You learn a process and then execute it with little subjective interpretation.
More advanced mathematics is about building relationships and connections
among disparate ideas.
The Fundamental Theorem of Calculus illustrates this:
It connects the ideas of differentiation (slope) and integration (area) in a single
result that then makes the process of computing areas under curves relatively easy.
Advanced mathematics is a creative process and that is why I do it and enjoy it.
While we will not be proving the Fundamental Theorem of Calculus in Math 135
(take Math 331 to do that!), we will be develop and hone
our mathematics creation skills by carefully reading the first five
chapters of our text: Logic, Sets, Induction, Relations, and Functions, and then doing
some additional reading on selected topics such as number theory or cardinality. Working with
these topics will improve your abilities to create, read, write, speak, and present mathematics,
all of which you will do in this course.
Another difference between Math 135 and the other math courses you have taken is
that this course will often be run as a seminar with very little of the class will devoted to my lectures.
Instead, there will be discussions, lots of student presentations, and small group work.
Most students find this format very challenging but I hope that you will find class sessions
and office hours supportive.
Goals and Outcomes
It is worth reiterating the main goals mentioned above.
First and foremost is to develop a familiarity with the precise language that is used throughout all
advanced mathematics.
This language will help you to carry out sophisticated and rigorous analyses and arguments (proofs)
in both written and oral work.
A second objective to develop a familiarity with certain basic mathematical structures (e.g., relations) and operations
that pervade a great deal of higher-level mathematics. But in the end, the ultimate goal is
to become an independent mathematician, discovering, creating, and confirming new ideas.
Text and Materials
We will use
Chapter Zero by Carol Schumacher as our text. You will need to be an active reader! The main
portion of the text consists of a series of Exercises. As the author states, "The students who use the book must
prove virtually all of the theorems themselves, so that in some ways the book is a long series of interconnected
exercises." At the same time, the author tries to take you "backstage" to explore the motivation behind definitions
and axioms and she provides practical tips about constructing proofs. However,
the key is to do the exercises as
you read the text. If you don't, you will learn almost nothing. Come in for help when you need it.
The Journal: This is a bit of an experiment for me this term. You will also need a notebook or composition book
(a "journal") to be used exclusively for
your active reading. (It should be different from your class notes.) This is where you should keep your answers
to or further questions about the exercises that are part of
the reading assignments in the text. Include notes from your conversations with classmates about the material.
You should date your entries which should roughly coincide with the assignment date. Also include the
exercise number from the text that identifies the entry. The entries should be roughly in numerical order.
I may occassionally collect these to make sure you are keeping up with work. I may also allow you to use them on
some quizzes.
To be honest, I am not sure if I would have liked keeping a journal for a math course when I was an undergraduate.
It may not fit everyone's learning style. But my reasons for trying it are four-fold: (1) I want you to keep up
with the daily active reading. That really is crucial. (2) It will help you to be organized and this is one way of
"enforcing" that. (3) It is way for you to generate and remember questions to raise in class. (4) When I collect
them, they will help me get a sense of what you are struggling with.
Class Attendance and Active Participation
Because of the nature of this course, its assignments,
and its assessment, your
attendance and participation is required.
Unexcused absences may adversely affect
your grade; certainly
more than three absences will significantly lower your grade.
Active participation is essential for understanding the material and doing well in the course.
For each class I will assign a small amount of reading in the text which should be done carefully. Read it through several
times.
Work through the examples, construct solutions to exercises, and proofs of the theorems. As mentioned,
the text provides only a small number of examples and proofs. This gives us the opportunity to discover and produce
the rest on our own. Working through these assignments is the most important part of this course.
Reading and associated practice exercises will be assigned at the beginning of each
class. I encourage working in groups of two or three on the practice exercises and practice proofs in the body of the
text. This can be very helpful in understanding the material. Answers to these practice problems should be
kept in your journals (separate from your class notes).
There will be two exams during the term and a
final exam. Each of the exams, including the final, will have take home and in-class components.
The in-class exams will be on Friday, October 7 and Monday, November 14. The take-home exams will be due on
or about the same dates. Make travel plans accordingly.
It is impossible to construct fair makeup exams
in mathematics. For your own protection, my policy is that there are no makeup examinations.
Your course grade will be calculated as follows.
| Active Participation | 48% |
| Test 1 | 16% |
| Test 2 | 16% |
| Final Exam | 20% |
Active participation includes
- Written assignments (~65%)
- There will be one or two such assignments per week, all work must be your own, they
should be done neatly in pencil (name on each page, stapled).
All mathematicians make too many mistakes and go up too many blind alleys to use pen.
