Math 1200 D Course Information


Instructor: Eli Brettler
Office: South 508 Ross
Telephone: 736-2100 Extension 66321
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.

  • Evaluation:

    ParticipationSee below10%
    Individual Investigation and Writing AssignmentsOne assignment to be handed in every other week30%
    Problem Solving Presentations See below15%
    Quizzes3 Fall, 3 Winter15%
    Final ExaminationWinter examination period30%

  • Participation: Participation is how you show your commitment to the course and to the other students taking the course with you. You are expected to share both of your mathematical knowledge and the feelings you have as you engage in doing mathematics.

    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.

  • Individual Investigation and Writing Assignments: Questions for investigation and solution will be assigned biweekly. Solutions are to be handed in. The following grading rubric will be used.

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

    Note that to receive full credit (4 points from 4) you must go beyond simply solving the problem as posed. Learn to think of your solutions as starting points.

    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.

  • Problem Solving Presentations: You are expected to continue working on the problems discussed in class and in the tutorials. Keep a record of your experiences and discoveries. You will be assigned one major problem solving investigation each term. A presentation documenting what you discover and what you experience is to be submitted at the end of each term. Due dates are December 3 for the Fall and April 8 for the Winter. More specific detail will be given in class.

  • Quizzes: There will be 6 in class quizzes, 3 per term.

    Here are some sample quiz question types:

    1. Given a problem and a sketch of a solution, formulate a more complete solution and present it with justification.
    2. Given a proof of some result, find any errors and correct them.
    3. Given various conjectures, find counterexamples if false, proofs if true.
    4. Provide simple proofs including direct proofs, indirect proofs, proofs by mathematical induction.
    The grade will be obtained by taking the average of the best 2 quiz grades from each of the terms. There will be no makeups for missed quizzes.

  • Final Examination: This will be a conventional timed, closed book exam, scheduled during the University Final Examination period. Question types would be similar to those examples given for the quizzes. Click to view the April 2011 examination and the April 2010 examination.

    Handouts and other resources:

  • Homework due September 28.
  • Tutorial Activity for September 19 and September 28.
  • Some notes related to the first lectures.
  • Homework due October 12.
  • Tutorial Activity for October 3 and October 12.
  • Homework due October 26.
  • Tutorial Activity for October 17 and October 26.
  • Fall Problem Solving Presentation.
  • Model Problem Solving Presentation.
  • Homework due November 7.
  • Tutorial Activity for November 7 and November 16.
  • Tutorial Activity for November 21 and November 30.
  • Homework due January 11.
  • Homework due January 25 and Polya excerpt.
  • Tutorial Activity for January 11and January 16.
  • Tutorial Activity for January 25 and January 30.
  • Homework due February 15.
  • Tutorial Activity for February 8 (moved to February 15) and 13.
  • Pascal's Triangle.
  • Winter Problem Solving Presentation.
  • Homework due March 11.
  • Tutorial Activity for March 1 and March 6.
  • A more complete discussion of Pascal's Triangle .
  • Tutorial Activity for March 15 and March 20.
  • Homework originally due March 29, now due March 28 at noon.
  • Lecture notes on paths for March 21.

    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