Notes and References (Part 2) A BRIEF GUIDE TO THE HISTORY OF MATHEMATICS FOR TEACHERS There are many aspects of the history of mathematics. As in the history of any subject, we can focus on the development of ideas, biography, civilisations, topics, institutions, etc., and single approach will not give a complete picture, but every aspect has a use and a value. Similarly, there are many ways teachers can approach the use of the history of mathematics, and what aspect you choose, and how you choose to incorporate it into your teaching will often depend on your personal interests and circumstances, and the background and context of the pupils with whom you are working.
No one is expecting you to be an expert, but there are some important guidelines that we need to consider that will assist you to plan and organise your lessons, and to avoid the pitfalls that so easily turn the exercise into a series of unsubstantiated ‘received opinions’.
The ‘Golden Rules’ were given in Notes and References Part 1. Here the remarks are specifically for teachers new to the area.
Internal and External Accounts
‘Internalist’ history of mathematics means the kind of history that has been written mostly by mathematicians looking at how the ‘big ideas’ in particular areas of mathematics developed. These pay little or no attention to the circumstances, situations and cultural contexts that gave rise to the mathematical ideas in question, and tell the story of ‘how it was done’. Internalist history also tends to interpret the mathematics of the past in terms of mathematics of the present.
In contrast ‘Externalist’ history of mathematics attempts to describe the situations, circumstances, cultural contexts and motivations of the mathematics of the past, and be more ‘true’ to the ideas that were developed, rather than ‘explaining’ them in modern terms. This is much more difficult, but more rewarding.
Both approaches have their uses. Internalist history can give a broad chronological idea of ‘what happened’ in the development of a particular topic, and who the principal actors may be, but not much idea about why or how. The Externalist approach attempts to get into the details and the situations of the actors, their problems, their motivations and concerns at the time. This is about the why and how, but necessarily focuses on more limited areas, or shorter time periods.
Chronology is not History
Chronology, a time-ordered list of achievements and publications, is very useful, and one way of getting a feel for the general development of ideas. However, it often gives the impression of an inevitable progress of a gradually improving ‘correct’ version of mathematics, and leaves out the more interesting (and often pedagogically useful) ‘whys’ and ‘hows’ of the innovations and advances in techniques and theories.
Credulity and Reliability
Don’t believe everything you read without question. Most textbooks on the history of mathematics published since about 1980 are more reliable, but there are still problems. It is often quite easy to find that older books have perpetrated errors by copying from each other, or not checked on available information that was available to them at the time.
Transmission of wrong or unreliable ideas can be found in books written for teachers and it can also happen for example, when researchers use history to theorise about cognitive development. For example, beware of someone who uses the ‘history of algebra’ as evidence for a particular way of presenting ideas in the classroom.
Priorities and Judgements
The tasks of the historian are to describe, interpret and explain so far as the evidence available allows. Claims about ‘who invented it first’ are usually futile. There are many cases where historians have been proved wrong when new evidence about the circumstances surrounding an event has been produced.
It may be that a writer can make a judgement about what might have happened on the strength of the evidence available, but the status of this judgement must be made clear to the reader.
Briefly, if the evidence is lacking, don’t speculate. Be honest and admit ignorance.
Speculation is not history
This applies to any events in the past when we do not have enough evidence to make a clear conclusion, and particularly to events that happened a long time ago. This is often found older editions of histories of mathematics and books on education. For example speculating about the origins of numbers and counting is a common example. It seems that the further into the past we go, some writers have had the temerity to make statements about what actually happened, whereas what little knowledge we really have, comes through the fog of time.
We often do not even have enough evidence to make reliable statements about some events that happened in the recent past, and it is always possible that our knowledge of the cultural and social contexts, or personal circumstances of individuals will change when new evidence appears.
Declare your sources
This is the Golden Rule in all cases. This is the way we avoid plagiarism, show our honesty, and can help build reliability and a resource base useful for others to follow.
A SHORT HISTORY OF MATHEMATICS BOOKLIST These books are arranged in categories, starting with general all-round history through to popular books which contain some history.
Where possible, I have suggested the ‘best buys’ and indicated prices based on the latest Amazon website trawl.
1. GENERAL HISTORIES Katz, Victor, J. (1999) (Second Edition, Corrected)
The best and most comprehensive and up-to-date history of mathematics available.
