Abstract
In writing this book, I have definitely stood on the shoulders of giants whose works on various aspects of mathematical visualization have enriched our understanding of how students actually learn mathematics. The impressive critical syntheses of research studies on visualization in mathematics by Presmeg (2006) and Owens and Outhred (2006), which have been drawn from the annual peer-reviewed proceedings of the International Group for the Psychology of Mathematics Education (IGPME) over a period that spans three decades (1976–2006), provide a comprehensive list of important contributors whose thoughts are reflected in various places in this book. Twenty years ago, the Mathematical Association of America published a visual-driven monograph edited by Zimmermann and Cunningham (1991) that consists of reflective essays by, including references to other, researchers who then began the exciting task of exploring ways to visualize abstract mathematical objects via the power of computer software tools that could support and mediate in the development of advanced mathematical concepts and processes.
This is awesome.
(Nikki, Grade 2, 7 years old)
As classroom observations and teacher interviews continued, it seemed that [the ten middle grades] teachers described their use of manipulatives as fun and distinct from their regular teaching of mathematics. Although these distinctions emerged subtly, a very clear indication of this occurred halfway through the year. Describing a lesson with manipulatives, Joan said, “Sometimes I think that they are just having fun, but I don’t mind because eventually we’ll get to the real math part” (interview 2). Later in the same interview Joan stated, “When we’re doing hands-on stuff they’re having more fun, so they really don’t think about it as being math” (interview 2). …. Not only did teachers appear to distinguish between “fun math” lessons where manipulatives were used and “real math” lessons where traditional paper-and-pencil methods were used, but they also made distinctions between parts of individual lessons. For example, the manipulatives may be used for exploration at the beginning or “fun math” part of a lesson, or they may be used in an activity or a game after the mathematics content was taught; but during the teaching of specific skills or content, paper-and-pencil methods were used to teach and practice the “real math.”
(Moyer, 2001, p. 187)
The lesson of the dichotomies should now be clear: they demand the “and” of intersection. Geometrical intuition never gets far without analytic abstraction, and vice versa.
(Wise, 2006, p. 80)
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Notes
- 1.
Situating the nature and context of thinking that was taking place in the 1990s outside the confines of mathematics education, Mitchell’s (1994) influential book, Picture Theory, cushioned the decade point within the “pictorial [or iconic or visual] turn in contemporary culture, the widely shared notion that visual images have replaced words as the dominant mode of expression in our time,” which should then necessitate “a new initiative called visual culture (the study of human visual experience and expression)” (italics added; Mitchell, 2005, p. 5). Mitchell, then, of course, was not recommending but problematizing the proposed wholesale shift from word to image by surfacing the reality-creating power of the latter via imaging technologies and the theories and attitudes people have about pictures/visuals/images and their relationships with the corresponding verbal representations. Brown’s (2002) thoughts below, which basically ground mathematics as being prior to imagination, offer a complementary perspective to Mitchell’s thoughts about the ontological power of images:
It is mathematics which inspires the imagination, as totally new forms and almost unbelievable patterns and structures are revealed. These fruits of the imagination, verified through the tough testing of logic and calculation, have what has been described as an “unreasonable effectiveness in the physical sciences.”
(Brown, 2002, pp. 157–158)
- 2.
Certainly the visual/symbolic divide, O’Halloran (2005) notes, “has a long history” (p. 129). O’Halloran’s trace begins with traditionalists during the time of Descartes who perceived the visual as a heuristic tool and the symbolic as the rigid road toward a valid proof. She then points to Davis (1974) as representative of mathematicians who hold the opposite view that visuals could be considered a valid form of proof leading to theorem-of-the-perceived kind of arguments. O’Halloran also draws on the work of Galison (2002) who sees the divide as an oscillating phenomenon and Shin (1994) who interprets mathematicians’ incredulity toward the visual medium as being all about constructing diagrams that are loaded with errors, imprecise, narrowly generalizable, and incomplete.
- 3.
In the US history of school mathematics, in particular, it is interesting to note how nineteenth century textbooks have found the use of manipulatives to be a valuable complementary pedagogical tool to the inductive method espoused in arithmetic and problem solving (Michalowicz & Howard, 2003). But then its status, including visualization, more generally, seemed to have been afloat in twentieth century algebra and geometry textbooks (Donaghue, 2003). Rousseau and Pestalozzi, as well as the emphasis on real-life problem solving, influenced mathematics pedagogy in the nineteenth century, which could not be upheld at least in the twentieth century context as a result of the mathematics community’s preoccupation toward developing an axiomatic structural approach to teaching mathematics.
- 4.
- 5.
A similar progressive view is emerging in accounts of growth in scientific knowledge. For example, I refer readers to Maienschein’s (1991) interesting analysis of drawings involving cell development, which depicts visualizations that progressively shift from photographs (as presented data) to diagrams (as represented data).
- 6.
Skemp’s progressive model reflects inductive approaches to word learning whereas more recent progressive accounts of Cobb and Gravemeijer ground their model in authentic (real and experientially real) mathematical activity that sees changes in notating, symbolizing, and abstracting as effects of mathematical reality construction.
- 7.
Peirce’s hypostasized abstraction, this “essential part of almost every really helpful step in mathematics” (1976, p. 160), is similar to Skemp’s (1987/1971) account of concept or word acquisition, which involves the formal “naming” of a category (i.e., the subject) that represents a set of instances having a shared common property or attribute (i.e., the predicate). For example, “twoness” is a term – a “new abstract object” (Otte, 2003, p. 219) – that hypostasizes concrete instances involving any two objects (indeed for Peirce cardinal numbers represent hypostasized abstractions drawn from a predicate of a collection. Peirce writes:
A term denoting a collection is singular, and such a term is an “abstraction” or product of the operation of hypostatic abstraction as truly as is the name of the essence. … Indeed, every object of a conception is either a signate individual or some kind of indeterminate individual. Nouns in the plural are usually distributive and general; common nouns in the singular are usually indefinite.
(Peirce, 1934, p. 299)
Otte (2003) provides additional examples such as the notion of a set “whose mode of existence depends on the existence of other fundamental things … which is based on the existence of its own elements” (p. 218). Imaginary numbers as an object were used to develop the theory of complex functions. “Again and again,” Otte (2003) notes about hypostasized abstractions, “a construction or an algorithmic procedure is taken as an object to be incorporated into another construction or procedure” (p. 220).
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Acknowledgment and Dedication
I wish to express my gratitude to the National Science Foundation (NSF) that provided funding for me to engage in longitudinal classroom work from 2005 to 2010 (under NSF Career Grant #0448649). Results that are reported in this book are all mine and do not reflect the views of the foundation. This book is dedicated to the students and teachers in my 2005–2010 study.
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Rivera, F.D. (2011). Introduction. 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_1
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