John Napier foiled a thief with the aid of logic and a black rooster. For this and other acts of creative problem solving, his servants and neighbors suspected him of witchcraft.

What does this have to do with mathematics?

Math was Napier’s favorite hobby. He invented logarithms to help people handle large numbers easily, and he even created a calculator out of a chessboard. [See how it works: addition, subtraction, multiplication.]

More stories from math history

Isaac Newton caused a UFO scare by flying a kite that carried a lantern. As a boy, Newton was a poor student who became a scholar only to show up the class bully.

Maria Agnesi solved math problems while sleepwalking. When she got stumped, she left the problem on her desk and went to bed. The next morning, she found the correct solution neatly written on her paper.

After teaching calculus to her younger brothers, Agnesi wrote what became Europe’s most popular calculus textbook for the next 50 years. But there was something she loved more than math --- she longed to become a nun, and she devoted much of her life to helping the poor and homeless.

Sofia (Sonya) Kovalevskaya became intrigued with math from reading her bedroom wall, papered with old calculus lecture notes. To escape Russia, where women were not allowed to study mathematics, she arranged a marriage of convenience. When her husband died, she struggled on as a single mother. Kovalevskaya won a prestigious prize for original mathematical research --- and her paper was so brilliant that the judges increased the prize to nearly double what they usually awarded.

The key to understanding

The story of mathematics is the story of interesting people. They faced the normal challenges of daily life as well as the creative challenges of mathematical imagination. For some, calculation and problem solving seemed as natural as breathing. Others worked for years in fits and starts before reaching a solution. Some had long and happy lives. Others died tragically young.

What a shame it is that our children see only the dry remains of these people’s passion. Worksheet exercises are the bare, abstract skeletons of what were once living puzzles.

As Victorian-era math professor James Glaisher said, “I am sure that no subject loses more than mathematics by any attempt to dissociate it from its history.”

Math and history --- what can they possibly have in common?

After all, history is all about kings and wars, while math is numbers and rules. Isn’t it?

“Biographical history, as taught in our public schools, is still largely a history of boneheads: ridiculous kings and queens, paranoid political leaders, compulsive voyagers, ignorant generals --- the flotsam and jetsam of historical currents,” according to popular math writer Martin Gardner. “The men who radically altered history, the great scientists and mathematicians, are seldom mentioned, if at all.”

It does not have to be that way for our children. By teaching math history, we can help our students build a mental picture of the ebb and flow of ideas through the centuries. They will see how men and women wrestled with concepts, made mistakes, argued with each other, and gradually developed the knowledge that today we take for granted.

“I will not go so far as to say that to construct a history of thought without profound study of the mathematical ideas of successive epochs is like omitting Hamlet from the play which is named after him. That would be claiming too much,” wrote Alfred North Whitehead, a pioneer of mathematical philosophy. “But it is certainly analogous to cutting out the part of Ophelia. This simile is singularly exact. For Ophelia is quite essential to the play, she is very charming… and a little mad.”

A “living books” approach

Most homeschool teachers, whatever our curriculum or schooling approach, understand the importance of teaching with “real” books. We read aloud biographies, historical fiction, or the classics of literature. We scour library shelves for the most creative presentations of scientific topics that interest our children. We encourage our high school students to go back to the original documents whenever possible.

And we teach math with a textbook.

One reason for this imbalance is that most of us never learned math history ourselves. We may not even be aware that math has a history. Our teachers made it seem like something handed down from on high, to be accepted and memorized --- and never to be challenged.

Fortunately, when we decide to embark on a tour of math history, we won’t have to go it alone. Several talented and knowledgeable guides are available. Some of them may be sitting on the shelf at your local library right now, just waiting to lead you along the way…

Math history in pictures

You can begin exploring the excitement of mathematics with your children through picture books. What's Your Angle, Pythagoras? offers a fanciful look at the childhood of that famous mathematician. The Librarian Who Measured the Earth tells how Eratosthenes calculated the circumference of the earth using sunlight and shadows. Dear Benjamin Banneker skips forward in history to examine the life of a self-taught African-American astronomer and mathematician.

