Math, M-242, Methods
of Proof, Fall
2011
This site updated Dec.1, 2011.
www.math.montana.edu/courses/m242 (Oddly, it needs the "www"
part.)
(The most recent update will be at
the top above the first horizontal line. )
Comments:
Section 7.1, B5. Negations of generalizations are existence
statements. Don't omit "there exists".
For example, the negation of "f is one-to-one" is not "f(x1)
= f(x2) and x1 = x2."
it is "There exist x1 and x2 such that "f(x1)
= f(x2) and x1 = x2."
Throughout the section, including B8 & B22. Suppose we ask you
to prove something about a recently-defined term you don't know
well, such as "one-to-one."
Does that make proofs hard because your "concept image" is not
well-formed? Perhaps. But it makes them easy in another way.
Proofs are sequences of prior results, and with few relevant prior
results you have only a few sentences (or, maybe only the
definition) you can use in a proof. So, you don't have to know a
lot. Here is what to do:
First, translate
the conclusion. Then, use one of our logical equivalences
to organize your (potential) proof. Begin in the place suggested by
the logical equivalence. Look among the prior results and hypothesis
for connections. If you can't find a connection, the conjecture may
well be false. Then, look for a counterexample.
B2 & B3: Good counterexamples give specific values to all the given letters.
For example: Conjecture: "If f: A -> A is onto A, then f is one-to-one."
You must give f and A.
Giving f without A is not
giving a counterexample. A "counterexample" such as "f(x) = x2" will not
do. What is A? Yes, it is not one-to-one on the reals. But the reals
can not serve as A, which is a critical part of the conjecture and
its hypothesis. This example does not satisfy the hypothesis.
Suppose you are to address a conjecture of the form "(A => B)
=> (C => D)." Note that "A" is NOT a hypothesis of the
conjecture. "C" is. Beginning with "A" probably indicates an
incorrect idea.
For example, to address "If g°f is one-to-one, then f is
one-to-one" do NOT begin with "Let g°f(x1) =
g°f(x2)" from "g°f is one-to-one." Begin with
"Let f(x1) = f(x2)" from the hypothesis in the
conclusion.
Nov. 21
Monday, Nov. 28:
There will be an Exam on
Chapter 4 and earlier material worth 100 points. It has
definitions, counterexamples, and proofs. It does not ask for
creative proofs--rather proofs that are very similar to those you
have seen. Study the forms of the proofs we have done and notice
what they have in common. Mastery of the definitions is
critical. So is mastery of alternative logical forms.
HW due Wednesday, Nov 30:
Read 5.1 closely, including the bad proof of Conjecture 9.
[Learn to recognize characteristics of bad proofs.] Hand in (5.1) B2, 3, 5, 8, 22
HW due Friday, Dec. 2:
(5.2) A1, 7, B1, 3, 4, 11, 14
Please bring all your old exams and quizzes to class Monday, Dec.
5. To prepare for the final exam we can look them over to
remind ourselves what we have covered.
HW due Monday, Dec. 5: (5.2) B8, 10, 12, 15, 19 has
two halves (thinking of "=" for sets as having two "subset" halves).
Resolve each half.
Create a list of advice to yourself about how to
do proofs. How would you approach a new proof in a new context? Make
it a concise list that is meaningful to you. Hand it in.
Here is a copy of
last year's final exam.
Wednesday, Dec. 7.
Lecture on 5.3. If
you did not hand in some previous HW, hand it in now.
Friday, Dec. 9. No HW
due. Review for the comprehensive final.
The final exam is 4:00-5:50 pm, Monday,
Dec. 12.
(You can find final times for other classes at: http://www.montana.edu/registrar/pdfs/FinalsSchedule.pdf
)
Nov. 18
Addition to homework for Monday, Nov. 21: Do as much
with this conjecture as you can.
Let S and T both have finite sups. Define R = {xy | x is in S and y
is in T}. Then sup(R) ___ sup(S)sup(T).
