The Methods, Benefits and Limitations of Indoctrination in Mathematics Education

Abstract

With mathematics education dominated by practices of standardized test alignment, the quest for mathematical understanding is adrift despite so much talk about the importance of STEM subject matters. To add to the confusion there has long been worries among educators about any curricular programs or instructional practices that can be labeled indoctrinative. The worse combination of practices seems afoot. Standardized testing has led to unmitigated indoctrination throughout much of public education yet the term is indoctrination is shunned as bad pedagogy. The argument here is that entry into any new discipline from times of antiquity to the present necessarily involves responsible and warranted indoctrination. Acknowledging the fruitful role of indoctrination early on should enable educators to benchmark when the practice has become excessive or otherwise inappropriate. Threshold moments demarcate between early instruction enabling the study of math from advanced and independent study wherein students’ development skills of reflection and evaluative deliberation relevant to mathematic thinking.

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References

  1. Aczel, A. (2015). Finding zero: A mathematician’s odessy to find the origin of numbers. New York, NY: St. Martin’s Press.

    Google Scholar 

  2. Adkins, P. (2004). Galileo’s finger: The ten great ideas of science. Oxford: Oxford University Press.

    Google Scholar 

  3. Alexander, A. (2014). The infinitesimals: How a dangerous mathematical theory shaped the modern world. London: Oneworld Pub.

    Google Scholar 

  4. Artstein, Z. (2014). Mathematics and the real world: The remarkable role of evolution in the making of mathematics. Amherst, NY: Prometheus Press.

    Google Scholar 

  5. Bailey, R. (2010). Indoctrination. In R. Bailey, R. Barrow, & D. Carr (Eds.), Sage handbook of philosophy of education. London: Sage.

    Google Scholar 

  6. Balter, M. (1998). Why settle down? The mystery of communities. Science, 20, 1442–1446.

    Google Scholar 

  7. Bartha, P. (2010). By parallel reasoning: The construction and evaluation of analogical arguments. Oxford: Oxford University Press.

    Google Scholar 

  8. Bradford, G. (2015). Achievement. New York, NY: Oxford University Press.

    Google Scholar 

  9. Boesch, C., & Tomasella, M. (1998). Chimpanzee and human cultures. Current Anthropology, 39, 591–614.

    Google Scholar 

  10. Burge, T. (2010). Origins of objectivity. New York, NY: Oxford University Press.

    Google Scholar 

  11. Butterworth, B. (1999). The mathematical brain. London: Macmillan.

    Google Scholar 

  12. Butterworth, B., & Kovas, Y. (2013). Understanding neurocognitive developmental disorders can improve education for all. Science, 340, 300–305.

    Google Scholar 

  13. Callan, E., & Arena, D. (2009). Indoctrination. In H. Siegel (Ed.), The Oxford handbook of philosophy of education. New York: Oxford University Press.

    Google Scholar 

  14. Chinn, C. A., & Samarapungavan, A. (2011). Distinguishing between understanding and belief. Theory Into Practice, 40(4), 235–241.

    Google Scholar 

  15. Chomsky, N., & Robichaud, A. (2014). Standardized testing as an assault on humanism and critical thinking in education. Radical Pedagogy, 11(1), 54–66.

    Google Scholar 

  16. Cobb, M. (2020). The idea of the brain: The future of neuroscience. New York, NY: Basic Books.

    Google Scholar 

  17. Cook, M. (2020). Sleight of mind: 75 ingenius paradoxes in mathematics, physics and philosophy. Cambridge, MA: MIT.

    Google Scholar 

  18. Crow, M. (1988). Ten misconceptions about mathematics and its history. In W. Aspray & P. Kitcher (Eds.), History and philosophy of modern Mathematics. Minneapolis: University of Minnesota Press.

    Google Scholar 

  19. Csibra, G., & Gergely, G. (2006). Social learning and cognition: The case for pedagogy. In Y. Munkata & M. Johnson (Eds.), Processes in brain and cognitive development. Oxford: Oxford University Press.

