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Why Should Historical Truth Matter to Teachers of Mathematics? Dispelling Myths while Promoting Maths Why Should Historical Truth Matter to Teachers of Mathematics? Dispelling Myths while Promoting Maths Judith V. Grabiner Pitzer College, Claremont, California [email protected] edu

1. First Myth: The social history of mathematics is easy; just determine what nation 1. First Myth: The social history of mathematics is easy; just determine what nation or group your mathematician comes from and generalize

2. All modern mathematics comes from Christian men in the Graeco-European tradition. 2. All modern mathematics comes from Christian men in the Graeco-European tradition.

Notable mathematicians of the past who are included on the 2009 MAA Poster, “Women Notable mathematicians of the past who are included on the 2009 MAA Poster, “Women of Mathematics” Hypatia of Alexandria (ca. 355 -415) Gabrielle du Châtelet (1706 -1749) Maria Gaetana Agnesi (1718 -1799) Caroline Herschel (1750 -1848) Marie-Sophie Germain (1776 -1831) Ada Lovelace (1815 -1852) Florence Nightingale (1820 -1910) Christine Ladd-Franklin (1847 -1930)

Sofia Kovalevskaia (1850 -1890) Charlotte Angas Scott (1858 -1931) Grace Chisolm Young (1868 -1944) Sofia Kovalevskaia (1850 -1890) Charlotte Angas Scott (1858 -1931) Grace Chisolm Young (1868 -1944) Emmy Noether (1882 -1935) Ann Johnson Pell Wheeler (1883 -1966) Dame Mary Cartwright (1900 -1998) Mina Rees (1902 -1997) Ruth Moufang (1905 -1977) Olga Taussky-Todd (1906 -1995)

Grace Hopper (1906 -1992) Emma Lehmer (1906 -2007) Cora Ratto de Sadosky (1912 -1981) Grace Hopper (1906 -1992) Emma Lehmer (1906 -2007) Cora Ratto de Sadosky (1912 -1981) Hanna Neumann (1914 -1971) Julia Bowman Robinson (1919 -1985) Olga Ladyzhenskaya (1922 -2004) Olga Arsen’enva Oleinik (1925 -2001) Etta Zubner Falconer (1933 -2002) Over 30% of all U. S. Ph. Ds. in math now are women. Biological change? Oh, sure.

You can still buy the colorful MAA “Women of Math” poster • http: //www. You can still buy the colorful MAA “Women of Math” poster • http: //www. maa. org/pubs/poster. W. pdf

Muhammad ibn Musa al-Khwarizmi (c. 780 – 850) Name Latinized as Algorismus Name then Muhammad ibn Musa al-Khwarizmi (c. 780 – 850) Name Latinized as Algorismus Name then confused with Arithmos Name & method became “Algorithm” _________________ His book: Al-kitab al-muhtasar fi hisab al-jabr wa’l’muqabala became “Algebra”

The picture illustrates a bow and bowstring, or cord, or (Greek) “chord” The picture illustrates a bow and bowstring, or cord, or (Greek) “chord”

The sine is the half-chord The sine is the half-chord

Since the sine is the half-chord: Sanskrit: jya - ardha (chord –half) Shortened into Since the sine is the half-chord: Sanskrit: jya - ardha (chord –half) Shortened into jya or jiva Transliterated into Arabic as: jiba RD WTHT VWLS: jaib = bay, inlet, cavity Translated into Latin as: Sinus

3. There wasn’t any real mathematics in the European Middle Ages. After the decline 3. There wasn’t any real mathematics in the European Middle Ages. After the decline of Greek mathematics, nothing important happened mathematically in Europe until the Renaissance.

