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History of Teaching Calculus

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Handbook on the History of Mathematics Education

Abstract

The history of teaching Calculus is recent. Still, shortly after the creation of Calculus by Newton and Leibniz, it was taught at both the university and secondary levels. This chapter mainly focuses on the European history of Calculus teaching but includes notices on the United States and Brazil. It documents some major approaches, suggestions, debates, and changes in this history from the eighteenth century to the relatively recent past, emphasizing key international movements and studies and simultaneously making use of these studies. Important sources for future research including the most influential Calculus textbooks are also listed.

This research is cofinanced by Fondazione CRTrieste and FSE, Regione Lombardia.

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Notes

  1. 1.

    For instance, see Russo (2004) and Katz (1995). For a careful study of these subjects from historical and epistemological points of view, see Schubring (2005) who also includes extensive bibliographical information.

  2. 2.

    Magnitude is a basic concept: We can briefly say that the magnitude of an object is “what we want to measure of it,” that is, the length (of a line), the extension (of a surface), the volume (of a solid body), and the duration (of an interval of time).

  3. 3.

    Leibniz considered curves defined by variables (i.e., y) depending on the abscissa (i.e., x). The slope of the tangent line was given by the differential ratio, which Leibniz denoted as dy/dx. This notation was called “Leibniz’s notation” and was also used for the limit of the incremental ratio of a function.

  4. 4.

    “When the values successively attributed to a particular variable indefinitely approach a fixed value in such a way as to end up differing from it by as little as we wish, this fixed value is called the limit of all the other values” (in Bradley and Sandifer 2009, p. 6).

  5. 5.

    For information on teaching calculus at the higher level in the eighteenth and nineteenth centuries and on relevant textbooks, see Gispert (2009), Grattan-Guinnes (2009), Pepe (2009), and Schubring (2005, 2009).

  6. 6.

    For instance, an Italian translation by Stanislao Canovai and Gaetano del Ricco was used at the end of the 1700s for teaching calculus at the University of Pavia (see Pepe 2009).

  7. 7.

    See Schubring (2009), who also includes a comparative evaluation of textbook production for analysis in Europe from the 1680s up to 1830.

  8. 8.

    The text for self-instruction by H. B. Lübsen (1855), Einleitung in die Infinitesimal-Rechnung zum Selbstunterricht is cited by Klein as an example of this conception. The text was based on the theory of series – at that time already outdated – which Lübsen considered more suitable for beginners.

  9. 9.

    It is of interest to mention that some female mathematicians actively took part in this conference including Charlotte Angas Scott. Also present was the American mathematician David Eugene Smith.

References

  • Artigue, Michèle. 1996. Réformes et contre-réformes dans l’enseignement de l’analyse au lycée. In Les sciences au lycée – Un siècle de réformes des mathématiques et de la physique en France et à l’étranger, ed. Bruno Belhoste et al., 197–217. Paris: Vuibert.

    Google Scholar 

  • Artigue, Michèle. 1998. L’évolution des problématiques en didactique de l’analyse. Recherches en Didactique des Mathématiques 18(2): 231–262.

    Google Scholar 

  • Austin, Joe Dan. 1979. High school calculus and first-quarter college calculus grades. Journal for Research in Mathematics Education 10(1): 69–72.

    Article  MathSciNet  Google Scholar 

  • Behrendsen, Otto, and Eduard Götting. 1911, 1st ed. 1908. Lehrbuch der Mathematik nach modernen Grundsätzen. Unterstufe. Leipzig/Berlin: Teubner.

    Google Scholar 

  • Behrendsen, Otto, and Eduard Götting. 1915, 1st ed. 1912. Lehrbuch der Mathematik nach modernen Grundsätzen. Oberstufe. Leipzig/Berlin: Teubner.

    Google Scholar 

  • Beke, Emanuel. 1914. Les résultats obtenus dans l’introduction du calcul différentiel et intégral dans les classes supérieures des établissements secondaires. L’Enseignement Mathématique 16: 245–284.

