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Visual Thinking and Diagrammatic Reasoning

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Toward a Visually-Oriented School Mathematics Curriculum

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

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Abstract

In Chapters 4 and 5, we specifically addressed contexts in which visuoalphanumeric symbols in school algebra and number sense could be interpreted as symbolic entities that have roots in structured visual experiences. We also discussed the significance of progressive symbolization relative to intra- and inter-semiotic transitions that occur from iconic and/to indexical and to symbolic representations. In this chapter, we focus our attention on diagrams that are purposefully constructed to convey visual relationships and mediate in students’ understanding of alphanumeric forms. Examples of such diagrams include tables in pattern generalization, graphs of functions, squares, sticks, and dots in adding and subtracting whole numbers, binary chips and number lines used in understanding integer operations, etc.

It is difficult to decide between the two definitions of mathematics: the one by its method, that of drawing necessary conclusions; the other by its aim and subject matter, as the study of hypothetical state of things.

(Peirce, 1956, p. 1779)

Mathematics requires an inter-subjectively given object. This is supplied in modern mathematics by conceptual systems and in Greek mathematics by the diagram.

(Netz, 1998, p. 38)

Plainly the movement to accord diagrams a substantial role in mathematics is crucial to a philosophy of real mathematics.

(Sherry, 2009, p. 60)

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Notes

  1. 1.

    The process of figural parsing and biasing shares many of the characteristics that Gal and Linchevski (2010) identified in relation to the operative apprehension of geometric figures (Duval, 1998), which involves engaging in an appropriate and purposeful “dimensional deconstruction of figures” that leads to “infer[ring relevant] mathematical properties in axiomatic geometry” (Gal & Linchevski, 2010, p. 180).

  2. 2.

    Swafford and Langrall (2000) found differing representational effects between presented and self-generated tables of values in patterning activity among 10 sixth-grade students prior to formal instruction. They note:

    Tables seemed to be more useful when students constructed them to make sense of the problem. However, when the interviewer provided a table for the student to complete or to examine, the table seemed to be more of a distraction than an aid, diverting students’ focus from the context of the problem to a string of numbers” (pp. 106–107).

    In my 3-year study with my middle school students, however, those student-constructed tables that merely show common difference (e.g., Fig. 5.3b) were problematic because they funneled my students to a particular, narrow form of numerical strategy that encouraged mostly constructive standard generalizations with no room for more creative and other complex forms of generalization (e.g., constructive nonstandard; deconstructive). The proposed inductive-structuring tables (e.g., Fig. 5.3a) as an alternative diagrammatic representation help overcome issues that Swafford and Langrall (2000) identified as unproductive and ineffective actions relevant to table use [“hinder(ing) their abilities to recognize and describe the relationship between dependent and independent variables implicit in the situation;” “cloud(ing) rather than clarify(ing) the students’ recognition of a relationship between the independent and dependent variables;” “draw(ing) students’ attention to the recursive relation between consecutive values of the dependent variable instead of the relation between the independent and dependent variables”; “forc(ing) an artificial relation between the numbers in the table with no regard for the context of the situation” (pp. 105–106)].

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  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). Visual Thinking and Diagrammatic Reasoning. 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_6

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