Fall 2014

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

All announcements will be posted in CourseWorks.

**Instructor:**
Michael Woodbury (x4-4988, 427 Mathematics, woodbury@math.columbia.edu)

**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:**

- September 2: First day of classes
- October 7: Tentative(!) date of Midterm 1
- October 7: Drop date (for most schools)
- November 13: Last day to Pass/Fail
- November 13: Tentative(!) date of Midterm 2
- December 4: Last day of classes
- Final Exam: TBD

- 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.)
- 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.)
- Ask questions in class, and utilize office hours if needed.
- 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. - 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.
- 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.

- Daily homework will be collected each day of class. It will be graded on neatness and completeness. Neatness means that in addition to being legible, you should not turn in pages with frayed edges (i.e. from a spiral bound notebook), and you should staple neatly. Completeness means that you make a reasonable effort to solve each problem. "Reasonable" means that you have a strategy that makes (at least) some progress towards a solution.
- Weekly homework will be chosen (by me) from the daily homework of the previous week. This must be typed. It will be completed in groups.

Some additional thoughts on homework:

- You are welcome--in fact, encouraged--to collaborate. On daily homework, each student must write his/her own solutions, and names of collaborators should be included.
- Late homework will not be accepted for a grade. (Late weekly homework will be evaluated for feedback, but a score of zero will be given.)

**Exams:** We will have 2 midterms, and one final

- See schedule below for dates. Note that these dates are tentative and may change.
- The date and time of the final exam will be announced at a later date. Midterm Exams will be take-home. The final exam will be given during the established exam time.
*There will be no make-up exams without a note from a doctor or a dean.**Let me know immediately if there's going to be a conflict*

**Grading:** The grading scheme is as follows:

- Participation: 25%
- Homework: 25%
- Midterms: 15% (each)
- Final: 20%

- Grading of Presentations. Your participation grade will be determined mostly by how many times you present. This means that getting up in front of the class is more important that getting up in front of the class and perfectly presenting a perfect solution. (Roughly speaking, the "average" number of times presenting will be a "B+".) In addition, I will keep track of your presentations by assigning a score according to the following table. (This score will have a mild impact on this component of your presentation grade.)

Grade | Criteria |
---|---|

1 | Minimal progress made |

2 | Significant gaps |

3 | minor technical errors |

4 | completely correct |

- Grading of in class participation. A portion of your participation grade will also come from comments that you provide in (and before) class. This includes participation during class and through the Discussion Boards on CourseWorks. You can get credit for good questions, comments, observations and explanations. I will assign points to your comments towards each problem according to the following scale. (Twelve (12) points earned in this way is equivalent to one time presenting a problem.)

Grade | Criteria |
---|---|

1 | Comment is on topic, but may not be particularly relevant or helpful to moving the discussion forward |

2 | Good comment. It addresses an important issue, and moves the discussion forward. |

3 | Great comment. It is especially insightful and illuminating. |

- Grading of Weekly Homework. Typically, the weekly homework will be comprised of eight problems to be completed by groups of four students. Each student will take charge of two problems. Each problem will be scored out of four possible points according to the table below. In calculating the student's overall score, both the scores on the problems he/she is directly in charge of, as well as the other problems will be used.

Grade | Criteria |
---|---|

1 | I don't understand this, but I see that you have worked on it. |

2 | There is some good intuition here but at least one serious flaw. |

3 | This is good but contains some mathematical or writing errors. |

4 | This is correct and well-written mathematics. |

- Grading of Daily Homework. Presumably there will be 18-20 total daily homework assignments. The lowest 3-4 scores will be dropped before computing the final grade. Each daily homework will be scored as either ✓+, ✓, or ✓- according to the following guidelines. If an assignment fails to be "neat" (meaning: either illegible, unstapled, or otherwise) then a reduction of score may occur.

Grade | Criteria |
---|---|

✓- | Minority of problems attempted |

✓ | Majority of problems attempted but not all |

✓+ | All problems have been attempted |

Date | Topics/Sections covered | Supplementary Text |
---|---|---|

Sept 2,4 | Intro to class and proofs; Sets (1.1-1.2) | Chapters 6-7 |

Sept 9,11 | More on Sets | Chapters 6-9 |

Sept 16,18 | Functions (1.3-1.6); Relations (2.1-2.3) | Chapters 14-17 |

Sept 23,25 | More Relations; Modular Arithmetic (2.5) | Chapters 10-11,27 |

Sept 30, Oct 2 | Propositional Logic (3.1-3.3) | Chapters 2-3 |

Oct 7, 9 | Midterm 1, Formulas and Quantifiers (3.3-3.4) | Chapter 18 |

Oct 14, 16 | Formulas, Quantifiers, Proof Strategies (3.3-3.5) | Chapter 18 |

Oct 21, 23 | Well Ordering and Induction (4.1-4.2); Polynomials (4.3) | Chapter 18 |

Oct 28,30 | Divisibility (7.1-7.4); Rings and Ideals (7.5) | Chapter 21, 27 |

Nov 6 | Cardinality (6.1-6.2) | Chapter 28 |

Nov 11, 13 | Midterm 2; More on Cardinality (6.3-6.4) | Chapters 22-24 |

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