    MATH  Google Scholar 

  • Bioche, Charles. 1914. L’organisation de l’enseignement du calcul des dérivées et des fonctions primitives dans les Lycées de France et sur les résultats obtenus. L’Enseignement Mathématique 16: 285–289.

    Google Scholar 

  • Bourlet, Carlo. 1896. Leçons d’Algèbre élémentaire. Paris: Colin.

    MATH  Google Scholar 

  • Bradley, Robert E., and Charles Edward Sandifer (eds.). 2009. Cauchy’s cours d’analyse. An annotated translation. Berlin/New York: Springer.

    MATH  Google Scholar 

  • Caramalho Domingues, João. 2008. Lacroix and the calculus. Basel: Birkhäuser.

    MATH  Google Scholar 

  • Carvalho, João Bosco Pitombeira de. 1996. Algumas considerações históricas sobre o ensino de cálculo na Escola Secundária. Cadernos CEDES (UNICAMP) 40: 62–81.

    Google Scholar 

  • Castelnuovo, Guido (ed.). 1909. Atti del IV Congresso Internazionale dei Matematici (Roma, 6–11 Aprile 1908), vol. 3. Roma: R. Accademia dei Lincei.

    Google Scholar 

  • Edmonds, Franklin Spencer. 1903. The central high school of Philadelphia. 1838–1902. The School Review 11(3): 211–226.

    Article  Google Scholar 

  • Fehr, Henri. 1905. La notion de fonction dans l’enseignement mathématique des écoles moyennes. L’Enseignement Mathématique 7: 177–187.

    MATH  Google Scholar 

  • Florêncio Aires, Ana Paula. 2006. O conceito de derivada no ensino secundário em Portugal ao longo do século XX: Uma abordagem histórica através dos planos curriculares e manuais escolares. Ph.D. dissertation, Universidad de Salamanca.

    Google Scholar 

  • Furinghetti, Fulvia. 2003. Mathematical instruction in an international perspective: The contribution of the journal L’Enseignement Mathématique. In One hundred years of L’Enseignement Mathématique. Moments of mathematics education in the twentieth century, Proceedings of the EM-ICMI symposium, ed. Daniel Coray et al., 19–46. Geneva: L’Enseignement Mathématique.

    Google Scholar 

  • Furinghetti, Fulvia, Marta Menghini, Ferdinando Arzarello, and Livia Giacardi. 2008. ICMI renaissance: The emergence of new issues in mathematics education. In The first century of the International Commission on Mathematical Instruction (1908–2008). Reflecting and shaping the world of mathematics education, ed. Marta Menghini et al., 131–147. Roma: Istituto della Enciclopedia Italiana fondata da Giovanni Treccani.

    Google Scholar 

  • Giacardi, Livia. 2006. L’insegnamento della matematica in Italia dall’Unità all’avvento del fascismo. In Da Casati a Gentile. Momenti di storia dell’insegnamento secondario della matematica in Italia, ed. Livia Giacardi, 1–63. Lugano: Lumières Internationales.

    Google Scholar 

  • Giacardi, Livia. 2009. The school as a ‘laboratory.’ Giovanni Vailati and the project for the reform of the teaching of mathematics in Italy. International Journal for the History of Mathematics Education 4(1): 5–28.

    Google Scholar 

  • Gispert, Hélène. 2007. Quelles lectures pour les conférences de mathématiques: savante, pédagogique, politique? In Science et enseignement. L’exemple de la grande réforme des programmes du lycée au début du XXe siècle, ed. Hélène Gispert, Nicole Hulin, and Marie-Claire Robic, 203–222. Paris: Vuibert/INRP.

    Google Scholar 

  • Gispert, Hélène. 2009. Les traités d’analyse et la riguer en France dans la deuxième moitié du XIXe siecle, des questions, des choix et des contextes. In Dalla pecia all’e-book. Libri per l’Università: stampa, editoria, circolazione e lettura. Atti del Convegno internazionale di studi, Bologna, 21–25 ottobre 2008, ed. Gian Paolo Brizzi and Maria Gioia Tavoni, 415–430. Bologna: CLUEB.

