Math 141 and Math 147, or permission of the instructor

Concepts which students are expected to know as a prerequisite:

1. Equivalence relations and their relationship to partitions

2. Mappings (functions), including injectivity (1 to 1 -ness) and surjectivity (ontoness)

3. Elementary set theory, including Cartesian products

4. Techniques of proof (at least some idea of how to approach a proof of a theorem)

5. Mathematical induction

6. Properties of the Integers (including divisibility, the Division Algorithm, the greatest common divisor, and prime numbers).

The text reviews each of these topics: See Chapters 6 and 12 and Appendices A, B and C. Some exercises from each of these chapters will be assigned (see below) for you to do.

In addition, if we get far enough, you will be expected to be familiar with the material on vector spaces - basis and dimension (see Chapter 28), and with a few additional properties of the integers (the Well Ordering Property, and the Fundamental Theorem of Arithmetic - i.e. the Unique Factorization Theorem) which are among the topics discussed in Chapters 21, 22 and 23.

The Course: It is my goal to cover the entire book, but I will be satisfied if we can get through Chapter 20, with some additional remarks on polynomial rings.

This text is probably very different from most which you have encountered until now. It uses what be described as a discovery approach. Each chapter does present some definitions and examples, but the meat of the theory is contained in the exercises, and you have to develop it for yourself, with my help and that of your group (see below).

Group work:

Each student in the class will be assigned to a group. Members of a group should sit together in class and are encouraged to get together outside of class to discuss the course material, to study and to do the homework (except as restricted below), with the proviso that all members of the group are to be active participants rather than just listeners or observers.

Some classes will begin with time set aside for the group to meet to review the previous night's homework and to select a problem which they would like me to do on the board. Sometimes groups will work together in class to try to develop new concepts and ideas.

Homework:

Numerous problems will be assigned each class to be done for the following class (see the attached problems list). Some of these problems will be designated as individual or as group "hand-in" problems, as discussed in the following sections of this syllabus. In general, however, other homework problems will not be collected (although you may submit them if you would like me to correct them). Nevertheless, it is expected that you attempt all of the problems in a timely manner, and that you keep up with the pace set in class. This is part of your responsibility to yourself as a student and to whoever is paying for your education. THE ONLY WAY TO LEARN MATHEMATICS IS TO DO MATHEMATICS.

Writing:

Since this is a W course, you will be expected to do a substantial amount of writing. Each week (more or less) some problems will be assigned to be handed in. A list of these problems may be found on page 6, below. These problems will be graded for written organization, style and grammatical correctness as well as for the mathematical content. The hypothetical audience for all written work will be the other students in the class. You must explain your reasoning to them and must convince them (and me) that your solution to the problem you are submitting is both correct and complete. While we are primarily interested in the organization and the persuasiveness of your arguments, correct grammar and spelling are not to be ignored.

"Hand-in" problems will be of two types: individual problems, and group projects. Group projects may be done with your group or, if you prefer, on your own. Individual problems, on the other hand, must represent your individual efforts and are not to be discussed with anyone other than me until after they have been submitted. Breaking this rule is a serious violation of the Student Code of Conduct and will be severely punished. (I caution you, based on past experience, that I almost always can tell when two students have worked together or when one has copied from the other, so I strongly advise you not to work with anyone and not to copy anyone else's work nor allow anyone to copy yours. This warning is not intended as a challenge but, rather, as a word to the wise.) If you need help with a hand-in problem, come to me.

Individual hand-in problems will be due the Tuesday after they are assigned. This same deadline applies to group projects which you decide to do on your own. However, group projects which are submitted as a group (one submission for the group is sufficient) will be not be due until the second Tuesday after they are assigned. (This should allow adequate time for the group to get together.) Problems listed on page 6 are considered to be assigned on the day we cover in class the chapter in which they appear. This applies even if I forget to mention the problem during class. If in doubt, ask. Late work will not receive credit.