Fill in the blank with the appropriate relationship and prove
it. If you can't prove it in general, add hypotheses that permit a
theorem to be stated and be clear about what you are then proving.
Illuminating examples are worthwhile, too.
Nov. 15
The final exam is
4:00-5:50 pm, Monday, Dec. 12 (Monday of exam week), as in the MSU Handbook.
(You can find final times for other classes at: http://www.montana.edu/registrar/pdfs/FinalsSchedule.pdf
)
HW
due Friday Nov. 18: (4.2)
B33, 35, 36, 41.
Monday, Nov. 21:
Read 5.1 at least through the definitions of "one-to-one" and
"onto". (5.1) A2, 11, 14, 17, 19
Here is last year's exam on Chapter 4.
Wednesday, Nov 23: This is the
day before Thanksgiving and there are no classes. Classes resume
Monday.
Monday, Nov. 28:
There will be an Exam on
Chapter 4 and earlier material worth 100 points. It has
definitions, counterexamples, and proofs. It does not ask for
creative proofs--rather proofs that are very similar to those you
have seen. Study the forms of the proofs we have done and notice
what they have in common. Mastery of the definitions is
critical. So is mastery of alternative logical forms.
Nov. 2
Due Monday, Nov. 7: (4.1) A12, 14, B14, 27,
34
HW due Wednesday, Nov. 9: (4.1) B35, B39, B41, B53,
Read Section 4.2 to at least the top of page 231. (4.2) A1, 2, 10,
12
(Friday, Nov 11, is Veteran's Day. No classes.)
HW due Monday, Nov. 14: (4.1) B54
(4.2) A17, 18, 25, 27, 35, 36, B1, 3, 12-15
HW due
Wednesday, Nov. 16: Read all of Section 4.2.
(4.1) B53. (4.2) A28, 29, 32, 45 (prove it), B16, 17,
30, 31, 32
Oct. 28
HW due Friday, Oct.
28: (3.5) B9, 12, 31
At the end of Friday I will
put on the web here a copy of last
year's exam on Chapter 3. Monday and Wednesday we will have
some time to discuss your questions about it.
For an extensive list of things you cannot do in
proofs, see here.
For an extensive list of things to do in proofs,
see here.
HW due Monday, Oct 31:
(3.6) A1, 2, 7, 8, 9, 10, 11, 12, 15, 19, B26, 33
Due Wednesday, Nov. 2:
Read 4.1. (4.1) Memorize the sentence-form definitions of the
set-theory terms. A2, B6, B11
Friday, Nov. 4: Exam on Chapter 3.
Oct. 17
HW due Monday, Oct. 24: Read
Section 3.5 and do (3.3) B64, (3.4) B8, 18, 23
Comment:
Much of low-level math is algorithmic. That is, there is
a method that will do the problems and you learn the method.
Then you do umpteen problems with the same method. Boring!
That is simply being a human calculator! (Value of a great
calculator: less than $150. On the web--free!)
Real math is more like exploration. You
don't know what you will find and you don't know how to get there.
Some people love exploration and discovery. Proof is more like that.
Real math is more like that. I sincerely hope that you enjoy trying
to put together proofs of things you have not seen before and don't
know how to prove.
By the end of Section 3.5 you will have seen all
the common methods of proof. You will have seen the different
way proofs are arranged. But you still need a lot more practice
to recognize when a given method or arrangment might apply.
For now, try anything that comes to mind-- but try it as
scratch work. Then, from all the ideas you have, select the ones
that work, and arrange them so each is true and prior and the result
follows from them.
Do not expect your work to be algorithmic. Proofs
are not algorithmic. They are often creative. Enjoy your
creativity!
Attitude
Goal: Be sure or be skeptical. Do not just
"do" steps. Pause at each to be
sure it is true. If you are not sure, don't write
that step (except as scratch work)! If it is not clearly
prior, be skeptical! Do not be accepting without
understanding. That is too passive and not the way mathematicians
think. Be aware that assertions can be false. Every "=" makes an
assertion. Every "<" makes an assertion. Every "=>" makes an
assertion. Be certain yours are true!
