Science & Education

, Volume 20, Issue 1, pp 1–35

# Using History to Teach Mathematics: The Case of Logarithms

• Evangelos N. Panagiotou
Article

## Abstract

Many authors have discussed the question why we should use the history of mathematics to mathematics education. For example, Fauvel (For Learn Math, 11(2): 3–6, 1991) mentions at least fifteen arguments for applying the history of mathematics in teaching and learning mathematics. Knowing how to introduce history into mathematics lessons is a more difficult step. We found, however, that only a limited number of articles contain instructions on how to use the material, as opposed to numerous general articles suggesting the use of the history of mathematics as a didactical tool. The present article focuses on converting the history of logarithms into material appropriate for teaching students of 11th grade, without any knowledge of calculus. History uncovers that logarithms were invented prior of the exponential function and shows that the logarithms are not an arbitrary product, as is the case when we leap straight in the definition given in all modern textbooks, but they are a response to a problem. We describe step by step the historical evolution of the concept, in a way appropriate for use in class, until the definition of the logarithm as area under the hyperbola. Next, we present the formal development of the theory and define the exponential function. The teaching sequence has been successfully undertaken in two high school classrooms.

## Keywords

Natural Logarithm Seventeenth Century Logarithmic Function Arithmetic Progression Geometric Progression
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Notes

### Acknowledgments

I wish to thank Professor Michael R. Matthews for great support and encouragement. I also wish to thank the four anonymous reviewers for their generous help in greatly improving the English in an early version of this paper and for their numerous suggestions and constructive criticism for improving both the style and content.