Page 6 also contains rules governing the format of problems to be handed in. It is important that you carefully read these rules. If you do not follow them, you will lose credit.

There will be three midterm exams, each worth approximately 16% of the final grade, and a cumulative final examination, worth approximately 22%. Grades for the written homework assignments will be worth approximately 25%; class participation will be worth approximately 5%.

Grades on exams will be assigned in the following manner: Each problem will be worth a certain number of points (which may or may not add up to a total of 100 on any one exam). Problems will be graded based on the indicated number of points and the total of these awarded points will be computed, giving a numerical score for the exam. These numerical scores will then be converted into letter grades in accordance with the following procedure: Prior to my totaling individual scores, I will determine what I deem to be appropriate cut-off level for each letter grade. For example, I might decide, "On this exam, in order to get an A-, a student should get a total of 72 points." After each student's grade is totaled, I may make small modifications in my predetermined cut-offs. For example, if my predetermined cut-off score for an A- is 72 points, and if there is one student with 71 points and the next highest point total is 54, I may decide to move the cut-off for A- to 71.

The grade for homework will be determined similarly, except that the letter grade will be based on the total points each student accumulates for the semester rather than on having a letter assigned for each assignment. (Most problems assigned will be graded on a basis of 5 points. Although I may modify this, past experience suggests that the cut-off for an A- will be a per problem average of 4 points out of 5. Thus, if, for example, 22 problems are assigned for the semester and if you get a total of 88 points, then your homework grade will be A-. Cut-offs for B-, C- and D- respectively will probably be 3+, 3- and 2. [Here, + represents 1/3, and - represents 1/3 off]. )

After a letter grade is assigned to each component of the grade, the letters will be converted to numbers according to a scale such as the following: A+ = 97; A = 94; A- = 90; B+ = 87; B = 84; B- = 80; C+ = 77; C = 74; C- = 70; D+ = 67; D = 64; D- = 60; F+ = 55; F = 50. The weighted average of the components will then be computed and reconverted to a letter grade using a slightly more liberal scale such as: A = 92; A- = 88; B+ = 85; B = 82; B- = 78; C+ = 75; C = 72; C- = 68; D+ = 65; D = 60; D- = 55.

As an example, suppose a student receives grades of A, B and A- on the three midterm exams, A- on the final; A- for the homework; and C+ for class participation. Then, using the scale above, these will be converted to 94, 84, 90, 90, 90 and 77 respectively. Using the weights indicated above, the weighted average would be (.16)94 + (.16)84 + (.16)90 + (.22)90 + (.25)90 + (.05)77 = 89.03. This would then be converted to a letter using the second scale. The resulting grade for the course would be A-.

Attendance and lateness:

It is expected that you attend class regularly and on time. I expect this not only because I want you to share in the benefit of my "words of wisdom" but also because a group with several members absent no longer can function as a group. A nationwide survey of employers indicates that in the vast majority of cases in which an employee is fired the primary reasons for this firing are absenteeism and especially tardiness.

While I do not have any hard and fast rules such as "five absences means an automatic F", I do take attendance, and you can be sure that excessive unexcused absence or lateness will affect your grade.

You are also expected to be present for each scheduled exam. If you determine in advance that you will not be able to be present on the date of a scheduled exam, I expect you to notify me immediately of that fact, so that we can discuss alternate arrangements. If a last minute emergency prevents you from being able to take a scheduled exam, I expect you to call my office AS SOON AS POSSIBLE. (If you are ill or your car won't start, I expect to hear from you on the morning of the exam. If you are in a coma, then when you emerge from the coma will be soon enough, assuming that you have a doctor's note attesting to the fact that you were in the coma!) If I am not available when you call, leave a message on my voice mail, stating your name, the class you are in, your reason for missing the exam and a phone number at which I can reach you later that day. If you don't follow these instructions, I am likely to treat your absence as unexcused.