Each assignment should be regarded as a short
paper in a writing course. I expect that you will go through several drafts before creating
the final draft to hand in. Your work will be graded on both mathematical content and form. The content
grade addresses whether you understood and used the appropriate mathematical ideas for the proof. The
form grade addresses whether your writing is clear, effective, complete, and grammatically accurate (both
English and mathematical).
- Quizzes (~10%)
- Very occasionally there will be brief quizzes (usually, but not always announced) that
will cover the readings; in some cases, I may let you use your journals for these.
- In-class presentations (~25%)
- Much of the class will consist of students presenting work to each other and you must do your share.
I will try to rely on volunteers to make presentations. This makes it possible for you to present work with
which you are both comfortable and confident. For this to work, all students must volunteer on a regular basis.
Each student must do at least two presentations. Presentations should be at least as well prepared as
written work. Any presenter who makes an error that he or she cannot correct at the board will get a second
chance without penalty during the next class period. Like written assignments, presentations will be given a
grade for mathematical content and a grade for the quality of the presentation. During a presentation, the
rest of the class also plays a critical (literally) part. They should be attentive and note where clarification
is needed, ask questions, propose alternate solutions, and add comments. Note: If there is additional
clarification needed which no one has mentioned, I am likely to ask questions of those seated rather than
of the presenter.
Academic Integrity
As mentioned, I encourage you to form a small group with whom you can discuss readings and exercises
that are not collected (journal work). Verbalizing your questions, explaining your mathematical
ideas, and listening to others will increase your understanding. BUT you should not feel free to
copy someone else's work or make your work available to someone else.
Copying constitutes plagiarism.
This is a violation of academic integrity that could result in failure in the course. There is, of course,
no collaboration or use of outside resources allowed on written assignments, quizzes, in-class, or take-home exams.
Finally, common courtesy demands that you be on time for class and that
you do not leave the room during class (unless you are ill). This will help you, your
classmates, and me to give our full attention to the course.
Caution
This course is intended for students who are seriously considering being a mathematics major or minor.
The course will be challenging and time consuming. I expect you to spend at least three to four hours of "homework" for
each hour of class work (i.e., 10 to 12 hours or more per week) on this course. I assume that this course has
your top priority for the term.
Here are a few simple things that you can do to be more successful in the course
- Come to all classes.
- Do the readings and practice problems carefully.
- Keep the answers to problems and notes on the readings in a journal.
- Review the notes from the last class before coming to class again.
Recopy the notes if appropriate. This makes studying for a test easy.
- Come in for help whenever you need it, preferably before you get too far behind or well before an assignment is due.
- Participate in class. Ask questions in class. Ask questions about the homework and readings.
This assumes a reasonable pace through the listed materials. We may need
to adjust this schedule based on our actual work in class.
- August 29-September 2: Syllabus, Introduction, Introduction to Logic. Chapter: 0.0-0.4,
1.1-1.5.
- September 5-9: Logic, Truth Tables, And-Or-Not, Existence Theorems. Chapter 1.5-1.9.
- September 12-16: Logic and Types of Proof.
Chapter 1.9-1.15.
- September 19-23: Set Basics, Indexed Sets. Chapter 2.1-2.3.
- September 26-30: The Algebra of Sets (De Morgan's Laws), the Power Set. Chapter 2.4-2.6.
- October 3-5: Introduction to Induction. Chapter 3.1-3.2.
- Friday, October 7: First Hour Test. Followed by Fall Break!
- October 12-14: Complete Induction. Chapter 3.3.
- October 17-21: More on Induction, Well-Ordering. Relations. Chapter 3.3, 4.1.
- October 26-28: Orderings. Chapter 4.2.
- October 31-November 4: Equivalence Relations. Chapter 4.3.
- November 7-11: Functions, Injections, Surjections, and Bijections. Chapter 5.1.
- Monday, November 14: Second Hour Test.
- November 16-18: More on Functions, Composition, and Inverses. Chapter 5.2-5.3
- November 21: Inverse Images and Images of Functions. Chapter 5.3.
- November 28-December 9: There are a few different options for the last two weeks of the
term, depending on whether we have been able to keep to our schedule.
- Option 1: Sequences, Cardinality, and Countability. Chapter 5.5, 7.1-7.3.
- Option 2: Binary Operations, Elementary Number Theory. Chapter 5.6 and as much of Chapter 6 as time allows.
- Option 3: Binary Operations, Construction of R. Chapter 5.6 and as much of Chapter 8 as time allows.
- Final Exam December 14 at 8:30 AM.