For beginning readers: A Fly on the Ceiling will make children laugh while they learn about René Descartes, the father of analytic geometry. Ben Franklin and the Magic Squares tells a lively story about one of old Ben's favorite pastimes.

Treat your older children (and yourself) to a couple of our family favorites. Archimedes and the Door of Science describes the life and discoveries of one of the greatest mathematicians who ever lived. Carry On, Mr. Bowditch follows the inspiring life of an 18th-century American hero whose mathematical studies saved the lives of countless sailors.

Edited to add: The Wonderful World of Mathematics will give your children a great overview of math in many cultures from Ancient Egypt to the Industrial Revolution. It is out of print, but used copies are available, or you may be able to get it through your library. [Thanks to Brian Foley for reminding me of this book in the comments below.]

Meet the men (and women) who made math history

Mathematicians Are People, Too [and Volume 2] features short, fictionalized vignettes for teachers to read aloud to elementary or middle school students. The authors have also written a three-volume reference series called Historical Connections in Mathematics: Resources for Using History of Mathematics in the Classroom [Volumes II, III], which offers basic facts and anecdotes about each mathematician, without fictional elaboration, and includes related worksheets.

Older students (and adults) will enjoy Famous Problems and Their Mathematicians. Another combination of anecdotes and activities, the book touches on many ideas that have challenged mathematicians for centuries.

If you want more complete biographies, start with Of Men and Numbers: The Story of the Great Mathematicians. For junior high or older students, try Men of Mathematics, a classic of math history that has been called over-romanticized, but I have not heard of anyone who didn’t enjoy it.

For balance, you may want to sample Women in Mathematics or Math Equals.

And those are just a few of the books available. Indeed, there is a lot more to the history of mathematics than most people ever suspect.

Edited to add: The World of Mathematics looks like a wonderful resource, and I can't wait to get it from my library. Editor James Newman has collected four volumes of articles by eminent mathematicians and other thinkers "from A'hmose the Scribe to Albert Einstein, presented with commentaries and notes." [Thanks to G Johnson for recommending this book in the comments below.]

And here are some of my all-time favorites

William Dunham writes for the general adult audience, but high school students will find his books pleasant reading. The Mathematical Universe offers an A-to-Z smorgasbord of math topics and people. Journey Through Genius follows the development of several discoveries by the great masters of mathematics.

Also for older readers, Keith Devlin provides insight into historical and modern math for a general audience in Mathematics: The Science of Patterns. This book covers a wide range of topics and gives readers an idea of what modern mathematicians do for a living.

Coming soon to "Let's Play Math!" blog

The Internet contains such a wealth of math history resources that they require a post of their own. I plan to work on that for next week [Now posted here.] --- and will, I think, be adding a math history section to my resources page as well.

Finally, my math newsletter always used to include historical tidbits and quotations. I have been republishing the Alexandria Jones stories, but I've fallen behind in the history department. Over the summer, I hope to catch up on my backlog.

:: :: :: :: :: :: :: :: :: :: ::

• Historical Tidbits: Alexandria Jones
• Egyptian Geometry and Other Challenges
• An Ancient Mathematical Crisis
• A Very Short History of Mathematics
• Rewriting the History of Math

First - the Science Blogs Book Club! They're chatting about Microcosm and plan to feature more in the future, with their fellow science bloggers chipping in with opinions. Great idea!

Secondly, if you noticed my earlier blog entry about the World Science Festival in New York City, then you really should be checking out the accounts of Nobel Intent, the Ars Technica's Science-Centric Journal. They've been doing a running series on the features of the festival, including:

World Science Fest aims young

Lawrence Krauss takes on the universe

World Science Fest: science gets you cool jobs

"... a number of the programs at the World Science Festival were targeted at younger audiences. Yesterday's program was no exception, as people doing interesting science-related jobs gave a approachable descriptions of how science got them work that pretty much everyone would recognize as cool."

One such presenter was Laurie Santos of Yale who works with lemurs, capuchins, and macaques - and in order to figure out how they think, she does magic tricks! There's a Bloggingheads.TV interview with her here.

