**Instructor:** Eli Brettler

**Office:** South 508 Ross

**Telephone:** 736-2100 Extension 66321

**E-mail:**
brettler@mathstat.yorku.ca

**WWW:** http://www.math.yorku.ca/Who/Faculty/Brettler/

**Normal Office hours:** By appointment. I am normally available Tuesday between 4:30 and 6:30 p.m., Thursday between noon and 2:00 p.m., Friday between noon and 2:00 p.m.

**Schedule of Classes and Tutorials (with some due dates shown):**

FULL CLASS MEETINGS, Thursdays 10:00 - 11:30, CLH 110

FALL: Sep 10, 17, 24, Oct 1 (Quiz 1), 8, 22, 29 (Quiz 2), Nov 5, 12, 19, 26 (Quiz 3), Dec 3,

WINTER: Jan 7, 14, 21, 28 (Quiz 4), Feb 4, 11, 25 (Quiz 5), Mar 4, 11, 18, 25 (Quiz 6), Apr 1

TUTORIAL 01, Fridays 10:30 - 11:30, RS 101A

Sep 11, 25, Oct 9, 30, Feb 13, 27, Jan 8, 22, Feb 5, 26, Mar 12, 26.

TUTORIAL 02, Fridays 10:30 - 11:30, RS 125

Sep 19, Oct 2, 23, Nov 6, 20, Dec 4, Jan 15, 29 Feb 12, Mar 5, 19, Apr 5 (Monday)

**Tutor:** Dorota Mazur

**Text:** Chartrand, Polimeni, Zhang: **
Mathematical Proofs, A Transition to Advanced Mathematics, Second
Edition**. We will cover Chapters 2 - 7.

**Supplementary Text:** John Mason with Leone Burton and Kaye Stacey,
**Thinking Mathematically**. This book gives an approach to problem solving and the problem solving experience. It is also a source for rich and varied problems.

**Statement of Purpose:**
This is a critical skills course. Here are some questions to consider.

Just what are the objects which you consider when you do mathematics? What is meant by the fraction one half? How does it represent a ratio? How does it represent a quantity? Are these conceptions different? Can you reconcile them?

How would you describe a triangle to someone (for example a blind person) who has never seen one.

How would you describe a circle?

What conventional conceptions do you have which inform your own thinking about these and other mathematical objects?What is meant by a proof? How you convince yourself, and how do you convince others that an answer is correct? What are the conventions for presenting concise mathematical proofs? How well does the presentation reflect the means by which a particular mathematical discovery was made? What does it mean for an ordinary language argument (mathematical or otherwise) to be valid? What is a counterexample? How does one make conjectures and how does one go about trying to assess whether they are correct?

It is pretty easy to convince oneself or others of the correctness of answers which seem intuitively correct. What is much harder is to convince when answers while correct are counterintuitive. An example some of you may have seen is the "Monty Hall Problem".Can you learn problem solving? Most of the problems you solved in High School were done mechanically or by mimicking solutions to similar problems in the textbook? What means are available to deal with problems which are genuinely novel?

The text, "Thinking Mathematically" by John Mason has a rich selection of problems for consideration. Most require minimal technical background but almost all require hard thinking. Mason suggests a way of working strongly grounded in self awareness both in terms of what you are doing, and how you feel while doing it.Are there techniques which extend your problem solving and proving capabilities? You will learn about combinatorial proofs which are arguments based on the analysis of situations rather the manipulation of formulas.

You will learn about recursive methods and mathematical induction as a tool in calculations and in proofs.

You will learn to use representations from other branches of mathematics (for example, geometric models to solve probability problems) to help obtain answers.

You will learn to present proofs and explanations which are concise and logically correct.What are expected outcomes of this course? You will learn to take risks as you engage with learning new mathematics and doing mathematical problem solving.

You will learn to express mathematical ideas with precision and clarity.

You will learn to ask questions whose consideration can lead to deeper understanding.

Participation | See below | 10% |

Individual Investigation and Writing Assignments | One assignment to be handed in every other week | 25% |

Investigation Projects | See below | 20% |

Quizzes | 3 Fall, 3 Winter | 15% |

Final Examination | Winter examination period | 30% |

Attendance at the weekly classes and at the tutorials is obligatory. You will lose 2 points from your course grade for each class or tutorial in excess of two which you miss each term.

The tutorial handout, which includes the problem or problems for consideration, will be posted prior to the tutorial meeting. This is to give you an opportunity to review it and consider (in the case of a problem) what might be involved in its solution. You are expected to actively participate in small group and whole tutorial group discussion.

Homework will be graded from 4 points. Grades will be assigned as follows:

- Level 4: (4 points from 4)
**Deep understanding of the problem.**Complete solution carefully presented. Provides multiple alternative solutions where possible. Considers variations based on the original question (with or without solutions). - Level 3: (3 points from 4)
**Good understanding of the problem.**Problem solved or a solution provided which can easily be completed, for example, one with a minor error which would be simple to correct. No evidence of engagement beyond finding an answer to the problem as posed. - Level 2: (2 points from 4)
**Incomplete understanding of the problem.**Limited progress to solution or a solution marred by major errors. - Level 1: (1 point from 4)
**Minimal understanding of the problem.**Work submitted shows little progress toward solution.

**Do your own work.** Don't look for a solution on the web or take one from another student's work unless you already have found your own solution and intend to review another to make a comparison. Work that is not original will be graded accordingly. Presenting someone else's work as your own without proper citation is academic dishonesty. You must cite any internet sources which you have consulted. You will be required to take the York University Academic Integrity Tutorial.

The lowest assignment grade each term will be dropped. Some assignments may be designed so that they can be handed in a second time with corrections.

These replace the Final Group Investigation Project which was assigned last year. Click here for an example of an A+ project submitted last year.

Here are some sample quiz question types:

- Given a problem and a sketch of a solution, formulate a more complete solution and present it with justification.
- Given a proof of some result, find any errors and correct them.
- Given various conjectures, find counterexamples if false, proofs if true.
- Provide simple proofs including direct proofs, indirect proofs, proofs by mathematical induction.

To read files in pdf format you can use the the free Acrobat reader.

**Note:** The last date to drop the course without
academic penalty is Feb 6.

Eli Brettler