The Planimeter as a Real and Virtual Instrument that Mediates an Infinitesimal Approach to Area

  • Ferdinando Arzarello
  • Daniele ManzoneEmail author
Part of the Mathematics Education in the Digital Era book series (MEDE, volume 8)


Drawing on a didactic gap detected between the elementary concept of area and the infinitesimal approach to it within the Italian secondary school curriculum, the notion of swept area is introduced in grades 10–11. The idea of swept area is introduced through the mediation of an artifact, the Polar Planimeter, both as a concrete physical-tool and as a virtual-object. It triggers and supports the semiotic productions of the students so that they can grasp the new concept. The notion of didactic cycle is used for designing students’ learning sequences. The activities in such sequences are of two types: sensory-motor and symbolic. The mediation of the artifact allows intertwining the two types so that the one can constantly be built on the other. Indeed, the practices mentioned above show a deep intertwining between their cultural and cognitive components.


Planimeter Swept area 


  1. Amsler, J. (1856). Amsler, über das Polar-Planimeter. Polytechnisches Journal, Band 140, Nr. LXXIII, 321–327.
  2. Archimedes (1912). The method of Archimedes recently discovered by Heiberg; a supplement to the Works of Archimedes. (T. L. Heath, Trans.). Cambridge: Cambridge University Press.Google Scholar
  3. Arzarello, F. (2006). Semiosis as a multimodal process [special issue]. Revista Latinoamericana de Investigacion en Matematica Educativa, 267–299.Google Scholar
  4. Arzarello, F., & Robutti, O. (2008). Framing the embodied mind approach within a multimodal paradigm. In Lyn D. English (Ed.), Handbook of international research in mathematics education (2nd ed., pp. 716–745). NY (USA), Abingdon (UK): Routledge, Taylor and Francis.Google Scholar
  5. Arzarello, F., Bartolini Bussi, M. G., Leung, A. Y. L., Mariotti, M. A., & Stevenson, I. (2012). Experimental approaches to theoretical thinking: artefacts and proof. In G. Hanna & M. de Villliers (Eds.), Proof and proving in mathematics education (pp. 97–137). Dordrecht Heidelberg London New York: Springer Science + Business Media.Google Scholar
  6. Arzarello, F., Robutti, O., & Soldano, C. (2015). Learning with touchscreen devices: a game approach as strategies to improve geometric thinking. In Proceedings of CERME 9, Prague, February 4–8, 2015.Google Scholar
  7. Bartolini Bussi, M. G., & Mariotti, M. A. (2008). Semiotic mediation in the mathematics classroom. In L. English, M. Bartolini Bussi, G. Jones, R. Lesh & D. Tirosh (Eds.), Handbook of international research in mathematics education (pp. 746–783). Lea, USA: Routledge.Google Scholar
  8. Bartolini Bussi, M. G., Taimina, D., & Isoda, M. (2010). Mathematical models as early technology tools in classrooms at the dawn of ICMI: Felix Klein and perspectives from different parts of the world. ZDM—The International Journal on Mathematics Education, 42(1), 19–31.CrossRefGoogle Scholar
  9. Boero, P., & Guala, E. (2008). Development of mathematical knowledge and beliefs of teachers: the role of cultural analysis of the content to be taught. In P. Sullivan & T. Wood (Eds.), International handbook of mathematics teacher education: Knowledge and beliefs in mathematics teaching and teaching development (Vol. 1, pp. 223–246). Rotterdam-Taipei: Sense Publ.Google Scholar
  10. Borwein, J. M., & Devlin, K. (2008). The computer as crucible: an introduction to experimental mathematics. Massachusetts: A K Peters.Google Scholar
  11. Castelnuovo, E. (1958). L’object et l’action dans l’enseignement de la géométrie intuitive. In C. Gattegno, W. Servais, E. Castelnuovo, J. L. Nicolet, T. J. Fletcher, L. Motard, L. Campedelli, A. Biguenet, J. W. Peskett, & P. Puig Adam (Eds.), Le matériel pour l’enseignement des mathématiques (pp. 41–59). Neuchâtel: Delachaux & Niestlé.Google Scholar
  12. Cavalieri, B. (1953). Geometria indivisibilibus continuorum nova quadam ratione promota. Bononiae: ex Typographia de Ducijs.Google Scholar
  13. Dewey, J. (1938). Logic: The Theory of Inquiry. In JA Boydston (Ed.), The Later Works 1925–1953, John Dewey, Vol. 12 (1986 edition ed.). (pp. 1–549).Google Scholar
  14. Douady, R. (1984). Jeux de cadres et dialectiqueoutil-objet dansl’enseignement des mathématiques. Thèsed’État, Univ. de Paris. Recherches en didactique des mathématiques, 7(2), 5–31, 1986.Google Scholar
