Introduction to Higher Mathematics, Spring 2011

Instructor: Andrew Obus
email: obus [at]
office: Mathematics 609
phone: 212-854-6240



MW 2:40-3:55, Mathematics 417.

Please ask questions if anything in lecture is unclear.  The class is small, and participation is strongly encouraged.  Lectures will run the entire 75 minutes. Please show up on time!

TA: TBA.  I will try to organize a weekly one hour recitation/Q&A, led by the TA.

Schedule of Lectures


Transition to Higher Mathematics, by Bob A. Dumas and John E. McCarthy. This is available at the Columbia Bookstore, on, and in the Mathematics Library.


There are three major goals for this class.  The first goal is to teach you how to reason like a mathematician.  The second goal is to teach you how to translate your reasoning into a clearly written proof.  The third goal is to give you a glimpse of some advanced mathematical topics (which will hopefully stir your enthusiasm for mathematics)!  Learning how to write a correct proof is difficult, and requires much practice.  But you will see steady improvement, and after taking this course you should feel considerable "ownership" of the mathematics you have learned.

The actual topics covered correspond to almost all of Chapters 1-7 of the book, with an additional smattering of topics from Chapters 8 and 9.  This is somewhat ambitious for one semester, and if we end up not covering absolutely everything, that is fine.

Expected Background: There are no formal prerequisites beyond high-school algebra and some exposure to calculus/limits (Calculus I is more than enough).  The biggest requirement is mathematical curiosity and the willingness to think hard about problems that are not necessarily straightforward.

If you have any questions concerning your background, please speak to me as soon as possible.

Office Hours

Mondays 4:00-5:00, Wednesdays 10:30-11:30. Mathematics 609 (my office). If these times do not work for you, we can try to set up an appointment.


Homework will be due on Wednesdays. It is due IN CLASS. If you are going to miss class for whatever reason, you must either submit the homework to my box outside of Mathematics 410 BEFORE the beginning of class, or have a friend submit your homework in class. Late or improperly submitted homework will never be accepted. If you know in advance you will be unable to turn in homework when it is due, you should plan to turn it in ahead of time. If you have a conflict with a religious holiday, you must let me know BEFORE the fact. I will not be sympathetic if you do not tell me in advance. I will drop your lowest homework score to allow for missed assignments or for assignments that pose special difficulty.

Homework must be neat, well-organized, and legible. In addition, it must be STAPLED or PAPER CLIPPED (no folding over the top-left corner or anything like that). Please write in paragraphs, sentences, and English words (oh my!) when they are called for. The TA should not have to decipher what you are doing--you should be clear and unambiguous about your methods on a homework problem.  You may work on the homework in groups (in fact, I encourage this), but your writeup must be done by yourself.  Please write down the names of classmates you have worked with on the homework.

Each homework assigment will have a problem or two marked as "bonus" problems.  You may choose either to do these and hand them in, or to give an oral presentation (see below).  Even if you choose to give a presentation, feel free to discuss and attempt the bonus problems!

Keep in mind that, unlike in other math classes you have taken, you will often not be able to solve a homework problem simply by a straightforward application of what has been covered in lecture.  Homework will require some thought and ingenuity!

Homework will be graded thoroughly (perhaps more thoroughly than you are used to)!  Every effort will be made to hand it back promptly.  Grades will be posted on Courseworks.

Schedule of Homework (will also be posted on Courseworks)

Oral Presentation

Instead of doing the bonus homework problems throughout the semester, you may instead choose to make a 10-15 minute oral presentation to me (privately) toward the end of the semester, where you will present some proof on the blackboard.  More guidelines on this will follow toward the middle of the semester.


Midterms will be in class on Wednesday February 16th, and Wednesday March 30th. If you have a conflict with one of these days, you must let me know now. Another exam on the same day is not considered a conflict.

The final exam is (tentatively) on Wednesday, May 11th, from 1:10PM-4:00PM.

Calculators are not permitted on exams.

Final Course Grades

25% Homework
15% Each midterm
30% Final Exam
10% Oral Presentation/Bonus HW
5%   Class Participation

Academic Dishonesty

I will tolerate no cheating, plagiarism, or other forms of academic dishonesty. Cheating on an exam will result in an automatic grade of zero for that exam. For more serious matters, the Office of Academic Affairs may be contacted.

Some Useful Links

Calculus Placement/Information Page

Columbia Undergraduate Math Page

Columbia Math Department

Extra Help


If you have (anonymous) comments for me about teaching style or anything related to the course, click here for a feedback form.