From the Mathematician's Study
to the AP Classroom:
300 Years of Learning and
Teaching Calculus
Fred Rickey
United States Military Academy
HPM, Pasadena, 24 October 2010
In the kingdom of knowledge it is
youth that inherits the land.
The calculus still holds out. Not in France,
Germany, Austria, England, Russia, Italy, Spain,
Norway, Sweden, Denmark, Holland, and
Belgium, where it has been incorporated into
the secondary school program for some years,
‒ but in the United States.
M. A. Nordgaard, 1928
The Nova methodus of 1684
The first paper on the differential calculus
Gottfried Wilhelm von Leibniz
(1646 – 1716)
Ways to Learn Calculus?
1.Teach yourself
2.Get a tutor
3.Take a class
Self Study
• Jakob and
Johann Bernoulli
• Malebranche
and his school:
Reyneau, . . .
Albrecht Durer (1471-1528)
Melancholia I, 1514
• etc.
Tutoring
• L’Hospital by
Johann Bernoulli
• Agnesi by Ramiro
Rampinelli, using
Reyneau’s Analyse
démontrée (1707)
Nicolas Neufchatel
• Etc.
Portrait of Johann Neudorfer and his Son, 1561
Lecturing
Portrait of Luca Pacioli by Jacopo de Barbari, 1495
Offered his energy to
experienced students
“especially in the new
and uncommon kind of
calculation, that is the
method of differentials
and integrals known
only to a few, by which
special inventions are
produced.”
Johann Bernoulli
First Calculus Course
1697-98
Isaac Newton
"So few went to hear
him and fewer that
understood him, that
ofttimes he did in a
manner, for want of
hearers, read to the
walls."
Benjamin Robins
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Frontispiece to Sprat's
History of the Royal Society
Studied on his own
Met Dr. Henry Pemberton
Moved to London
Studied more mathematics
Traveled to the continent
Elected FRS, age 21
Became a teacher
Authors Robins Studied
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Apollonius
Archimedes
Fermat
Huygens
DeWitt
Sluse
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Gregory
Barrow
Newton
Taylor
Cotes
From Teacher to Professor ?
• Robins hoped to be the first
mathematics professor at
Woolwich
• Planned a course on
fortifications and gunnery
• Walpole was displeasure
with Robins’ attacks
• Mr. Derham became
the first professor of
mathematics at Woolwich,
1741-1743.
Mathematics at Woolwich, 1741
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That the second Master shall teach the Science of Arithmetic, together with the
principles of Algebra and the Elements of Geometry, under the direction of the
Chief Master.
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That the chief Master shall further instruct the hearers in Trigonometry and the
Elements of the Conick Sections.
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To which he shall add the Principles of Practical Geometry and Mechanics, applied
to raising and transporting great Burthens;
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With the Knowledge of Mensuration, and Levelling, and its Application to the
bringing of water and the draining of Morasses;
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And lastly, shall teach Fortification in all its parts.
But no calculus
• Preface
– 55 pages
• Ch I: Internal ballistics
– 65 pages
• Ch2: External ballistics
– 30 pages
• Total:
150 pp
• Published 1742
Euler
1745
• Frederick the Great asks
about the best book on
gunnery
• Euler magnanimously
recognizes Robins
• Euler starts researching
Robins’s results
• Euler adds annotations
2400
English translation of Euler, 1777
From Euler’s Preface
Some are of the opinion that
fluxions are applicable only in
such subtle speculations as can
be of no practical use. . .
But what has been just now said
of artillery is sufficient to
remove this prejudice.
Mathematics at Woolwich, 1772
1. The Elements of Euclid
2. Trigonometry applied to Fortification, and the
Mensuration of Superficies and Solids
3. Conic Sections.
4. Mechanics applied to the raising and transporting
heavy bodies, together with the use of the lever
pulley, wheel, wedge and screw, &c.
5. The Laws of Motion and Resistance, Projectiles,
and Fluxions.
Now some calculus!
The Impact of Ballistics on Mathematics:
Calculus was taught in artillery schools
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Piedmont-Savoy in 1750s
Royal Artillery and Military Academy, Turin
Prussian Artillery Corps
French regimental school at Auxonne
Austrian Artillery Academy
Ecole Polytechnique, 1794
West Point, 1810, 1815, 1823 to date.
George Wentworth and David Eugene Smith
School Arithmetics: Primary Book, 1919, p. 52
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Teaching Calculus in the US
• Harvard, founded 1636
– 1719 undergraduate thesis on fluxions
• William and Mary, 1693
– Thomas Jefferson studied Emerson’s Doctrine of
Fluxions (1743), with William Small, 1760 - 1762
• Yale, 1701
– 1758
February 1815
Andrew Ellicott
The first
calculus class
In the US ?
Charles Davies
• 1823 Calculus a
regular course at WP
for yearlings.
• Lacroix, Traité
élémentaire
• Davies’ Calculus
(1834). Used 53
years
International Commission
On the Teaching of Mathematics
• ICM Rome 1908
• Reports published in the next decade,
12,000 pages!
• National Committee on Curriculum Reform,
US, 1923
– Four HS curricula: All recommended calculus
The First Colleges to teach
Freshman Calculus
• MIT
– F. S. Woods and F. H. Bailey
• University of Rochester
– A. S. Gale and C. W. Watkeys
– Both were about 1907
Nordgaard 1928
The conventional college text will never do
• Ellen Hayes, Calculus with Applications, an
Introduction to the Mathematical Treatment of
Science, 1900.
• William A. Granville and Percey Smith, Elementary
Analysis, 1910.
• Gale and Watkeys, Elementary Functions and
Applications, 1920.
• Griffin, Introduction to Mathematical Analysis, 1921.
• Mullins and Smith, Freshman Mathematics, 1927
• William F. Osgood, Elementary Calculus, 1921
Nordgaard 1928
Changes in Teaching Math Since 1910
‒ NCTM Yearbook #3 (1929)
• Some HS calculus:
• No HS calculus:
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Germany
England
Germany
Holland
USA
Austria
Czechoslovakia
France
Hungary
Italy
Japan
Russia
Scandinavia
Switzerland
William Anthony Granville
• Text used at West Point
1907-1948 and
1953-1963
Calculus in US High Schools
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John Swenson, Wadleigh HS, NYC,
Ethel Durst, University HS, Oakland, CA, 1925
Frank Austin, University HS, Iowa City, 1927
Robert Goff, New Britain HS, CT, 1922
The Advanced Placement Program
• General Education in School and College, 1952
• The Kenyon Plan
• 1954: First AP exams
• 27 schools in 1955
• Educational Testing Service began teacher
training programs
To be continued . . .
after much more research.