REINHARD
LAUBENBACHER Center for Quantitative Medicine University of Connecticut Health Center Farmington, CT 06030 laubenbacher@uchc.edu |
DAVID
PENGELLEY Dept. of Mathematical Sciences New Mexico State University Las Cruces, NM 88003 USA davidp@nmsu.edu |

Bienvenidos! Here we offer information and materials on using original historical sources in teaching mathematics. This includes our own experiences and materials, and those of others who are teaching with original sources. We welcome suggestions for other links to include here, and comments and suggestions for improvements.

Below we provide information on:

- Our own odyssey on teaching with original sources.
- Our lower division course and first book: Mathematical Expeditions: Chronicles by the Explorers.
- Our upper
division course and second book:
Mathematical
Masterpieces:
Further Chronicles by the Explorers (coathored with Arthur Knoebel and Jerry Lodder).

- Our graduate course on The Role of History in Teaching Mathematics.
- Our student projects for Teaching Discrete Mathematics and Computer Science (and now more!) via Primary Historical Sources.
- Our articles on and about history of mathematics and its role in teaching.
- Other resources and inspiration for teaching with original sources and using history in teaching mathematics.

Our journey towards utilizing original texts as the primary
object of study in undergraduate and graduate courses began at
the senior undergraduate level. In 1987 we read
William Dunham's article *A "Great Theorems" Course in
Mathematics *(*American Mathematical Monthly* **93**
(1986), 808-811), in which he describes a course based on
mathematical masterpieces from the past, viewed as works of
art. His ideas and materials went on to become the well
known best-seller *Journey Through Genius: Great Theorems of
Mathematics.* We were inspired to develop a similar
course, at the senior level, but with one crucial difference:
Whereas Dunham presents his students with his own modern rendition
of these masterpieces, our idea was to use the original texts
themselves. With assistance from New Mexico State
University's honors program, dean, and mathematics department, we
developed and team taught the course *Great
Theorems: The Art of Mathematics*, and it has now
found a successful and permanent niche in the university's
curriculum, serving as a lively capstone course for students
majoring in a number of diverse disciplines. It is the only
mathematics course certified to meet the university's "Viewing a
Wider World" upper division general education requirement. Our
experiences with this senior level course convinced us that
teaching with original sources could be both successful and
inspiring for us and our students. The course is described
in detail in Mathematical
Masterpieces:
Teaching With Original Sources (html) (or dvi
or ps)
(in *Vita Mathematica: Historical Research and Integration with
Teaching*, R. Calinger (ed.), MAA, Washington, 1996, pp.
257-260). We also involved other faculty in teaching and
contributing material for this course. Our four author second book Mathematical Masterpieces: Chronicles by
the Explorers has emerged from this course, written with
two of these colleagues at New Mexico State University, Arthur
Knoebel and Jerry Lodder.

We came to believe that this approach to teaching and learning
could also help provide the motivation, perspective, and overview
so lacking in typical lower division courses, since it is being
increasingly recognized that an historical point of view can
address these deficiencies. As Niels Henrik Abel observed: "**It appears to me that if one wants to make
progress in mathematics, one should study the masters and not
the pupils.**" We have written an article Recovering
Motivation in Mathematics: Teaching with Original Sources (html)
(or
dvi or ps) (*UME
Trends ***6**, September 1994) espousing our reasons and
philosophy for this teaching approach. We were inspired to
try to use the study of original texts as a teaching pedagogy
introducing lower division students to important currents of
mathematical thought.