  15. Duval, R. (1995). Quelcognitifretenir en didactique des mathématiques? Actes de l’Écoled’été, 1995.Google Scholar
  16. Edwards, A. W. F. (2003). Human genetic diversity: Lewontin’s fallacy. BioEssays, 25, 798–801. doi: 10.1002/bies.10315.CrossRefGoogle Scholar
  17. Epp, S. (1994). The role of proof in problem solving. In A. H. Schoenfeld (Ed.), Mathematical Thinking and Problem Solving (pp. 257–269). Hillsdale, NJ: Lawrence Erlbaum Associates, Inc., Publishers.Google Scholar
  18. Eves, H. (1991). Two surprising theorems on Cavalieri congruence. The College Mathematics Journal, 22, 118–124, March 2, 1991.Google Scholar
  19. Galileo, G. (1661). The systeme of the world in four dialogues. (T. Salusbury, Trans.) (pp. 219–220). London (Original work published 1632). Retrieved from
  20. Goldin-Meadow, S. (2003). Hearing gestures: How our hands help us think. Chicago: Chicago University Press.Google Scholar
  21. Goodstein, D. L., & Goodstein, J. R. (1996). Feynman’s lost lecture: the motion of planets around the sun. New York: W.W. Norton & Co.Google Scholar
  22. Greeno, J. (1994). Comments on Susanna Epp’s chapter. In A. Schoenfeld (Ed.), Mathematical thinking and problem solving (pp. 270–278). Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar
  23. Hall, R., Nemirovsky, R. (2012). Introduction to the special issue: modalities of body engagement in mathematical activity and learning. Journal of the Learning Sciences, 21(2).Google Scholar
  24. Hanna, G. (1996). The ongoing value of proof. In: Proceedings of the International Group for the Psychology of Mathematics Education, Valencia, Spain (Vol. I).Google Scholar
  25. Hanna, G. (2000). Proof, explanation and exploration: an overview. Educational Studies in Mathematics, 44, 5–23.CrossRefGoogle Scholar
  26. Hasan, R. (2002). Semiotic mediation, language and society: Three exotropic theoriesVygotsky, Hallyday and Bernstein. Retrieved from
  27. Hintikka, J. (1999). Inquiry as inquiry: A logic of scientific discovery. Springer Science + Business Media Dordrecht.Google Scholar
  28. Horgan, J. (1993). The death of proof. Scientific American, 93–103.Google Scholar
  29. Kepler, J. (1609). Astronomia Nova ΑΙΤΙΟΛΟΓΗΤΟΣ seu physica coelestis, tradita commentariis de motibus stellae Martis ex observationibus G.V. Tychonis Brahe. Heidelberg: Voegelin.Google Scholar
  30. Kepler, J. (1615). Nova stereometria doliorvm vinariorvm [New solid geometry of wine barrels]. Retrieved from
  31. Lakoff, G., & Nunez, R. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York: Basic Books.Google Scholar
  32. McNeill, D. (1992). Hand and mind: What gestures reveal about thought. Chicago: University of Chicago Press.Google Scholar
  33. National Council of Teachers of Mathematics. (1989). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  34. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  35. Newton, I. (1704) Tractatus de quadratura curvarum (J. Harris, Trans.). London. (Original work published 1710) from Latin. Retrieved from
  36. Papert, S. (1980). Mindstorms: Children, computers, and powerful ideas. New York: Basic Books.Google Scholar
  37. Rabardel, P. (1995). Les hommes et les technologies [people and technology]. Paris: Armand Colin.Google Scholar
  38. Radziszowski, S., & McKay, B. (1995). R(4,5) = 25. Journal of Graph Theory, 19(1995) 309–322. Retrieved from
  39. Ruthven, K. (2008). Mathematical technologies as a vehicle for intuition and experiment: A foundational theme of the International Commission on Mathematical Instruction, and a continuing preoccupation. International Journal for the History of Mathematics Education, 3(2), 91–102.Google Scholar
  40. Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15, 3–14.CrossRefGoogle Scholar
  41. Smith, D. E. (1913). Intuition and experiment in mathematical teaching in the secondary schools. In Proceedings of the Fifth International Congress of Mathematicians (Vol. II, pp. 611–632).Google Scholar
  42. Tall, D. (1989). Concept images, generic organizers, computers and curriculum change. For the Learning of Mathematics, 9(3), 37–42.Google Scholar
  43. Wu, H.-H. (1996). The role of Euclidean geometry in high school. Journal of Mathematical Behavior, 15, 221–237.Google Scholar

Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  1. 1.Dipartimento di Matematica “G. Peano”Università di TorinoTurinItaly
  2. 2.Istituto Sociale and Scuola Secondaria di I grado E. ArtomTurinItaly

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