Spicing Up Your Math Classes With History

V. Frederick Rickey

West Point

• 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

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

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

Gottfried Wilhelm von Leibniz

(1646 – 1716)

and the product rule

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!

• 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 Product Rule: 21 November 1675

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

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

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

• 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.

Cartoons provide an opportunity to speak of many things:

• Projectile motion• History of Ballistics• Ethics

Bernard Bolzano and the Intermediate Value Theorem

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.

• 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.

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 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.

• Early work on infinite sets!

• Defined continuity.

• Constructed a nowhere differentiable function.

• Assumptions?

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?

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

• ConsiderF(Θ) = T (Θ) – T(Θ + π)

• F(0) and F(π) have opposite signs.

ConsiderF(Θ) = T (Θ) – T(Θ + π)

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

• Bernhard Bolzano proved the Intermediate Value Theorem in 1817.

Our Hero !

• 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)

The Archimedes Palimpsest

The Repaired

Text

• Do you see any mathematics?

The Stomachion

Although “Euler” is pronounced “Oil-er”,

it does not follow that “Euclid” is pronounced

“Oi-clid.”

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

Solve y k4d4y

dx4 0.

Factor 1 k4p4 0 :1 k p1 kp1 k2p2The solution is :

y xk C

xk D E Cosx

k F Sinx

k

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

A change of Ontology:

Study functions

not curves

VIII. Trig Functions

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

• 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

E366

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

Laplace

as quoted by Libri, 1846

Abstract

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.