**Joan D. Lukas**

**Professor Emerita of Mathematics **

**617-287-6454**

**Joan.Lukas@umb.edu**

** **

We will address such questions as: ÒWhat
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?Ó 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)

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 http://www.faculty.umb.edu/joan_lukas/Honors252

**Week 1: September 8 What
is a Number?**

What do we consider numbers? ÒWhereÓ 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 ÒNumber
is the substance of all things?Ó What did they mean by ÒnumberÓ? 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, ÒNumerology or, what
Pythagoras WroughtÓ, 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 ÒnumberÓ**

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**

** **

** **

**Accommodations:**

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 http://www.umb.edu/student_services/student_rights/code_conduct.html.

** **

**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 1^{st} version and the experience of your in-class
presentation.