(1.3, part 3) B2 (1.4, part
2) B1 (do a good job), 2, 6, 10 (1.5) A2, A10,
25 [One more problem] You have heard
the phrase "cruel and unusual punishment." Where does it come
from? How is the word "and" used in it, that is, which
possible interpretation is intended?
[To be continued at
the top of this page.]
Course policies for Methods
of Proof, M-242, at Montana
State University.
Time and Room: 10:00-10:50
am, MWF, in Wilson 1-139 (Spring 2013)
Goals: You will learn to read, write,
and think like an advanced mathematician. You will
learn to read symbolic mathematics with comprehension,
express mathematical thoughts clearly, reason logically,
recognize and employ common patterns of mathematical
thought, and read and write proofs.
Instructor: Dr. Warren
Esty, 994-5354, Wilson 2-238 (East wing, South wall).
westy at math dot montana dot edu Phone calls
and e-mails are both fine. Appointments are easier to
arrange on the phone.
Office hours: I love
the material and am happy to help.
Mondays and Fridays: 9:00-9:50
MWF and many other hours. You are more than welcome whenever
I am in the office. If you want to arrange to meet some
other hour, just ask in class, call (994-5354), or drop
in.
Required text: Proof: Introduction to Higher
Mathematics, fifth edition, available
at the bookstore, by Warren W. Esty and Norah C. Esty.
This course has almost nothing to do with
calculation, so no calculator is required.
Bring your text to class every day. We
will use it in class.
Course Content:
We will proceed straight through the text, covering every
section through Chapter 5.
Chapter 1: Preview of proof, sets, logic for mathematics
(including truth tables and important logical equivalences
that provide alternative forms).
Chapter 2: generalizations, existence statements, negations,
reading symbolic mathematics with full comprehension,
logical form and deduction, and practice with alternative
forms in the context of rational and irrational
numbers.
Chapter 3: Proof of theorems about inequalities and
absolute values, theory of proofs, proofs by contradiction
or contrapositive, proofs by mathematical induction, and
common types of mistakes in proofs.
Chapters 1 through 3 complete the theory.
The rest of the course provides practice in several content
areas of mathematics.
Chapter 4 is Set Theory including bounds and suprema.
Chapter 5 is about the concepts of one-to-one and onto,
functions applied to sets, and cardinality.
We will cover through Chapter 5.
Prerequisite:
Math 182 (two semesters of calculus). The mathematical
sophistication provided by additional mathematics such as
Math 221 (Matrix Theory) and Math 224 (Calculus of Functions
of Several Variables) would be very welcome, but the
material covered in those courses is not a prerequisite. In
fact, the material in Calculus is not a prerequisite
either--we just want you to have read a lot of mathematics.
This course is primarily for students who
wish to be math teachers or math majors, and others, such as
computer science students, who need to grasp proof. It
is a through discussion of the most important types of
thought processes in mathematics.
I will:
- Have passion for the material
- Enjoy the class
- Give you your work back promptly
- Give you lots of helpful criticism and feedback on your
work
- Listen and respond to your concerns
- Realize that students have a life outside of class and
not make unreasonable demands on you
- Take questions seriously
- Help outside class if you are trying hard and want help
You will:
- Enjoy the class
- Appreciate learning during class time
- Read all the material in the text outside class
- Do the homework--almost always on time
- Show up for (almost) every class, be on time,
and be prepared
- Try hard to get good at math
- Ask for help when trying isn't working
- Observe proper etiquette
Etiquette. Proper etiquette is required. During class,
students will not engage in any potentially distracting
behavior such as reading a newspaper, text-messaging, or
whispering about non-math subjects. Cell phones must be turned
off and unavailable. Pagers or watches that make a sound,
however quietly, must have the sound off. No type of earphones
is allowed.
Attendance:
Attendance every day is expected. More than a
couple unexcused absences is unacceptable. Of course,
excuses for academic reasons, illness, participation in
university sporting events, and significant life events
will be accepted. Every day in class you will learn about
common mistakes and how to avoid them. It is not possible to
recognize your own errors in logic, so you must take every
opportunity to see deceptive errors in reasoning explained and
to get feedback about your own and your classmates errors in
reasoning. Students who miss a day are missing a significant
lesson that cannot easily be recovered from the text alone.
