Math 110:
Discovering in Mathematics

   Department of Mathematics and Computer Science
   Hobart and William Smith Colleges

   Fall, 2008.

   Instructor:  David J. Eck.

   Monday, Wednesday, Friday, 3:00--3:55 PM.
   Room Eaton 110.

About This Course

The title of this course is Discovering in Mathematics, and a major goal of the course is to allow you to experience math as a working mathematician does: as a voyage of discovery in a world that requires imagination and creativity to appreciate fully. Many people encounter mathematics as a process of memorization of facts and formulas that have little meaning or motivation. There is little to love in such a discipline, and it is no wonder that some people come to dislike math or to believe that they are not good at it.

For a typical mathematician, on the other hand, the mathematical world is full of meaning and beauty. The mathematician seeks to understand this world and to discover and understand new aspects of it. But this type of exploration is not limited to mathematicians. Any time you solve a mathematical problem by understanding the problem and using that understanding to invent a solution, you are engaged in an act of mathematical discovery, which can be both rewarding and fun. It is my hope that this course will help you experience mathematics as an interesting and creative endeavor.

The textbook for the course is Symmetry, Shape, and Space: An Introduction to Mathematics through Geometry, by L. Christine Kinsey and Teresa E. Moore (ISBN 1930190093). This textbook has an unusual format, with many exercises interspersed throughout the reading. You really have to think about the exercises to get the most out of the reading. Some of the exercises might take a lot of thought and exploration to solve. Some you might not solve at all. The goal is not to find a "correct" solution to every exercise. In this course, it's the exploration itself that is important. We will do many of the exercises in class, with the class either working as a whole or in small groups. Some of the exercises will be left for individual work outside of class. For some of the exercises, I will ask you to write up solutions and turn them in for grading. We will cover substantially less than half of the book.

In addition to the textbook, there will be readings from two other books that you will need to buy. The readings from these books are meant to convey something of the nature of mathematics and of its history. The first book is The Math Instinct: Why You're a Mathematical Genius (Along with Lobsters, Birds, Cats, and Dogs), by Keith Devlin (ISBN 156025839X). The Math Instinct makes the argument that people have a natural mathematical ability, and it tries to explain why people with such a natural ability often fail nevertheless at school math. A large part of the book is about the natural mathematical ability of animals -- we will skip most of that material and will concentrate on what the book has to say about people. The second book is Zero: The Biography of a Dangerous Idea, by Charles Seife (ISBN 0140296476). Zero covers the discovery of the number zero and the surprising slow acceptance of the concept. It also has a lot to say about infinity, and why infinity and zero are so closely related. We will skip the last three chapters, which deal mostly with zero as it comes up in physics.

Finally, I will hand out a few additional readings, most notably a handout on the mathematics of voting that we will read around the time of the presidential election on November 4.

Assignments and Quizzes

There will be a homework assignment almost every week. In general, I will collect assignments on Wednesday of the following week. The majority of assignments will begin with group work in class. You will then be responsible for finishing up any work that was not finished in class and for writing up clear and complete solutions for all the exercises. You are welcome to continue group work outside of class, if you prefer to do that, but you will be responsible for writing up your own clear and complete solutions to all homework exercises. The homework that you turn in should be your own work, written in your own words and not copied from other people in your group. All solutions should be written in full English sentences and paragraphs and should explain your reasoning. You will not get credit for writing down an answer without any justification.

Over the course of the semester, there will probably be a few ten-minute quizzes, which might or might not be announced in advance. Quizzes can cover material from assigned readings before we discuss the readings in class. Grades from quizzes will be added to your assignment grade, and the combined grade for assignments and quizzes will count for 40% of the total grade for the course.

Final Project

In addition to the regular weekly assignments, you will complete a final project. This project will count for 12% of the total grade for the course. Two types of project are possible. One option is to write a five-to-seven page research paper on some mathematical topic. Another option is to build some sort of object or create some work of art that illustrates a mathematical principle, along with a short, one or two page paper describing what you have done. Most projects of the second type will be based on chapters from Symmetry, Shape, and Space that we do not cover in class.

The final project will be due at the end of the semester, but there is nothing to stop you from getting an early start. More information and a list of possible topics can be found at


Many of the exercises that you will work on in class will require extra material or supplies of some kind. Some of this will be supplied to you, but you are responsible for having a few basic supplies and bringing them to class when required. Things that you will have to supply yourself include at least regular paper, pen, pencil, eraser, ruler, and scissors. Any necessary supplies that you should bring to a class will be announced in advance. For example, for the second week of class, you will need a ruler and a pencil with eraser.

