Abstract
In some circles of mathematics education, repetition and rote are somehow conflated in terms of their pedagogical uses and ramifications. In this paper, I argue for the separation of the two, relying upon a framework suggested by Martin Buber’s I–Thou ontology. In the presentation of Buber’s ideas, I highlight the notion of will-as-would-join-with-grace, to be contrasted with plain will. The merit of repetition in teaching and learning, as I argue, is not in automaticity—the common rationale—but in fostering and supporting a deepened sense of connection and/or intimacy to the object under study.
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Notes
Naturally, one could argue that each is part of a whole and cannot be separated from the other and that, further, the iterative nature of their interaction makes such a distinction contrived, or worse, misleading. Yet, for the sake of organization and analysis, I have decided to consider the two separately.
By “passive,” I do not mean something necessarily negative, such as in a fear-based passivity that would be equated more with withdrawal. Perhaps “receptive” gives a better sense of what is meant, although I will use the two words interchangeably, for a primary definition of “passive” is “(adj.) acted upon rather than acting or causing action”—which gives more of the neutral tinge that I intend. In connection, Wong (2002) argues that educational research and policy have neglected this “passive” element of learning to their own detriment.
This is not to say that “will” is to be construed necessarily as self doing unto other, but more simply, as self doing, or self bringing forth. For example, a person can bring forth an attitude of receptivity so as to receive other. In this case, there is no doing unto other, but simply the doing of the embodying of receptivity. While not a doing unto other, it is what one brings forth into the interaction.
Note that I am using the word “preparation” in a way different from the idea of planning and anticipating for, thinking ahead, or rehearsing. Instead, I mean something closer to “precondition.” Only, I prefer the word “preparation” because the word implies an active element, whereas precondition feels too static for the act of readying oneself for the undergoing.
As I have already argued, the role of will is not only in preceding grace, but that there is instead an iterative relationship between the two. If one takes will as the affective aspect that a learner brings to problem solving and maps grace to the cognitive undergoing of understanding, one is reminded of Goldin’s (2002) comments, that “When individuals are doing mathematics, the affective system is not merely auxiliary to cognition—it is central. However, affect as a representational system is intertwined with cognitive representation [italicized in the original]” (p. 60).
This is not to say that all students will necessarily reify such understandings, but that the understandings can more likely be experienced due to the simpler computational nature of the problems, and as Watson and Mason describe, “through discussion, [they can be] brought to articulation” (p. 106).
This particular attitude toward repetition as being potentially different from what US educators may think of as rote is seen among teachers in Japan, China, and Hong Kong (Dahlin & Watkins, 2000). Similarly, Marton, Wen, & Fong (2005) found that many Asian students distinguish between meaningful and rote memorization, the former being characterized by an accompanying understanding.
References
Bereiter, C., & Scardamalia, M. (1989). Intentional learning as a goal of instruction. In L. B. Resnick (Ed.), Knowing, learning, and instruction: Essays in honor of Robert Glaser. Hillsdale: Lawrence Erlbaum.
Buber, M. (1970). I and Thou (translated by W. Kaufman). New York: Charles Scribner.
Dahlin, B., & Watkins, D. (2000). The role of repetition in the processes of memorising and understanding: A comparison of the views of German and Chinese secondary school students in Hong Kong. British Journal of Educational Psychology, 70, 65–84.
Dewey, J. (1934). Art as experience. New York: Perigree.
Gattegno, C., Powell, A., Shuller, S., & Tahta, D. (1981). A seminar on problem solving. For the Learning of Mathematics, 2(1), 42–46.
Goldin, G. A. (2002). Affect, meta-affect, and mathematical belief structures. In G. C. Leder, E. Pehkonen, & G. Törner (Eds.), Beliefs: A hidden variable in mathematics education? (pp. 59–72). Dordrecht: Kluwer.
Handa, Y. (2003). A phenomenological exploration of mathematical engagement: Approaching an old metaphor anew. For the Learning of Mathematics, 23(1), 22–28.
Handa, Y. (2011). What does understanding mathematics mean for teachers? Relationship as a metaphor for knowing. New York: Routledge.
Hawkins, J. N. (1994). Issues of motivation in Asian education. In H. F. O'Neil & M. Drillings (Eds.), Motivation: Theory and research (pp. 101–118). Hillsdale: Laurence Erlbaum.
Heller, A. (1979). A theory of feelings. Assen: Van Gorcum.
Hess, R. D., & Azuma, H. (1991). Cultural support for schooling: Contrasts between Japan and the United States. Educational Researcher, 20(9), 2–8. 12.
Ingarden, R. (1961). Aesthetic experience and aesthetic object. Philosophy and Phenomenological Research, 21(3), 303–323.
James, W. (1890/1950). The principles of psychology (vol. 1). New York: Dover (Original work published 1890).
Kang, W., & Kilpatrick, J. (1992). Didactic transpositions in mathematics textbooks. For the Learning of Mathematics, 12(1), 2–7.
Khisty, C. J. (2002). The significance of dialogue in problem solving using Martin Buber's triad—"I–Thou–We". Paper presented at the 26th Conference of the International Group for the Psychology of Mathematics Education, Norwich, UK.
Ko, P. Y., & Marton, F. (2004). Variation and the secret of the virtuoso. In F. Marton & A. B. M. Tsui (Eds.), Classroom discourse and the space of learning (pp. 72–93). Mahwah: Erlbaum.
Krause, E. (1975/1986). Taxicab geometry: An adventure in non-Euclidean geometry. New York: Dover (Original work published 1975).
Langer, E. (1989). Mindfulness. Reading: Addison-Wesley.
Marton, F., Wen, Q., & Wong, K. C. (2005). 'Read a hundred times and the meaning will appear…' Changes in Chinese University students' views of the temporal structure of learning. Higher Education, 49, 291–318.
Neyland, J. (2004). Towards a postmodern ethics of mathematics education. In M. Walshaw (Ed.), Mathematics education within the postmodern (pp. 55–73). Greenwich: Information Age.
Owen, E., & Sweller, J. (1985). What do children learn while solving mathematics problems? Journal of Educational Psychology, 77, 272–284.
Resnick, L. B., & Neches, R. (1984). Factors affecting individual differences in learning ability. In R. J. Sternberg (Ed.), Advances in the psychology of human intelligence (Vol. 2, pp. 275–323). Hillsdale: Lawrence Erlbaum.
Schoenfeld, A. (2004). The math wars. Educational Policy, 18(1), 253–286.
Tuckey, C. (1904). Examples in algebra. London: Bell.
Winchester, I. (1990). Introduction—Creativity, thought, and mathematical proof. Interchange, 21(1), i–vi.
Watson, A., & Mason, J. (2006). Seeing an exercise as a single mathematical object: Using variation to structure sense-making. Mathematical Thinking and Learning, 8(2), 91–111.
Wong, D. (2002). The opposite of control: Deweyan aesthetics, motivation, and learning. Paper presented at the American Educational Research Association, New Orleans, LA.
Wu, H. (1997). The mathematics education reform: Why you should be concerned and what you can do. The American Mathematical Monthly, 104, 946–954.
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Portions of this manuscript are excerpted from Handa (2011).
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Handa, Y. Teasing out repetition from rote: an essay on two versions of will. Educ Stud Math 79, 263–272 (2012). https://doi.org/10.1007/s10649-011-9343-0
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DOI: https://doi.org/10.1007/s10649-011-9343-0