| REINHARD
LAUBENBACHER Virginia Bioinformatics Institute Virginia Polytechnic and State University Blacksburg, VA 24061 USA 540-231-7506; 540-231-2606 (FAX) reinhard@vbi.vt.edu |
DAVID
PENGELLEY Dept. of Mathematical Sciences, 3MB New Mexico State University Las Cruces, NM 88003 USA 505-646-3901; 505-646-1064 (FAX) davidp@nmsu.edu |
Below we provide information on:
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.
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.
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:
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.
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. 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.
We welcome those who would like to use or test our student projects.
Here is part of a draft teaching module for discrete mathematics,
titled Sums
of numerical powers in discrete mathematics: a two-thousand year journey.
The initial part is about sums of squares, based on primary source
material by Archimedes.
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
Contact us at reinhard@math.vt.edu
or davidp@nmsu.edu
Last revised Jan. 24, 2010.