Joan D. Lukas

Professor Emerita  of Mathematics



Course Description

 We will address such questions as: “What are the mathematical abilities of human infants? Of non-human animals? What is a number? How have various cultures--such as the Mayan, for example--answered this question?”  Students will learn that mathematical concepts are not an unchanging unity, but have developed over centuries, often with much disagreement. We will treat the biological and cultural development of mathematics.



A background in high school mathematics and an interest in pursuing these ideas are the only prerequisites. Students will be expected to attend every class session and participate in discussions.

1)    Term paper    (30% of grade)   & brief class presentation                        (10% of grade)  

Due Dates:

o      Proposal                                                                                                                                                                                                                                                                                                                                        October 25                                                                                                                       

o      Complete Paper                                                                                                                                                                                                                                                                                                                        November 29

o      Presentation                                                                                                                                                                                                                                                                                                                            December 6-15

o      Revised Paper                                                                                                                                                                                                                                                                                                                            December 20

2)    Midterm exam: October 18                                                                                                                                                                                                                                                                                                                                                           (20% of grade)

3)    Final exam: Week of December 17-23                                                                                                                                                                                                (30% of grade)

4)    Class participation                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  (10% of grade)


Course Texts


Stanislas Dehaene, The Number Sense: How the Mind Creates Mathematics.

Oxford University Press (paperback edition), 2000.


George Gheverghese Joseph, The Crest of the Peacock: Non-European Roots of Mathematics. Princeton University Press, 2000.


Robert Kaplan, The Nothing That Is: A Natural History of Zero. Oxford University Press (paperback edition), 2000.


This syllabus is available electronically at








Week 1: September 8                                                                         What is a Number?

What do we consider numbers?  “Where” are they? Are mathematical concepts discovered or invented?  How do they change as individuals and societies develop?



Week 2: September 13, 15                              Biological Origins of Number Concepts            

Where do our ideas of number begin? To what extent do infants and non-human beings exhibit number sense? What is the earliest evidence of numerical reasoning in human culture?

Readings: Dehaene: Chapters 1 and 2; Joseph Chapters1and 2


Week 3: September 20, 22                           Mathematics in Ancient Mesopotamia and Egypt                                                                                                                              

How were numbers and numerical operations treated in these societies?  What kinds of problems were solved?

Readings: Joseph, Ch. 3-5; Dehaene  Chapter 4; Kaplan  Ch 0-1


Week 4:  September 27, 29                          Number and Mathematics in Ancient Greece

What did the Pythagoreans mean by “Number is the substance of all things?” What did they mean by “number”?   How did mathematical concepts evolve into axiomatic systems?

Readings: Lloyd Motz and Jefferson Hane Weaver, The Story of Mathematics, Avon Books, 1993. Chapter 1

Dudley Underwood, “Numerology or, what Pythagoras Wrought”, Mathematical Association of America, 1997, Chapter 2.

Kaplan, Chapter 2



Week 5: October 4, 6                                    Recognizing and Representing Large Numbers

Why do we need to represent large numbers?  What are the challenges and solutions?

Readings: Kaplan, Chapter 3; Dehaene, Chapters 3 and 4




 Week 6: October 13                                                                                                                         Mathematics in Ancient China

How did mathematics develop in this society? How did it interact with other parts of the world?

Readings: Joseph, Chapters 6


Week 7:  October 18, 20                               Midterm Exam, Discussion of term papers



Week 8: October 25, 27                                               Indian and Arabic Contributions

How did the Hindu-Arabic numeral system develop?

Readings: Joseph, Chapters 8, 10; ; Kaplan, Chapters 4-6



Week 9: November 1, 3                                                                         Zero as a Number

Readings: Kaplan, Chapters 7,8

How were zero and negative numbers introduced? Why were they incorporated by some cultures centuries before others? What has been made by different cultures of the idea of infinity?

Term paper proposal due.                                                                                                   



Week 10: November 8, 10                                      Hindu- Arabic Numerals in Europe           

How were the new numerals introduced to the West?  Where and why were they resisted?

Readings: Georges Ifrah, The Universal History of Numbers, Wiley, 2000. Ch. 26.

Karl Menninger, Number Words and Number Symbols: A Cultural History of Numbers, Dover, 1992, pp. 422-445.

Kaplan, Chapter 9.



Week 11:  November 15, 17                                                                                                             Renaissance Mathematics & Nature


Kaplan Chapters 10-14.


Week 12: November 22, 24                                                                                                              Mathematics and the Brain

 How are mathematical abilities based in the brain? How can we use our knowledge of mathematics/brain connections to improve mathematics education?

Readings: Dehaene   Chapters 5 - 8,

Oliver Sacks, The Man Who mistook His Wife for a Hat, Harper & Row, 1987. Ch. 23





Week 13: November 29, December 1 Modern extensions and refinements of  “number”

How do we decide what counts as a number?  Is infinity a number?

Readings: Dehaene   Chapter 9; Kaplan Chapters 15, 16

Term paper due.



Weeks 14, 15: December 6, 8, 13, 15   Student Presentations on Term Papers


December 17-23:  Final Exam week:  Revised paper due December 20




Section 504 of the Americans with Disabilities Act of 1990 offers guidelines for curriculum modifications and adaptations for students with documented disabilities.  If applicable, students may obtain adaptation recommendations from the Ross Center for Disability Services, M-1-401 (617-287-7430).  The student must present these recommendations and discuss them with each professor within a reasonable period, preferably by September 10, the end of the Add/Drop period.


Student Conduct:

Students are required to adhere to the University Policy on Academic Standards and Cheating, to the University Statement on Plagiarism and the Documentation of Written Work, and to the Code of Student Conduct as delineated in the Catalog of Undergraduate Programs, pp. 44-45 and 48-52.  The code is available online at



Term papers:

Your paper should involve a deeper investigation of a topic touched on in class or the readings or an exploration of a related topic involving mathematical expressions in different cultures.  You should consult several sources and analyze and compare their approaches.  The paper should be approximately 10 pages in length.

Sample topics:

1.              Historical struggles over the introduction of negative numbers.

2.              Treatment of fractions in several early cultures.

3.              Interactions among mathematics, astronomy, and astrology.

4.              Interactions between mathematics and religious beliefs.

5.              Mathematics in the French Revolution.

6.              Relationships between historical and individual development of number concepts.

7.              Relationships between social organization and development of mathematics.


Paper proposal due October 24: 

A one-paragraph description of your topic along with a list of resources you plan to consult.


Paper due November 29: 

Approximately 10 pages with full bibliography.


In-class presentation December 6, 8, 13, 15:

You will give a presentation of 10-15 minutes on your work during the last 2 weeks of the semester.  You should prepare a one-page handout for your classmates as part of this presentation


Revised paper due December 20: 

Revisions may be based both on my comments on your 1st version and the experience of your in-class presentation.