Calculating Aids in Mathematics Education Before the Advent of Electronic Calculators: Didactical and Technological Prospects

  • Dragana MartinovicEmail author
Part of the Mathematics Education in the Digital Era book series (MEDE, volume 11)


This chapter presents a synopsis of ideas, movements, and reforms that influenced education in the twentieth century before the electronic calculators started being routinely used in mathematics classrooms around the world. This was an era of rapid industrialization which required efficient skilling of the population; the new approaches to how mathematics is taught and learnt were influenced by the succession of psychological theories and technological innovations. Along the way, some didactical and technological approaches were abandoned, while others kept reappearing, although altered. The readers are introduced to selected technological innovations of the period (e.g., teaching machines, thinking machines, and calculating/computing machines), along with the questions, expectations, and disappointments they gave rise to.


Teaching machines Thinking machines Computing machines Aspects of design and use in schools Place in education Psychological theories 


  1. Adler, I. 1961. Thinking machines: A layman’s introduction to logic, Boolean algebra, and computers. New York, NY: The John Day Company.Google Scholar
  2. Aikins, H. A. 1913. Educational appliance. Retrieved from
  3. Aleven, V., B.M. McLaren, J. Sewall, and K.R. Koedinger. 2009. A new paradigm for intelligent tutoring systems: Example-tracing tutors. International Journal of Artificial Intelligence in Education 19 (2): 105–154.Google Scholar
  4. Altman, G. G. 1897. Apparatus for teaching arithmetic. Retrieved from
  5. Beck, L.L. 1960. A report on the use of calculators. The Arithmetic Teacher 7 (2): 103.Google Scholar
  6. Benjamin, L.T. 1988. A history of teaching machines. American Psychologist 43 (9): 703–712.Google Scholar
  7. Blikstein, P. 2013. Seymour Papert’s legacy: Thinking about learning, and learning about thinking. Available at
  8. Borba, M., and M. Bartolini Bussi. (2008). Resources and technology throughout the history of ICMI (working groups—reports). In The first century of the international commission on mathematical instruction (1908–2008). Reflecting and shaping the world of mathematics education, ed. by M. Menghini, F. Furinghetti, L. Giacardi, and F. Arzarello, 289–300. Roma: Istituto della Enciclopedia Italiana.Google Scholar
  9. Botts, T., and L. Pikaart. 1961. Mathematics from the modern viewpoint. The Mathematics Teacher 54 (7): 498–504.Google Scholar
  10. Brownell, W.A. 1935. Psychological considerations in the learning and the teaching of arithmetic. The National Council of teachers of mathematics. The Tenth Yearbook. The Teaching of Arithmetic, 1–31. New York, NY: Teachers College, Columbia University.Google Scholar
  11. Bruner, J.S. 1960. The process of education. Cambridge, MA: Harvard University Press.Google Scholar
  12. Bruner, J.S. 2007. On learning mathematics. The Mathematics Teacher 100: 48–55 (Paper presented before The National Council of Teachers of Mathematics, Salt Lake City, Utah, August 1960). Reprinted from The Mathematics Teacher 53: 610–619.Google Scholar
  13. Burkhardt, H. 1985. Computer aware curricula: Ideas and realisation. In The Influence of computers and informatics on mathematics and its teaching, proceedings from a symposium held in Strasbourg, France in March 1985, ed. by R. F. Churchhouse et al., 147–155. Cambridge etc: Cambridge University Press.
  14. Cech, J.P. 1970. The effect the use of desk calculators has on attitude and achievement in ninth-grade general mathematics classes. Unpublished thesis at School of Education, Indiana University, Bloomington.Google Scholar
  15. Clements, D.H., and J. Sarama. (1997). Research on LOGO: A decade of progress. In Logo: A retrospective, ed. by D. Maddux Cleborne, and D. LaMont Johnson, 9–46. Binghamton, NJ: The Haworth Press, Inc.Google Scholar
  16. Commission on Post-War Plans of the NCTM. 1945, May. Second report of the commission of post-war plans. The Mathematics Teacher 38: 195–221.Google Scholar