    Google Scholar 

  • Godfrey, Charles. 1909. The teaching of mathematics in English public schools for boys. In Atti del IV Congresso Internazionale dei Matematici (Roma, 6–11 Aprile 1908), vol. 3, ed. Guido Castelnuovo, 449–464. Roma: R. Accademia dei Lincei.

    Google Scholar 

  • Godfrey, Charles, and Arthur Warry Siddons. 1918, 1st ed 1913. Elementary algebra. Volumes I and II complete. Cambridge: Cambridge University Press.

    Google Scholar 

  • Grattan-Guinnes, Ivor. 2009. Instruction in the calculus and differential equation in Britain, 1820s–1900s. In Dalla pecia all’e-book. Libri per l’Università: stampa, editoria, circolazione e lettura. Atti del Convegno internazionale di studi, Bologna, 21–25 ottobre 2008, ed. Gian Paolo Brizzi and Maria Gioia Tavoni, 443–454. Bologna: CLUEB.

    Google Scholar 

  • Howson, Albert Geoffrey. 1984. Seventy five years of the International Commission on Mathematical Instruction. Educational Studies in Mathematics 15: 75–93.

    Article  Google Scholar 

  • Howson, Albert Geoffrey. 2009. The school mathematics project: Its early years. International Journal for the History of Mathematics Education 4(1): 111–139.

    MathSciNet  Google Scholar 

  • Kahane, Jean-Pierre. 2003. L’enseignement du calcul différentiel et intégral au début du vingtième siècle. In One hundred years of L’Enseignement Mathématique. Moments of mathematics education in the twentieth century, Proceedings of the EM-ICMI symposium, ed. Daniel Coray et al., 167–178. Geneva: L’Enseignement Mathématique.

    Google Scholar 

  • Katz, Victor J. 1995. Ideas of calculus in Islam and India. Mathematics Magazine 68(3): 163–174.

    Article  MATH  MathSciNet  Google Scholar 

  • Klein, Felix. 1924. Elementarmathematik vom höheren Standpunkte aus. I. Band. Arithmetik – Algebra – Analysis. Berlin: Springer (1st ed. 1908).

    Google Scholar 

  • Klein, Felix. 1925. Elementarmathematik vom höheren Standpunkte aus. II. Band. Geometrie. Berlin: Springer (1st ed. 1908).

    Google Scholar 

  • L’Enseignement Mathématique 7. 1905.

    Google Scholar 

  • L’Enseignement Mathématique 16. 1914.

    Google Scholar 

  • L’Enseignement Mathématique 28. 1929.

    Google Scholar 

  • L’Enseignement Mathématique 29. 1930.

    Google Scholar 

  • L’Enseignement Mathématique 32. 1933.

    Google Scholar 

  • L’Enseignement Mathématique 36. 1937.

    Google Scholar 

  • L’Enseignement Mathématique 37. 1938.

    Google Scholar 

  • Marchi, Maria Vitttoria, and Marta Menghini. 2011. La matematica nel Liceo Scientifico. Archimede 2: 87–94.

    Google Scholar 

  • Mezynski, Karen, and Julian C. Stanley. 1980. Advanced placement oriented calculus for high school students. Journal for Research in Mathematics Education 11(5): 347–355.

    Article  Google Scholar 

  • Moise, Edwin. 1962. The new mathematics programs. The School Review 70(1): 82–101.

    Article  Google Scholar 

  • Neelley, J.H. 1961. A generation of high school calculus. The American Mathematical Monthly 68(10): 1004–1005.

    Article  MathSciNet  Google Scholar 

  • Nordgaard, Martin A. 1928. Introductory calculus as a high school subject. In Selected topics in the teaching of mathematics. Third yearbook of the National Council of Teachers of Mathematics. New York: Bureau of Publications, Teachers College, Columbia University.

