This document comes in TWO versions:

• Blackboard has a PDF version which was posted at the start of the semester. It is not meant to be updated unless a serious error was detected or a shift in departmental policy requires such an update.
• The HTML version can be found by following the links from my home page (http://www2.math.binghamton.edu/p/people/mfochler/start). The direct link is http://www.math.binghamton.edu/mfochler/math-330-2015-fall/html/math-330-syllabus.html You should check frequently for updates to the HTML version.

Course home page:     Math 330 Home

PREREQUISITES:     Math 222 (Calculus 2) on a C- level or permission of the instructor.

If you did not pass Math 222 with a grade better than D this does not automatically disqualify you but you must see me asap! You need some Calculus background in the second half of the course and you may have to catch up on some topics to follow the course. You will be dropped from this course unless you can convince me that you have knowledge or are able to learn quickly about limits, continuity, power series, derivatives as limits of difference quotients and integrals as limits of Riemann sums.

NOT a prerequisite but STRONGLY RECOMMENDED     Basics of linear algebra:

vector spaces and subspaces, linear independence, (linear) span, basis, Euclidean space Rⁿ as a vector space and matrices as linear mappings between Euclidean spaces,

Textbook:     The Art of Proof: Basic Training for Deeper Mathematics, by Matthias Beck and Ross Geoghegan (Springer, 2010) (REQUIRED).

Additional material:     The HTML version of this syllabus has a clickable link for Course material (REQUIRED).

Course description:

You will learn in this course how to think like a mathematician:

• Understand the nature of a rigorous mathematically proof
• Learn to write such proofs

To do so means you must acquire some knowledge in the following areas:

• logic: direct proofs vs proofs by contradiction, logical quantifiers
• the difference between axioms, definitions and propositions
• sets, functions and relations
• induction and recursion

The subject matter used to teach you the above will be primarily taken from the theory of number systems: an axiomatic approach to the properties of natural numbers, integers, fractions and real numbers will be presented. You will also learn how to compare the sizes of infinite sets (cardinality).

A solid 20% of the course will be devoted to the foundations of real analysis. Most of that material cannot be found in the text book and you will have to consult the additional course material. See the section on lectures below.

Lectures:

The text has three parts:

• Part 1: The Discrete - Ch.1 - 7
• Part 2: The Continuous - Ch.8 - 13
• Part 3: Further Topics - Appendices A - G

The first half of the course will be about part 1 plus additional material on sets and functions. Also a portion of ch. 13 on cardinality will be discussed, some of it without complete proofs.

The second half of the course will deal with part 2 and appendix A but there will be significant additions to the material:

• lim inf and lim sup of sequences
• Basics of metric spaces: neighborhoods, open and closed subsets
• Sequence continuity and ε - δ continuity
• Uniform continuity and uniform convergence
• Sequence compactness and the Heine-Borel theorem
• Zorn's Lemma and its use to prove that every vector space has a basis

Success in this course:

A. General advice:

Success in this course depends largely on your attitude and effort. Attendance and participation in class is critical. It is not effective to sit and copy notes without following the thought processes involved in the lecture. For example, you should try to answer the questions posed by your lecturer. Students who do not actively participate have much more difficulty. However, be aware that much of the learning of mathematics at the university takes place outside of the classroom. You need to spend time reviewing the concepts of each lecture before you attempt homework problems.

As with most college courses, you should expect to spend a minimum of 2 hours working on your own for every hour of classroom instruction (at least 8 hours per week). It can also be very helpful to study with a group. This type of cooperative learning is encouraged, but be sure it leads to a better conceptual understanding. You must be able to work through the problems on your own. Even if you work together, each student must turn in his or her own work, not a copied solution, on any collected individual assignment.

B. Advice specifically for this class:

This is probably your first math course which is not focused on applying math theorems to solve problems such as "What is the derivative of the function ..." or "Compute the probability that ...". Rather, the focus is on reading and writing proofs for those math theorems. This requires a very high degree of mathematical abstraction and you will have to do a lot of hard work to develop the ability to do this kind of mathematical thinking.

My goal is to help you develop this ability. This requires that you attend class regularly, work through the material presented in class and do the reading and homework assignments.

Each homework assignment will specify what material you must prepare for by which date. You will be asked to review material before it is taught in class. Of particular importance is that you memorize the axioms and definitions beforehand. Be sure to read this part as soon as a new assignments has been posted! Moreover be aware that not all theorems and propositions in the text come with proofs: For some of those I shall give the proof, others will be given as homework assignments. You are encouraged, alone or in a group, to try to figure out missing proofs, even if they are not given as homework problems. There may be cases where you are not able to do this. That's fine but be sure to understand what each one of the theorems and proposition means and be able to reproduce it from memory. Of course you need not remember it word for word but, mathematically speaking, your version must mean exactly the same as the original item.

It is your responsibility to keep informed of all announcements, syllabus adjustments, or policy changes, regardless of whether they were made by email, on Blackboard, or during class.

