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Visual Thinking in Basic Analysis

Marcus Giaquinto

in Visual Thinking in Mathematics

This chapter explores the vexed question of visual thinking in analysis. Is visualizing in analysis just a facilitator? Or can it have a non-redundant role in discovering truths of analysis? It is ... More

This chapter explores the vexed question of visual thinking in analysis. Is visualizing in analysis just a facilitator? Or can it have a non-redundant role in discovering truths of analysis? It is argued that its role in discovery is necessarily highly restricted; but several other important functions are fulfilled by visual means in analysis. The chapter also discusses Rolle's Theorem and Bolzano's Theorem.Less

Mental Number Lines

Marcus Giaquinto

in Visual Thinking in Mathematics

The main concern of this chapter is the association of numbers with a visually represented line. The association is usually acquired in junior school, and is then extended to provide an integrated ... More

The main concern of this chapter is the association of numbers with a visually represented line. The association is usually acquired in junior school, and is then extended to provide an integrated representation of integers, fractions, and irrational numbers, called the real number line. However, the nature and origin of mental number lines, and the way they are embedded in our thinking, are not obvious. This chapter discusses mental number lines given the evidence to date. It shows how innate and cultural factors interact to determine the nature and role of mental number lines in basic numerical thinking. It also underscores their importance in more advanced mathematics.Less

Introduction

Marcus Giaquinto

in Visual Thinking in Mathematics

This introductory chapter outlines the main themes of the book. It then describes visual thinking in mathematics during the 19th century. An overview of the chapters in the book is presented.

Visual Aspects of Calculation

Marcus Giaquinto

in Visual Thinking in Mathematics

This chapter describes the roles of visual thinking in calculation, with the aim of showing that in many cases it is not peripheral. It does this by setting the visual aspects of calculation within ... More

This chapter describes the roles of visual thinking in calculation, with the aim of showing that in many cases it is not peripheral. It does this by setting the visual aspects of calculation within an account of the operations involved and the cognitive resources used. The chapter also considers the epistemology of basic arithmetic.Less

Symbol Manipulation

Marcus Giaquinto

in Visual Thinking in Mathematics

This chapter explores the nature and uses of visual thinking with symbols in mathematics. It examines visual symbolic thinking, which is more varied than one might expect, to see how it can ... More

This chapter explores the nature and uses of visual thinking with symbols in mathematics. It examines visual symbolic thinking, which is more varied than one might expect, to see how it can contribute to discovery, security, illumination, and generality. It also looks at the roles of symbolic thinking in certain algebraic examples.Less

Visualizing in Mathematics

Marcus Giaquinto

in The Philosophy of Mathematical Practice

Visual thinking in mathematics (thinking with representations in visual imagination or external diagrams) has a plurality of uses. This chapter assesses the epistemic standing of major uses of visual ... More

Visual thinking in mathematics (thinking with representations in visual imagination or external diagrams) has a plurality of uses. This chapter assesses the epistemic standing of major uses of visual thinking, concentrating on proving (and following a proof), and discovering, i.e. coming to believe a truth by one's own lights in a way that is reliable and involves no violation of epistemic rationality. Taking into account the sorts of error to which visual thinking is especially vulnerable, it is argued that visual thinking can have a non-superfluous role in proving. The chapter then discusses and illustrates the possibility of discovering mathematical truths visually. Finally, other uses of visual thinking in mathematics are surveyed.Less

Diagrams in Geometric Proofs

Marcus Giaquinto

in Visual Thinking in Mathematics

Presentations of geometric proofs are often accompanied by diagrams for quick and easy comprehension. But to many people, it seems clear that diagrammatic reasoning cannot be a part of the argument ... More

Presentations of geometric proofs are often accompanied by diagrams for quick and easy comprehension. But to many people, it seems clear that diagrammatic reasoning cannot be a part of the argument itself, otherwise it would be prey to the very insecurity that we want to eliminate — insecurity from visual thinking — and so the argument would not be able to justify its conclusion; it would not be a proof. This is the line of thought that most strongly supports the widespread belief that diagrams can have no epistemological role in proof. The main aim of this chapter is to investigate this negative view and the argument for it presented in this book.Less

The Vast Range of Visual Worlds

Laura Otis

in Rethinking Thought: Inside the Minds of Creative Scientists and Artists

The insights of five strikingly different visually gifted thinkers shape this chapter: cell biologist Barry Shur, engineer Temple Grandin, painter Mary J. Welty, neuroscientist Hugh Wilson, and U. S. ... More

The insights of five strikingly different visually gifted thinkers shape this chapter: cell biologist Barry Shur, engineer Temple Grandin, painter Mary J. Welty, neuroscientist Hugh Wilson, and U. S. Poet Laureate Natasha Trethewey. Their introspections confront a narrative of how, since the 1880s, neuroscientists and psychologists have tried to learn how visual mental imagery works, including experiments by Francis Galton, Allan Paivio, Roger Shepard, Stephen Kosslyn, and Zenon Pylyshyn. The participants’ insights support cognitive neuroscientist Maria Kozhevnikov’s finding that spatial visualization (focused on dimensions, orientation, velocity, and direction of motion) differs markedly from object visualization (focused on shapes, colors, textures, and visual details.) Their introspections illustrate Kozhevnikov’s claim that there is no unified “visual” way of thinking, and that spatial and object visualization constitute two distinct thinking modes. At the same time, their insights trouble her idea, because their mental worlds differ so greatly that two kinds of visual thinking seem inadequate to describe them.Less