Thus we developed the course *Spirit and
Evolution of Mathematics*, again with support from
the New Mexico State University mathematics department and honors
program, allowing us to team teach the course while under
development. It provides an "introduction to great problems
of mathematics" for students with a good high school background in
mathematics, and is intended both to attract and retain
mathematics majors, and to give non majors a rich experience in
the nature and content of mathematical thought, satisfying a lower
division university mathematics general education requirement (the
course is one of only a handful certified for this). In
fact, the true prerequisite is a certain level of mathematical
maturity and ability, rather than courses with specific
content. Thus, a much broader audience has access to an
interesting course with serious mathematical content. Our
experiences, after teaching this course numerous times, have shown
that with careful selection of original texts, supplemental prose
readings, and appropriate format for classroom activities and
assignments, this approach can be a tremendous success.
Students find the study of original sources fascinating,
especially when combined with prose readings supplying cultural
and historical context, giving the course something of an
interdisciplinary flavor. The benefits for instructors and
students alike are a deepened appreciation for the origins and
nature of modern mathematics, as well as the lively and
stimulating class discussions engendered by the interpretation of
original sources. The course is described in detail in our
article Great
Problems
of Mathematics: A Course Based on Original Sources (html)
(or dvi
or ps)
(*American Mathematical Monthly* **99** (1992), 313-317).
Our first book Mathematical
Expeditions: Chronicles by the Explorers grew out of this course.

Since then we have expanded the use of original sources into high
school courses as well as graduate courses. Work with high
school students during two summer workshops at Colorado College
with Mike Siddoway is described in Great
Problems
of Mathematics: A Workshop for High School Students (html)
(or dvi
or ps)
(*College Mathematics Journal* **25** (1994),
112-114). We also conducted a graduate course at New Mexico
State University for high school teachers on using original
sources in the high school curriculum. Our graduate students
showed great interest in this, and it has evolved into a regular
graduate course The
Role of History in Teaching Mathematics, providing part
of a growing mathematics education component in the mathematics graduate program
at New Mexico State University. The paper A
graduate course on the role of history in teaching mathematics
describes the course and its origins. The
course syllabus considers the use of history, in particular
original sources, throughout the mathematics curriculum. Our
graduate students in this course develop and critique
major teaching units based on history, often on original sources,
and we now have quite a collection of the historical
teaching
modules they have written. A number of these have been
tested in the classroom. Their level ranges from middle
school through the advanced undergraduate curriculum. Write to us
if you want copies of any of these. Our long-term dream is that
the entire mathematics curriculum should be historically based,
with original sources playing a role throughout, and we ourselves
are endeavoring to incorporate both history and original sources
into all the courses we teach.

More recently David has teamed up with other colleagues from mathematics and computer science in applying our approach to the teaching of discrete mathematics, broadly conceived. We are combining the pedagogy of student projects (introduced into our calculus classes years ago) with the pedagogy of using original historical sources, in a NSF-funded program to develop and test student projects written using primary sources for teaching discrete mathematics.

Teaching with historical sources has also led us to several research projects in the history of mathematics, as shown in our articles listed below.

Mathematical Expeditions: Chronicles by the Explorers

Our first book of annotated original sources

The cover features portraits of five mathematicians whose original writings are at the heart of our five chapters, overlain with Sophie Germain's handwriting from a letter she wrote to Gauss in May of 1819 on her work on Fermat's Last Theorem, also featured in the book. See if you can read what Germain wrote to Gauss, or identify the people in the portraits. The book includes translations of Germain's letter and manuscripts, and ninety-four portraits, mosaics, artwork, facsimiles of handwritten manuscripts and letters, and figures.

From the back cover

*This book contains the stories of five mathematical journeys
into new realms, told through the writings of the explorers
themselves. Some were guided* *by mere
curiosity and the thrill of adventure, while others had more
practical motives. In each case the outcome was a vast
expansion of the known mathematical world and the realization
that still greater vistas remained to be explored. The
authors tell these stories by guiding the reader through the
very words of the mathematicians at the heart of these events,
and thereby provide insight into the art of approaching
mathematical problems.*

*The book can be used in a variety of ways. The five
chapters are completely independent, each with varying levels of
mathematical sophistication. The book will be enticing to
students, to instructors, and to the intellectually curious
reader. By working through some of the original sources
and supplemental exercises, which discuss and solve -- or
attempt to solve -- a great problem, this book helps the reader
discover the roots of modern problems, ideas, and concepts, even
whole subjects. Students will also see the obstacles that
earlier thinkers had to clear in order to make their respective
contributions to five central themes in the evolution of
mathematics.*

Mathematical
Expeditions **is suitable for several types of college
courses:**

Mathematical Expeditions
has been reviewed
by the Mathematical Association of America. And there is also a review
in Mathematical Reviews.

