Mathematics 104:  Mathematical Ideas
Spring 2005

Professor:        Jeff Johannes                                    Section 1    MWF 2:30-3:20p    Sturgis 113
Office:            South 326A                    
Telephone:      245-5403
Office Hours:    Monday 10:30 - 11:20a, 1:30 - 2:20p; Tuesday 2 - 3p, 4 - 5p; Wednesday 8 - 9p and by appointment or visit
Email Address:

    The Heart of Mathematics:  An invitation to effective thinking, Burger & Starbird
    What is Mathematics Really?  Reuben Hersh

    To develop an understanding of the nature of mathematics and its relevance in daily life.  

    In this course we will examine the question "What is mathematics and what is it good for?".  We will learn by exploring mathematics that is not frequently studied in high school or undergraduate mathematics courses.  We will come to the (perhaps surprising) conclusion that mathematics is not primarily about computing or measuring, but rather about a style of thought.  The important applications of mathematics are more about making life decisions and solving problems than paying mortgage, finding the area of fabric, or determining the speed of a cannonball.  We will also directly explore the philosophy and history of mathematics.  Why does mathematics exist at all?  

   We have two very different books for this course.  The Heart of Mathematics is a fun coffee-table type book.  The book is about as easy to read as a light magazine.  The main point of this book is that mathematics helps us to think about our daily life and the world around us.  This will be the source of our mathematical content and most of the projects.  Homework will primarily come from this book.  The exams will cover material in this book.  
    What is Mathematics Really? is much more a philosophy book than a mathematics book.  Occasionally the reading is rough, but the mathematics is almost always very simple.  The references and discussions may feel obscure at times.  Hersh addresses this eventually, saying something like "If a reference is unfamiliar, it's probably not important."  We will discuss Hersh's book for 30-45 minutes occasionally.  For those days you are required to bring reading reactions to class.  These reading reactions must include reactions (items you particularly identified with, disagreed with, or do not understand but would like to discuss) to at least five topics in the reading.  They must be written in intelligible English.  Each one will be evaluated out of 5 points, with points deducted for fewer than five points being addressed.

Course Content
    We will begin the course by reading Chapter One of Heart of Mathematics.  We will also read Chapters 1 - 5 and 13 of What is Mathematics, Really?  The remaining course content will be determined based on student preference indicated on forms distributed on the first day.  The most popular sections will be discussed in class January 31 - March 11 and April 4 - 15.  Other sections will be assigned to paired students as projects.  I will also present Hersh's brief summary of calculus to give a different perspective on material that is commonly presented in mathematics courses.  

    Your grade in this course will be based upon your performance on these items:
        In-class exams             15% each
        Reading Reactions        10%
        Homework                  15%
        Colloquium Report      10%
        Project                       15%
        Final Exam                 20%
Colloquium Report
    Attend one of the department colloquium talks.  Write a report.  In the report, describe the content of the talk (you do not need to explain all the details, but it is necessary to include the main points that the speaker was attempting to convey).  In addition to your description of the talk, also write how this talk added to your understanding of the nature of mathematics.  

    Each student is responsible for completing a project as part of a pair.  A project will consist of reading a section of one of our textbooks and completing all the exercises in the chosen part (there may be supplemental exercises for projects chosen from Hersh).  Projects will be presented during the last two weeks of class.  Writeups are due on the last day of class.  

Homework Exercises
    There will be homework exercises assigned from each section that we cover in class.  The first homework assignment will be finalised after we decide on the course content.  Following that each homework assignment will be announced on the day that the previous assignment is due.  You are encouraged to consult with me outside of class on any questions toward completing the homework.  You are also encouraged to work together on homework assignments, but each must write up their own well-written solutions (a simple rule - discuss homework, but do not look at each other's writing).  Each question will be counted in the following manner:
    0 – missing or copied question
    1 – question copied
    2 – partial question
    3 – completed question (with some solution)
    4 – completed question correctly and well-written
Each entire homework set will then be graded on a 90-80-70-60% (decile) scale.  Late items will not be accepted.  Homework will be returned on the following class day.

In-class Exams
    In class exams will check your understanding of the mathematical content of the course.  They will have questions directly from the "Solidifying Ideas" questions pertaining to the sections we have discussed. 

Final Exam
    Half of the final exam will be in the same form as the in-class exams.  A quarter of the exam will require you to state the main idea of several projects other than your own.  The last quarter of the exam will require you to summarise your understanding of Hersh's book and to explain your reaction to his ideas.  

     Occasionally you will be given anonymous feedback forms.  Please use them to share any thoughts or concerns for how the course is running.  Remember, the sooner you tell me your concerns, the more I can do about them.  I have also created a web-site which accepts anonymous comments.  If we have not yet discussed this in class, please encourage me to create a class code.  This site may also be accessed via our course page on a link entitled anonymous feedback.  Of course, you are always welcome to approach me outside of class to discuss these issues as well. 

Religious Holidays
    It is my policy to give students who miss class because of observance of religious holidays the opportunity to make up missed work.  You are responsible for notifying me no later than January 28 of plans to observe the holiday.  


January 19    Course Introduction
January 21    Burger & Starbird - Fun and Games

January 24  Fun and Games / Course Plan and Projects Assigned
January 26   Hersh Preface
January 28    2.4

January 31    2.4
February 2    2.5
February 4    2.5

February 7    Hersh 1
February 9     4.3 / HW1 due
Febuary 11    4.3

February 14   4.3 / 4.7
February 16   4.7
February 18   4.7

February 21   Hersh 2 / HW2 due
February 23   review
February 25   XM1

February 28  6.5
March 2      6.5
March 4      6.5 / Hersh 3

March 7     7.6
March 9     7.6
March 11   7.6 / HW3 due

March 21   Hersh Calc
March 23   Hersh Calc
March 25   Hersh Calc

March 28   Hersh 4 / HW4 due
March 30   review
April 1      XM2

April 4    7.2
April 6    7.2
April 8    7.2 / Hersh 5

April 11    7.3
April 13    7.3
April 15    7.3

April 18    Hersh 13 / HW5 due
April 20    Projects
April 22    Projects

April 25    Projects
April 27    Projects
April 29    Projects

May 2        Review / Project writeups due

Friday, May 6   3:30 - 6;30p    Final XM

Complete 4 of Chapter 1 Mindscapes (follow all directions and answer all particular questions in section 1.4).  Select one each from these groups:  1 - 3, 4 - 6, 7 - 9, 10 - 15

2.4   31, 35, 37, 38
2.5   4 of 16 - 20
4.3    16, 17, 19, 20, 22
4.7    16, 20, 21, 24

6.5   26, 28, 29, 32, 36
7.6    21, 23, 26, (29 or 30), (one of 31, 33, 35).
Calculus    1.  Why does 6 / 3 = 2?  Why does 3 / 0 have no answer?  Why does 0 / 0 have any answer?
                2.  Following Hersh's discussion, consider a stone traveling by the equation h(t) = -16t^2 + 10t + 5, compute the speed at time t by computing the distance change in a moment H near time t divided that by moment H.
                3.  What is the slope of a curve at a point where it reaches a maximum or a minimum?

7.2    26, 27, 35, (one of 36, 38, 39)
7.3    27, 31, 34, (one of 37, 39, 40)