When you read (and you must read), if you are not
certain a line is true, work with it until you are. Do not accept
steps just because they are in print. Be sure or be skeptical.
Due Wednesday, Oct.
26: (3.5) A2, 4, B2, 14
Oct. 17
HW due Wednesday, Oct. 19: Section
3.2: B14 if you didn't get it before (using B13 works, but
there is another way), 16, 17, 18, 22. Read
Section 3.3 (of course). Also hand in: (3.3) A1, 2, B1, 2, 5,
11, 12, 15, 17.
HW due Friday, Oct. 21: (3.3) B58. Read Section 3.4. Be sure
you understand Example 1. Then also hand in (3.4) A1, 3, B2,
4, 5
Oct. 14
HW due Monday, Oct. 17: (3.2) B7, 8, 11, 12, 14,
37
One goal of Chapter 3 is to instill an attitude change. You should become skeptical.
Some things you know well and are certain about.
Good. However, anytime something new and similar appears, you must
develop the attitude that it might be false. Perhaps you can
prove the new variant from your list of prior results. Then
you know it is true. Or, perhaps it is false and you can prove
it false with a counterexample.
You need to learn how proofs work. There are many
deceptive arguments that can masquerade as proofs. In order to make
sure you do not accept false assertions as true, you need to learn
to recognize errors in proofs. Read the proofs in Chapter 3
with an eye toward both prior results and logic (the two components
of proof). By the end of Chapter 3, you will have studied examples
of all the major types of proofs. Then Chapters 4 and 5 will provide
practice.
How to do proofs:
There is no simple algorithm. We will compile advice as we
progress through the course. Here is one important idea:
Use scratch paper. Write
down ideas and definitions, especially of the conclusion, on scratch
paper without expecting that what you write will be the final
organization. After you have written enough that you can see most of
the proof, create a final copy by selecting and organizing the
useful parts from all that you have written.
Oct. 12
HW due Friday, Oct. 14: (3.1) B10,
Prove Theorem 12 using Definition 3, B21, B24. (3.2) A2, 5, 8, 10,
13, 19, 25.
Begin to read 3.2. In Section 3.2, pay attention
to the order in which the results are listed. Each one joins the
list of prior results for purposes of proving the next. The
inequality facts from Section 3.1 are also prior.
On prior results and
citations. At some point you get to stop citing old
and well-known results. For example, when doing a calculus proof
you do not need to cite results on inequalities from 3.1 by
number. By then, you are supposed to know what is true and what
is not, and inequalities are supposed to be so much lower level
and so far prior to calculus that we can assume you use them
correctly, even without citation. (I know, from experience, that
this is an incorrect assumption because some students manipulate
inequalities and absolute values incorrectly in the context of
calculus. Nevertheless, I will not expect citations of prior
results that are much lower level than the current work.) In
practice, that means, in this course, you are expected to cite prior results from the current
and immediately previous sections, but not from
sections on lower-level material studied long ago.
Oct. 10
HW due Wednesday, Oct. 12: Read much of Section
3.1. Study the given proofs line-by-line. Make
sure you know the reason for each assertion. Every "=" requires a
reason. Every ">" requires a reason. Every sentence requires a
reason. For this section, the work you need to do is to read
carefully. You need to seek justifications for every step and
realize how details are important. The homework to be handed in
is only a small part of what you are to do. Read (slowly and
thoroughly) first. Then apply the lessons to the homework. (You may
use lower-numbered results to prove higher-numbered results.)
If you find anything hard to read, I hope that you will ask
questions about it in class. Hand
in: (3.1) B1, 2, 4, 8, 9,
18, 19, 20 [In proofs, cite a prior result for every
assertion.]
On
the first day I handed out this handout.
It should make more and more sense as we go along. Look at it again
and see where you are making progress and what of it you do not yet
understand.
For example, the first entry is "true or
false". What does this mean to you?