In addition to this, NY Times Op-Ed Contributor Brian Greene (apparently he spoke several times at the festival, introducing lectures and conversations among scientists and other cultural denizens) has written a great essay:

The reason science really matters runs deeper still. Science is a way of life. Science is a perspective. Science is the process that takes us from confusion to understanding in a manner that’s precise, predictive and reliable — a transformation, for those lucky enough to experience it, that is empowering and emotional. To be able to think through and grasp explanations — for everything from why the sky is blue to how life formed on earth — not because they are declared dogma but rather because they reveal patterns confirmed by experiment and observation, is one of the most precious of human experiences.

Speaking of experiences, this was sent to me by R: What Dictionaries and Optical Illusions Say About Our Brains, from Science America:

"Cognitive scientist Mark Changizi does not bother with how the brain accomplishes a task, but rather why it performs the function in the first place... The prolific Changizi recently published two papers: one that sets out to explain how our lexical systems evolved and another that suggests how the brain's visual system is adapted to anticipate the future a fraction of a second before we actually see it. (See related slideshow here."

And remember how I started looking at the link between science and juggling a while back? How about the science of fun? A great little overview on what Martin Gardner is up to now and the event named after him - Gathering For Gardner - or G4G, which brings together experts from the world of maths, magic and puzzles:

Gardner became interested in maths through "mathematical" magic tricks - and magicians, not mathematicians, formed his main social circle as a young adult. He liked magic, he said, because it gave rise to a sense of wonder about the world. "You see a woman levitated and that reminds you that it is just as miraculous that she falls to the ground by gravity... you don't realise that gravity is just as mysterious as a woman levitating." Does maths give Gardner that same wonder? "Absolutely, yes."

There's a few accounts from this years conference too - an Australian!

Inside Rodgers' house, where hundreds of items from his puzzle collection were on display, Colin Wright, an Australian who lives in the Wirral, was holding court. With his schoolboyish, ginger hair and glasses, he looks just how you might expect a mathematician to look - in fact, he is a juggler, too. "It seemed like the obvious thing to do after I learned to ride a unicycle," he said.

He has helped develop a mathematical notation for juggling, which has electrified the international juggling community. It turns out that, with a language, jugglers have been able to discover tricks that had eluded them for thousands of years. "Once you have a language to talk about a problem, it aids your thought process," Wright said as he took out some bean balls to demonstrate a recently invented three-ball juggle. "Maths is not sums, calculations and formulae. It is pulling things apart to understand how things work."

Oh, if you're wondering about the gender ratio...

At university level and above, maths is a very male affair, although at GCSE girls now outperform boys. At the G4G, fewer than 20% of the participants were women. Some of them presented talks in which they applied high level maths to crochet, knitting, needlework and quilting. It turns out that "mathematics and the fibre arts" can actually convey deep mathematical ideas in a novel way - such as what a hyperbolic space might look like, which is something that has baffled mathematicians for centuries. Carolyn Yackel, one of the genre's pioneers, gave a talk on how to knit a pair of hyperbolic trousers. (You knit an octagon in hyperbolic space and then join the sides together.)

Para empezar vamos a sumar la siguiente serie: 1+ 1/2 + 1/4 + ... + 1/2^n-1... ¡Pero qué haces! Intentar sumar la serie término a término sí que es una tarea sobrehumana, no, hagámoslo de otra manera, con un dibujo, (¿con un dibujo?), sí, con un dibujo, no me interrumpas:

Si el área del cuadrado total es de 1, es intuitivo ver que 1/2 + 1/4 + ... es igual a 1, que sumado con el primer uno nos da un total de dos. Aunque esta no es una demostración muy contundente, pero bueno, para andar por casa sirve.

Llama la atención que la suma de términos infinitos dé algo finito, algo tan insignificante como 2, pero así es. Estas cosas son las que hacen tan bella a las matemáticas, y ésta está repleta de ellas... Pero no nos desviemos.

