Educational Studies in Mathematics

, Volume 49, Issue 3, pp 283–312 | Cite as

Establishing a custom of proving in american school geometry: evolution of the two-column proof in the early twentieth century

  • Patricio G. Herbst


Having high school students prove geometrical propositions became the norm in the United States with the reforms of the 1890's — when geometry was designated as the place for students to learn the ‘art of demonstration.’ A custom of asking students to produce and write proofs in a ‘two-column format’ of statements and reasons developed as the teaching profession responded to the demands of reform. I provide a historical account for how proving evolved as a task for students in school geometry, starting from the time when geometry became a high school subject and continuing to the time when proof became the centerpiece of the geometry curriculum. I use the historical account to explain how the two-column proof format brought stability to the course of studies in geometry by making it possible for teachers to claim that they were teaching students how to prove and for students to demonstrate that their work involved proving. I also uncover what the nature of school geometry came to be as a result of the emphasis in students' learning to prove by showing that students' acquisition of a generic notion of proof was made possible at the expense of reducing students' participation in the development of new ideas. I draw connections between that century-old reform and current reform emphases on reasoning and proof. I use observations from history to suggest that as we carve a place for proof in present-day school mathematics we must be leery of isolating issues of proving from issues of knowing.


High School Twentieth Century High School Student Teaching Profession School Mathematics 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Baker, J.: 1969, 'Minority report to the national council of education', in National Education Association, Report of the Committee on secondary school studies, Arno Press, New York, pp. 56–59. (Original work published 1893)Google Scholar
  2. Ball, D.L. and Bass, H.: 2000, 'Making believe: The collective construction of public mathematical knowledge in the elementary classroom', in D. Phillips (ed.), Constructivism in Education: Yearbook of the National Society for the Study of Education, University of Chicago Press, Chicago, pp. 193–224.Google Scholar
  3. Beman, W. and Smith, D.E.: 1899, New Plane and Solid Geometry, Ginn, Boston.Google Scholar
  4. Carson, G.: 1913, Essays on Mathematical Education, Ginn, London.Google Scholar
  5. Chauvenet, W.: 1870, A Treatise on Elementary Geometry with Appendices containing a Collection of Exercises for Students and an Introduction to Modern Geometry, Lippincot, Philadelphia.Google Scholar
  6. Chauvenet, W.: 1898, Treatise on Elementary Geometry, W. Byerly (ed.), Lippincot, Philadelphia. (Original work published 1887)Google Scholar
  7. Davies, C.: 1850, The Logic and Utility of Mathematics, with the Best Methods of Instruction Explained and Illustrated, Barnes, New York.Google Scholar
  8. Dewey, J.: 1903, 'The psychological and the logical in teaching geometry', Educational Review 25, 387–399.Google Scholar
  9. Donoghue, E.: 1987, The Origins of a Professional Mathematics Education Program at Teachers College, Unpublished doctoral dissertation. Columbia University Teachers College.Google Scholar
  10. Eliot, C. et al.: 1969, 'Report of the Committee of Ten to the National Education Association', in National Education Association, Report of the Committee on secondary school studies, Arno Press, New York, pp. 3–5. (Original work published 1893)Google Scholar
