Abstract
In this chapter, we explore cultural and blind-specific issues and their implications to visual thinking in mathematics. Issues on culture refer to those patterns of knowledge and skills, tool use, thinking, acting, and interacting that are favored by, and specific to, groups that support them. It is interesting to consider the possibility that students’ developing explicit knowledge in a lifeworld-dependent context interacts with macroconditions such as taken-as-shared cultural practices, which are likely to influence the way they perform visualization without them being aware of them.
Every culture, with its privileges or taboos of seeing, shapes a certain way of thinking, as it is in turn shaped by norms or vetoes of looking.
(Beltin, 2008 p. 189)
It is, of course, the norm rather than the exception that more than one sense modality contributes to the perception of an object. We hear, see, smell, and feel the same object. That is not because all senses provide the same information, but because the contributions from different modalities converge and overlap sufficiently to be felt as “same” rather than as “different.”
(Millar, 1994 p. 47)
Overall, neuropsychological reports, neuroimaging and behavioral findings support the view that visuo-spatial processes and numerical representation are intimately related. This relationship may constitute an early and fundamental link which, although partially shaped by our cultural constraints, remains an essential component of our cognitive architecture.
(de Hevia, Vallar, & Girelli, 2008 p. 1370).
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- 1.
For Baudrillard (1981/1972), the rationality of signs is seen in its ability to distinguish between signifier and signified and
functions as the agent of abstraction and universal reduction of all potentialities and qualities of meanings that do not depend on or derive from the respective framing, equivalence, and specular relation of a signifier and signified. This is the directive and reductive rationalization transacted by the sign – not in relation to an exterior, immanent “concrete reality” that signs would supposedly recapture abstractly in order to express, but in relation to all that which overflows the schema of equivalence and signification; and which the sign reduces, represses, and annihilates in the very operation that constitutes it. … The rationality of the sign is rooted in its exclusion and annihilation of all symbolic ambivalence on behalf of a fixed and equational structure.
(Baudrillard, 1981/1972, p. 149).
I see social and cultural practices as providing conditions that engender the possibility of reduction, repression, and annihilation of sign ambivalence. The well-defined character of signs used in formal and institutional mathematical knowledge indicates this necessary elimination of ambivalence.
- 2.
The term culture has many meanings and is certainly more complicated than the one that I describe in this chapter [cf. De Abreu, 2000; Lerman, 2001; Nunes 1992; Rivera & Becker, 2007b; special issue of the Journal of Intercultural Studies on culture in mathematics education edited by FitzSimons (2002)]. Byrne et al. (2004), for example, identify six dimensions of culture that merit individual analysis – that is, culture as pattern (social transmission processes); a sign of mind (solidified cognitive habits drawn from social transmission); a bonus (knowledge and practices resulting from social transmission); inefficiency (negative consequent actions resulting from social transmission); physical products (cultural actions and tools); and meaning (rituals). My characterization of culture is pragmatic; it has useful elements that allow me to explain the ideas I pursue later in relation to cultural forms of seeing in mathematics.
- 3.
See footnote 4 in Chapter 2 (p. 55).
- 4.
Zebian’s (2008) “ecologically-informed approach to numeric cognition” involves coordinating social and cognitive approaches. The cognitive approach “offers a way of thinking about how surface notation is recognizable and decoded and how these processes are related to semantically-based processes” (p. 362). The sociocultural approach “is acutely sensitive to the socially situated cognitive demands of numeric practices” (p. 362).
- 5.
Were (2003) provides a thorough and engaging discussion of the object of kapkap, a patterned-shell ornament in the Western Island Melanesia, which he uses to demonstrate the rigor of mathematical thinking that comes with understanding its social and cultural significance. While the object conveys to these people certain “everyday and ritual performances,” it is the “translation from mental to material form that mobilizes mathematical thinking and spatial reasoning” (p. 42).
- 6.
Raju (2001) also notes that Buddhists value inference as a valid means of validation but reject authoritative testimony, while Naiyayikas value all three (empirical, inference, authoritative testimony) means, including analogy (p. 328).
- 7.
Ginsburg et al. (2003) also suggested the hypothesis that the teachers involved in the study influenced what the children were doing during free play. But this perspective was discounted due to the fact that the teachers observed were not seen interacting with the children during free play.
- 8.
Morin became fully blind at age 6.
- 9.
Females were excluded as the authors’ way of coping with less complicated data at the time of the study (i.e., holding the sex variable constant).
- 10.
Ginns’s (2005) meta-analysis of 43 experimental studies prior to 2004 shows an overall strong modality effect (with moderation effects in some aspects), that is, “across a broad range of instructional materials, age groups, and outcomes, students who learned from instructional materials using graphics with spoken text outperformed those who learned from a graphics with printed text” (p. 326). A nonmathematical context that involves the use of nonvisual modality modes of learning among blind subjects involves studies in mobility (e.g., use of echoes, guide dogs, and long canes; sound cues; felt cues; electronic travel aids and ultrasonic echolocating prosthesis; cf. Strelow, 1985; Veraart & Wanet-Defalque, 1987).
- 11.
Giroux was completely blind at age 11.
- 12.
While recent haptic interface technologies in virtual reality provide externally induced compensatory strategies for blind learners due to loss in visual ability, they are also meant for them to develop useful and detailed spatial cognitive maps in long-term memory that will help improve their mobility and orientation skills (cf. Lahav, 2006).
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Rivera, F.D. (2011). Cultural and Blind-Specific Issues and Implications to 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_7
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