Merton mean-speed theorem If the velocity is changing uniformly, Distance covered in time t Merton mean-speed theorem If the velocity is changing uniformly, Distance covered in time t = distance covered by average speed (Vmax+ Vmin)/2 in the same time Stated by William of Heytesbury, 1335, at Merton College, Oxford

Diagram is first drawn by Nicole Oresme, 1350 Area under velocity graph = distance Diagram is first drawn by Nicole Oresme, 1350 Area under velocity graph = distance covered in time t = distance covered by average speed (Vmax+ Vmin)/2 in the same time

Oresme’s diagram Oresme’s diagram

Galileo’s diagram, from his epoch-making book Two New Sciences, 1633 Galileo’s diagram, from his epoch-making book Two New Sciences, 1633

1/2 + 2/4 + 3/8 + 4/16 + 5/32 + … k/2 k + 1/2 + 2/4 + 3/8 + 4/16 + 5/32 + … k/2 k + … Sum? 1/2 + 1/4 + 1/8 + 1/16 + 1/32 +… = 1/2 + 1/8 + 1/16 + 1/32 +… = 1/4 + 1/16 + 1/32 +… = 1/8 + 1/32 +… = 1/16, etc. Right column adds up to 2 So the sum is 2. This solution is due to Richard Swineshead (Suiseth), 14 th century, nicknamed “Calculator. ”

Proof that the harmonic series diverges: Nicole Oresme, 14 th Century 1/2 + 1/3 Proof that the harmonic series diverges: Nicole Oresme, 14 th Century 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + … = 1/2 + (1/3 + 1/4) + (1/5 + 1/6 + 1/7 + 1/8) + … ≥ 1/2 + (1/ 4 + 1/ 4) + (1/8 + 1/8) + … = 1/2 + 1/ 2 which exceeds any given quantity. + . . .

4. Newton invented the calculus just so he could do his physics. 4. Newton invented the calculus just so he could do his physics.

D. T. Whiteside, ed. , The Mathematical Papers of Isaac Newton, 8 volumes, Cambridge D. T. Whiteside, ed. , The Mathematical Papers of Isaac Newton, 8 volumes, Cambridge University Press, 1967 – 1981 The papers on the discovery of the calculus are in Volume I, covering 1664 – 1666 (in Latin – sorry!)

5. Serious statistical thinking in the sciences begins in the natural sciences; the social 5. Serious statistical thinking in the sciences begins in the natural sciences; the social sciences learned this from natural science and copied it so they’d look scientific.

Adolphe Quetelet, 1796 -1874 Adolphe Quetelet, 1796 -1874

Quetelet: “Curve of possibilities”: 1830 s Quetelet: “Curve of possibilities”: 1830 s

Quetelet, heights of French conscripts, early 19 th century (dip in graph greatly exaggerated) Quetelet, heights of French conscripts, early 19 th century (dip in graph greatly exaggerated) 1. 57 meters N = 100, 000. < 1. 57 m, 2275 more than predicted Between 1. 570 and 1. 624 m, 2114 fewer than predicted

6. The mathematical approach can solve any problem. 6. The mathematical approach can solve any problem.

 • Augustin-Louis Cauchy (1789 - 1857) • Auguste Comte (1798 - 1857) • Augustin-Louis Cauchy (1789 - 1857) • Auguste Comte (1798 - 1857)

Check it out! Victor J. Katz, A History of Mathematics: An Introduction (a 950 Check it out! Victor J. Katz, A History of Mathematics: An Introduction (a 950 -page Introduction) Best is the Third Edition, Addison-Wesley, 2009

“Convergence”: Using historical materials to teach mathematics http: //mathdl. maa. org/math. DL/46/ History and “Convergence”: Using historical materials to teach mathematics http: //mathdl. maa. org/math. DL/46/ History and Pedagogy of Mathematics Newsletter: http: //www. clab. edc. uoc. gr/hpm/News. Letters. htm

Let’s ask again: Why should historical truth matter to teachers of mathematics? 1. 2. Let’s ask again: Why should historical truth matter to teachers of mathematics? 1. 2. 3. 4. Examples Mathematical practice In general, truth matters. Mathematics evolves.