Bueno, ahora sí, vayamos con un ejemplo de tarea sobrehumana. Al filósofo James F. Thompson se le debió encender una lucecita en la cabeza cuando se le ocurrió la paradoja de la lámpara, una máquina infinita que es encendida durante el primer minuto, apagada durante medio minuto, encendida otra vez durante un cuarto de minuto, apagada nuevamente durante... y así infinitas veces. Esta serie de encendidos y apagados dura en total dos minutos, como hemos deducido más arriba... Creo que es evidente por qué a esta tarea se la llama sobrehumana: se trata de realizar infinitas acciones en un tiempo finito.

La pregunta es: ¿estará la lámpara encendida o apagada cuando hayan transcurrido los dos minutos? Si asociamos a cada numero natural, por orden, con cada uno de los encendidos y apagados, tenemos que en cada pulsación de turno impar la lámpara será encendida y en las de turno par apagada... Pero sabemos que no hay ningún número natural que sea el último. Por lo tanto, ¡no hay manera de saber si la lámpara estará encendida o apagada transcurrido el tiempo!

Esta es una paradoja que aún no está resuelta; no hay acuerdo. Puedes entrar en más detalles en el libro ¡Ajá! Paradojas que hacen pensar de Martin Gardner o en El tamiz.

Se han hecho otras versiones de la paradoja, por ejemplo el filósofo Max Black ha presentado esa paradoja imaginando una máquina del infinito que transfiere una bola de la bandeja A a la B en un minuto, durante el medio minuto siguiente, deposita nuevamente la bolita en A...

Many students and undergraduates seem to think of mathematicians as old, white, middle-class men who are obsessed with their subject, lack social skills and have no personal life outside maths. The student’s views of maths itself included narrow and inaccurate images that are often limited to numbers and basic arithmetic.

From my university years, I can attest to the existence of mathematicians with no social skills. From undergraduates who spent their entire degree in the library to professors who, terrified of communication with students, delivered their lectures on topology or analysis entirely to the blackboard, it is true that extreme examples exist. Films such as "A Beautiful Mind" and mathematicians such as Grigori Perelman do little to dispel this image.

But of course well-adjusted mathematicians, who play frisbee or drink beer or dabble in stand-up comedy, also exist in large numbers. In the mathematics department I had friends, lecturers and professors who did not conform at all to this stereotype. And it can't be the case that all senior school maths teachers are crotchety old bags.

I would suggest that the image problem runs deeper - that maths is unpopular because the subject itself is seen as dull. And no wonder, when you consider just what is involved in compulsory school mathematics. Times tables and long division, with the best will in the world, will never be exciting. They are necessary skills, in the same way that reading and writing are. But while it's hard to find anyone who will admit to illiteracy, many wear their "no head for figures" with pride.

For me, maths was a school subject I found easy, but never interesting. Until one day someone handed me a book by Martin Gardner. I found it hard to believe that this collection of logic puzzles, board games and instructions for intricate paper models bore any relation to the sheets of long multiplication I did every day in school. Of course, at the age of seven or so, an awful lot of the detail was lost on me. But in subsequent years, I revisited that book and others like it. My interest in final-year university topology can be directly traced to the picture of a Mobius strip on the cover of that book.

And that's what's missing in school mathematics. Yes, there are people for whom arithmetic will forever present a problem. But not nearly as many as claim that to be the case. Perhaps a little more time spent on the many, varied and interesting aspects of mathematics (without the technical detail, of course) in the years leading up to GCSE would persuade more people to continue their studies of mathematics past the age of sixteen. The persistent narrowing of the curriculum certainly isn't working.

Learning to read is boring. But with basic skills, a person can work their way up to Shakespeare. Learning arithmetic is boring. But mathematics isn't.

Para el que no lo sepa, la fórmula mencionada es la siguiente:

1²+ 2² + 3² + … + n² = (2n + 1)n(n+1)/6

Y a continuación la demostración (desde luego, lo que se muestra es el caso particular para n = 4, pero el lector se dará cuenta fácilmente de que el proceso es válido para cualquier n):

Si sumamos tres veces 1²+ 2² + 3² + … + n², obtenemos un rectángulo de base 2n + 1 y altura 1 + 2 + 3 + … + n (¿por qué?):

Por lo tanto, usando la fórmula para la suma de los n primeros números naturales obtenemos que la suma que buscamos es:

1²+ 2² + 3² + … + n² = 1/3 (2n +1) n (n + 1).1/2, que evidentemente es el mismo resultado que el de arriba.

