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Spicing Up Your Math Classes With History. V. Frederick Rickey West Point.

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Spicing up your math classes with history

Spicing Up Your Math Classes With History

V. Frederick Rickey

West Point

Spicing up your math classes with history

  • One day a monk leaves at sunrise to climb a mountain. He walks at a leisurely pace, sometimes stopping to enjoy the view, even retracing his path to look again at a pretty flower. He arrives at the summit at sundown, spends the night meditating, and starts home down the same path the next day at sunrise, arriving home at sunset.

  • Was there a time of day when he was exactly at the same point on the trail on the two days?

Emmy noether born march 23 1882

Emmy Noether born March 23, 1882

Someone once described her as the daughter of the mathematician Max Noether.

To this Edmund Landau replied ``Max Noether was the father of Emmy Noether. Emmy is the origin of coordinates in the Noether family.''

Thoralf skolem died 23 march 1963

Thoralf Skolem died, 23 March 1963

This Norwegian logician was the first to introduce non-standard models of the natural numbers.

Spicing up your math classes with history

Gottfried Wilhelm von Leibniz(1646 – 1716)and the product rule

The product rule 11 november 1675

The Product Rule: 11 November 1675

  • Leibniz introduces the dx notation and asks “Is dx . dy = d(x . y)?”

  • He lets x = cz + d and y = z2 + bz and computes dx.dy = c (2z +b).

  • “But you get the same thing if you work out d(x.y) in a straightforward manner.”

  • Later that day: Whoops!

The product rule 21 november 1675

The Product Rule: 21 November 1675

  • dx . y = d (x . y) – x . dy.

  • “Now this is a really noteworthy theorem.”*

  • It took Leibniz 10 days to correct his error, and, my dear students, if you don’t master this in 10 days, you will not do well on the next exam.

  • From the zeroeth Putnam Exam (1933): When is dx . dy = d(x . y)?”

* The Early Mathematical Manuscripts of Leibniz, ed. by J. M. Child (1920/2005), p. 107

Spicing up your math classes with history

Bonaparte rightly said that many of the decisions faced by the commander-in-chief resemble mathematical problems worthy of the gifts of a Newton or an Euler.

  • Carl von Clausewitz, On War, 1832

Spicing up your math classes with history

I must study politics and war that my sons may have liberty to study mathematics and philosophy. My sons ought to study mathematics and philosophy, geography, natural history, naval architecture, navigation, commerce, and agriculture, in order to give their children a right to study painting, poetry, music, architecture, statuary, tapestry, and porcelain.

  • John Adams

Alan turing

Alan Turing

  • I.J. Good, a wartime colleague and friend, has aptly remarked that it is fortunate that the authorities did not know during the war that Turing was a homosexual; otherwise, the Allies might have lost the war.

  • Peter Hilton, from "Cryptanalysis in World War II -- and Mathematics Education,“ Mathematics Teacher, Oct. 1984.

Spicing up your math classes with history

  • Cartoons provide an opportunity to speak of many things:

  • Projectile motion

  • History of Ballistics

  • Ethics

Bernard bolzano and the intermediate value theorem

Bernard Bolzano and the Intermediate Value Theorem

Bernhard bolzano 1781 1848

Bernhard Bolzano (1781-1848)

  • Son of an art dealer who founded an orphanage.

  • Studied philosophy, physics and mathematics at Prague.

  • Ordained a Catholic Priest in 1804.

Spicing up your math classes with history

  • Appointed to the new chair of philosophy of religion at the University of Prague

  • An unsuitable position given his unorthodox religious and political ideas.

  • Dismissed in 1819 and placed under house arrest.

Spicing up your math classes with history

Purely analytic proof of the theorem that between any two values which give results of opposite sign there lies at least one real root of the equation

Abraham gotthelf k stner 1719 1800

Abraham Gotthelf Kästner (1719-1800)

  • Proved statements which were commonly regarded as evident in order to make clear the assumptions on which they are based.

  • Influenced both Bolzano and Gauss.

Spicing up your math classes with history

  • Early work on infinite sets!

  • Defined continuity.

  • Constructed a nowhere differentiable function.

Spicing up your math classes with history

  • Assumptions?

Spicing up your math classes with history

Is there a direction I can point such that the temperature at the boundary of the State is the same in that direction and in the opposite direction?

Spicing up your math classes with history

  • T(Θ) = Temperature at border of state when you are in Columbus and looking in direction Θ.

  • Consider

    F(Θ) = T (Θ) – T(Θ + π)

  • F(0) and F(π) have opposite signs.

Spicing up your math classes with history


F(Θ) = T (Θ) – T(Θ + π)

Does it matter whether you are in Albany or West Point or Selden?

Our hero

Our Hero !

  • Bernhard Bolzano proved the Intermediate Value Theorem in 1817.

Spicing up your math classes with history

  • One can invent mathematics without knowing much of its history. One can use mathematics without knowing much, if any, of its history. But one cannot have a mature appreciation of mathematics without a substantial knowledge of its history.

  • Abe Shenitzer quoted in "Thinking the Unthinkable: The Story of Complex Numbers (with a Moral)," by Israel Kleiner, Mathematics Teacher, Oct. 1988.

Archimedes died 212 bce

Archimedes (died 212 BCE)

The archimedes palimpsest

The Archimedes Palimpsest

The repaired text

The Repaired Text

  • Do you see any mathematics?

The stomachion

The Stomachion

Although euler is pronounced oil er it does not follow that euclid is pronounced oi clid

Although “Euler” is pronounced “Oil-er”, it does not follow that “Euclid” is pronounced “Oi-clid.”

Euler about 1737 age 30

Euler about 1737, age 30

  • Painting by J. Brucker

  • 1737 mezzotint by Sokolov

  • Black below and above right eye

  • Fluid around eye is infected

  • “Eye will shrink and become a raisin”

  • Ask your ophthalmologist

  • Thanks to Florence Fasanelli

Euler creates trig functions in 1739

Euler creates trig functions in 1739

Spicing up your math classes with history

Often I have considered the fact that most of the difficulties which block the progress of students trying to learn analysis stem from this: that although they understand little of ordinary algebra, still they attempt this more subtle art. From the preface of the Introductio

Chapter 1 functions

Chapter 1: Functions

A change of Ontology:

Study functions

not curves

Viii trig functions

VIII. Trig Functions

Spicing up your math classes with history

He showed a new algorithm which he found for circular quantities, for which its introduction provided for an entire revolution in the science of calculations, and after having found the utility in the calculus of sine, for which he is truly the author . . .

Eulogy by Nicolas Fuss, 1783

Spicing up your math classes with history

  • Sinus totus = 1

  • π is “clearly” irrational

  • Value of π from de Lagny

  • Note error in 113th decimal place

  • “scribam π”

  • W. W. Rouse Ball discovered (1894) the use of π in Wm Jones 1706.

  • Arcs not angles

  • Notation: sin. A. z

Institutionum calculi integralis 1769

Institutionum calculi integralis, 1769


Read euler read euler he is our teacher in everything

Read Euler, read Euler, he is our teacher in everything.


as quoted by Libri, 1846



How might one use history in the classroom? A variety of examples will be provided:  If the topic of the day is the product rule, then we ought to let students know what Leibniz can teach them on this topic. If the Intermediate Value Theorem is the topic, don't fail to mention Bolzano. A good way to start a class is to talk about what happened on this date. How about using quotations to make a point about mathematics. Which famous mathematicians were born on this date? The latest Smithsonian magazine has an article about the Archie, so lets mention that famous palimpsest. Finally since this year is the 300th anniversary of Euler's birth, let's talk about his work on trigonometry.

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