  17. Ellis, M.W., and R.Q. Berry III. 2005. The paradigm shift in mathematics education: Explanations and implications of reforming conceptions of teaching and learning. The Mathematics Educator 15 (1), 7–17.Google Scholar
  18. Evans, J.L. 1965. Programming in mathematics and logic. In Teaching machines and programmed learning II, data and directions, ed. R. Glaser, 371–440. Washington, D.C.: National Education Association.Google Scholar
  19. Fehr, H. 1947. The place of multisensory aids in the teacher training program. The Mathematics Teacher 40: 212–216.Google Scholar
  20. Furinghetti, F., M. Menghini, F. Arzarello, and L. 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. by M. Menghini, F. Furinghetti, L. Giacardi, and F. Arzarello, 131–147. Roma: Istituto della Enciclopedia Italiana.Google Scholar
  21. Grouws, D. A., and K. L. Cebulla. 2000. Elementary and middle school mathematics at the crossroads. In American education: Yesterday, today and tomorrow, Part II, ed. by T. L. Good, 209–255. Chicago, IL: University of Chicago Press.Google Scholar
  22. Hansen, D.N. 1966. Computer assistance with the educational process. Review of Educational Research 36 (5): 588–603.Google Scholar
  23. Hartley, S. S. 1977. Meta-analysis of the effects of individually paced instruction in mathematics. Unpublished doctoral dissertation, University of Colorado.Google Scholar
  24. Hartmann, G.F. 1966. Gestalt psychology and mathematical insight. The Mathematics Teacher 59 (7): 656–661.Google Scholar
  25. Hoffman, R.I. 1968. The slow learner—Changing his view of mathematics. NASSP Bulletin 52 (87): 86–97.Google Scholar
  26. Hoffman, R.R., and L.G. Militello. 2009. Perspectives on cognitive task analysis: historical origins and modern communities of practice. New York, NY: Psychology Press.Google Scholar
  27. Horton, E.M. 1937. Calculating machines and the mathematics teacher. The Mathematics Teacher 30 (6): 271–276.Google Scholar
  28. Kieren, T.E. 1977. Mathematics education research in Canada: A prospective view. In Educating teachers of mathematics: The Universities’ responsibility. Proceedings of the Canadian Mathematics Education Study Group (CMESG), ed. by A.J. Coleman, W.C. Higginson, and D.H. Wheeler, 2–21. Kingston, ON: Queen’s University.Google Scholar
  29. Kilpatrick, J. 2008. The development of mathematics education as an academic field. In The first century of the International Commission on Mathematical Instruction (1908–2008). Reflecting and shaping the world of Mathematics education, 25–39. Roma: Istituto della Enciclopedia Italiana.Google Scholar
  30. Klein, F. 1932. Elementary mathematics from an advanced standpoint: Arithmetic-algebra-analysis. Translated from the third German edition by E. R. Hedrick and C. A. Noble. London, England: Macmillan and Co.Google Scholar
  31. Kline, M. 1966. A proposal for the high school mathematics curriculum. The Mathematics Teacher 59 (4): 322–330.Google Scholar
  32. Kline, M. 1973. Why Johnny can’t add: The failure of the new math. New York: St. Martin’s Press.Google Scholar
  33. Kulik, J.A., P.A. Cohen, and B.J. Ebeling. 1980a. Effectiveness of programmed instruction in higher education: A meta-analysis of findings. Educational Evaluation and Policy Analysis 2 (6): 51–64.Google Scholar
  34. Kulik, J.A., C.-L.C. Kulik, and P.A. Cohen. 1980b. Effectiveness of computer-based college teaching: A meta-analysis of findings. Review of Educational Research 50(4): 525–544.Google Scholar
  35. Lambdin Kroll, D. 1989. Connections between psychological learning theories and the elementary mathematics curriculum. In New Directions for Elementary School Mathematics: 1989 Yearbook, ed. P.R. Trafton, 199–211. Reston, Virginia: National Council of Teachers of Mathematics.Google Scholar
  36. Larrivee, J.A. 1958a. A history of computers I. The Mathematics Teacher 51 (6): 469–473.Google Scholar
  37. Larrivee, J.A. 1958b. A history of computers II. The Mathematics Teacher 51 (7): 541–544.Google Scholar
  38. Lumsdaine, A.A. 1962. UNESCO educational media conference recommendations. Audio Visual Communication Review 10 (6): 338–342.Google Scholar
  39. McCorduck, P. 1979. Machines who think: A personal inquiry into the history and prospects of Artificial Intelligence. San Francisco, CA: W. H. Freeman and Company.Google Scholar
