TEACHING
WITH ORIGINAL HISTORICAL SOURCES IN MATHEMATICS
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:

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Our odyssey of teaching with original sources

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.

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Mathematical Expeditions: Chronicles by the Explorers

Our first book of annotated original sources
Cover photo of Mathematical
                Expeditions
The Authors
emerged from the lower division Spirit and Evolution of Mathematics course. This book was written  with support from the National Science Foundation's Division of Undergraduate Education, and is available from Springer in paperback or hardcover in their Undergraduate Texts in Mathematics/Readings in Mathematics series.

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:

  • a course for strong liberal arts majors,
  • a history of mathematics course,
  • a course to attract and retain mathematics majors.

  • For  more information on these possibilities and other aspects of using the book in teaching, look at the answers to some frequently asked questions about our book.


    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 below are for illustration purposes only, not for reproduction or class use.
    Electronic copyright 1998, Reinhard Laubenbacher and David Pengelley.
    (The excerpts are mostly .dvi files. If other formats are needed, let us know.)
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    Mathematical Masterpieces: Further Chronicles by the Explorers

    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 Cover of Mathematical Masterpiecescontains 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:

  • a history of mathematics course at the upper division,
  • a capstone course for mathematics majors,
  • upper division enrichment for majors in secondary mathematics education, engineering, or the sciences.

  • 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. 

    We welcome hearing from you about the book.

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    Teaching Discrete Mathematics and Computer Science 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. 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.

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    Our articles on and about history of mathematics and its role in teaching
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    Other resources for teaching with original sources and for history of mathematics

    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; Hlne 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 Guzmn
    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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    Return to main Department of Mathematical Sciences page at New Mexico State University
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    Contact us at reinhard@math.vt.edu or davidp@nmsu.edu
    Last revised Feb. 9, 2014.