There are many other histories of mathematics with similar titles, dating back to the 1960s. Many have gone through a number of revisions and new editions, but are still unreliable, and I consider Katz’s now to be the ‘hard-to-beat’ standard.
Amazon price for the Paperback is $ 67.70 At this price, you are paying for quality.
There is a ‘short’ version, which is cheaper, but if you can afford it, ……
Kline, Morris. (1990)
Mathematical Thought from Ancient to Modern Times
Oxford Oxford University Press
I recommended this for my undergraduate students when I was teaching my History of Mathematics courses in the ‘70s and ‘80s because it was the best general book around at the time.
It has useful summaries reviewing the different periods in history, and from the 17th century, reviews the achievements in each century.
Originally published as a single volume hardback in 1972, this now appears at Amazon as two volumes in Paperback at $22.50 each.
Kline produced a number of other books worth looking out for - among them are:
Mathematics in Western Culture was a Penguin paperback and is still available - but beware his limited interpretations and the development of perspective as a purely mathematical exercise; Mathematics the Loss of Certainty, about questions and crises in 20th century mathematics and Mathematics and the Physical World.
Boyer, C.B., Merzbach, Uta, C. and Asimov, I.(1991)
A History of Mathematics
New York and London Wiley
This is the last revised edition of what was the standard reference for the original Open University History of Mathematics course and was one of the best available before Kline and Katz (above). The last chapters have been brought up to date in 1989 and 1991, though there is still little on China, India and Arabian mathematics, and omissions (like three pages in logic and computing and nothing on statistics).
The Appendix contains a useful chronological table of mathematical events, from the origin of man, to the 20th century.
The Paperback edition is available from Amazon at $26.37
Resnikoff, H. L. and Wells, R.O. (1973) Mathematics in Civilization
[1984 edition enlarged and corrected] New York: Dover Publications.
A new supplement, 'Twentieth-Century Mathematics,' an index to the supplement, and solutions to the exercises at the end of chapters have been added.
Struik. D.J. (1987)
A Concise History of Mathematics. Dover
This is the last edition of a book by the ‘Grand Old Man’ of History of Mathematics who died a few years ago at the age of 100. Struik was a Marxist, and tells his story in about 200 pages. About half the book deals with mathematics up to the 17th century and the rest devotes a chapter each to the rest up to the mid 20th century. It is still recommended for a quick overview as an introduction, before you pass on to something more substantial.
It’s found on Amazon at $ 9.95
Rouse-Ball, W. W. (1960)
A short account of the history of mathematics. Macmillan, London, 1888.
Not all older books are useless. This was published in (1888 !) by an English mathematician and educator, and contains much information not found in later books. His ‘short account’ is contained in 500 pages with a detailed index and is still being reprinted. However, it is an example of ‘Internalist’ history where mathematics does not really begin until the Greeks and is seen as a subject inevitably progressing to greater and better heights.
Smith, David Eugene (1860-1944) History of Mathematics. Two volumes. Boston, 1923-1925. Reviewed: Isis 6, 440-444. Reprints: Ginn, Boston, 1951-1953 Dover, New York, 1958.
This contains a good collection of original resources many of which cannot be found in later books. His two volume Sourcebook in Mathematics is also available in reprint.
David Eugene Smith was primarily responsible for introducing the history of mathematics as a course for teachers in the Teachers College at Columbia University in the early part of the 20th century.
2. SOURCEBOOKS These are collections of original sources in translation, for an English - speaking audience. Sometimes fairly specialist in their collections of papers and extracts from books, and moat can be found in libraries.
Fauvell, J. and Gray, J. (1987) The History of Mathematics: A Reader.
This is the Open University book for the second History of Mathematics course taught there. A huge variety of readings from Aristotle to Computers. Probably still the best and most relevant for teachers.
Stedall, Jacqueline. (2008) Mathematics Emerging: A Sourcebook (1540-1900)
This book is the outcome of the History of Mathematics undergraduate course taught at Oxford University. O. U. P.
In spite of the title, it does cover some essential material from early mathematics, the beginnings of geometry, the theory of numbers and early algebra and then concentrates on improvements in notation, analytic geometry and early calculus. Its later sections concentrate on the development of Power series, Functions, Convergence and Completeness and the emergence of Structural and Linear Algebra, and Complex Analysis. It has notes on People, Institutions and Journals and a good Bibliography.
Struik, D. J. (1969) A Source Book in Mathematics 1200-1800.