## References

1. Alexanderson, G. L. (2003). Major donation announced for new MAA conference center. Focus, 23(2), 3–5.Google Scholar
2. Atkinson, K. E. (1989). An introduction to numerical analysis. New York: Wiley.Google Scholar
3. Ayoub, R. (1993). What is a Napierian logarithm? American Mathematical Monthly, 100, 351–364.
4. Bachelard, G. (1938/1983). La Formation De l’ Esprit Scientifique. Paris: J. Vrin.Google Scholar
5. Bartolini-Bussi, M. G., & Bazzini, L. (2003). Research, practice and theory in didactics of mathematics: Towards dialogue between different fields. Educational Studies in Mathematics, 54, 203–223.
6. Bartolini-Bussi, M. G., & Sierpinska, A. (2000). The relevance of historical studies in designing and analysing classroom activities. In J. Fauvel & J. Van Maanen (Eds.), History in mathematics education, The ICMI Study (pp. 154–161). Dordrecht: Kluwer.Google Scholar
7. Bishop, A. J. (1991). Mathematical enculturation. Dordrecht: Kluwer.Google Scholar
8. Boyé, A. (2006). Des Logarithmes Ordinaires aux Logarithmes Naturels. In Histoire de Logarithmes (pp. 217–232). Commission Inter-IREM, Paris: EllipsesGoogle Scholar
9. Brousseau, G. (1983). Les obstacles epistemologiques et les problemes en mathematique. Recherches en Didactiques des Mathematiques, 4(2), 164–198.Google Scholar
10. Brousseau, G. (1997). Theory of didactical situations. Dordrecht: Kluwer.Google Scholar
11. Brown, R. G. (1992). Advanced mathematics. Boston: Houghton Mifflin Company.Google Scholar
12. Buck, R. C. (1980). Sherlock Holmes in Babylon. The American Mathematical Monthly, 87, 338–345.Google Scholar
13. Burn, B. (1998). Napier’s logarithms. Mathematics in School, 27, 32–33.Google Scholar
14. Burn, B. (2000). Gregory of St. Vincent and the rectangular hyperbola. The Mathematical Gazette, 84, 480–485.
15. Burn, R. P. (2001). Alphonse Antonio de Sarasa and Logarithms. Historia Mathematica, 28, 1–17.
16. Cajori, F. (1913). History of the exponential and logarithmic concepts. The American Mathematical Monthly, 20(1), 5–14; 20(2), 35–47; 20(6), 173–182.Google Scholar
17. Cajori, F. (1994). History of the logarithmic slide rule and allied instruments. Mendham, NJ: Astragal Press.Google Scholar
18. Chevallard, Y. (1985). La transposition didactique: Du savoir savavt au savoir enseigne. Grenoble: La Pensée Sauvage.Google Scholar
19. Churchill, R. V., Brown, J. W., & Verhey, R. F. (1974). Complex variables and applications. London: Mc Graw-Hill.Google Scholar
20. Coolidge, J. L. (1950). The number e. American Mathematical Monthly, 57, 591–602.
21. Coolidge, J. L. (1990). The mathematics of great amateurs. Oxford: Claredon.Google Scholar
22. Dhombres, J. (1993). Is one proof enough? Travels with a mathematician of the Baroque period. Educational Studies in Mathematics, 24, 401–419.
23. Dreyfus, T. (1991a). On the status of visual reasoning in mathematics and mathematics education. In F. Furinghetti (Ed.), Proceedings of the fifteenth international conference for the psychology of mathematics education (Vol. 1, pp. 32–48). IPC, PME 15, Assisi, Italy.Google Scholar
24. Dreyfus, T. (1991b). Advanced mathematical thinking processes. In D. Tall (Ed.), Advanced mathematical thinking (pp. 25–41). Dordrecht: Kluwer.Google Scholar
25. Edwards, C. H. (1979). The historical development of the calculus. New York: Springer.Google Scholar
26. Euler, L. (1748/1990). Introduction to analysis of the infinite, books I and II (J. D. Blanton, Trans.). New York: Springer.Google Scholar
27. Euler, L. (1770/1984). Elements of algebra, (John Hewlet, Trans.). New York: Springer.Google Scholar
28. Fauvel, J. (1991). Using history in mathematics education. For the Learning of Mathematics, 11(2), 3–6.Google Scholar
29. Fauvel, J. (1995). Revisiting the history of logarithms. In F. Swetz, J. Fauvel, O. Bekken, B. Johanson, & V. Katz (Eds.), Learn from the masters! (pp. 39–48). USA: MAA Notes.Google Scholar
30. Fauvel, J., & Gray, J. (1987). The history of mathematics: A reader. London: Macmillan.Google Scholar
31. Fauvel, J., & van Maanen, J. (2000). History in mathematics education, The ICMI Study. Dordrecht: Kluwer.Google Scholar
32. Flegg, H. G., Hay, C. M., & Moss, B. (Trans.) (Ed.) (1985). Nicolas Chuquet, renaissance mathematician. Boston: Reidel.Google Scholar
33. Fowler, D. H. (1992). A final-year university course on the history of mathematics: Actively confronting the past. American Mathematical Monthly, 76, 46–48.Google Scholar
34. Freudenthal, H. (1973). What groups mean in mathematics and what they should mean in mathematical education. In A. G. Howson (Ed.), Developments in mathematical education (pp. 101–114). Cambridge: Cambridge University Press.Google Scholar
35. Freudenthal, H. (1984). The implicit philosophy of mathematics history and education. In Proceedings of the ICMI 1983 (Vol. 2, pp. 1695–1709). Warszawa/Amsterdam: PNW/North Holland.Google Scholar
36. Friedelmeyer, J.-P. (1990). Teaching 6th form mathematics with a historical perspective. In J. Fauvel (Ed.), History in the mathematics classroom: The IREM papers (pp. 1–16). Leicester: Mathematical Association.Google Scholar
37. Furinghetti, F., & Radford, L. (2002). Historical conceptual developments and the teaching of mathematics: From phylogenesis and ontogenesis theory to classroom practice. In L. English (Ed.), Handbook of international research in mathematics education (pp. 631–654). New Jersey: Lawrence Erlbaum.Google Scholar
38. Guedj, D. (1998). Le théorème du perroquet. Paris: Editions du Seuil. (F. Wynne, Trans., 2000: The Parrot’s theorem: A novel. New York: St. Martin’s Press).Google Scholar
39. Gulikers, I., & Blom, K. (2001). A historical angle, a survey of recent literature on the use and value of history in geometrical education. Educational Studies in Mathematics, 47, 223–258.
40. Hadamard, J. (1954). The psychology of invention in the mathematical field. New York: Dover.Google Scholar