Tests and Grading

There will be two one-hour in-class tests, which will be given on Friday, October 3 and Friday, November 7. There will also be a final exam, which will be given at the officially scheduled time, 8:30 AM on Thursday, December 18. The final exam will be designed to be only a little longer than the two in-class tests. Please make plans to be present for all three tests.

The grades for this course will be weighted as follows:

          First test:            16%
          Second test:           16%
          Final Exam:            16%
          Final project:         12%
          Assignments/Quizzes:   40%


I will take attendance in class almost every day. I expect you to be present and on time for all classes. There might be extraordinary circumstances that force you to miss a few classes. You should discuss any such cases with me at the earliest possible time. In the absence of such extraordinary circumstances, you can expect that having more than a few absences will lower your grade for the course.

Office Hours, E-mail, and Web

My office is room 313 in Lansing Hall. My office phone extension is 3398. I am on campus most days, and you are welcome to come in anytime you can find me there. On most weekdays, I will be on campus at least between 10:00 AM and 3:00 PM (but I am less likely to be in on Thursday). I have class Monday, Wednesday, and Friday from 12:20 to 1:15 and from 3:00 to 3:55. My official office hours are 1:30 to 2:50 on Monday, Tuesday, Wednesday, and Friday. Office hours are times when I promise to try my best to be in my office. I do not generally make appointments during my office hours, since they are times when I am available to students on a first-come, first-served basis. When necessary, I am happy to make appointments for meetings outside my scheduled office hours.

My e-mail address is E-mail is good way to communicate with me, since I usually answer messages the day I receive them.

The Web page for this course is at I will post weekly readings, assignments, and other information on that page.

Math Intern

The Department of Mathematics and Computer Science employs a "math intern" who is available to help students in calculus and precalculus courses. The math intern might be able to offer some help in Math 110 as well, although you should not expect him to know what is going on in the course or to be prepared to answer questions about everything that we cover. The math intern this semester is Dave Brown, and he will hold office hours in room Lansing 310 at the following times (subject to change):

Tentative Schedule

Here is a very tentative weekly schedule of readings and activities for the course. We will probably follow this schedule exactly for the readings from The Math Instinct and from Zero. But the schedule for Symmetry, Shape, and Space might change as we get experience with using the book. A few extra readings are noted in this schedule; there might be a few more on other topics.

Dates Readings, Etc.
Sep. 1, 3, 5 Introduction to the course;
     review of angles, slopes, polygons, and area.
Reading for Friday: The Math Instinct, Chapter 1.
Sep. 8, 10, 12 Symmetry, Section 2.1 "Billiards"
     and Section 2.2 "Celtic Knots."
Reading for Friday: The Math Instinct, Chapters 2 and 3.
Sep. 15, 17, 19 Symmetry, Section 4.1 "Tilings."
Handout on "Number Systems."
Reading for Friday: The Math Instinct, Chapter 10.
Sep. 22, 24, 26 Symmetry, Section 4.2 "Irregular Tilings."
First encounter with M. C. Escher's Art.
Reading for Friday: The Math Instinct, Chapter 11.
Sep. 29; Oct. 1, 3 Symmetry, Section 5.1 "Kaleidoscopes."
TEST on Friday, October 3.
Oct. 6, 8, 10 Computer lab on Symmetry.
Symmetry, Section 5.2 "Rosette Groups."
Reading for Friday: Zero, Chapters 0 and 1.
Oct. 15, 17 No class on Monday because of Fall Break.
Symmetry, Section 5.3 "Frieze Patterns"
Reading for Wednesday: Zero, Chapter 2.
Oct. 20, 22, 26 Symmetry, Section 5.4 "Wallpaper Patterns."
In-class viewing of the video Flatland.
Reading for Friday: Zero, Chapter 3.
Oct. 27, 29, 31 Symmetry, Section 6.1 "Flatlands."
Handout on "Voting Paradoxes."
Nov. 3, 5, 7 Continuing with "Voting Paradoxes."
Symmetry, Section 6.2 "The Fourth Dimension."
TEST on Friday, November 7.
Nov. 10, 12, 14 Computer lab on Fractional Dimensions.
Symmetry, Section 10.1 "Perspective."
Reading for Friday: Zero, Chapter 4.
Nov. 17, 19, 21 Symmetry, Section 10.2 "Optical Illusions"
     and Section 11.1, "Non-Euclidean Geometry."
Reading for Friday: Zero, Chapter 5.
Nov. 24 Continue with non-Euclidean Geometry.
No class on Wednesday or Friday (Thanksgiving).
Dec. 1, 3, 5 Symmetry, Sections 7.1 and 7.2 "Polyhedra."
Reading for Friday: Zero, Chapter 6.
Dec. 8, 10, 12 Wrapping up the course.
Reading for Friday: The Math Instinct, Chapters 12 and 13.
Dec. 18 Final Exam
8:30 AM