Cambridge, Mass. Harvard University Press
In its time, a classic containing much material not widely seen before.
Amazon Paperback (used & new) £18.00
Dunham, W. (2005) The Calculus Gallery: Masterpieces from Newton to Lesbegue.
Princeton and Oxford. Princeton University Press
This book is different in the sense that each of the extracts is commented on and the language and notation is often adapted for the modern reader. With that warning, it is quite useful in introducing some of the important ideas in the development of the calculus and making them more accessible.
Amazon Paperback £ 11.35
3. SOME SELECTED TOPICS Menninger, K. (1992)
Number Words and Number Symbols: A Cultural History of Numbers
Boston MIT Press
Comprehensive and reliable, originally translated from the German and published by MIT press in 1970 as a Hardback this popular book has been reprinted many times. Now in Paperback can be found at Amazon for about $11.50 - a bargain!
Ifrah, G. (1999)
The Universal History of Numbers: From Prehistory to the Invention of the Computer
London Harvill Press
Translation from the original French edition of 1994
Two volumes bound as one.
Paperback 600+ pages. $15.60 from Amazon
The following are really good ‘popular’ history books on specific topics. They are good, well written, and well researched.
A History of π
St Martin’s Press N.Y.
Maor, E. (1998)
Princeton University Press
Maor, E. (2007)
The Pythagorean Theorem: A 4,000-year History (Hardcover)
Princeton University Press
Maor, E. (1994) "e", The Story of a Number (Paperback)
Princeton University Press
Maor, E. (1987) Paperback
To Infinity and Beyond: A Cultural History of the Infinite
Princeton University Press
Beckmann, P. A History of Pi (Paperback)
These five books (by Eli Maor, who teaches history of mathematics in Chicago, and and Beckmann) are really worth by having in the school/college library. They are a mine of information, generally historically accurate and full of ideas and examples that can be used in the classroom at different levels.
Singh, S. Fermat’s Last Theorem
The latest edition describes the solution by Andrew Wiles
Derbyshire, J. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics.
A very good popular book giving an exposition of a significant and important problem in mathematics.
A good popular exposition. Take what you want/need out of this book, but some parts ar hard work if the subject matter is not familiar.
‘The Story of Algebra.’ Solution of Equations, Discovery of Imaginary Numbers, Vecor Spaces, Galois, Groups and Rings, Algebraic Geometry, Abstract Algebra.
4. MATHEMATICAL CULTURES Zaslavsky, Claudia (1999)
Africa Counts: Number and Pattern in African Cultures
Amazon Paperback $11.53
Originally published in the 1970s this was the first book to try to raise awareness of the contributions of non-Europeans to traditional views of mathematical culture.
Ascher, Marcia. (2004)
Mathematics Elsewhere: An Exploration of Ideas Across Cultures
Woodstock, Oxfordshire Princeton University Press
First edition 2002 Paperback can be found at Amazon for $ 19.95
A pioneer in bringing our attention to mathematics in other cultures.
Joseph, G., G. (1991)
The Crest of the Peacock: The Non-European Roots of Mathematics
London First published by I.B. Tauris
The original hardback edition immediately became popular and was then published by Penguin a year later in 1992. Joseph forcefully makes the point that many accounts of the history of mathematics have been dominated by a ‘Euro-Centric’ attitude and he shows how much of what we claim as ‘European’ actually came from other much older civilizations.
Katz, V. (2007) The Mathematics of Egypt Mesopotamia, China, India and Islam.
Princeton University Press
Four internationally renowned authors write on their own subjects of expertise.
This is the real background for the ‘elementary’ mathematics taught in school:
Arithmetic, practical geometry, proto-algebra and trigonometry. Amazon about £40.
Princeton. Princeton University Press
Highly Recommended for your school library.
5. BIOGRAPHIES Bell, E. T. (1883-1960). Men of mathematics. New York, Simon and Schuster  Eric Temple Bell was an English mathematician and this book became very popular. Useful as a quick read, but not very reliable. There is a Dover edition still available.
Osen, Lynn M. Women in mathematics. M.I.T. Press, Cambridge, MA, 1974.
Amazon various prices from £ 3.07 (Paperback used) to new £12.55
Grinstein, Louise and Campbell, P.J. (1987)
Women of Mathematics: a Bibliographic Sourcebook.
A good resource, but out of print at the moment.
