Advanced History of Mathematics

(This course has been developed but has not yet been offered.)

Prerequisite: Advanced Trigonometry/Precalculus or the equivalent

Course Description

This course will survey mathematical ideas from a historical perspective. Students will not only be asked to engage with mathematical concepts and apply these concepts to problem-solving, they will also be required to read, write about, and explain their historical developments. In roughly chronological order and thematically, the course will explore all of the different cultures around the world that have contributed to the advancement of mathematics. Students will gain an appreciation for mathematics as a human achievement and endeavor, and this will culminate in a research paper on a mathematician of the student's choosing - consisting of both biographical information and a thorough discussion of their contribution to mathematics.

Grading for Semester Course

Per marking period (quarter):

Homework – 20%

Reading Quizzes/Math Quizzes – 30%

Quarter Exam – 25%

Mathematician Presentation/Research – 25%

Textbooks

We will be using two textbooks, one of them will provide the overarching narrative and overview (Berlinghoff and Gouvea) while the other (Stillwell) will allow us to delve into some of the mathematical details:

1. Mathematics and its History by John Stillwell (3rd Edition).

2. Math Through the Ages: A Gentle History for Teachers and Others by William P. Berlinghoff and Fernando Q. Gouvea (2nd Edition)

Most of the assigned problem sets will be from Stillwell's book. These will be opportunities for students to engage with the mathematical ideas and will require creative approaches to problem-solving in order to develop one's mathematical maturity. Difficult as it may be initially, persist, and you will come to appreciate the beauty of an elegant argument. 

Very important: learning math is about doing problems. That is how you get better. You need to give the problems a try by yourself first. We will have class time dedicated to discussion of the assigned problems. 

Mathematician Research

Click on the following document for instructions: Mathematician Research Project

If a mathematician is not on the list, you may still choose that mathematician to do research on as long as you receive instructor approval. There are many mathematicians that can be found in both of the textbooks. In Berlinghoff's book, there is a section entitled "When They Lived" - another list you can choose from. There will be one day during the first quarter when you will be allowed the class period to begin your research. You need to prepare a short 10-minute presentation on a scheduled day at the end of the first quarter. There will be one class day when the rough draft of your paper is due so that peer editing and revisions can take place. More details will be provided in class... For your sources, please consider using the following as one of them for your paper: 

The MacTutor History of Mathematics Archives

Lecture Notes

Students will discuss the readings and problems in class. Though there will be mini lectures throughout the semester to establish context, background, and ideas, the details will be from the readings and discussions. Please note: the posted lecture notes were originally intended for the instructor's purposes only and may not be complete. By no means are the posted lecture notes meant to serve as a substitute for attendance in class and the reading of the text. It is not guaranteed that they are without error either. Should there be any errors, notify and inform the instructor of the error immediately or as soon as possible. 

1. Pythagoras' Theorem

2. Pythagorean Triples and Rational Parametrization of the Circle

3. Greek Geometry

4. Apollonius and the Conic Sections

5. Greek Number Theory

6. More Greek Number Theory

7. Infinity in Greek Mathematics

8. Number Theory and Algebra in Asia

9. Hindu-Arabic Number System

10. Europe: Polynomial Equations

11. More on Tartaglia and Cubics

12. Analytic Geometry

13. The Nature of the Continuum

14. Projective Geometry

15. Calculus

16. The Number Theory Revival

17. Non-Euclidean Geometry

1 Class Day (First Quarter) – Begin Work for Mathematician Research

1 to 2 Class Days (First Quarter) – Preparation, Short 10-Minute Student Presentations

1 Review Day for Quarter Exam (First Quarter)

1 Class Day for Quarter Exam (First Quarter)

1 Class Day (Second Quarter) – Draft of Research Paper Due – Peer Review and Revision

1 Review Day for Quarter Exam (Second Quarter)

1 Class Day for Quarter Exam (Second Quarter)

2 Class Days (Second Quarter) – Film: The Man Who Knew Infinity

*There will be short math quizzes, reading quizzes, and written responses or reflections for the reading throughout. Students need to complete the problems from each section of the text after a thorough reading of Stillwell and reviewing of class discussion notes. These problems will be checked – students will discuss the readings and problems in class. Though there will be mini lectures throughout the semester to establish context, background, and ideas, the details will be from the readings and discussions. 

The subject matter is vast and is beyond the scope of any first course in the history of mathematics, providing room for further independent study of topics of interest. Here are some other books to consider as supplementary literature in addition to the chosen books. 

1.  A Journey Through Genius: The Great Theorems of Mathematics by William Dunham

2. A History of Mathematics: An Introduction (3rd Edition) by Victor J. Katz

3. A History of Mathematics (3rd Edition) by Carl B. Boyer and Uta C. Merzbach

4. The History of Mathematics: An Introduction (7th Edition) by David Burton