  11. Eliot, C.: 1905, 'The fundamental assumptions in the report of the Committee of Ten (1893)', Educational Review 30, 325–343.Google Scholar
  12. Fawcett, H.: 1938, The Nature of Proof - The National Council of Teachers of Mathematics Thirtheenth Yearbook, Bureau of Publications of Teachers College, Columbia University, New York.Google Scholar
  13. Greenleaf, B.: 1858, Elements of Geometry with Practical Applications to Mensuration, Robert Davis, Boston.Google Scholar
  14. Halsted, G.: 1893, 'The old and the new geometry', Educational Review 6, 144–157.Google Scholar
  15. Harris, W.T.: 1894, 'The committee of ten on secondary schools', Educational Review 7, 1–10.Google Scholar
  16. Hilbert, D.: 1971, Foundations of Geometry, L. Unger, Trans., P. Bernays, Rev.. Open Court, La Salle, IL. (Original work published in German in 1899)Google Scholar
  17. Hill, F.: 1895, 'The educational value of mathematics', Educational Review 9, 349–358.Google Scholar
  18. Howson, G.: 1982, A History of Mathematics Education in England, Cambridge University Press, Cambridge.CrossRefGoogle Scholar
  19. Jones, P.: 1944, 'Early American geometry', The Mathematics Teacher 37, 3–11.Google Scholar
  20. Kilpatrick, J.: 1992, 'A history of research in mathematics education', in D. Grouws (ed.), Handbook of Research in Mathematics Teaching and Learning, Macmillan, New York, pp. 3–38.Google Scholar
  21. Kliebard, H.: 1986, The Struggle for the American Curriculum 1893-1958, Routledge and Kegan Paul, Boston.Google Scholar
  22. Kline, M.: 1965, 'View of the new math', in E. Moise, A. Calandra, R. Davis, M. Kline and H. Bacon (eds.), “New Math”, (Council for Basic Education, Washington, DC, pp. 13–16.Google Scholar
  23. Krug, E.: 1964, The Shaping of the American High School, Harper and Row, New York.Google Scholar
  24. Lakatos, I.: 1976, Proofs and Refutations: The Logic of Mathematical Discovery, J. Worrall and E. Zahar (eds.), Cambridge University Press, Cambridge.CrossRefGoogle Scholar
  25. Lakatos, I.: 1978, 'A renaissance of empiricism in the recent philosophy of mathematics?' in J. Worrall and G. Currie (eds.), Imre Lakatos. Mathematics, Science and Epistemology: Philosophical Papers, Cambridge University Press, Cambridge, Vol. 2, pp. 24–42. (Original work published in 1967)CrossRefGoogle Scholar
  26. Legendre, A.-M.: 1819, Elements of Geometry, J. Farrar (ed. and trans.), Hilliard and Metcalf, Cambridge, New England.Google Scholar
  27. Legendre, A.-M.: 1841, Elements of Geometry, J. Farrar (ed. and trans.), Hilliard Gray, Boston.Google Scholar
  28. Legendre, A.-M.: 1848, Elements of Geometry and Trigonometry, D. Brewster (trans.), C. Davies (rev. and ed.), Barnes, New York.Google Scholar
  29. Moore, E.H.: 1926, 'On the foundations of mathematics', in C. Austin, H. English, W. Betz, W. Eells and F. Touton (eds.), A General Ssurvey of Progress in the Last Twenty-Five Years: First Yearbook, National Council of Teachers of Mathematics, Washington, DC, pp. 32–57. (Original speech delivered in 1902)Google Scholar
  30. NCTM: 2000, Principles and Standards for School Mathematics, Author, Reston, VA.Google Scholar
  31. Newcomb, S. et al.: 1893, 'Reports of the conferences: Mathematics', in National Education Association, Report of the Committee on secondary school studies, Arno Press, New York, pp. 104–116. (Original work published 1893)Google Scholar
  32. Nightingale, A. et al.: 1899, 'Report of the Committee on College entrance requirements', in National Education Association, Report of Committee on College Entrance Requirements - July, 1899, NEA, Chicago, pp. 5–49.Google Scholar
  33. Playfair, J.: 1860, Elements of Geometry; Containing the First Six Books of Euclid, with a Supplement on the Quadrature of the Circle, and the Geometry of Solids. To which are Added Elements of Plane and Spherical Trigonometry, Collins and Hannay, New York. (Original work published 1795)Google Scholar
