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:

You will:

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.