Abstract
In this closing chapter, I should point out that we certainly have come a long way since the time Klotz (1991) asserted that “visualization has a more important role to play in mathematics education” and the need to be “willing to ask hard questions about this approach” (p. 103). Research data from a variety of sources at least in the last 20 years or so should help us deal with a nagging issue on visualization that Taussig (2009) has succinctly captured for us in the following way: “But to get to basics, why draw?1” (p. 265).
Simply put, the visual teaches us to think with the body.
(Sherwin, Feigenson, & Spiesel, 2007, p. 147)
The advantage of verbal formulations is that they can conform to semantic or logical rules, but a preoccupation with syntactical features of representation means that we still lack an understanding of how visual thinking works in conjunction with language-based reasoning.
(Gooding, 2006, p. 41)
I see the growth of mathematical knowledge as a process in which an unrigorous reasoning-practice, a scattered set of beliefs about manipulations of physical objects, gives rise to a succession of multi-faceted practices through rational transitions, leading ultimately to the mathematics of today.
(Kitcher, 1983, p. 226)
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Notes
- 1.
Taussig’s (2009) reflection about the picture/text hierarchy below is worth noting in light of our goal in this book of suturing the split:
[T]he hiatus or no-man’s land between picture and text in the anthropological tradition raises a further question as to the general devaluation of drawing in relation to reading and writing in modern Western cultures and maybe in many other cultures as well. We do everything to get children to read, write, and speak well. But why not draw too? Shortly after I wrote this I drove to the supermarket close to where I live in upstate New York past a sign on the road. It read: “Summer Reading Camp.”
(Taussig, 2009, p. 268; italics added for emphasis)
- 2.
I borrowed these terms from Tucker (2006) who used them in the context of the history of visual representations in science, in particular, nineteenth century scientific photographs. The use of pretty pictures conveys the need to popularize scientific discourses and “the packaging of scientific concepts for mass audiences” (p. 117). By illustration fallacy, it refers to “the mistake of assuming that illustrations produced ‘outside’ of professional science lack scientific significance or value” (p. 117).
- 3.
I have intentionally left out research results from studies involving statistics, geometry, and technology in mathematics learning. But there are interesting overlaps in terms of findings and implications. Statistics and geometry in the school mathematics curriculum fundamentally rely on visual representations, so the visual status in these content strands is not as problematic as the case with the algebra and number sense strands. Certainly, there is much instructional and psychological knowledge that could be gained from research done in these two areas, so I refer readers to exemplary syntheses on geometry understanding (which includes spatial understanding) and statistics learning (e.g., Battista, 2007; Clements & Battista, 1992; Gattis, 2001; Owens & Outhred, 2006; Shaughnessy, 1992, 2007). Concerning technology in mathematics learning, I refer readers to several syntheses of research that also discuss the central role of visual representations in mathematical knowledge acquisition (Ferrara, Pratt, & Robutti, 2006; Laborde, Kynigos, Hollebrands, & Strässer, 2006; Kaput, 1992; Mariotti, 2002; Yerushalmy & Chazan, 2002; Zbiek, Heid, Bume, & Dick, 2007).
- 4.
Gregory (2005) also surfaces the continuous relationship between image and knowledge and construction (i.e., what “it” is and what “it” is for). He writes:
We see a table as something to support things, and as made of hard scratchable inflammable wood we have learned about from years of interactive experiments. A picture of a table calls up this object-knowledge so it looks almost real – yet pictures are very odd. … When we look at a familiar object we interact with it by appreciating its potential uses or functions. Its picture calls up the knowledge of functions; but at the same time we are warned, by noting that it is merely a picture, not to expect anything materially useful from it.
(Gregory, 2005, p. 120)
- 5.
I am using the term here in a metaphorical manner. But as an aside I refer readers to Gergen’s (2002) very interesting article in which he explored the phenomenon of absent presence – “present but simultaneously rendered absent … erased by an absent presence” (p. 227) – resulting from our “diverted or divided consciousness invited by communication technology” (p. 227) and cyber-driven culture.
- 6.
The terms generalization, abstraction, and gradual abstracting are conceptually related to the three types of abstracting activity offered by Brook (1997), that is, abstracting out, abstracting away, and building abstraction, respectively.
- 7.
Radford’s (2009) notion of sensuous cognition encompasses the emotional dimension that Peirce has noted about the term sensuous relative to abductive action. As an “alternative approach to classical mental [and rational] views of cognition,” Radford characterizes sensuous modes of thinking in mathematics in “multimodal material” terms, which involve “a sophisticated coordination” of speech and symbols, gestures, body, and actions performed on cultural tools (p. 111) in order to deepen our understanding of how such “ephemeral symptoms [do not merely] announc(e) the imminent arrival of abstract thinking, but genuine constituents of it” (p. 123). A good example of this sensuous cognitive action to pattern generalization is shown in Fig. 5.5. Second grader Dexter explained the structure of his generalization by pointing out the necessary parts using his right hand. In this book, I addressed and emphasized various aspects of what I might classify as visual rational cognitive action in the context of sociocultural learning despite the appearance of similarities in the manner Radford (2009) developed his sensuous cognitive perspective. Both perspectives are complementary, of course. Finally, I note Thagard’s (2010) initial exploration of what he calls embodied abduction that appears to share many of the same features of sensuous cognition. Embodied abduction involves the generation of explanatory hypotheses by a convolutary conceptual process that combines information drawn from two or more modality sources.
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Rivera, F.D. (2011). Instructional Implications: Toward Visual Thinking in Mathematics. 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_8
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