    Google Scholar 

  20. Curry, A. (2008). Seeking the roots of ritual. Science, 319, 278–280.

    Google Scholar 

  21. Darling-Hammond, L. (2007) Evaluating “no child left behind.” The Nation. Retrieved from http://www.thenation.com/article/evaluating-no-child-left-behind

  22. D’Agostino, S. (2020). How to free your inner mathematician. New York, NY: Oxford University Press.

    Google Scholar 

  23. Dedekind, R. (1888). Was sind und was sollen die Zablen Braunsch-weig. Viewig: P.III.

    Google Scholar 

  24. DeFelipe, J. (2011). The evolution of the brain, and human nature of cortical circuits, and intellectual creativity. Frontiers in Neuroanatomy, 5, 1–17.

    Google Scholar 

  25. Dehaene, S. (2011). How the mind creates numbers. New York, NY: Oxford University Press.

    Google Scholar 

  26. Dehaene, S. (2020). How we learn: Why brains learn better than any machine. New York, NY: Viking.

    Google Scholar 

  27. Diaconis, P., & Skyrms, B. (2018). Ten great ideas about chance. Princeton, NJ: Princeton University Press.

    Google Scholar 

  28. Dreyfus, H., & Taylor, C. (2015). Retrieving realism. Cambridge, MA: Harvard University Press.

    Google Scholar 

  29. Elgin, C. Z. (2017). True enough. Cambridge, MA: MIT Press.

    Google Scholar 

  30. Ellenberg, J. (2014). How not to be wrong: The power of mathematical thinking. New York: Penguin Press.

    Google Scholar 

  31. Erwig, M. (2017). Once upon an algorithm. Cambridge, MA: MIT Press.

    Google Scholar 

  32. Feferman, A., & Feferman, S. (2008). Alfred Tarski: Life and logic. Cambridge, MA: Cambridge University Press.

    Google Scholar 

  33. Feynman, R. (1985). Surely you are joking Mr. Feynman. New York, NY: W.W. Norton.

    Google Scholar 

  34. Fortnoy, L. (2013). The golden ticket. P, NP and the search for the impossible. Princeton: Princeton University Press.

    Google Scholar 

  35. Gallistel, C. R., Gelman, R., & Cordes, S. (2006). The cultural and evolutionary history of the real numbers. In S. C. Levinson & P. Jaisson (Eds.), Evolution and culture. Cambridge: MIT Press.

    Google Scholar 

  36. Gatchel. (1972). The evolution of a concept. In I. Snook (Ed.), Concepts of indoctrination (pp. 2–22). New York, NY: Routledge.

    Google Scholar 

  37. Gigerenzer, G. (2002). Calculated risk: How to know when numbers deceive you. New York, NY: Simon & Schuster.

    Google Scholar 

  38. Glimcher, P. (2003). Decisions: Uncertainty and the brain. Cambridge, MA: MIT Press.

    Google Scholar 

  39. Goldstein, R. (2005). Incompleteness: The proof and paradox of Kurt Godel. New York: W. W. Norton.

    Google Scholar 

  40. Gopnik, A., Meltzkoff, M. A., & Kuhl, P. (1999). The scientist in the crib: What early learning tells us about the mind. New York: William Morrow Co.

    Google Scholar 

  41. Gopnik, A., & Graf, P. (1988). Knowing how you know: Young children’s ability to identify and remember the sources of their beliefs. Child Development, 59(5), 1366–1371.

    Google Scholar 

  42. Gordon, P. (2004). Numerical cognition without words: evidence from Amazonia. Science, 306(5695), 496–499.

    Google Scholar 

  43. Graham, L., & Kantor, J. (2009). Naming infinity: A true story of religious mysticism and mathematical curiosity. Cambridge, MA: The Belnap Press of Harvard.

