Math 1200 A Course Information


  • (Dec 1) Lily will not be available in the MathLab next week. If you want some last minute assistance with the Fall Problem Solving Presentation, email me - Eli Brettler - for an appointment.
  • (Nov 25) There will be a drop-in tutorial for all students on Wednesday, Nov 30, at 10:30. We'll meet in S 101 Ross. This will be an opportunity to discuss concerns (general or particular) concerning the Problem Solving Presentation.
  • (Nov 2) Based on the experience this morning in Tutorial 1, one additional tutorial needs to be dedicated to Square Take-away. The problem which I had planned for Tutorial 5 will now be the question for Homework 5. I will hand out this homework in class on Tuesday. It will be due on November 21 and there will be one last Fall homework assignment with due date December 5.

    At this point, you know enough about Square Take-away to begin work on the Problem Solving Presentation. Start writing now, waiting, if necessary, until after the last tutorial meetings to fill in detail and further refine your work.

  • (Oct 24) The directions for the Fall Problem Solving Presentation will be handed out in class tomorrow. The due date is December 6 which is a change from that which originally appeared on the course outline.
  • (Oct 24) For the notes on Proofs and Proving which are to form the basis of tomorrow's lecture click here.
  • (Oct 17) The quizzes will be returned in class tomorrow morning. It was graded from 20 with the BONUS worth an additional 5 points. Here is a list of all the grades.
        1.0    2.0    3.0    3.0    4.0    4.0    7.0    8.0    8.0    9.0   11.0 
       11.0   11.0   12.0   12.0   13.0   13.0   13.0   15.0   15.0   18.5   19.0 
       20.0   21.5   22.0   23.0   24.0   24.0   25.0   25.0  
    The mean grade was 13.23 from 20. The median grade was 12.50/20.

  • (Oct 17) Starting this week, there will be a drop-in tutorial for students whose regular (required) tutorial does not meet. Details will be given in class.
  • (Sep 26) Please read Strogatz: Division and Its Discontents. For the handout for Tutorial 2 click here.
    Some further discussion of Mathematical Induction is available here.
  • (Sep 20) For a discussion of some of what was covered in the first two classes click here. For an excerpt from Rosen, Discrete Mathematics on Mathematical Induction click here. I will post my own discussion next week.
  • (Sep 13) The tutorial rooms and the due date for Homework 1 have been corrected.
  • (Sep 9) Click here for Homework due September 19.

    Instructor: Eli Brettler
    Office: South 508 Ross
    Telephone: 736-2100 Extension 66321
    Normal Office hours: By appointment. I am normally available Tuesday and Wednesday afternoon.

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

    FULL CLASS MEETINGS, Tuesdays 10:00 - 11:30, TEL 0005
    FALL: Sep 13, 20, 27, Oct 4 (Quiz 1), 18, 25, Nov 1 (Quiz 2) , 8, 15, 22, 29(Quiz 3), Dec 6
    WINTER: Jan 3, 10, 17(Quiz 4), 24, 31, Feb 7, 14(Quiz 5), 28 , Mar 6, 13, 20 (Quiz 6), 27.

    TUTORIAL 01, Wednesdays 10:30 - 11:30, Ross S 101
    Sep 14, 28, Oct 19, Nov 2, 16, Jan 4, 18, Feb 1, 15, Mar 7, 21.
    TUTORIAL 02, Wednesdays 10:30 - 11:30, Ross S 125
    Sep 21, Oct 5, 26, Nov 9, 23, Jan 11, 25, Feb 8, 29, Mar 14, 28.


    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 6 for the Fall and April 2 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 19.
  • Tutorial for September 14 and September 21.
  • Homework due October 3.
  • Summary of discussion in first two classes.
  • Excerpt from Rosen on Mathematical Induction.
  • Further discussion of Mathematical Induction.
  • Tutorial for September 28 and October 5 .
  • (Double) Homework due October 24.
  • Tutorial for October 19 and October 26.
  • Homework due November 7.
  • Fall Problem Solving Presentation Assignment.
  • Proofs and Proving notes.
  • Tutorial for November 2 and November 9
  • Homework due November 21
  • Excerpt from Krantz for Proof Reading Exercise
  • Materials for Quiz 3
  • Homework due December 5
  • Excerpt from Krantz with Commentary
  • Excerpt on pictorial proofs from the book Street-Fighting Mathematics.
  • Homework due January 9
  • Handout for first winter tutorials
  • Homework due January 23
  • Homework due February 6
  • Homework due March 5
  • Winter Problem Solving Project Assignment

    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 10.

    Eli Brettler