    Google Scholar 

  • Pepe, Luigi. n.d. Libri elementari per le Università e per le scuole. In Mathematica Italiana. Pisa: Scuola Normale Superiore. http://mathematica.sns.it/opere/182/. Accessed 9 Aug 2011.

  • Pepe, Luigi. 2009. Sulla via del rigore. I manuali di calcolo differenziale e integrale nell’Ottocento in Italia. In Dalla pecia all’e-book. Libri per l’Università: stampa, editoria, circolazione e lettura. Atti del Convegno internazionale di studi, Bologna, 21–25 ottobre 2008, ed. Gian Paolo Brizzi and Maria Gioia Tavoni, 393–414. Bologna: CLUEB.

    Google Scholar 

  • Perry, John. 1897. The calculus for engineers. London: Arnold.

    MATH  Google Scholar 

  • Perry, John (ed.). 1902. Discussion on the teaching of mathematics which took place on September 14th, at a joint meeting of two sections: Section A., mathematics and physics; Section L., education. London/New York: Macmillan.

    Google Scholar 

  • Poincaré, Henri. 1904. Les définitions générales en mathématiques. L’Enseignement Mathématique 6: 257–283.

    MATH  Google Scholar 

  • Rash, Agnes M. 1977. Is calculus an appropriate high school course? The High School Journal 60(6): 277–283.

    Google Scholar 

  • Rau, Heinz. 1954. Weg und Ziel der mathematischen Ausbildung. In Der mathematische Unterricht für die sechzehn- bis einundzwanzigjährige Jugend in der Bundesrepublik Deutschland, ed. Heinrich Behnke, 42–72. Göttingen: Vandenhoek & Ruprecht.

    Google Scholar 

  • Rozenberg, Noah Bryan. 1921. The place of the elementary calculus in the senior high-school mathematics. And suggestions for a modern presentation of the subject. New York: Teachers College, Columbia University.

    Google Scholar 

  • Rüping, Heinrich. 1954. Kurzer Überblick über die Entwicklung des höheren Schulwesens und des mathematischen Unterrichts seit dem Erscheinen des IMUK-Berichtes von 1911. In Der mathematische Unterricht für die sechzehn- bis einundzwanzigjährige Jugend in der Bundesrepublik Deutschland, ed. Heinrich Behnke, 36-42. Göttingen: Vandenhoek & Ruprecht.

    Google Scholar 

  • Russo, Lucio. 2004. The forgotten revolution: How science was born in 300 BC and why it had to be reborn. Berlin/Heidelberg/New York: Springer (Italian ed. 1996).

    Google Scholar 

  • Schubring, Gert. 1985. Essais sur l’histoire de l’enseignement des mathématiques, particulièrement en France et en Prusse. Recherches en Didactique des Mathématiques 5: 343-385.

    Google Scholar 

  • Schubring, Gert. 1996. Changing cultural and epistemological views on mathematics and different institutional contexts in 19th century Europe. In L’Europe mathématique – Mythes, histoires, identités. Mathematical Europe – Myths, history, identity, ed. Catherine Goldstein, Jeremy Gray, and Jim Ritter, 361–388. Paris: Éditions de la Maison des Sciences de l’Homme.

    Google Scholar 

  • Schubring, Gert. 2003. L’Enseignement Mathématique and the first International Commission (IMUK): The emergence of international communication and cooperation. In One hundred years of L’Enseignement Mathématique. Moments of mathematics education in the twentieth century, Proceedings of the EM-ICMI symposium, ed. Daniel Coray et al., 47–65. Geneva: L’Enseignement Mathématique.

    Google Scholar 

  • Schubring, Gert. 2005. Conflicts between generalization, rigor and intuition. Number concept underlying the development of analysis in 17–19th century France and Germany. New York: Springer.

    Google Scholar 

  • Schubring, Gert. 2007. Der Aufbruch zum ‘funktionalen Denken’: Geschichte des Mathematikunterrichts im Kaiserreich. NTM 15: 1–17.

