About Math 135
This course is likely to be very different from any other math course you may have taken, and you
may find it unsettling at first.
Math 135 has a reputation as a "proofs course," but it is more than that. Think of it as a
language course.
Mathematics has its own precise language and
it is this precise language that permits rigorous proofs and
reveals the beauty and elegance of mathematics.
So this course is also about
aesthetics. The ultimate goal is for you
to be able to communicate mathematical ideas effectively to other mathematicians. The emphasis in
Math 135 is on
doing mathematics: In the process you will produce, create, and discover mathematics.
Your previous mathematics courses emphasized solving variations on
problems that were illustrated in class or in your text.
In fact, many students enjoy this rote aspect of elementary mathematics:
You learn a process and then execute it with little subjective interpretation.
More advanced mathematics is different; it requires building relationships and making 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 makes the process of computing areas under curves relatively easy.
Advanced mathematics is a very creative process and that is why I do 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 develop and hone
our mathematics creation skills by carefully reading the first five
chapters of our text: Logic, Sets, Induction, Relations, and Functions. If we have time we may discuss cardinality
and infinite sets. Working with
these topics will improve your ability 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 little of the class devoted to my lectures.
Instead, there will be discussion, 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.
Prerequisites, Goals, and Outcomes
The nominal prerequisite for this class is that you have earned at least a C in Calculus II (Math 131), or earned AP credit for the class.
A more important prerequisite is that you are seriously considering being a mathematics major or minor.
The first and foremost goals are to develop the tools and 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 is to develop a familiarity with certain basic mathematical structures and operations
that pervade higher-level mathematics. The ultimate goal is
to become an independent mathematician, discovering, creating, and confirming new ideas.
This course serves as the foundation for all later courses in mathematics. As such, this course should be your
main priority this semester. You should expect to spend 10 to 12 hours per week outside of class
on work associatied with the course.
Text and Supplies
Text: We will using most of the first five chapters in
Chapter Zero by Carol Schumacher as our primary 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.
The key is to do the exercises as
you read the text. If you don't, you will learn almost nothing. Come in for hints and help when you need it.
Supplies: As always in mathematics you will need the usual pencils, erasers, and paper. (Mathematicians
work in pencil! Avoid pen for assignments.) In addition you will need
a composition book to use as a journal (spiral notebooks and three-ring binders will not be accepted)
and a highlighter.
Class notes should be kept in a different notebook of your choice, not in your journal.
Please obtain access to a stapler. Collected assignments of more than one page should be stapled (not clipped)
before submission to ensure that pages of your work do not become lost or separated.
Reading and practice exercises will be assigned for each
class. There will be three types of assignments.
Check the course
course website:
http://math.hws.edu/~mitchell/Math135F11/index.html
daily for the latest assignments.
- Journal Assignments:
- Journal assignments will be based on readings and exercises in the text or additional exercises that
I will provide.
Journal assignments should be completed before the next class as preparation for the next
day's work. The readings will usually be only a few pages long, but should be reviewed several times. In your journal,
work out examples, keep a list of new terms (definitions), work the exercises, and prove (or try to prove)
the theorems.
Include any additional notes or your questions about the material; these can be useful when you see me
for additional help. I encourage working in groups of
two or three on some of these journal exercises. This can be very
helpful in understanding the material. Lansing 310 is a good place to work together.
I will collect your journal at each exam, but you should work in your journal daily.
(The word journal means "a daily record of events.")
I will be looking for solutions to selected exercises, as well as the overall effort put
into engaging the course material. To help
me grade your work, please follow these guidelines.
- Keep exercises, problems, and proofs in the order they are assigned.
- Clearly label the problem or proof you are working on and
use a highlighter on each label, so they are easy to identify. Copy the exercise question, if needed.
- If you are unable to finish a problem or a proof, show how you attempted to do it and
explain why your attempt(s) did not work. [Seek help in office hours, too!]
- Mention any partners with whom you worked on the problem.
- Record new vocabulary and definitions as you come to them. (A great study tool.)
- Remember: Do not put your class notes in your journal.
- Reflections:
- There will be four or five "reflective" writing exercises during the term. They will be relatively short
and you will generally have a week to complete them. Some reflections will be based
on chapters from the book Letters to a Young Mathematician by Ian Stewart. I have
placed three copies on 3-hour reserve in the library.
Each reflection will be a short typed paper (usually 2 to 3 pages).
As with any college writing assignment, each reflection
will be evaluated on how well it is written as well as creativity and expression of thought.
The first reflection is an essay about yourself that will help me get to know you.
- Collected Homework:
- Most weeks there will be two written assignments: one due
on Tuesday and one on Friday. These are opportunities for you
to carefully and precisely express mathematical ideas and will include writing proofs of
theorems. Your proofs will be graded on both mathematical content and form.
The content score reflects your mastery of the mathematical concepts required for the
proof and your use of appropriate proof methods. The
form grade addresses whether your writing is clear, effective, complete, and grammatically
accurate (both English and mathematical).