1) CASÁS FERREÑO, N. (2007): Deducción por inducción, Suma, 55, 55-60.

• 1.- La semana pasada conseguí apagar la luz de mi dormitorio y meterme en la cama antes de que la habitación quedase a oscuras. Hay tres metros desde la cama al interruptor de la luz. ¿Cómo pude apañármelas?

• 2.- Siempre que mi tía viene a visitarme a mi estudio tiene que bajar del ascensor cinco plantas antes, y subir andando por la escalera hasta mi piso. ¿Puedes explicar por qué?

• 3.- Una noche, aunque mi tío estaba leyendo un libro apasionante, su mujer le apagó la luz. La sala estaba oscura como el carbón, pero mi tío siguió leyendo sin inmutarse. ¿Cómo es posible?

• 4.- Esta mañana se me cayó un pendiente en el café. Y, aunque la taza estaba llena, el pendiente no se mojó. ¿Y eso?

• 5.- A mi padre, que iba sin paraguas ni sombrero, lo pilló ayer un chaparrón. La ropa se le empapó, pero pese a llevar la cabeza descubierta, no se mojó ni un pelo. ¿Cómo lo explicas?

• 6.- En una línea de ferrocarril, el tendido tiene doble vía excepto en un túnel, que no es lo bastante ancho para acomodar ambas. Por ello, en el túnel la línea es de vía simple. Una tarde entró un tren en el túnel marchando en un sentido, y otro tren entró en el mismo túnel, pero en sentido contrario. Ambos iban a toda velocidad; sin embargo no llegaron a colisionar. ¿Cómo es posible?

• 7.- Un preso fugado iba caminando por una carretera comarcal cuando vio acercarse velozmente un auto de la policía. Aunque la intención del fugado era huir hacia el bosque, echó a correr 10 metros en dirección al vehículo que se acercaba. ¿Hizo esto para mostrar su desdén por las fuerzas del orden, o pudo tener otra razón más poderosa?

Il seguente grafico (tratto da Wikipedia) mostra l'andamento della curva delle probabilità all'aumentare delle persone.

Si vede come oltre le 60 persone la probabilità è praticamente del 100% (questo vuol dire che, ad esempio, in una tipica aula universitaria si ha praticamente la certezza che almeno due persone hanno lo stesso compleanno).
Un evidente esempio del paradosso è dato dalle date di nascita e di morte dei presidenti degli Stati Uniti. Esaminando le date (43 date di nascita e 39 date di morte) si vede che Polk e Harding nacquero il 2 novembre mentre Carter e Heisenhower il 14 ottobre; Truman e Ford morirono il 26 dicembre, Polk e Buchanan il 15 giugno e ben tre presidenti, Jefferson, Adams e Monroe, morirono il 4 luglio.

Sounds like Fermat's Last Theorem, and became dramatically popular in an extremely short span. While I do not agree with the Professor's philosophy about life, success etc; it still is a masterpiece delivery at such a timely moment. Whats intriguing about the lecture (and the persona of the orator) are the simple mindedness, honesty, and the self involved. We need drama like these- and even if we agree or not- we need the appraisal of such drama. Bored of the set examples and repetitive preachings- look for it. [googlevideo=http://video.google.com/videoplay?docid=-5700431505846055184&q=randy+pausch&total=38&start=0&num=10&so=0&type=search&plindex=0]

Martin Gardner & Friends
I am a big fan of Martin Gardner, and I think that being his fan is my biggest achievement so far. (People like Persi Diaconis attributed their life's success to him). Take a look & feel the thunder. Click here, QuickTime player needed.

Also see (mind you this is by a Stanford Mathematician)[googlevideo=http://video.google.com/videoplay?docid=6283160772183401751&q=persi+diaconis&total=2&start=0&num=10&so=0&type=search&plindex=0]
Life after Poincare: Grigori Perelman
Everybody knows who is Perelman, but nobody knows whats he doing now. Check 'em out (part 1 of n)

G103

A (surreal) day in the life of an undergraduate on the 4-year M.Math degree at the University of Warwick.[googlevideo=http://video.google.co.uk/videoplay?docid=-4035423643441275899]

Zeitgeist

I like such stuff. Buck up.[googlevideo=http://video.google.com/videoplay?docid=-594683847743189197&hl=en]

A series of video lectures on cutting edge science. User friendly. Worth a look.