And here are brief biographies of ourselves.

Here you can also view the book's preface (which discusses teaching uses for the book), the table of contents, some chapter synopses, and some excerpts from various sections. (The figures and photos don't show up here, the page numbers don't match those in the table of contents, and page breaks and spacing are different from the actual published book.) We welcome your questions, or requests for further excerpts you would like to see. We will add other synopses or excerpts from time to time.

(The excerpts are mostly .dvi files. If other formats are needed, let us know.)

- Preface (html) (or pdf or dvi or ps)
- Table of Contents (html) (or pdf or dvi)
- Introduction to the Geometry chapter (dvi) (or pdf or ps)
- Euclid's Parallel Postulate (dvi) (or pdf or ps)
- Poincare's Euclidean model for Non-Euclidean Geometry (dvi) (or pdf or ps)
- Introduction to the Set Theory chapter (dvi) (or pdf or ps)
- Cantor's Infinite Numbers (dvi) (or pdf or ps)
- Introduction to the Analysis chapter (dvi) (or pdf or ps)
- Archimedes' Quadrature of the Parabola (dvi) (or pdf or ps)
- Archimedes' Method (dvi) (or pdf or ps) (of Treating Mechanical Problems)
- Leibniz's Fundamental Theorem of Calculus (dvi) (or pdf or ps)
- Introduction to the Number Theory chapter (dvi) (or pdf or ps)
- Euclid's Classification of Pythagorean Triples (dvi) (or pdf or ps)
- Germain's General Approach (dvi) (or pdf or ps) (to Fermat's Last Theorem)
- Introduction to the Algebra chapter (dvi) (or pdf or ps)
- Euclid's Application of Areas and Quadratic Equations (dvi) (or pdf or ps)
- Cardano's
Solution
of
the Cubic (dvi) (or pdf
or ps)
or pdf
with
images

- Bibliography (html) (or pdf or dvi or ps)

Together with our colleagues Arthur
Knoebel and Jerry Lodder,
and with further support from the National Science Foundation, we
have written an elder sibling for Mathematical
Expeditions. The new book Mathematical Masterpieces contains annotated original sources from our upper
division course *Great Theorems: The
Art of Mathematics*, presented as a capstone
for the undergraduate mathematics curriculum. The book is
available now from Springer, in
hardcover or paperback, in their Undergraduate Texts in
Mathematics/Readings in Mathematics series.

The cover features portraits of mathematicians whose original writings are at the heart of our four chapters. See if you can identify the people in the portraits. The cover also shows a figure by Huygens from the construction of an evolute in his Horologium Oscillatorium (The Pendulum Clock), in our chapter on the development of the concept of curvature. And we display Chinese text by Qin Jiu-Shao on approximating roots of polynomials, from our chapter on numerical solutions of equations. The book has many portraits, artwork, facsimiles of original works, and figures.

From the back cover

Experience the
discovery of mathematics by reading the original work of some
of the greatest minds throughout history. Here are
the stories of four mathematical adventures, including the
Bernoulli numbers as the passage between discrete and
continuous phenomena, the search for numerical solutions to
equations throughout time, the discovery of curvature and
geometric space, and the quest for patterns in prime numbers.
Each story is told through the words of the pioneers of
mathematical thought. Particular advantages of the historical
approach include providing context to mathematical inquiry,
perspective to proposed conceptual solutions, and a glimpse into
the direction research has taken. The text is ideal for
an undergraduate seminar, independent reading, or a capstone
course, and offers a wealth of student exercises with
a prerequisite of at most multivariable calculus.