Bonus credit will be given for creative approaches to proofs or multiple proofs.
Your work should be done neatly in pencil (name on each page, stapled).
You should go through several drafts before creating
the final draft to hand in. Review the online
tips for writing proofs.
- Friday assignments (generally due by 4:00 PM in my office):
-
You may discuss "Friday assignments" with a partner or two. Be sure to note your partners
clearly on the assignment. Though you may
discuss Friday assignments with others, your write-up must be completed entirely on your
own. So make notes while discussing ideas with your partner(s), but be sure the final
write-up is yours.
- Tuesday assignments (generally due by 3:30 PM in my office):
- These assignments
must be done individually without help from any other people or outside resources except your book,
notes, and me. Treat them like a take home exam.
All assignments should be turned in on paper, not by email. Even though assignments are
due outside of class, you are always
welcome to turn them in at the beginning of class before they are due. The point value of each written assignment
will be determined by the length and complexity of the assignment. No late assignments will be accepted.
- Exams:
- There will be two evening exams during the term and a
final exam.
The evening exams will be from 7:00 to
9:00 PM on Thursday, October 6 and Sunday, November 13.
The final exam is on Friday, December 16 at 1:30 PM.
Adjust your plans accordingly. The final may include a take-home portion due at the start of the
in-class exam. It is impossible to construct fair make-up exams. For your own protection,
my policy is that there are no make-up examinations.
Your course grade will be calculated as follows.
| Active Participation | 52% |
| Tests 1 and 2 | 15% each |
| Final Exam | 18% |
Active participation is essential for understanding the material and doing well in the course. It includes
- Written assignments (~70%)
- See remarks above.
- Quizzes (~5%)
- Very occasionally there will be brief quizzes (usually, but not always announced) that
will cover the readings.
- In-class presentations (~20%)
- As the course progresses, students will be presenting work to each other and you must do your share.
I will try to rely on volunteers to make presentations. This way you will be able to present work with
which you are comfortable and confident. For this to be successful, all students must volunteer on a regular basis.
Each student must make at least two presentations, and at least one of them must be a formal proof. Presentations
should be at least as well prepared as
written work. Any presenter who runs into difficulty that he or she cannot correct at the board will have a second
chance without penalty during the next class period to complete the work. Like written assignments, presentations will be given a
grade for mathematical content and form and a grade for the quality of the presentation. During a presentation, the
rest of the class also plays a critical (literally) role. You should be note where clarification
is needed, ask questions, propose alternative 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. Bonus points can be earned for each additional presentation. Review the online
tips for presentations at the course website.
- Seminars: (~5%)
- In addition to regular class time, you must attend two mathematics/computer
science seminar talks during the semester, at least one of which must be on mathematics.
Seminars usually begin between 3 and 5 PM and last an hour. You must be present
and attentive for the entire talk to receive
full credit. (This may be reduced to one seminar, depending on the number of talks scheduled this term.)
Bonus: You may earn bonus points for each additional mathematics/computer science seminar talk you attend.
- Attendance:
-
Because of the nature of this course, its assignments,
and its assessment, your attendance and participation is required.
More than three unexcused absences will lower your grade by a full letter.
The greater the number of absences, the greater the reduction.
Excused absences require documentation such as a letter from a dean.
On the other hand, if you have perfect attendance in the
course and always arrive on time, I will add two points to your course grade.
Academic Integrity
As mentioned, I encourage you to form a small group with whom you can discuss readings, journal work,
and "Friday assignments." 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 quizzes, in-class or take-home exams, or the
final write-ups of assignments.
Classroom Courtesy
It is impolite to arrive late to class or leave the classroom while class is in session unless it is an emergency
or you are ill.
Common courtesy requires your attention (and mine) on the class session. Phones should be turned off.
Caution
Math 135 is intended for students seriously considering being mathematics majors or minors.
The course will be challenging and time consuming. I expect you to spend at least three to four hours of work for
each hour of class work (i.e., 10 to 12 hours or more per week). I assume that this course has
your top priority for the term.
- Come on time to all classes.
- Do the readings and journal problems carefully.
- Review the notes from the last class before coming to class again.
Recopy the notes if appropriate. This makes preparation for a test easy.
- Start assignments early so that you have time to see me, if necessary. Come in for help whenever you need it,
preferably before you get too far behind and well before an assignment is due.
- Participate and ask questions in class.
Disability Accommodations: If you are a student with a disability for which you may need
accommodations, you should self-identify and register for services with the Coordinator of
Disability Services at the Center for Teaching and Learning (CTL), and provide documentation
of your disability. Disability related accommodations and services generally will not be
provided until the registration and documentation process is complete. The guidelines for
documenting disabilities can be found at the following website:
http://www.hws.edu/disabilities.
Please direct questions about this process or Disability Services at HWS to David Silver,
Coordinator of Disability Services, at silver@hws.edu or x3351.