Algebra Without Context Is Empty, Visualizations Without Concepts Are Blind

  • Rainer Kaenders
  • Ysette Weiss
Part of the ICME-13 Monographs book series (ICME13Mo)


In the acquisition and formalization of mathematical concepts, the transition between algebraic and geometric representations and the use of different modes of representation contextualizes abstract algebra. Regrettably, the role of geometry is often limited to the visualization of algebraic facts and figurative memory aids. Such visualizations are blind for the underlying concepts, since transitions between concepts in different representations assume the existence of symbols, language, rules and operations in both systems. The history of mathematics offers contexts to develop geometrical language and intuition in areas currently being taught in school in a purely algebraic fashion. The example of the determination of zeros of polynomials shows how reflecting on posing a problem in ancient Greek mathematics, engineering mathematics (19th century) and paper folding (beginning of the 20th century) can help to develop geometrical concepts, language and intuition stemming from an algebraic context.


Engineering Greek mathematics Algebraic/geometric representations Pictorial/symbolic visualizations Horner’s scheme Lill’s method Paper folding Historical context 



The authors thank Carl-Peter Fitting for alerting them to the method of Captain Lill that turned out to be such a fruitful topic when using history to foster concept development. They are grateful for the elaborate and constructive comments of the referees that helped to improve this chapter considerably.


  1. Anonyme. (d’après M. Lill). (1868). Résolution graphique des équations algébriques qui ont des racines imaginaires. Nouvelles Annales de Mathématiques, 7(2), 363–367.Google Scholar
  2. Beloch, M. P. (1936). Sul metodo del ripiegamento della carta per la risoluzione dei problemi geometrici. Periodico di Mathematiche, 16(4), 104–108.Google Scholar
  3. Burton, D. (2011). The history of mathematics: An introduction (7th ed.). New York: McGraw-Hill.Google Scholar
  4. Courant, R., & Robbins, H. (1941). What is mathematics? An elementary approach to ideas and methods. London: Oxford University Press.Google Scholar
  5. Descartes, R. (1954). The geometry of René Descartes (Second Book). Mineola, NY: Dover.Google Scholar
  6. Herrmann, A. (1927). Das Delische Problem (Verdopplung des Würfels). Leibzig: B. G. Teubner.CrossRefGoogle Scholar
  7. Horner, W. G. (1819). A new method of solving numerical equations of all orders, by continuous approximation. Philosophical Transactions of the Royal Society of London, 109, 308–335. Google Scholar
  8. Hull, T. C. (2011). Solving cubics with creases: The work of Beloch and Lill. American Mathematical Monthly, 118(4), 307–315.CrossRefGoogle Scholar
  9. Jahnke, T. (2012). Die Regeldetri des Mathematikunterrichts. In M. Ludwig & M. Kleine (Eds.), Beiträge zum Mathematikunterricht, 46th meeting of the GDM (pp. 413–416). Münster: WTM.Google Scholar
  10. Jankvist, U. T. (2009). A categorization of the “whys” and “hows” of using history in mathematics education. Educational Studies in Mathematics, 71(3), 235–261.CrossRefGoogle Scholar
  11. Kalman, D. (2008). Uncommon mathematical excursions: Polynomia and related realms. Washington, DC: The Mathematical Association of America.Google Scholar
  12. Kasahara, K. (2004). The art and wonder of origami. Gloucester: Quarry Books.Google Scholar
  13. Klein, F. (1897). Famous problems of elementary geometry: The duplication of the cube; the trisection of an angle; the quadrature of the circle (W. W. Beman & D. E. Smith, Trans.). Boston: Ginn & Co. (Authorized Translation of F. Klein’s Vorträge über Ausgewählte Fragen der Elementargeometrie, ausgearbeitet Von F. Tägert.).Google Scholar
  14. Klein, F. (1926). Elementarmathematik von höheren Standpunkte aus, II: Geometrie. Berlin: Springer.Google Scholar
  15. Klein, F. (2016). Elementary mathematics from an advanced standpoint: Vol. II Geometry (G. Schubring, Trans.). Heidelberg: Springer.Google Scholar
  16. Kvasz, L. (2000). Patterns of change—Linguistic innovations in the development of classical mathematics. Basel: Birkhäuser.Google Scholar
  17. Lill, E. (1867). Résolution graphique des équations numériques de tous les degrés à une seule inconnue, et description d’un instrument inventé dans ce but. Nouvelles Annales de Mathématiques, 6(2), 359–362.Google Scholar
  18. Österreichisches Biographisches Lexikon und Biographische Dokumentation (ÖBLBD). (1971). ÖBL 1815–1950 (Vol. 5). Wien: Verlag der Österreichischen Akademie der Wissenschaften (Lieferung 23).Google Scholar
  19. Tabachnikov, S. (2017). Polynomials as polygons. The Mathematical Intelligencer, 39(1), 41–43.CrossRefGoogle Scholar
  20. Wang, L., & Needham, J. (1954). Horner’s method in Chinese mathematics: Its origins in the root-extraction procedures of the Han dynasty. T’oung Pao, International Journal of Chinese Studies, 43(1), 345–401.CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Hausdorff Center for MathematicsMathematisches Institut, Rheinische Friedrich-Wilhelms-Universität BonnBonnGermany
  2. 2.Abteilung MathematikdidaktikInstitut für Mathematik, Johannes Gutenberg-Universität MainzMainzGermany

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