  34. Poincaré, H.: 1899, 'La logique et l'intuition dans la science mathématique et dans l'enseignement', L'Enseignement Mathématique 1, 157–162Google Scholar
  35. Quast, W.G.: 1968, Geometry in the High Schools of the United States: An Historical Analysis from 1890 to 1966, Unpublished doctoral dissertation. Rutgers - The State University of New Jersey, New Brunswick.Google Scholar
  36. Rav, Y.: 1999, 'Why do we prove theorems?' Philosophia Mathematica 7, 5–41.CrossRefGoogle Scholar
  37. Ravitch, D.: 2000, Left Back: A Century of Failed School Reforms, Simon and Shuster, New York.Google Scholar
  38. Richards, E.: 1892, 'Old and new methods in elementary geometry', Educational Review 3, 31–39.Google Scholar
  39. Schoenfeld, A.: 1987, 'On having and using geometrical knowledge', in J. Hiebert (ed.), Conceptual and Procedural Knowledge: The Case of Mathematics, Erlbaum, Hillsdale, NJ, pp. 225–264.Google Scholar
  40. Schultze, A.: 1912, The Teaching of Mathematics in Secondary Schools, MacMillan, New York.Google Scholar
  41. Schultze, A. and Sevenoak, F.: 1901, Plane and Solid Geometry, MacMillan, New York.Google Scholar
  42. Schulze, A. and Sevenoak, F.: 1913, Plane Geometry, A. Schultze, (rev.), MacMillan, New York.Google Scholar
  43. Shibli, J.: 1932, Recent Developments in the Teaching of Geometry, Author, State College, PA.Google Scholar
  44. Shutts, G.: 1892, 'Old and new methods in geometry', Educational Review 3, 264–266.Google Scholar
  45. Simson, R.: 1756, The Elements of Euclid, viz. the First Six Books together with the Eleventh and Twelfth. In this edition, the Errors, by which Theon, or others, have long ago vitiated these Books, are corrected, and some of Euclid's Demonstrations are Restored, Foulis, Glasgow.Google Scholar
  46. Sizer, T.: 1964, Secondary Schools at the Turn of the Century, Yale University Press, New Haven.Google Scholar
  47. Slaught, H. et al.: 1912, 'Final report of the National Committee of Fifteen on geometry syllabus', The Mathematics Teacher 5, 46–131.Google Scholar
  48. Smith, D.E.: 1911, The Teaching of Geometry, Ginn, Boston.Google Scholar
  49. Smith, E.R.: 1909, Plane Geometry Developed by the Syllabus Method, American Book Co, New York.Google Scholar
  50. Stanic, G.M.A.: 1983, Why Teach Mathematics? A Historical Study of the Justification Question, Unpublished doctoral dissertation. University of Wisconsin, Madison.Google Scholar
  51. Stanic, G.M.A.: 1987, 'Mathematics education in the United States at the beginning of the twentieth century', in T. Popkewitz (ed.), The Formation of School Subjects: The Struggle for Creating an American Institution, Falmer, New York, pp. 145–175.Google Scholar
  52. Wells, W.: 1887, The Elements of Geometry, Leach, Shewell, and Sanborn, Boston.Google Scholar
  53. Wells, W.: 1908, New Plane and Solid Geometry, Heath, Boston.Google Scholar
  54. Wentworth, G.: 1878, Elements of Geometry, Ginn and Heath, Boston.Google Scholar
  55. Wentworth, G.: 1888, A Text-Book of Geometry, Ginn, Boston.Google Scholar
  56. Wentworth, G.: 1899, Plane and Solid Geometry, Ginn, Boston.Google Scholar
  57. Young, J.W.A. et al.: 1899, 'Report of the Committee of the Chicago section of the American Mathematical Society', in National Education Association, Report of Committee on College Entrance Requirements - July, 1899, NEA, Chicago, pp. 135–149.Google Scholar
  58. Young, J.W.A.: 1906, The Teaching of Mathematics in the Elementary and the Secondary School, Longmans, Green, and Co, New York.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Patricio G. Herbst
    • 1
  1. 1.School of EducationThe University of MichiganAnn ArborUSA

Personalised recommendations