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. You also must list the names of everyone you have collaborated with.

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 be prepared for by which date. You will be asked to review the 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 of your homework as soon as a new assignments has been posted! Moreover be aware that not all theorems and propositions in the course material come with proofs: For some of those I will 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:

There is a substantial positive correlation between success in this course and success in "engineering math" classes such as your standard Math 223 - Math 227 calculus sequence or the 400 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 take a look at ch.3 of the MF doc (The Axiomatic Method) which is, by the way, what I teach in the first two lectures. This is deliberate: You will find out before the add/drop deadline for the semester whether you want to drop this course in favor of another one.
  » Compare this with the material in your Stewart Calculus book or even your upper division level 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 Beck/Geoghegan 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!
• Here are some statistics for this class from the Fall 2015 semester: 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!
• The following is the most important advice I can give you to do all right in this course: Start your homework early so you can turn in more than just one iteration.! Homework counts for 40% of the grade, and you can work in groups.

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!