Are you looking at mathematical sentences and
judging them as true or false? Are you more aware that
sentences can be false? Are you able to look at a false assertion
and be confident it is false? Do you know how to prove it false? Are
you more aware of the role of the hypothesis in a conditional
sentence? Are you more aware that it is critical that open sentences
can be false? (Or else they do not have information about the value
of the variable.)
We have not covered much about proofs or prior
results or justification yet, but that comes next.
Look at the part of the handout entitled
"Important things you will eventually understand." If there is
anything we have covered (mostly logic, so far) that you are not
comfortable with, please come get help.
Sept. 23
HW due Friday, Sept. 30:
(2.4) A2, 10, 12, 16, 18, 19,
B1, 2, 12, 14, 15
HW due Monday, Oct. 3:
(2.5) A1, 3, B1, 2, 14, 17, 18
HW due Wednesday, Oct. 5:
Review for the exam by doing and handing in the previous
version of Exam 2 (handed out in class Friday, and here) and coming to class with
questions.
Exam 2, Friday, Oct. 7, on Chapters
1 and 2. Memorize the Quadratic Theorem and the
definitions related to bounded, increasing, even, and
rational. Be sure you know the logic of quantifiers, and the
technical terms we have discussed (e.g. placeholder).
For Monday. Oct. 10,
after the exam, read some of 3.1 and do: 3.1, A2, 5, 9.
Sept. 23
HW due Friday: none, this
one time.
HW due Monday, Sept. 26: (2.2) A1, 2,
3, 6, 12, 17, 22, B1, 2, 10, 15 19, 21, 30, (2.3)
A1, 2, 3, 8, 12, B3
If you mean "there exists" do not omit
saying it. For example, the negation of "If |x|
> 5, then x > 5" is not "|x| > 5 and x ≤ 5."
It is "There exists
x such that "|x| > 5 and x ≤ 5."
It is common to suppress "for all" when it is intended,
but it is not okay to omit "there exists" when it is intended.
Suppose you are to give the negation of "If ab = ac,
then b = c." It is not "ab = ac and b ≠
c." The negation is "There
exists a, b, and c such that ab = ac and b ≠
c." Never omit "there exists" if you mean it!
HW due Wednesday, Sept. 28:
(2.3) A15, 64, B1, 7, 10,
14, 27, 40
Sept. 15
Some solutions to exercises in Chapter 2 have been placed on line.
Ask in class for the URL or adapt the one you used for Chapter 1 to
have a "2" where there was a "1".
Ideally, students should do the
homework themselves first and then be able to rapidly check their
answers. Wrong answers should provoke thought and reconsideration.
I am aware that some students will abuse the gift
by copying answers. Of course, reproducing answers does not promote
learning and you can bet that any lack of learning will be
discovered by exams, so I really hope you use that gift in this
order:
1) Do the work yourself first
2) Check to see if you got it right
3) Reconsider mistakes thoughtfully
4) Ask in class or my office to clarify any remaining
concerns.
P.S. If you find an error and are the first to report it to
me, there will be a minor reward.
Sept. 12
HW due
Wednesday, Sept. 14: (1.5) A7, 11, 15, 20, 25,
30, B1, 2, 3,
7, 17
HW due Friday, Sept. 16:
(1.5)
B4 (1.6) A1-7, B1, 10, 15, 27, 32, 51 (2.1) A1, 2, 3, 8
Learn the names of the results from logic (pages
86-88).
HW due Monday, Sept. 19:
(2.1)
A10-11, 18, B1, 2, 3, 4, 6, 8, 18, C2
Here is a link to Exam
1 from last Fall.
Wednesday, Sept. 21: No HW
due. Exam in class on Chapter
1.
Aug. 26
Read the course policies.
First day of class: Monday, August
29, 2011. We will cover Section 1.1 and discuss the
course.
Homework due Wednesday, Aug. 31: Read all of Section 1.1.
