From the Mathematicians Study to the AP Classroom

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 • • 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 • • • Apollonius Archimedes Fermat Huygens De. Witt Sluse • • • 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 • 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. • That the chief Master shall further instruct the hearers in Trigonometry and the Elements of the Conick Sections. • To which he shall add the Principles of Practical Geometry and Mechanics, applied to raising and transporting great Burthens; • With the Knowledge of Mensuration, and Levelling, and its Application to the bringing of water and the draining of Morasses; • And lastly, shall teach Fortification in all its parts. But no calculus

• Preface – 55 pages • Ch I: Internal ballistics – 65 pages • Ch 2: 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 • • Piedmont-Savoy in 1750 s 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 • •

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, Traite e le mentaire • 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: • • • Germany England Germany Holland USA • No HS calculus: • • • 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 • • 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.