For user friendly lectures in cutting edge Mathematics, see here.

For user friendly lectures in cutting edge Physics, see here.

What teacher hasn't heard a student complain, "When am I ever going to have to use this?" Didn't most of us ask it ourselves, once upon a time? And unless we choose a math-intensive career like engineering, the truth is that after we leave school, most of us will never again use most of the math we learned. But if math beyond arithmetic isn't all that useful, then what's the point?

If you or your student is singing the Higher Math Blues, here are some quotations that may cheer you up --- or at least give you the strength of vision to keep on slogging.

We study mathematics...

To understand Creation

I don't want to convince you that mathematics is useful. It is, but utility is not the only criterion for value to humanity. Above all, I want to convince you that mathematics is beautiful, surprising, enjoyable, and interesting. In fact, mathematics is the closest that we humans get to true magic. How else to describe the patterns in our heads that --- by some mysterious agency --- capture patterns of the universe around us? Mathematics connects ideas that otherwise seem totally unrelated, revealing deep similarities that subsequently show up in nature.

--- Ian Stewart
The Magical Maze

That vast book which stands forever open before our eyes, the universe, cannot be read until we have learnt the language in which it is written. It is written in mathematical language, and the letters are triangles, circles, and other geometrical figures, without which means it is humanly impossible to comprehend a single word.

--- Galileo Galilei
quoted by Clifford Pickover, A Passion for Mathematics

To train our minds

The investigation of mathematical truths accustoms the mind to method and correctness in reasoning, and is an employment peculiarly worthy of rational beings.

--- George Washington
quoted by William Dunham, The Mathematical Universe

I told myself, "Lincoln, you can never make a lawyer if you do not understand what demonstrate means." So I left my situation in Springfield, went home to my father's house, and stayed there till I could give any proposition in the six books of Euclid at sight. I then found out what "demonstrate" means, and went back to my law studies.

--- Abraham Lincoln
quoted by William Dunham, The Mathematical Universe

To understand history

In most sciences, one generation tears down what another has built, and what one has established another undoes. In mathematics alone, each generation adds a new story to the old structure.

--- Herman Henkel
quoted by Noah benShea, Great Quotes to Inspire Great Teachers

Biographical history, as taught in our public schools, is still largely a history of boneheads: ridiculous kings and queens, paranoid political leaders, compulsive voyagers, ignorant generals --- the flotsam and jetsam of historical currents. The men who radically altered history, the great scientists and mathematicians, are seldom mentioned, if at all.

--- Martin Gardner
quoted by G. Simmons, Calculus Gems

I will not go so far as to say that constructing a history of thought without profound study of mathematical ideas is like omitting Hamlet from the play named after him. But it is certainly analogous to cutting out the part of Ophelia. For Ophelia is quite essential to the play, she is very charming. . . and a little mad.

--- Alfred North Whitehead
quoted in The Viking Book of Aphorisms

To appreciate the beauty

The mathematician does not study pure mathematics because it is useful, he studies it because he delights in it, and he delights in it because it is beautiful.

--- Henri Poincaré
quoted by Theoni Pappas, More Joy of Mathematics

A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. The mathematician's patterns, like the painter's or the poet's, must be beautiful. The ideas, like the colors or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics.

--- Godfrey H. Hardy
A Mathematician's Apology

To enjoy the mental challenge

At age eleven, I began Euclid, with my brother as tutor. This was one of the great events of my life, as dazzling as first love. I had not imagined there was anything so delicious in the world.

--- Bertrand Russell
The Autobiography of Bertrand Russell

:: :: :: :: :: :: :: :: :: :: ::

• Math Quotes VI: Beauty in Mathematics
• How Can We Teach Problem Solving?
• Quotations XII: Mathematicians at Play
• Quotations XIV: The Joy of Mathematics
• Quotations XV: More Joy of Mathematics