**Mathematical Masterpieces
is suitable for several types of college courses:**

Mathematical Masterpieces has been reviewed by the Mathematical Association of America.

Here you can read the book's preface (which discusses teaching
uses for the book), the table of contents, and the chapter
introductions and some sample sections of the book. The article The bridge
between the continuous and the discrete via original sources
describes one of the chapters, and the article Curvature
in
the Calculus Curriculum discusses how source material from
our curvature chapter has been used in teaching calculus.

- Preface
- Table of Contents
- The Bridge Between Continuous and Discrete
- Solving Equations Numerically: Finding Our Roots
- Curvature and the Notion of Space
- Patterns in Prime Numbers: The Quadratic Reciprocity Law

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Teaching Discrete Mathematics and Computer Science (and now more!) via Primary Historical Sources

David is part of a team of mathematicians and computer scientists
at this and other universities, who are applying this approach to
the teaching of discrete mathematics, broadly conceived. We are
melding the pedagogy of teaching with student projects (introduced
into our
calculus
classes years ago) with the pedagogy of using original
historical sources, in a NSF-funded program to develop, test,
evaluate, and disseminate student projects written using primary
historical sources for courses in discrete mathematics,
combinatorics, abstract algebra, logic, and algorithmic thought in
computer science (and now even more!). See our web pages Teaching
Discrete Mathematics via Primary Historical Sources
for the pedagogy and results of our Phase I NSF pilot
grant, including the classroom projects developed and published
under that grant through year 2006. See our web pages Learning
Discrete Mathematics and Computer Science via Primary Historical
Sources for the work commencing in year 2008 under our Phase
II NSF expansion grant, including the many new projects being
created under that grant. And see the pages Transforming
Instruction in Undergraduate Mathematics via Primary Historical
Sources (TRIUMPHS) for continuing work commencing in 2015 to
develop primary source projects (PSPs) and mini-PSPs for the
content of all regular courses for mathematics majors, pre-service
teachers, and other STEM discipline majors. We welcome those who
would like to use or test our student projects.

Our articles on and about history of mathematics and its role in teaching

- Great
Problems
of
Mathematics: A Course Based on Original Sources (html) (or
pdf
or dvi
or ps),
*American Mathematical Monthly***99**(1992), 313-317. - Great
Problems
of Mathematics: A Workshop for High School Students (html)
(or pdf
or dvi
or ps),
*College Mathematics Journal***25**(1994), 112-114. - Recovering
Motivation
in
Mathematics: Teaching with Original Sources (html) (or pdf
or dvi
or ps)
(with M. Siddoway),
*UME Trends***6**, September 1994. - Mathematical
Masterpieces:
Teaching With Original Sources (html) (or pdf
or dvi
or ps),
in
*Vita Mathematica: Historical Research and Integration with Teaching*, R. Calinger (ed.), MAA, Washington, DC, 1996, pp. 257-260. - Gauss,
Eisenstein,
and the `Third' Proof of the Quadratic Reciprocity Theorem:
Ein Kleines Schauspiel (html) (or pdf
or dvi
or ps),
*Mathematical Intelligencer***16**(1994), 67-72. - Eisenstein's
Misunderstood
Proof of the Quadratic Reciprocity Theorem (html) (or pdf
or dvi
or ps),
*College Mathematics Journal***25**(1994), 29-34. - Lagrange and the Solution of Numerical Equations (with G.
McGrath),
*Historia Mathematica***28**(2001), 220-231. - ``Voici ce que j'ai trouve'': Sophie Germain's Grand Plan to Prove Fermat's Last Theorem (January 2010 revision), to appear in Historia Mathematica.
- The
bridge between the continuous and the discrete via original
sources, in
*Study the Masters: The Abel-Fauvel Conference*, Kristiansand, 2002 (ed. Otto Bekken et al), pp. 63-73, National Center for Mathematics Education, University of Gothenburg, Sweden, 2003. - A
graduate course on the role of history in teaching mathematics,
in
*Study the Masters: The Abel-Fauvel Conference*, Kristiansand, 2002 (ed. Otto Bekken et al), pp. 53-61, National Center for Mathematics Education, University of Gothenburg, Sweden, 2003. - Dances
between
continuous
and discrete: Euler's summation formula (pdf) or (dvi),
*Euler 2K+2 conference, Rumford, Maine, 2002*(editors Robert Bradley and Ed Sandifer), and revised invited paper for the Euler Festschrift 2007 (eds., Larry d'Antonio, Robert Bradley and Ed Sandifer), Mathematical Association of America, in press. - Arthur Cayley and the first paper on group theory, in From Calculus to Computers: Using the Last 200 Years of Mathematical History in the Classroom (eds. R. Jardine and A. Shell), pp. 3-8, Mathematical Association of America, 2005.
- Teaching and Learning Mathematics from Primary Historical Sources (Janet Barnett, Jerry Lodder, David Pengelley), in PRIMUS (Problems, Resources, and Issues in Mathematics Undergraduate Studies), in press.
- Teaching Discrete Mathematics Entirely from Primary Historical Sources (Janet Barnett, Guram Bezhanishvili, Jerry Lodder, David Pengelley), in PRIMUS (Problems, Resources, and Issues in Mathematics Undergraduate Studies), in press.