    Article  MATH  MathSciNet  Google Scholar 

  • Schubring, Gert. 2008. The origins and early incarnations of ICMI. In The first century of the International Commission on Mathematical Instruction (1908–2008). Reflecting and shaping the world of mathematics education, ed. Marta Menghini et al., 113–130. Roma: Istituto della Enciclopedia Italiana fondata da Giovanni Treccani.

    Google Scholar 

  • Schubring, Gert. 2009. The way from the combinatorial school to the reception of Weierstrassian analysis. In Dalla pecia all’e-book. Libri per l’Università: stampa, editoria, circolazione e lettura. Atti del Convegno internazionale di studi, Bologna, 21–25 ottobre 2008, ed. Gian Paolo Brizzi and Maria Gioia Tavoni, 431–442. Bologna: CLUEB.

    Google Scholar 

  • Schubring, Gert, Hélène Gispert, Nikos Kastanis, and Livia Giacardi. 2008. The emergence of mathematics as a major teaching subject in secondary schools (panel discussion). In History and epistemology in mathematics education, Proceedings of the 5th European summer university, ed. Evelyne Barbin, Nad’a Stehlíková, and Constantinos Tzanakis, 719–730. Prague: Vydavatelsky Press.

    Google Scholar 

  • Seyfarth, Friedrich. 1924. Zur Entwicklung der mathematischen Unterrichtsreform in Deutschland. In Elementarmathematik vom höheren Standpunkte aus. I Band. Arithmetik - Algebra - Analysis, Felix Klein, Zusatz 1, 291–303. Berlin: Springer.

    Google Scholar 

  • Silva, Circe Mary Silva da. 1996. O conceito de derivada no ensino da matemática no Brasil do século XIX. ICME-8 satellite meeting HPM, Braga, 1, 80–87.

    Google Scholar 

  • Sonnet, Hippolyte. 1869. Premiers Éléments du Calcul Infinitésimal, à l’usage des jeunes gens qui se destinent à la carrière d’ingénieur. Paris: Hachette.

    Google Scholar 

  • Sorge, D.H., and G.H. Wheatley. 1977. Calculus in high school – At what cost? The American Mathematical Monthly 84(8): 644–647.

    Article  MathSciNet  Google Scholar 

  • Swenson, John A. 1934. A course in the calculus for secondary schools. Ann Arbor: Edwards Brothers.

    Google Scholar 

  • Tannery, Jules. 1903. Notions de Mathématiques. Paris: Delagrave.

    MATH  Google Scholar 

  • The Paris report on calculus in secondary schools. 1914. The American Mathematical Monthly 21(10): 324–327.

    Google Scholar 

  • Vita, Vincenzo. 1986. I programmi di matematica per le scuole secondarie dall’unità d’Italia al 1986: rilettura storico-critica. Bologna: Pitagora.

    Google Scholar 

  • Zuccheri, Luciana, and Verena Zudini. 2007. On the influence of cognitive theories in the teaching of calculus in Austrian secondary schools at the beginning of the 20th century. Rendiconti dell’Istituto di Matematica dell’Università di Trieste 39: 347–357.

    Google Scholar 

  • Zuccheri, Luciana, and Verena Zudini. 2008. The ‘Jacob Method’: An example of application of cognitive theories in the period of the introduction of calculus in Austrian secondary mathematics instruction. The International Journal for the History of Mathematics Education 3(2): 57–64.

    Google Scholar 

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Authors and Affiliations

  1. University of Trieste, Trieste, Italy

    Luciana Zuccheri

  2. University of Milano-Bicocca, Milan, Italy

    Verena Zudini

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  1. Luciana Zuccheri
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  2. Verena Zudini
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Correspondence to Luciana Zuccheri .

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Editors and Affiliations

  1. Teachers College, Columbia University, New York, New York, USA

    Alexander Karp

  2. Universität Bielefeld Institut für Didaktik der Mathematik, Bielefeld, Germany

    Gert Schubring

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Zuccheri, L., Zudini, V. (2014). History of Teaching Calculus. In: Karp, A., Schubring, G. (eds) Handbook on the History of Mathematics Education. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-9155-2_24

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