If we have not made arrangements for you to take a make-up exam before the scheduled exam, you should be prepared to take a make-up on the day you return to campus. In general, I find it difficult to compose two exams which are truly comparable, so I prefer not to. For an excused absence, I may decide to give a make-up or I may decide to disregard the exam and to increase the percentages which your other exams and/or writing grades contribute your final grade. (This is my decision, not yours.)

Exams which are missed due to unexcused absence will receive the grade of zero.

Students who miss more than one exam may receive a zero for the second exam, regardless of the reason for the absence, unless advance arrangements have been made.

Tentative examination dates:

Tuesday, February 29

Thursday, March 30

Thursday, April 27

Final examination:

Tuesday, May 9

Final comments:

What I discuss in class is not likely to coincide exactly with what is in the text. There will be material in the text which I do not have an opportunity to discuss in class, and there will be material which I discuss in class which may not be in the text. You are responsible for both. While exams will concentrate most heavily on what I cover in class, you are also responsible for knowing what is contained in any sections of the text which are not specifically excluded. (This includes material discussed in all assigned exercises.)

I suggest the following routine for this course (as well as for any other mathematics course you may take):

1. Try to read the next scheduled section of the text BEFORE it is covered in class. It is not necessary that you understand it completely, but try to get an idea of what it is about and where any difficulties you may have in understanding it lie.

2. Pay attention in class and try to take as careful notes as you can. ASK QUESTIONS about material you do not understand.

3. AFTER class, reread the text and your notes, and try to fill in any gaps that may exist in your notes. Make a list of items you do not understand and ASK about them during the following class or during my office hours.

4. Attempt all of the assigned problems. Answers or solutions to some of the problems are contained at the back of the book; but do not rely on this as a crutch. If you need to consult the answers to guide you through some problems, that is OK; but, if you find that you have to "massage" most of your work to get the answers in the back of the book, then it is time to make an appointment to see me. No solutions or answers will be available to guide you when it comes time to take a test.

5. When studying for each test, review your notes and the text and the problems which have been assigned. If you have been keeping up with the work and doing all of the above as you go along, it should not be necessary to "cram" for the test; a brief review is all that should be needed.

Overnight assignments (homework to be done and discussed in class and on the bulletin board but not collected):

Review (should be completed no later than February 1)

Appendix B (pp 351-352): 1,2,7,8

Appendix C (p 354): 1-6

Chapter 6 (pp 62-65): A all; B 1,2; C all; D 1,2,5,6; E 1,2,3,5,6; F 1,5;

Chapter 3 (pp 29-35): A all; B 2,3,5; C all; D all; read* G;

Chapter 4 (pp 39-43): A 1-5; B all; C 1,4,5,7; D 1-5,8; E all; F all; H 1-5;

Chapter 5 (pp 48-55): A 1,4,5; B 1,2,5; C 1,2,6 and 7 as it applies to 6; D 3,5,7,8; E all; F all; read G & H;

Chapter 7 (pp 75-79): A all; B all; C 2,3; D all; F 1,2; G 2,3; H 1,2;

Chapter 8 (pp 86-89): A all; B 1-3; C all; E 1,2;

Chapter 9 (pp 97-102): A all; C 1,4; D 1,2; E 1-5; F 1,2; H all; J all;

Chapter 10 (pp 108-110): B all; D 1-5; E 1,2,3,4,6; F all;

Chapter 11 (pp 115-118): A 1-4,6; B 1-5; C 1-6; D 1,2,6; E all;

Chapter 13 (pp 130-135): A all; B 1,2,5,6,7; C 1-5; D 2-4; E 1-5; read I, J & K;

Chapter 14 (pp 141-146): A 1-5; B all; C all; D 1-3,5; E 1,2,4,5; F 1-3; G all;

Chapter 15 (pp 152-156): A 1-5; B 1,2; C 1,2; E all; read F,G,H;

Chapter 16 (pp 160-168): A 1-3,5; C all; G all; J all; read H,K,L,M,N,O,P,Q;