Original source materials available

Excerpts
on
the Euler-Maclaurin summation formula, from *Institutiones
Calculi Differentialis* by Leonhard Euler (pdf format),
or in (dvi
format), and at the Euler
Archive.

Excerpt from
a letter of Monsieur Lame to Monsieur Liouville on the question:
Given a convex polygon, in how many ways can one partition it
into triangles by mean of diagonals?: Lame's elegant
geometric solution to finding the one step recursion relation
solving Euler's decomposition problem, leading to the factorial
formula for Catalan numbers.

Other courses based on original sources

History
of
Mathematics with Original Sources; Gary Stoudt, Indiana
University of Pennsylvania

Work of
Great Female Mathematicians; Hélène Barcelo, Arizona State
University

History of
Mathematics; Fred Richman, Florida Atlantic University

Development
of
Mathematical Ideas; Man-Keung Siu, University of Hong Kong

History
of
Mathematics; Phill Schultz, University of Western Australia

Bibliographies for using history in teaching
mathematics

Some
selected resources for using history in teaching mathematics; D.
Pengelley

Bibliography
of
Collected Works of Mathematicians; Cornell University
Mathematics Library

Articles on using history in teaching mathematics

Origin
and Evolution of Mathematical Theories: Implications for
Mathematical Education; Miguel de Guzmán

The ABCD of using history of mathematics in the (undergraduate)
classroom; Man-Keung Siu

(in dvi
format) (in pdf
format) (see also our bibliography
for reprintings).

Other resources

Fred
Rickey's
home page on history of mathematics and teaching

History and Pedagogy of
Mathematics (HPM); International Study Group

History
and Pedagogy of Mathematics (HPM); America's Section

British Society for
the History of Mathematics

Canadian Society for the History
and Philosophy of Mathematics

Convergence: Where
Mathematics, History and Teaching Interact, MAA

History of
Mathematics Special Interest Group, Mathematical Association of
America (HOMSIGMAA)

History
of
Mathematics; David E. Joyce, Clark University

History
of
Mathematics; David R. Wilkins, Trinity College, Dublin

History
of Mathematics - Mathematics Archives, Univ. of Tennessee

The Math Forum Internet
Mathematics Library

MathWeb
History: American Mathematical Society

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*Contact us at *laubenbacher@uchc.edu or davidp@nmsu.edu

*Last revised April 12, 2017.*