  40. Milech, D.K., G.R. Kirsner, and B. Waters. 1993. Applications of psychology to computer-based tutoring systems. International Journal of Human-Computers Interaction 5 (1): 23–40.Google Scholar
  41. Nwana, H.S. 1990. Intelligent tutoring systems: An overview. Artificial Intelligence Review 4: 251–277.Google Scholar
  42. Nwana, H.S. 1993. An approach to developing intelligent tutors in mathematics. Computers & Education 20 (1): 27–43.Google Scholar
  43. O’Shea, T. 1979. A self-improving quadratic tutor. International Journal of Man-Machine Studies 11 (1): 97–124.Google Scholar
  44. Papert, S. 1980. Mindstorms: Children, computers, and powerful ideas. New York, NY: Basic Books.Google Scholar
  45. Räsänen, P. 2015. Computer assisted interventions on basic number skills. In The Oxford handbook on numerical cognition, ed. by R. Coohen Kadosh and A. Dowker, 745–766. Oxford, UK: Oxford University Press.Google Scholar
  46. Schlauch, W.S. 1940. The use of calculating machines in teaching arithmetic. The Mathematics Teacher 33 (1): 35–38.Google Scholar
  47. Schoenfeld, A.H. 2001. Mathematics education in the twentieth century. In Education across the century: The centennial volume. Part I, ed. by L. Corno, 239–278. Chicago, IL: National Society for the Study of Education.Google Scholar
  48. Shult, D. 1981. Calculator use in schools. In Calculators in the classroom: With applications for elementary and middle school teachers, ed. D. Moursund, 181–183. New York: Wiley.Google Scholar
  49. Shute, V.J., and J. Psotka. 1994. Intelligent tutoring systems: Past, present and future. Interim Technical Paper for Period April 1M3–April M4. Air Force Materiel Command Brooks Air Force Base, Texas. Retrieved from
  50. Skinner, B.F. 1965. Reflections on a decade of teaching machines. In Teaching machines and programed learning II, data and directions, ed. R. Glaser, 4–20. Washington, D.C.: National Education Association.Google Scholar
  51. Skinner, B.F. 1986. Programmed instruction revisited. Phi Delta Kappan 68 (2): 103–110.Google Scholar
  52. Spencer, D.D. 1968. Computers: Their past, present, and future. The Mathematics Teacher 61 (1): 65–75.Google Scholar
  53. Spiers, G.F. 1996. An analogical reasoning based mathematics tutoring system. Unpublished doctoral dissertation, Computing Department Lancaster University, Lancaster, Great Britain.Google Scholar
  54. Stolurow, L.M., and D. Davis. 1965. Teaching machines and computer-based systems. In Teaching machines and programed learning II, data and directions, ed. R. Glaser, 162–212. Washington, D.C.: National Education Association.Google Scholar
  55. Suppes, P., and M. Jerman. 1969. Computer assisted instruction at Stanford. Educational Technology 9: 22–24.Google Scholar
  56. Tobin, M.J. 1968. Teaching machines in programmed instruction. A contribution to the UNESCO seminar on Programmed Instruction, Varna, Bulgaria, August 1968.Google Scholar
  57. Trafton, P.R., and M.N. Suydam. 1975. Computational skills: A point of view. The Arithmetic Teacher 22 (7): 528–537.Google Scholar
  58. Uhr, L. 1969. Teaching machine programs that generate problems as a function of interaction with students. In ACM '69: Proceedings of the 1969 24th national conference, 125–134.Google Scholar
  59. VanLehn, K. 1990. Mind Bugs: The origins of procedural misconceptions. Cambridge, MA & London, UK: The MIT Press, A Bradford Book.Google Scholar
  60. Voogt, J., P. Fisser, J. Good, P. Mishra, and A. Yadav. 2015. Computational thinking in compulsory education: Towards an agenda for research and practice. Education and Information Technologies 20: 715–728.Google Scholar
  61. Vygotsky, L. 1978. Mind in society. Boston, MA: Harvard University Press.Google Scholar
  62. Weaver, F.J., and M.N. Suydam. 1972. Meaningful instruction in mathematics education. ERIC Information Analysis Center for Science, Mathematics, and Environmental Education, Columbus, Ohio.Google Scholar
  63. Winograd, T. 1991. Thinking machines: Can there be? Are we? In The boundaries of humanity: Humans, animals, machines, ed. J.J. Sheehan, and M. Sosna, 198–223. Berkeley, CA: University of California Press.Google Scholar
  64. Woody, C. 1935. Arithmetic. Review of Educational Research 5 (1): 14–30.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.University of WindsorWindsorCanada

Personalised recommendations