6. BOOKS FOR TEACHERS Baumgart, J. K. (1989)(Ed.) Historical Topics for the Mathematics Classroom
31st Yearbook of the (American) National Council of Teachers of Mathematics
This is a revised edition of a book which first appeared in 1969.
Highly Recommended. Amazon price is $ 42.70
Contains a number of brief survey-histories of different areas: e.g. arithmetic and calculation, geometry, algebra; and many ‘capsules’ on typical topics found in the curriculum which give basic information and references for further investigation. Answers questions like “why do we call them sin, cos and tan? why 360 degrees in a circle? or who invented matrices?” A real mine of information for teachers of pupils of all ages. Amazon sell ‘used’ copies which are often as good as new.
There is an earlier edition (1969) still available from Amazon.
Bunt, L. N. H., Jones, P. H. and Bedient J. D. (1988)
The Historical Roots of Elementary Mathematics
Originally published as a Hardback in 1976, and now available as a Dover Paperback
Egyptian, Babylonian and Greek mathematics with a chapter on the later development of arithmetic. This deals with most of the arithmetic, early algebra and geometry found in school curricula. This book is aimed at teachers and has a number of sections devoted to the pedagogical and didactic discussion of aspects of mathematics and its development.
Amazon $ 14.95 Very useful. A good resource book.
Katz Victor J. (2000)
Using History to Teach Mathematics: An International Perspective (New Mathematical Library) (Paperback)
Victor Katz is the author and editor of a number of books on using history of mathematics in the classroom, and the principal organizer of the ‘Convergence - Loci’ website of the MAA mentioned in Part 1 of these notes.
Amazon £ 26.59
Amy Shell-Gellasch (2007)
Hands on History: A Resource for Teaching Mathematics
Mathematical Association of America (MAA)
Amazon $53.95 (out of stock at present)
Berlinghoff William P. and Gouvea F.Q.
Math Through the Ages: A Gentle History for Teachers and Others.
Expanded Edition with Classroom Resource Materials (Hardcover)
7. POPULAR BOOKS WITH SOME HISTORICAL CONTENT These are fairly typical, and can all be found on Amazon, often quite cheaply. Enter the title of one of these books and you will find quite a collection that you can inspect. In most cases you can ‘look inside’ to see the contents list and get a feel of the book from the introduction. They are intended for ‘the intelligent reader’, and more and more are coming on to the market. They contain a variable amount of historical material which these days is fairly reliable, and ‘popularise’ the mathematics involved. The way the stories are told can be quite useful for introducing anecdotes about particular mathematicians or topics for lessons.
Nahin, P. An Imaginary Tale, the story of “i” (The Square Root of Minus One)
Amazon £15.80 (16+ and A Level)
Seife, C. Zero: The Biography of a Dangerous Idea (Paperback)
Kaplan R. and Kaplan, E.The Nothing That Is: A Natural History of Zero (Paperback)
Clegg, B. A Brief History of Infinity: The Quest to Think the Unthinkable (Paperback)
Mazur, B. Imagining Numbers: (Particularly the Square Root of Minus Fifteen) (Paperback)
Kaplan, R. and Kaplan, Ellen The Art of the Infinite: Our Lost Language of Numbers (Paperback)
8. WEBSITES The “Mactutor” website is the best available. Still growing, and being updated and corrected, it is the most comprehensive easily navigable website for the History of mathematics.
The latest update contains new biographies and many new entries of “additional material”
http://www-history.mcs.st-and.ac.uk/history/ “Convergence” is the official History of Mathematics website of the Mathematical Association of America (MAA)
You will get a window with the four sites of the Math Digital Library.
LOCI - Convergence is one of these - it contains historical material but also accounts of using history in the classroom and many other ideas.
The British Society for the History of Mathematics has its website at:
The British Society for the History of Mathematics exists to promote research into the history of mathematics and its use at all levels of mathematics education.
Just started and being constructed is the BSHM site for education which has its own address, but is a link from the main BSHM site:
http://www.bshmeducation.org/education.html All these websites provide recommended links to other sites which have been monitored for accuracy, consistency and historical value.
Biographies of Women Mathematicians can be found at:
http://www.agnesscott.edu/lriddle/women/women.htm For Mathematicians of the 17th and 18th Centuries go to:
http://www.maths.tcd.ie/pub/HistMath/People/RBallHist.html The History of Mathematical Words, Symbols and Notation
Here there are links to images of mathematicians on postage stamps and words that are ambiguously defined in school books.