- (Mar 20) Homework 9 (which was due on March 29, Good Friday) will now be due one day earlier, at noon on March 28.
- (Feb 13) The tutorial which was to meet on Friday, Feb 8, has been rescheduled for Friday, Feb 15, SAME TIME AND PLACE.
- (Feb 7) Homework 7 (originally due on Feb 15) will be due at noon on Monday, February 25.
- (Jan 30) For the Winter Problem Solving Presentation you will be required to present careful and complete solutions to 3 of the 5 tutorials. You will receive an additional problem after the reading week break to solve and include in the presentation.
- (Jan 21) Homework due February 15 has been posted here.
- (Jan 21) The tutorial dates have been corrected below. In particular the remaining meetings are
**TUTORIAL 01 (corrected)**, Wednesday 10:30 - 11:30, Ross S 125

Jan 30, Feb 13, Mar 6, 20.**TUTORIAL 02 (corrected)**, Friday 10:30 - 11:30, Ross S 125

Jan 25, Feb 8, Mar 1, 15.

- (Jan 10) Homework due January 25 has been posted here. The appended excerpt from Polya is posted here.
- (Nov 27) Homework due January 11 has been posted here.
- (Nov 22) As announced in class, the quiz scheduled for Nov 29 has been cancelled. Class will meet and we will continue discussion of proof methods. The quiz grade will be calculated using the best 3 of the 5 quizzes to be written.
- (Oct 24) Homework due November 7 has been posted here.
- (Oct 10) Homework due October 26 has been posted here.
- (Sep 26) Homework due October 12 has been posted here.
- (Sep 19) An introduction to the method of mathematical induction can be found in Chapter 6, Section 1, of the text. Practice problems are 6.4, 6.5, 6.6, 6.8(b), 6.9, 6.11 on page 150.
- (Sep 19) Some notes related to the first few lectures have been posted here.
- (Sep 9) Homework due September 28 has been posted here.

**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 Monday and Wednesday morning after 10:30, Tuesday and Thursday afternoon until 2:30.

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

FULL CLASS MEETINGS, Thursdays 8:30 - 10:00 Vari Hall 3009

FALL: Sep 13, 20, 27, Oct 4 (Quiz 1), 11, 18, 25, Nov 8 (Quiz 2), 15, 22, 29(Quiz 3).

WINTER: Jan 10, 17, 24, 31(Quiz 4), Feb 7, 14, 28, Mar 7 (Quiz 5), 14, 21, 28 (Quiz 6), April 4.

TUTORIAL 01 (corrected), Wednesday 10:30 - 11:30, Ross S 125

Sep 19, Oct 3, 17, Nov 7, 21, Jan 16, 30, Feb 13, Mar 6, 20.

TUTORIAL 02 (corrected), Friday 10:30 - 11:30, Ross S 125

Sep 28, Oct 12, 26, Nov 16, 30, Jan 11, 25, Feb 8, Mar 1, 15.

**Tutors:** TBA

**Text:** Gary Chartrand, Albert D. Polemeni and Ping Zhang, *Mathematical Proofs: A Transition to Advanced Mathematics*.

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

** Online Resource:** Steven Strogatz on the Elements of Math (New York Times, Opinionator Blog). For access to his posts click here. You can hear Strogatz on NPR (National Public Radio) by clicking here.

** And just for fun:** Tom Lehrer singing, That's Mathematics.

**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 what you experience 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.

You will discover for yourself that mathematics is as much about thinking as about doing. A polemic by Paul Lockhart on the current state of mathematics in schools is available here.

Participation | See below | 10% |

Individual Investigation and Writing Assignments | One assignment to be handed in every other week | 30% |

Problem Solving Presentations | See below | 15% |

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. You are expected to actively participate in small group and whole 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 ask the tutors to solve the problems or copy 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 receive 0 points. Presenting someone else's work as if it is your own (i.e., without proper citation) is academic dishonesty. You must cite any sources which you have consulted. You will be required to take the York University Academic Integrity Tutorial.

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 15.

Eli Brettler