    Google Scholar 

  44. Gray, J. (2008). Plato’s ghost: The modernist transformation of mathematics. Princeton: Princeton University Press.

    Google Scholar 

  45. Greene, T. F. (1972). Philosophical essays. In I. A. Snook (Ed.), Concepts of indoctrination (pp. 32–45). New York, NY: Routledge.

    Google Scholar 

  46. Hacker, A. (2016). The math myth: And other STEM delusions. New York, NY: New Press.

    Google Scholar 

  47. Hand, D. (2014). The improbability principle. New York: Scientific American/Farrar, Straus & Giroux Press.

    Google Scholar 

  48. Harris, M. (2015). Mathematics without apologies. Princeton: Princeton University Press.

    Google Scholar 

  49. Harvey, C. W. (2004). Liberal indoctrination and the problem of the community. Synthese, 111(1), 15–16.

    Google Scholar 

  50. Hornstein, L. (2011). The Tarskian turn: Deflationism and the axiomatic turn. Cambridge, MA: MIT.

    Google Scholar 

  51. Ifrah, G. (1985). From one to zero: A universal history of numbers. New York, NY: Viking.

    Google Scholar 

  52. Kjeldsen, T., et al. (2014). The role of history and philosophy in university mathematics education. In M. Matthews (Ed.), International handbook of research in history, philosophy and science teaching (pp. 837–871). Dordrecht: Springer.

    Google Scholar 

  53. Kovas, Y., Haworth, C. M., Petrill, S. A., & Plomin, R. (2007). Mathematical ability of 10-year old boys and girls. Journal of Learning Disabilities., 40, 554–567.

    Google Scholar 

  54. Koretz, D. (2008). Measuring up: What educational testing really tells us. Cambridge MA: Harvard University Press.

    Google Scholar 

  55. Landsman, J., & Gorski, P. (2007). Countering standardization. Educational Leadership, 64(8), 40–44.

    Google Scholar 

  56. Levitin, J. (2014). The organized mind. New York: Dutton.

    Google Scholar 

  57. Lewis-Williams, D., & Pearce, D. (2005). Inside the neolithic mind. London: Thames and Hudson.

    Google Scholar 

  58. Lockhart, P. (2017). Arithmetic. Cambridge, MA: The Belnap Press of Harvard.

    Google Scholar 

  59. Lyons, I. M., & Beilock, S. L. (2011). Numerical ordering ability mediates the relation between number-sense and arithmetic competence. Cognition, 121(2), 256–261.

    Google Scholar 

  60. Mankiewicz, R. (2000). The story of mathematics. Princeton: Princeton University Press.

    Google Scholar 

  61. Maor, E. (1991). To infinity and beyond: A cultural history of the infinite. Princeton: Princeton University Press.

    Google Scholar 

  62. Massey, D. (2002). A brief history of human society. American Sociological Review, 67, 1–29.

    Google Scholar 

  63. Mazur, J. (2005). Euclid in the forest; discovering universal truth in logic and math. New York, NY: Pi Press.

    Google Scholar 

  64. Mazur, J. (2016). Fluke: The math and myth of coincidence. New York: Basic Books.

    Google Scholar 

  65. Mlodinow, L. (2015). The upright thinkers: The human journey from living in trees to understanding the cosmos. New York: Pantheon.

    Google Scholar 

  66. Morris, I., & Macerdo, S. (2015). Foragers, farmers and fossil fuels: How human values evolve. Princeton: Princeton University Press.

    Google Scholar 

  67. Nichols, S., Glass, G., & Berliner, D. (2012). High-stakes testing and student achievement: Updated analyses with NAEP data. Education Policy Analysis Archives, 20(20), 1–30.

