Irrational images – the visualization of abstract mathematical terms (original) (raw)
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Geometrical Representation of Irrational Numbers
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In this snapshot the author contrasts the differential behavior of three familiar parameterized curves for rational and irrational values of the parameter. Students can explore a rich diversity of curves through investigating the parameter space. Through visualizing the difference between curves of rational and irrational parameters, students can better appreciate the distinction and come to gain more sophisticated intuitions about functions. We discuss here three families of curves that all possess the same property: each of them has a closed bounded symmetric form at rational values of a ratio of some principal parameters of the system from which they originate, and is an infinite bounded curve that never follows any finite segment of itself, if this ratio is irrational. For each of these three families, this mathematical fact is easily proven, and provides a creative possibility for visualizing the difference between rational and irrational numbers and between the processes that depend on them. The proof for one simple case discussed here, is reproduced in this paper, since such proofs are widely available for more general cases (see, e.g., Galperin & Zemliakov, 1990; Maor, 1998; Thomas and Finney, 1980). For all three curves, the rational numbers are related to only by their definition as a ratio of two integers. We do not refer to other properties,
Mathematics and Mathematical Thinking in Visual Perspective
Everybody learns that, if we stand on a long straight railway track, and look along the track as it recedes from us, the rails seem to come closer and closer. Everybody is taught that parallel lines meet at the horizon at a mysterious vanishing point. Is this true? And how can we use mathematical diagrams to create physically accurate 2-dimensional pictures of 3-dimensional objects? In my experience, these issues are usually taught poorly or inaccurately in Art classes, and not taught at all in Mathematics classes. This paper shows, amongst other fascinating details, how to draw with correct artists’ perspective!
Irrational Numbers A Constructive approach at Elementary and High School
Author's Edition, 2013
Irrational numbers, subject of this ebook, represent a sophisticated mathematical idea, focused on little intuitive theoretical aspects, with few connections to sensorial world. A crucial point lies on the fact that talking about irrational numbers necessarily leads to discuss the tenuous and inherent connection with real numbers. As an initial resource, we highlight a literature review from Arcavi et al. (1987), Fischbein; Jehian; Cohen (1995), Soares; Ferreira; Moreira (1999), Ripoll (2001), Rezende (2003), Leviathan (2004), Zazkis;Sirotic (2004), Sirotic;Zazkis (2007), Barthel (2010), Silva (2011) and Voskoglou e Kosyvyas (2011), among some others, researchers who conducted studies with respect to irrational and real numbers. For this composition, we initiated a research considering didactical, epistemological and historical resources to discuss the problem of introducing this numeric field in a significant way at Elementary Mathematics. Facing the assessments made through the argumentations pointed out, we highlight the limitations of operative, deterministic and exact aspects to present irrational numbers presentation, a feature that is common in Mathematics’ teaching. The relationship of tension and interaction between some axes – finite&infinite, exact&approximate and discrete&continuous – allowed us to conceive a metaphoric and suggestive place that we call 'the Space of Meanings'. We consider ‘the Space of Meanings’ was a metaphoric field that allows understanding and guiding an irrational numbers approach at Elementary Mathematics, in a more comprehensive and significant context, subject which will be discussed throughout this text.
In this text I develop the thesis that geometrical diagrams are depictions, not symbols; they depict geometrical objects, concepts or states of affairs. Besides developing this claim, I will defend it against three recent challenges from (Sherry 2009), Macbeth (2009, 2010, 2014) and (Panza 2012). First, according to Sherry, diagrams are not depictions, because no single depict can depict more than one thing, yet a single geometrical diagram can represent different geometrical figures in different contexts. I will argue that, once we recognize that resemblance underdetermines depiction, we can see that pictures can indeed depict different thing in different contexts and, consequently, there is nothing surprising about a single diagram depicting different geometrical figures. Next, I will defend it against a similar argument by Macbeth, according to which diagrams can represent different geometrical figures, even within the context of a single geometrical proof. Finally, I will defend it against Panza’s argument that there are essentially spatial features that geometrical objects have only insofar as they inherit them from the diagrams that represent them, and this is incompatible with the hypothesis that geometrical diagrams are depictions for depictions inherit their visual and spatial properties from the objects they represent, and not the other way around. In response, I will argue that once we understand the sense in which subjects are metaphysically prior to their depiction, we will see that my claim that Euclidean diagrams are depictions is not incompatible with Panza’s thesis.
Graphic Insight into Mathematical Concepts1
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The human brain is powerfully equipped to process visual information. By using computer graphics it is possible to tap this power to help students gain a greater understanding of many mathematical concepts. Furthermore, dynamic representations of mathematical processes furnish a degree of psychological reality that enables the mind to manipulate them in a far more fruitful way than could ever be achieved starting
Neoplatonism in Late Antiquity
Chapter 8 considers the role of the imagination as it appears in Proclus’ commentary on Euclid, where mathematical or geometrical objects are taken to mediate, both ontologically and cognitively, between thinkable and physical things. With the former, mathematical things share the permanence and consistency of their properties; with the latter, they share divisibility and the possibility of being multiplied. Hence, a geometrical figure exists simultaneously on four different levels: as a noetic concept in the intellect; as a logical definition, or logos, in discursive reasoning; as an imaginary perfect figure in the imagination; and as a physical imitation or representation in sense-perception. Imagination, then, can be equated with the intelligible or geometrical matter that constitutes the medium in which a geometrical object can be constructed, represented, and studied.
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In this paper we discuss visualizations in mathematics from a historical and didactical perspective. We consider historical debates from the 17th and 19th centuries regarding the role of intuition and visualizations in mathematics. We also consider the problem of what a visualization in mathematical learning can achieve. In an empirical study we investigate what mathematical conclusions university students made on the basis of a visualization. We emphasize that a visualization in mathematics should always be considered in its proper context.