MATH V2000.001: Introduction to Higher Math
  Fall 2014

Lecture: TR 10:10am - 11:25pm, Mathematics 520


All announcements will be posted in CourseWorks.

Course Syllabus

Instructor:  Michael Woodbury (x4-4988, 427 Mathematics,

Office Hours: Tuesday 9-10am; Thursday 11:30am-12:30pm; Friday 9-10am

Teaching Assistants:
Graduate TA: Remy van Dobben de Bruyn (Remy works in the helproom),
Undergrad TA: Linus Hamann (Linus also works in the helproom.)
If you have questions about the mathematics, you can get help in the helproom. You don't have to look specifically for our TAs.
If you have questions about grading of homework or exams, you should ask me.

Text: Dumas and McCarthy Transition to Higher Mathematics: Structure and Proof, McGraw-Hill, 2007.

Supplementary Text: Daepp and Gorkin Reading, Writing, and Proving: A Closer Look at Mathematics, Springer, 2011. (It is possible that the link to access an online copy of this text will work only if you are accessing it through the Columbia network. Please let me know if you experience problems.)

Course description: Introduction to understanding and writing mathematical proofs. Emphasis on precise thinking and the presentation of mathematical results, both in oral and in written form. Intended for students who are considering majoring in mathematics but wish additional training. We will cover chapters 1-5 and 7 of the textbook (not necessarily in that order.)

Prerequisites: Strictly speaking there are no prerequisites for this course. If you have tried to understand proofs in a linear algebra or calculus class previously, you may be better prepared than someone who has not seen anything of this nature, but the material of such courses is not directly applicable.

Important Dates:

  1. September 2: First day of classes
  2. October 7: Tentative(!) date of Midterm 1
  3. October 7: Drop date (for most schools)
  4. November 13: Last day to Pass/Fail
  5. November 13: Tentative(!) date of Midterm 2
  6. December 4: Last day of classes
  7. Final Exam: TBD
Advice: Here are some suggestions to help you succeed:

  1. Don't procrastinate. Research has shown to we learn better if we are consistent in our efforts to learn something new. (To help you in your efforts to not procrastinate, homework will be collected daily.)
  2. Be an active learner. Watching me, a TA/tutor or a fellow student do math without doing work on your own will not be enough to learn the material. (Participation in class is crucial. Even when you aren't sure whether your solution is correct, or even if you only have an inkling of what to do, you'll learn from presenting your unpolished thoughts.)
  3. Ask questions in class, and utilize office hours if needed.
  4. Read the book. I won't be lecturing from the book (except on occasion), but you are required to learn it. You will have to read the book. (This type of classroom style is called inverted classroom.) Learning to read mathematics is an important goal of the course. Another goal is to learn to write proofs; so, in your reading of the text, you are not looking for tricks to do the problems, but rather, you want to find what the important definitions and theorems are so that you can apply them to the homework. Moreover, the book can help you become fluent in the type of language and arguments that are involved in proofs. You should read it with the intent to discover how such arguments are constructed.
  5. Do your best on the daily homework. As discussed below, from a grade standpoint, you need only try all of the problems to "do well" on the daily homework. However, if you think deeply about the problems before class, the classroom discussion will be so much more valuable.
  6. Realize that it's okay to be stuck--this is what math is about: getting stuck and trying to invent ways to work around the obstacles. Talk with others (myself, a TA, fellow students) about where you seem to be stuck, and what you might possibly do to make progress. Often, we learn just as much from dead ends and wrong directions.

Resources: My office hours and the helproom will give you the chance to talk to someone who has a "big picture view." I also encourage you to use your classmates as a sounding board. Collaboration is highly recommended.

Homework: There are two types of written homework.

Some additional thoughts on homework:

Exams: We will have 2 midterms, and one final

Grading: The grading scheme is as follows:

1Minimal progress made
2Significant gaps
3minor technical errors
4completely correct
1Comment is on topic, but may not be particularly relevant or helpful to moving the discussion forward
2Good comment. It addresses an important issue, and moves the discussion forward.
3Great comment. It is especially insightful and illuminating.
1I don't understand this, but I see that you have worked on it.
2There is some good intuition here but at least one serious flaw.
3This is good but contains some mathematical or writing errors.
4This is correct and well-written mathematics.
✓-Minority of problems attempted
Majority of problems attempted but not all
✓+All problems have been attempted

Class and Reading Schedule

This schedule is tentative and I expect that it WILL change. The best way to know what is going on is by coming to class, but I will try to keep this up to date as well.
DateTopics/Sections coveredSupplementary Text
Sept 2,4Intro to class and proofs; Sets (1.1-1.2)Chapters 6-7
Sept 9,11More on SetsChapters 6-9
Sept 16,18Functions (1.3-1.6); Relations (2.1-2.3)Chapters 14-17
Sept 23,25More Relations; Modular Arithmetic (2.5)Chapters 10-11,27
Sept 30, Oct 2Propositional Logic (3.1-3.3)Chapters 2-3
Oct 7, 9Midterm 1, Formulas and Quantifiers (3.3-3.4)Chapter 18
Oct 14, 16Formulas, Quantifiers, Proof Strategies (3.3-3.5)Chapter 18
Oct 21, 23Well Ordering and Induction (4.1-4.2); Polynomials (4.3)Chapter 18
Oct 28,30Divisibility (7.1-7.4); Rings and Ideals (7.5)Chapter 21, 27
Nov 6Cardinality (6.1-6.2)Chapter 28
Nov 11, 13Midterm 2; More on Cardinality (6.3-6.4)Chapters 22-24

Course Details | Schedule | Top | Home