41. Hallenberg, A. (Ed.) (1989/1969). Historical topics for the mathematics classroom. In Thirty-first yearbook of NCTM. Washington, DC: The Council.Google Scholar
42. Heath, T. L. (1956). The thirteen books of the elements (Vol. 1, 2, 3). New York: Dover.Google Scholar
43. Heiede, T. (2000). A Babylonian tablet. In J. Fauvel & J. Van Maanen (Eds.), History in mathematics education, The ICMI Study (pp. 255–257). Dordrecht: Kluwer.Google Scholar
44. Jahnke, H. H., Knoche, N., & Otte, M. (1996). History of mathematics and education: Ideas and experiences. Göttingen: Vandenhoeck und Ruprecht.Google Scholar
45. Katz, V. (1995). Napier’s logarithms adapted for today’s classroom. In F. Swetz, J. Fauvel, O. Bekken, B. Johanson, & V. Katz (Eds.), Learn from the masters! (pp. 49–55). USA: MAA Notes.Google Scholar
46. Klein, F. (1945). Elementary mathematics from an advanced standpoint: Arithmeticalgebraanalysis (E. R. Hedrick & C. A. Noble, Trans.). New York: Dover Publications.Google Scholar
47. Kline, M. (1972). Mathematical thought from ancient to modern times. New York: Oxford University Press.Google Scholar
48. Kouteynikoff, O. (2006). Inventions de Nombres: Calculs ou Resolutions? In Histoire de Logarithmes (pp. 11–37). Commission Inter-IREM, Paris: Ellipses.Google Scholar
49. Kronfellner, M. (1996). The history of the concept of function and some implications for classroom teaching. In R. Calinger (Ed.), Vita mathematica: Historical research and integration with teaching (pp. 317–320). Washington: MAA.Google Scholar
50. Lakatos, I. (1976). Proofs and refutations. Cambridge: Cambridge University Press.Google Scholar
51. Lubet, J-P. (2006). Dans le Traités d’ Analyse: Logarithmes et Exponentielles au Gré de la Rigueur. In Histoire de Logarithmes (pp. 233–268).Commission Inter-IREM, Paris: Ellipses.Google Scholar
52. McBride, C. C., & Rollins, J. H. (1977). The effects of history of mathematics on attitudes toward mathematics of college algebra students. Journal for Research in Mathematics Education, 8(1), 57–61.
53. Menghini, M. (2000). On potentialities, limits and risks. In J. Fauvel & J. Van Maanen (Eds.), History in mathematics education, The ICMI Study (pp. 86–90). Dordrecht: Kluwer.Google Scholar
54. Ofir, R., & Arcavi, A. (1992). Word problems and equations: An historical activity for the algebra classroom. The Mathematical Gazette, 76(475), 69–84.
55. Pierce, R. C. (1977). Sixteenth-century astronomers had prosthaphaeresis. The Mathematics Teacher, 70, 613–614.Google Scholar
56. Pólya, G. (1954/1990). Induction and analogy in mathematics. Princeton: Princeton University Press.Google Scholar
57. Pólya, G. (1981). Mathematical discovery (Vol. II). New York: Wiley.Google Scholar
58. Radford, L., & Guérette, G. (1996). Second degree equations in the classroom: A Babylonian approach. In V. Katz (Ed.), Using history to teach mathematics: An international perspective (pp. 69–75). Washington: MAA.Google Scholar
59. Resnikoff, H. L., & Wells, R. O., Jr. (1984). Mathematics in civilization. New York: Dover.Google Scholar
60. Rowe, D. E. (1996). New trends and old images in the history of mathematics. In R. Calinger (Ed.), Vita mathematica: Historical research and integration with teaching (pp. 3–16). Washington: MAA.Google Scholar
61. Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational studies in mathematics, 22, 1–36.
62. Sfard, A. (1995). The development of algebra: Confronting historical and psychological perspectives. Journal of Mathematical Behavior, 14, 15–39.
63. Sierpinska, A. (1994). Understanding in mathematics. London, Washington, DC: The Falmer.Google Scholar
64. Silver, E. A. (1997). Fostering creativity through instruction rich in mathematical problem solving and problem posing. Zentrallblatt fur Didactik der Mathematik, 3, 75–80.
65. Smith, D. E. (1915). The law of exponents in the works of the sixteenth century. In C. G. Knott (Ed.), Napier tercentenary memorial volume (pp. 81–91). London: Longmans Green and Co.Google Scholar
66. Smith, D. E. (1923/1958). History of mathematics (Vol. 2). New York: Dover Publications.Google Scholar
67. Smith, D. E. (1929/1959). A source book in mathematics. New York: Dover Publications.Google Scholar
68. Sternberg, R. J. (1988). The nature of creativity: Contemporary psychological perspectives. New York: Cambridge University Press.Google Scholar
69. Struik, D.J. (1948/1987). A concise history of mathematics. New York: Dover Publications.Google Scholar
70. Swade, D. (1991). Charles Babbage and his calculating engines. London: Science Museum.Google Scholar
71. Swetz, F. (1984). Seeking relevance? Try the history of mathematics. Mathematics Teacher, 77(1), 54–62.Google Scholar
72. Swetz, F. (1987). Capitalism and arithmetic: The new math of the 15th century. La Salle: Open Court.Google Scholar
73. Swetz, F. (1995). To know and to teach: Mathematical pedagogy from a historical context. Educational Studies in Mathematics, 29, 73–88.
74. Van der Waerden, B. L. (1961). Science awakening. New York: Holt, Rinehart and Winston.Google Scholar
75. Van Looy, H. (1984). A chronology and historical analysis of the mathematical manuscripts of Gregorius a Sanctio Vincentio. Historia Mathematica, 11, 57–75.
76. Vasco, C. E. (1995). History of mathematics as a tool for teaching mathematics for understanding. In D. N. Perkins, et al. (Eds.), Software goes to school (pp. 56–69). New York: Oxford University Press.Google Scholar
77. Waldeg, G. (1997). Histoire, Epistemologie et Methodologie dans la Recherche en Didactique. For The Learning of Mathematics, 17(1), 43.Google Scholar
78. Whiteside, D. T. (1960–1962). Mathematical thought in the later 17th century. Archive for the History of Exact Science, 1, 179–387.Google Scholar
79. Whiteside, D. T. (1967–1976). The mathematical papers of Isaac Newton (Vol. I, II). Cambridge: Cambridge University Press.Google Scholar