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Visuoalphanumeric Representations in Pattern Generalization Activity

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Toward a Visually-Oriented School Mathematics Curriculum

Part of the book series: Mathematics Education Library ((MELI,volume 49))

Abstract

In Chapter 4, we drew on a few examples from my own classroom work in discussing a progressive account of symbol formation in school mathematics. A visually grounded approach provides an alternative and effective route that could assist students in understanding mathematics better. Otte (2007) notes how mathematical knowledge seems to be already “everything [that] just is and thus means itself” (p. 243). The more pressing issue appears to be “not that of rigor but the problem of the development of meaning” (Rene Thom quoted in Otte, 2007, pp. 244–245).

How general is general?

(Bastable & Schifter, 2008, p. 166).

The most important operation of the mind is that of generalization

(Peirce, 1960, p. 34).

(A)lgebra … not as symbol manipulation, not as arithmetic with letters, not even as the language of equations, but as a succinct and manipulable language in which to express generality and constraints on that generality

(Mason, 2008, p. 77).

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Notes

  1. 1.

    Parker and Baldridge (2004) emphasize the need for tables to have real (or experientially real) and predictable contexts with a sufficient number of particular instances in order to generate a reasonable algebraic expression (or formula). For example, the table below with the context about rainfall could not be assessed correctly, that is, even if the expression appears to take the algebraic form (1/2)h, “there is no reason why the rainfall will continue to be given by that expression, or any expression. This question cannot be answered” (p. 90). Other examples include stock market prices and gas prices, where tables could be generated but oftentimes do not lead to correct and justifiable algebraic expressions.

  2. 2.

    This notion conceptually shares Thagard’s (1978) interpretation of Peirce’s abduction as “cover[ing] both the act of arriving at plausible new hypotheses and the act of entertaining them for the sake of further investigation” (p. 166). The end result (i.e., at least in provisional terms) of progressive abduction is an inference to the best explanation (“IBE;” Hartman, 1965).

  3. 3.

    JRB stands for Joanne Rossi Becker, interviewer.

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Authors and Affiliations

  1. Department of Mathematics, San Jose State University, 1 Washington Square, San Jose California, 95192, USA

    Dr. Ferdinand D. Rivera

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Rivera, F.D. (2011). Visuoalphanumeric Representations in Pattern Generalization Activity. In: Toward a Visually-Oriented School Mathematics Curriculum. Mathematics Education Library, vol 49. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0014-7_5

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