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Toward a Visually-Oriented School Mathematics Curriculum pp 59–97Cite as

Visual Roots of Mathematical Cognitive Activity

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

Abstract

Fischbein (1977) in the opening epigraph is certainly correct in pointing out our natural predisposition toward constructing images in order to make sense of some knowledge that appears to us perhaps initially in either linguistic or alphanumeric form. In the case of Gemiliano, he liked mathematics despite his many struggles with its symbolic aspect because in most cases he understood what was happening, at least visually. I should note that visualizing facts and images does not necessarily imply the use of pictures alone. They could also be routed propositionally, that is, in either linguistic or algebraic form. But whether those images take the shape of pictures or language, I underscore a basic problem some learners have in the case of school mathematical objects, concepts, and processes, that make sense despite the absence of any natural mapping with the real world.

I can do this!

(Gemiliano, Grade 2, 7 years old)

The human mind is inclined, naturally, to visualize facts. We think more easily with images because we are used to thinking of material objects.

(Fischbein, 1977, p. 155)

More generally, to see means to see in relation.

(Arnheim, 1971, p. 54)

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Notes

  1. 1.

    Two points are worth noting. First, Miller (1990) spoke about visualizing that arises from “non-veridical perception … simply the experience of visualizing something … in the absence of an actual object, scene, or person” (p. 4). For example, in (pure) mathematics, it is common among mathematicians to visualize objects that have no corresponding physical existence. However, their reification rests on an axiomatic, deductive system, which then allows us to classify them under structured visual representations. Second, the manner in which we characterize Level 0 visualizing is different from Nemirovsky and Ferrara’s (2009) notion of mathematical imagination. I share their view that imagination causes individual learners to “entertain possibilities for action” (p. 159), including the distinction they developed between pure and empirical possibilities with the latter referring to actions drawn from empirical evidence and the former to actions based on an unconditional acceptance of some assumptions. But then they went a step further in situating “mathematical imagination” under pure possibilities that have all the qualities of “logical necessity in all its deductive and inductive modalities” (Nemirovsky & Ferrara, 2009, p. 160). In my classification, what they consider to be mathematical imagination is a negotiated form of visualization, which could be either formational or transformational in context.

  2. 2.

    Gibson (1986/1979), of course, was referring to visual perception involving everyday objects in a person’s (or an animal’s) environment. Land, for example, naturally affords particular kinds of animals to physically move or pose in ways that adapt to their intentions. In the case of mathematical objects, visual affordances of objects require an interpretive act of discerning and constructing some intention relevant to the objects.

  3. 3.

    Begle’s (1979) notion of mathematical objects had him classifying various kinds of mathematical ideas, as follows: (1) facts (notation-based and deduced ones); (2) concepts; (3) operations (assignment of meaning from object to object); and (4) principles (relationships). Leushina (1991) also wrote about mathematical objects but in the context of visual aids. Leushina’s classification of aids depends on “how the surrounding reality is reflected,” as follows: (1) natural; (2) representational; and (3) graphic. My notion of mathematical objects does not conflate object and the meanings, associations, and ontological sources we derive from them. I leave aside conceptual meanings and intentions in favor of what I consider to be the raw dimension in which inscriptions and physical or material manifestations generally are reified forms of their ideal referents.

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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 Roots of Mathematical Cognitive 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_3

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