    Google Scholar 

  68. Nieder, A. (2019). A brain for numbers. Cambridge, MA: MIT Press.

    Google Scholar 

  69. Pearl, J., & Mackenzie, D. (2018). The book of why: The new science of cause and effect. New York, NY: Basic Books.

    Google Scholar 

  70. Perkins, D. N., Jay, E., & Tishman, S. (1993). New conception of thinking from ontology to education. Educational Psychologist, 28, 67–85.

    Google Scholar 

  71. Pica, P., Lemer, C., Izard, V., & Dehaene, S. (2004). Exact and approximate; arithmetic in an Amazonian indigene group. Science, 306(5695), 499–503.

    Google Scholar 

  72. Posamentier, A. S. (2020). The joy of geometry. New York, NY: Prometheus Books.

    Google Scholar 

  73. Poundstone, W. (1988). The labyrinths of reason: Paradoxes, puzzles, and the frailty of knowledge. New York, NY: Doubleday.

    Google Scholar 

  74. Putnam, H. (2015a). Naturalism, realism and normativity. Journal of the American Philosophical Association, 1(2), 312–328.

    Google Scholar 

  75. Putnam, H. (2015b). Intellectual autobiography of Hilary Putnam. In R. Auxier, D. Anderson, & L. Hahn (Eds.), The philosophy of Hilary Putnam. Chicago: Open Court.

    Google Scholar 

  76. Ravitch, D. (2010). The death and life of the great American school system: How testing and choice are undermining education. New York, NY: Basic Books.

    Google Scholar 

  77. Rowlands, S. (2014). Philosophy and the secondary school mathematics classroom. In M. Matthews (Ed.), International handbook of research in history, philosophy and science teaching (pp. 705–730). Dordrecht: Springer.

    Google Scholar 

  78. Secolsky, C., Judd, T., Magaram, E., Levy, S., Kossar, B., & Reese, G. (2016). Using think-aloud protocols to uncover misconceptions and improve developmental math instruction: An exploratory study. Numeracy, 9(1), 6.

    Google Scholar 

  79. Siegel, H. (1988). Educating reason: Rationality, critical thinking, and education. New York: Routledge.

    Google Scholar 

  80. Siegel, H. (1989). Relativism refuted: A critical of contemporary epistemological relativism. British Journal of Philosophy of Science, 40(3), 419–427.

    Google Scholar 

  81. Siegel, H., & Smith, M. U. (2019). Must evolutionary education that aims at belief be indoctrinating. Science and Education, 28(9–10), 1235–1247.

    Google Scholar 

  82. Sigman, M. (2017). The secret life of the mind. New York, NY: Little, Brown & Co.

    Google Scholar 

  83. Smith, G. (2014). Standard deviation: Flawed assumptions tortured data and other ways to lie with statistics. London: Overlook Duckworth.

    Google Scholar 

  84. Snook, I. A. (1970). The concept of education. Studies in Philosophy of Education, 7, 65–108.

    Google Scholar 

  85. Snook, I. A. (Ed.). (1972). Concepts of indoctrination. London: Routledge & Kegan Paul.

    Google Scholar 

  86. Sober, E. (2015). Ockham’s razors: A user’s manual. Cambridge: Cambridge University Press.

    Google Scholar 

  87. Sokal, A. (2009). Beyond the hoax: Science, philosophy and culture. Oxford: Oxford University Press.

    Google Scholar 

  88. Soni, J., & Goodman, R. (2017). A Mind at Play: How Claude Shannon invented the information age. New York, NY: Simon & Schuster.

    Google Scholar 

  89. Sosa, E. (2011). Reflective knowledge: Apt belief and rational knowledge. Oxford: Oxford University Press.

    Google Scholar 

  90. Spelke, E. S., Breinlinger, B., Macomber, J., & Jacobson, K. (1992). Origins of knowledge. Psychological Review, 99, 605–632.

    Google Scholar 

  91. Stewart, I. (2013). Visions of infinity. New York, NY: Basic Books.

    Google Scholar 

  92. Strogatz, S. (2019). Infinite powers: The calculus reveals the secrets of the universe. New York, NY: Houghton Mifflin.