Mathematics is a written language (much more than it is a
spoken language). Therefore, this
is a reading and writing course. You learn to read
and write by reading and writing. Read the
text thoroughly. Every section, learn the meaning of the
terms listed at the end of the section's conclusion (just
above the "exercises." If you don't recall where a term was
introduced in the text, use the index to look it up). For this
section, hand in written homework at the beginning of class
Wednesday:
Bring
your text to class every day. We will use it in class.
Due Wednesday, Sept. 1:
(1.1) A1, 3, 4,
6, 8, 11, B1, 5, 6, 8, 13, 14, 21, 22, 29.
Label your homework with your name at the
very top of the page and the section number (e.g. "1.1") of
the homework just below it.
Also for Wednesday, read: "Is the internet making us stupid?"
http://www.npr.org/templates/story/story.php?storyId=91543814
(Really, it is short, so read it!)
The original article, "Is Google making us stupid?", in The Atlantic magazine is not
short (I don't expect you to read it, but I did):
http://www.theatlantic.com/doc/200807/google
It provoked quite a buzz, so search on the title will get many
hits.
Additional comments: Do you multitask well? Here
is a summary of some surprising research
on multitasking.
The first day in class you will get an
important handout listing some special emphases of this
course. Here is a link to a copy in HTML (here
is a copy in MS Word.)
This
web site will be updated frequently. Upcoming HW will be listed,
quiz and exam dates will be posted, mathematical advice will be
posted, and homework listed as due in the future might be
modified or postponed if it will improve the class.
Due
Friday, Sept.
2: Section 1.2: A2, 5, 12, 26, 30,
B1, 5.
In this course you will learn how
to learn math by reading it. One way we teach you to read is
to require you to do it. You
learn to read by reading. Therefore, many days
there will be homework due on material we have not yet
covered in class. You will read the section and learn how to
answer the questions. Then the following lecture will
clarify any remaining issues.
If you have tried, but are still uncertain about a problem,
on your HW put a big question mark, ?, in the
margin. Also, put the problem number on the side chalk
board before class and I will try to make sure it is covered in
class. If it is not covered in class, when I mark papers I
will note those problems and possibly devote time during the
next class to them.
Monday,
Sept. 5 is Labor Day and there are no classes.
Due
Wednesday,
Sept. 7: (1.2, part 2): B3, 4, 17, 21, 22,
26, 42, (1.3) A2, 5, 12, 13, 17.
Quiz Wednesday Sept. 7 on
1.1-1.3. Know the terms.
Be sure you can pronounce all the mathematical expressions and grasp
all the "grammar" exercises. Little typographical differences can
make a big difference in the referent (the thing being named or
referred to). For example, there is a major difference between a and A and a major difference
between (, [, and {.
Here is the quiz that
was given last year.
Mathematics is a written language. To
get good at math, you must read it. Read!
If you are coming to class and feel yourself slipping
even the slightest bit behind, please come see me in the
office. I want to help!
Fortunately, we will use the language of mathematics every day
and we never drop any topic, so you will see every usage and
hear every pronunciation again and again. Pay attention and
notice what is giving you trouble. Let me know and I will
help.
HW due Friday, Sept. 9:
(1.3, part 2) B1 (1.4) A1-4, 6, B17.
HW due
Monday, Sept. 12: (1.3, part 3) B2 (1.4,
part 2) B1 (do a good job), 2, 6, 10 (1.5) A2, A21
[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: 1:10-2:00
pm, MWF, in Wilson 1-139 (Fall 2011)
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: 9:00-10:50
MTWF and just before and after class, 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 or call
(994-5354).
Required text: Proof: Introduction to Higher
Mathematics, fifth
edition,
available at the bookstore, by Warren W. Esty and Norah C.
Esty. There is no solutions manual.
This course has almost nothing to do with
calculation, so no calculator is required.
Bring your text to class every day.
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.
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.
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 abscences
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 throughly, 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
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 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 4:00-5:50 pm, Monday, Dec. 12
(Monday of exam week), as
in the MSU Handbook. Arrange your winter-break 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/pdfs/FinalsSchedule.pdf
)
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.