If you miss a day, I will not be able to recreate
the class experience for you. Find a friend who can help you
catch up, read the text thoroughly, and then I will be glad to
help you with specific questions
Cheating: I give
you permission to work on homework jointly with others in the
class. In fact, I encourage you to work with others because
math is a language and learning to communicate in the language
helps meet the goals of this course. In this course, learning
by working with others is not cheating. However, you must hand
in your own work and copying someone else's work to get the
homework done is unacceptable. The purpose of homework is not
"to get it done," rather, "to learn how to do it." If
your homework results in learning, that is all I can ask of
it.
In contrast, exams must be entirely your
own work.
Homework. There
will be homework due almost every day. It is important
that it be attempted on time. The work you hand in need not be
all correct, but it must display serious effort. More than a
few late homeworks is not acceptable. I will give you
important and useful feedback on all the HW you do on time.
You may work on homework with other
current students. You are even encouraged to work with
others. You may ask previous students about individual
problems. But obtaining work from others and presenting it as
your own is unethical and forbidden.
You are
expected to work, on average, about two hours outside class
for each class hour.
You must read
the assigned sections. Learning to read math with
full comprehension is one of your goals, and you learn to read
by reading. Reading is part of those two hours.
Bring
your text to class every day. We will use it in class
regularly.
Exams and Grading. There will be unit exams,
frequent quizzes, regular homework, class participation, and a
comprehensive final.
To receive full credit, daily homework must be
handed in on time. Homework handed in late will receive
three-fourths credit.
Exam dates will be announced on this site.
Homework and its due dates will be announced on this site.
Your course letter grade will be based primarily on exams and
quizzes. Homework is necessary. Not regularly handing in the
homework, or handing in work that displays little appropriate
effort, will lower your letter grade. However, homework is
intended to help you learn and its impact on your grade is
primarily that it serves as evidence of your attempt to learn.
Getting a few problems wrong or incomplete will not lower your
grade if you display appropriate effort. I want to help if you
have difficulties.
The final exam is 8:00 - 9:50 April 30
(Tuesday of exam week), as
in the MSU schedule. Arrange your summer schedule so you can
take the final at the scheduled time.
(You can find final exam times for other classes at:
http://www.montana.edu/registrar/exams/
)
Conflicts. You are required to take all exams and
the final exam at the scheduled hours (unless you have another
exam or class scheduled at that hour, in which case we will
make arrangements). Any exceptions must be approved well
in advance, and in no case will exceptions be made for
two exams.
Attitude. Some students think math
is merely a list of procedures--a succession of algorithms
for "how to do" things. Proof is a major part of
mathematics that is not at all like that conception of
mathematics. So, you may need to change your attitude about
what math really is. It is hard for anyone to change their
attitude about anything, so this part may be difficult for
you.
Mathematics is a written language (much
more so than a spoken one). One goal is to have you learn to
read with comprehension. Then you will be able to grasp what
mathematical sentences really say (They probably say more than
you think!) and learn without relying on the teacher. How
can we help you reach this goal? By making you read and
work with material even before there is a lecture on it. You learn to read by reading. So,
expect to learn by reading. Lectures will clarify things, but not
always introduce
things.
Success.
Higher mathematics requires a significantly different
way of thinking. There is a much greater focus on the
truth, or falsehood, of statements and connections
between facts. There is much less focus on algorithms
(methods for doing problems).
Here
is advice about how to learn math.
Read each section. Do
not skip the harder parts. In fact, when the going gets rough
you need to slow down and read it several times until it makes
sense. If it remains unclear, ask me!
This is hard!
But, you will be learning an extremely valuable
skill.
Don't skim.
Don't expect that only high points are important (Don't
read only the bold parts).
Don't skip the rest of the paragraph because you want
to move along to the next high point.
Really do read the next paragraph in the text.
Mathematics is a written
language and you learn it by reading (and writing),
not by listening in class.
Homework:
If something on your homework is wrong, I will mark it
wrong at the place where it goes wrong. Please make sure
you understand why. Do not treat your homework as just
part of your grade. Treat it as an occasion to learn.
Anything you got wrong must be looked at again and
studied much harder than anything you easily got right.
Some things are easy. It is not much of an
accomplishment if you can learn the easy stuff. Some
things are harder. Put substantial effort into making
sure you understand the harder stuff too.
This
site will be updated frequently as the course
progresses.
The end. Check
for updates at the top of the page.
The converse and a
counterexample in the comics!