    Google Scholar 

  93. Su, F. (2020). Mathematics for human flourishing. New Haven, CN: Yale University Press.

    Google Scholar 

  94. Tammet, D. (2012). Thinking in numbers: How maths illuminates our lives. UK: Hoddei & Stoughton.

    Google Scholar 

  95. Tomasella, M. (2014). A natural history of human thinking. Cambridge, MA: Harvard University Press.

    Google Scholar 

  96. Tversky, B. (2019). Mind in motion. How action shapes thought. New York, NY: Basic Books.

    Google Scholar 

  97. Wagner, P. (1982). The philosopher as teacher: Philosophy in mathematics education. Metaphilosophy, 13(1), 79–90.

    Google Scholar 

  98. Wagner, P. (1986). Philosophical praxis. Teaching Philosophy, 9(4), 291–299.

    Google Scholar 

  99. Wagner, P. (1990). Will education contain fewer surprises for students in the future? In V. Howard (Ed.), Varieties of thinking: Recent research from the harvard philosophy of education research center (pp. 176–211). New York: Routledge.

    Google Scholar 

  100. Wagner, P. (2006). Probability, decision theory, and a curricular approach to developing critical thinking. Journal of Thought, 42(2), 23–38.

    Google Scholar 

  101. Wagner, P. (2011). Isolationism and the possibility of truth. Interchange, 42(4), 389–408.

    Google Scholar 

  102. Wagner, P. (2017). Warranted indoctrination in science education. In M. Matthews (Ed.), Handbook in History and philosophy of science education. Dordrecht, Holland: Springer.

    Google Scholar 

  103. Wagner, P., & Dede, C. (1983). Disciplinary paradigm shifts: A new frontier for future research. World Futures Bulletin, 17(2), 25–29.

    Google Scholar 

  104. Wagner, P., Johnson, D., Fair, F., & Fasko, D. (2017). Focus on thinking. Lanham, MD: Rowman & Littlefield.

    Google Scholar 

  105. Wagner, P., Johnson, D., Fair, F., & Fasko, D. (2018). Thinking ahead. Lanham, MD: Rowman & Littlefield.

    Google Scholar 

  106. Wagner, P., & Lopez, G. (2010). The great conversation and the ethics of inclusion. Multicultural Perspectives, 12(3), 167–172.

    Google Scholar 

  107. Wagner, P. A., & Fair, F. (2020). Education for knowing: Theories of knowledge for effective student building. New York, NY: Rowman & Littlefiled.

    Google Scholar 

  108. Wang, H. (1987). Reflections on Kurt Godel. Cambridge, MA: MIT.

    Google Scholar 

  109. Wheeler, M., & Feghali, I. (1983). Much ado about nothing: Preservice elementary school teachers’ concept of zero. Journal of Research in Math Education, 14, 147–155.

    Google Scholar 

  110. Wiley, R. (2015). Noise matters: The evolution of communication. Cambridge, MA: Harvard University Press.

    Google Scholar 

  111. Wittgenstein, L. (1975). In C. Diamond (Ed.), Lectures in the foundations of mathematics. Ithaca, New York: Cornell University Press.

    Google Scholar 

  112. Yau, S. (2020). The shape of a life: One mathematician’s struggle for the universe’s hidden geometry. New Haven, CN: Yale University Press.

    Google Scholar 

  113. Zellini, P. (2020). The mathematics of the gods and the algorithms of men: A cultural history. New York, NY: Pegasus.

    Google Scholar 

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Wagner, P.A. The Methods, Benefits and Limitations of Indoctrination in Mathematics Education. Interchange 52, 41–56 (2021). https://doi.org/10.1007/s10780-021-09415-7

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Keywords

  • Bench-marking
  • Indoctrination
  • Mathematical understanding
  • Disciplinary thresholds
  • Autonomy
  • Ritualized signaling