Fifty Years of Thinking About Visualization and Visualizing in Mathematics Education: A Historical Overview
This chapter surveys meanings given to the term “visualization” in mathematics, mathematics education, and psychology, and considers the evidence for the oft-heard assertion that mathematics learners tend to prefer to think algorithmically rather than vis
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Fifty Years of Thinking About Visualization and Visualizing in Mathematics Education: A Historical Overview McKenzie (Ken) A. Clements
Abstract This chapter surveys meanings given to the term “visualization” in mathematics, mathematics education, and psychology, and considers the evidence for the oft-heard assertion that mathematics learners tend to prefer to think algorithmically rather than visually. The analysis reveals that students who do very well on penciland-paper “visualization” tests often prefer not to use visual methods when attempting to solve mathematical problems; and those who do not do well on standard visualization tests often describe themselves as “visual thinkers”, and prefer to use visual methods when attempting to solve mathematics problems. The influence of various mathematics educators, and especially Alan Bishop—who thought of visualization in terms of a person’s use of visual images when posing and solving mathematics problems—of Norma Presmeg, and of a group of mainly Israeli mathematics educators who developed the construct “concept image”, is also examined. Views of some mathematicians are also taken into account. In the early 1990s, Zimmermann and Cunningham (Visualization in teaching and learning mathematics, 1991) wrote of how David Hilbert had spoken of two tendencies in mathematics—one that sought to crystallize logical relations, and the other to develop intuitive understanding, especially through “visual imagination” (p. 2). In addressing that theme, Ted Eisenberg and Tommy Dreyfus (Visualization in teaching and learning mathematics, pp. 25–37, 1991) spoke of mathematics students’ preference for “algorithmic over visual thinking” (p. 25). The paper draws special attention to the work of two lesserknown mathematics education researchers, Nongnuch Wattanawaha and Stephanus Suwarsono. It was Suwarsono who devised and applied a method whereby learner preferences for visual or verbal thinking, as well as the “visualities” of the mathematics tasks themselves, could be measured and calibrated on the same scale, using item response theory. Keywords Visual image · Visualization · Mathematics learning · Problem solving · Item response theory · Verbal-visual
M.A. Clements (B) Department of Mathematics, Illinois State University, Campus Box 4520, Normal, IL 61790-4520, USA e-mail: [email protected] M.N. Fried, T. Dreyfus (eds.), Mathematics & Mathematics Education: Searching for Common Ground, Advances in Mathematics Education, DOI 10.1007/978-94-007-7473-5_11, © Springer Science+Business Media Dordrecht 2014
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M.A. Clements
More than two centuries ago, Johann Friedrich Herbart, a German philosopher, psychologist and educator, emphasized the need for teachers to try to become aware of links and preferences in individual learners’ minds. Herbart urged teachers to design lessons that took into account what learners were likely to be thinking about when they attempted to solve problems (Ellerton and Clements 2005). During the twentieth century there gradually developed, among math
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