Chapter 17 (pp 174-180): A 1,3-5,7; B all; C all; D all; F all; H all; I all; J all; read K,L M;

Chapter 18 (pp 185-189): A 1,2,4-6; B 1-6,9; C 1-7; D all; E 1-4,7; F 1-4,6,7; H 1,3,4,read 5;

Chapter 19 (pp 195-199): A all; B all; C 1,2; D 2; E 2-4; H 1,2;

Chapter 20 (pp 205-207): A 1; B all; C all; D all;

Chapter 24 (pp 246-250): A all; B 1,3,5,6; C 1-7; D 1,2,6; F 1-3; read G,H I,J;

Chapter 25 (pp 255-257):

Chapter 26 (pp 265-269):

Chapter 27 (pp 276-281):

Chapter 28 (pp 289-291):

Chapter 29 (pp 297-300):

Chapter 30 (pp 307-310):

Chapter 31 (pp 317-322):

Chapter 32 (pp 330-333):

Chapter 33 (pp 342-344):

*: For problems marked "read", it is not necessary to do the problem. Just read it for general interest and, if it concludes by proving a theorem, know the statement of the theorem (but not the proof).

Hand-in assignments:

Instructions for written work:

1. Each problem is to begin on a new page.

2. You are to write on one side of the page only, leaving ample blank space for me to write comments.

3. Pages of a problem may be stapled or clipped together. However, different problems are NOT to be stapled to each other.

4. Your name is to appear on the top of the first page of each problem, and your name or initials are to appear at the top of each subsequent page.

5. Everything you write is to be in grammatically and idiomatically correct English. This does not mean that you may not use equations and other forms of mathematical shorthand. What is does mean is that, when read aloud, everything you write, including equations etc., should read as complete and correct sentences. (Thus, it is important to know how to pronounce an equation in English.) Transitions and paragraph formation are also important considerations.

6. Although there is no specific requirement to this effect, word processed or typed work is preferred, with work written in blue or black pen preferable to work done in pencil, even though you may have to cross out rather than erase mistakes. Work done in red will not be accepted. In any case, all work submitted must be clearly legible.

Review (due Tuesday, February 3, 1998)

Appendix A ((pp 347-348): 11, 24

Chapter 6 (pp 62-65): Group project: G all;

Chapter 3 (pp 29-35): B 4, F all;

Chapter 4 (pp 39-43): G all;

Chapter 5 (pp 48-55): B 4; C 4 and 7 as it applies to 4; D 1;

Chapter 7 (pp 75-79): E all;

Chapter 8 (pp 86-89): G all; Group project: H all;

Chapter 9 (pp 97-102): B all; G 1,2,3; H1; Group project: I all;

Chapter 10 (pp 108-110): D6; Group project: G all;

Chapter 13 (pp 130-135): E 7; F all (See 131bottom-132top); Group project: E 6; H all;

Chapter 14 (pp 141-146): C 3,8; D 6; H all; Group project: I 1-9;

Chapter 15 (pp 152-156): C 7; Group project: D 1,3,4;

Chapter 16 (pp 160-168): E all; I all; Group project: D all; F all; pick

one from among K, L, M, and N

Chapter 17 (pp 174-180): A 2,6; G all; Group project: E all;

Chapter 18 (pp 185-189): C 8,9; F 5; J all; Group project: G all;

Chapter 19 (pp 195-199): D 1,3; Group project: F all;

Chapter 20 (pp 205-207): E 4,5; Group project: F all;

Chapter 24 (pp 246-250): E 4;

Chapter 25 (pp 255-257):

Chapter 26 (pp 265-269):

Chapter 27 (pp 276-281):

Chapter 28 (pp 289-291):

Chapter 29 (pp 297-300):

Chapter 30 (pp 307-310):

Chapter 31 (pp 317-322):

Chapter 32 (pp 330-333):

Chapter 33 (pp 342-344):