Avoid failure:

Of course there is a substantial positive correlation between success in "engineering math" classes such as your standard Math 221, Math 222 calculus sequence or the 300 level statistics classes. Nevertheless, be mindful of the following:

• This course requires you to not just "sort of" know the axioms, definitions, theorems and major propositions involved but to know them precisely. When you do your homework, you can look up the precise statements; during quizzes and exams you must remember them.
• You need to work on a much higher level of abstraction. The best way to see what I mean is to compare a couple of pages in the Beck/Geoghegan text with some in your Stewart Calculus book or your statistics book. You'll see that the latter are about mastering a lot of "cookbook recipes", understanding under which circumstances they can be applied to solve an application problem and do so quickly and without computational errors.
      Contrast that with a problem such as proposition 2.18(iii) on p.19 of the text where you are asked to prove that if you take any number k = 1, 2, 3, 4, ... then the expression k³ + 5k can be evenly divided by 6. Solving such a problem, even completely understanding the proof if it is given to you to such an extent that you will be able to solve similar problems, requires a very different set of skills.
      Some of you will have to work a lot harder than others to master the course material.
• If you are not a Math major and this course is just one of several options available to you, I advise you to only take it if you are seriously interested in doing abstract mathematics, especially if you are worried about not bringing down your GPA. You should have some understanding of how you will be doing when you get close to the add/withdraw deadline. Even if you are only a little bit unsure, come see me!
      I taught this class in Fall 2015. Of initially 26 students, 24 were left after the Withdraw deadline. Of those one got an A and three got a B+. Each one of those four did at least three attempts for almost every homework problem. The point: It will require a lot of work on your part to get a decent grade!

Unless you are very certain that you will do sufficiently well in this class, keep your options open! Be sure that you can withdraw without losing your financial support or, if you are an international student, your student status!

Exams:

There will be two exams and one final exam. No notes, books, cell phones, or laptops are allowed for tests. Each standard exam will last 50 minutes and is worth 100 points. Make an effort to show up 10 minutes early for those exams so they can start on time.

Exam dates can be found on the course home page. Make all arrangements necessary to take the tests at those dates as it is extremely unlikely that they will be changed.

Final exam:

The final exam counts for 200 points and it will last two hours. The final exam counts for 200 points and it will last two hours. Date and time are not known at the time of writing of this syllabus and I shall publish the info in the course home page once it becomes available.

Date and time for all finals are set by the registrar and there is no flexibility. Do not make travel arrangements that will have you leave campus prior to the exam. You can request a makeup final only if you have another final at the same time (direct conflict) or you have three final exams scheduled within 24 hours. It will be given after the regular final, not before. If you want to request to take the alternate final then you must do so before Thursday, December 3 by sending me an email.

Quizzes & Homework:

Quizzes:

There will be at least 10 quizzes. The sum of points will be adjusted to 200. The number of quizzes depends on how the class is doing in knowing the axioms, definitions, main propositions and theorems as checking for this will be the main purpose of the quizzes. Additional quizzes will be given if the class needs to do better. Quizzes will often not be announced.

Homework:

Homework counts for 40% of the grade and will be graded in iterations: You will have a total of up to 3 iterations (i.e., a total of 4 submissions) for most of those assignments. The final submission date will be noted on the homework assignment and it will usually be two weeks after the date when the homework is posted. You will have less than two full weeks during the last two weeks of the semester and you may get additional time when holidays fall into that period.

Especially at the beginning I shall grade your homework according to the "red line" method: I stop grading when I see a major flaw and I'll mark that spot with a red line. I may comment on the nature of the problem or you may have to figure it out on your own.

You will learn from the textbook and from my presentations how to write a proof, but here are some purely technical requirements you should be clear about from the start: Write your proofs very neatly. Use lined paper so that your text is written in straight lines. Leave margins of at least 1/2 inch to the left and at least 1 1/2 inches to the right. Write your homework double-spaced so you can insert some missing items in a neat and orderly fashion. Write legibly! I'll be spending a lot of time looking at your homework as it is and I won't have time to carefully deliberate whether your variable was, e.g., a "u" or a "v".

Attendance:

Attendance will not be taken but if you miss a quiz, announced or not, then you miss it.

Blackboard:

Grades will be recorded in the Blackboard online gradebook. I'll make an effort to grade and record quizzes and exams quickly.

Homework grades are problematic as there usually is a lapse time of two weeks between posting an assignment and knowing for sure how many points you got. Also there will be such a large number of problems to keep track of that posting to Blackboard will be at irregular times, probably not more than once every three weeks.

Visit blackboard frequently!

Record keeping:

Keep all your records! I cannot change a grade for a test, quiz or homework assignment if there is no evidence! I advise you to maintain a file or spreadsheet with your grades so that it is easy to see whether my postings and your records have discrepancies! If your grade is incorrect you should notify me immediately, especially as the semester winds down: I won't change any grades after the date of the final. Again, visit blackboard frequently!

Your grade:

You can earn a total of 1,000 points in this course:

Attendance Policy and Make-up Policy:

Registration in this course obligates the student to be regular and punctual in class attendance. Make-up exams and quizzes will only be given in response to an excused absence. Excused absences include illness, religious holiday, a major tragedy in the family and participation in official BU athletic events. To be excused, absences should be properly documented, for example with a doctor's note. Bring documentation to me asap. The document should be issued to you at the day of the test. For example, if you missed a test on Friday, you should provide a Friday’s doctor’s note. The makeup will be scheduled within one week from the missed exam. You must request a make-up in writing by sending an email to me.

Students will NOT be given the opportunity to complete old assignments at the end of the semester to improve their grades.

Academic Honesty:

Incidents of academic dishonesty will be dealt with severely. There is precedent for giving an "F" for the course to a student who attempts to advance his/her grade illegally. Dishonesty includes, but is not limited to: copying another student's work, letting someone copy your work, lying to or intentionally misleading an instructor, signing someone else's name to a document. To eliminate suspicion, only writing/erasing utensils will be permitted on desks during an exam.

Best wishes for a successful semester!

Michael Fochler