Visuospatial Reasoning

Stephen K. Reed

in Cognitive Skills You Need for the 21st Century

Visual thinking has aided many scientific discoveries and is also useful in everyday reasoning. The Animation Tutor provides animation feedback to help students improve their ability to estimate and ... More

Visual thinking has aided many scientific discoveries and is also useful in everyday reasoning. The Animation Tutor provides animation feedback to help students improve their ability to estimate and calculate answers to problems. Examples include calculating the average speed of a round trip and using spatial relationships as a substitute for deriving algebraic solutions. Computer simulations of human thinking have emphasized rule-based reasoning, but these simulations now include a visual buffer to model visuospatial reasoning. It is often difficult to discover new information in visual images such as reinterpreting an ambiguous figure although people are more successful in mentally combining figures to create useful objects. Applications of research on cognitive geography include improving spatial information, geographic education, map design, urban planning, and landscape design.Less

Drawing as Thinking: A Visual Response to To the Lighthouse

Sarah Blake

in Interdisciplinary/Multidisciplinary Woolf

This chapter examines dialogues through art and across time by making a reading of Virginia Woolf's To the Lighthouse through drawing. Focusing on the relationship between drawing and reading, it ... More

This chapter examines dialogues through art and across time by making a reading of Virginia Woolf's To the Lighthouse through drawing. Focusing on the relationship between drawing and reading, it argues that both acts can be creative, even playful, activities that enable spontaneity and continuing dialogue. It highlights a correspondence between the fluidity of Mrs. Ramsay's experience of reading in To the Lighthouse and ideas of indeterminacy and open-ended exploration in drawing practice. It also considers conversational drawing as visual thinking and as a technique that encourages a collaborative approach to mark-making. Finally, it suggests that To the Lighthouse can be read as an exploration of two opposing ways of thinking, and of how these contrasting mind-sets coexist and communicate with one another.Less

Introduction

Ildar Garipzanov

in Graphic Signs of Authority in Late Antiquity and the Early Middle Ages, 300-900

The first section discusses definitions of the graphic sign and its typologies, and provides an overview of relevant academic literature. The second section highlights major historiographic trends in ... More

The first section discusses definitions of the graphic sign and its typologies, and provides an overview of relevant academic literature. The second section highlights major historiographic trends in the study of graphic signs in the humanities from the early twentieth century to the present day. The next section outlines the relation of graphic signs to a wider corpus of graphic non-figurative data in the late antique Mediterranean and early medieval Europe with reference to the overarching methodological framework of visual thinking and graphic visualization and the related concept of early graphicacy, focusing particularly on the latter’s general cognitive aspects and intrinsic connection to the late antique and early medieval cultural system of visual representation. The concluding section defines the book’s subject, namely graphic signs of authority, outlines their functional usage in early medieval political culture, and summarizes the content of the following chapters.Less

Picturing Difference and Disability in Our Classrooms

Keri Watson

in Teaching the Works of Eudora Welty: Twenty-First-Century Approaches

Eudora Welty’s photographs from the 1939 Mississippi State Fair offer an exciting opportunity to teach students visual literacy and introduce them to performance, critical race, gender, and ... More

Eudora Welty’s photographs from the 1939 Mississippi State Fair offer an exciting opportunity to teach students visual literacy and introduce them to performance, critical race, gender, and disability studies. Seeing, reading, and thinking critically are important skills for any student, and close looking at Welty’s photographs of sideshow banners, including “Mule Face Woman,” “Headless Girl,” “Rubber Man,” and “Hypnotized,” in coordination with close reading of her short stories “Keela, the Outcast Indian Maiden” and “Lily Daw and the Three Ladies” facilitates a richer understanding of diversity.Less

Avoid Being Abstract When Making Policies on the Welfare of Animals

Temple Grandin

in Species Matters: Humane Advocacy and Cultural Theory

This chapter examines how animal advocacy can take different forms in diverse academic disciplines, in this case animal science. The chapter discusses the importance of staying in touch with what is ... More

This chapter examines how animal advocacy can take different forms in diverse academic disciplines, in this case animal science. The chapter discusses the importance of staying in touch with what is actually happening on the ground, in farms and in slaughter plants, and the use of animals for food based on experience from working for thirty-five years designing better equipment to improve the treatment of cattle and pigs and the perspective of a reformer who wants to improve the livestock industry. It also explains a particular approach to animal welfare issues and how visual thinking changes this approach to improving animal welfare. Finally, it considers policies that have unintended bad consequences and may make animal welfare worse, along with a dispute with animal rights advocates who want to abolish the use of animals for food. The chapter argues in favor of humane advocacy for animals—at least for sentient, affect-bearing animals capable of feeling pain—while defending the right to continue eating meat from animals raised and slaughtered humanely.Less

Visuality through and through

Hentschel Klaus

in Visual Cultures in Science and Technology: A Comparative History

This chapter moves on from the discussion of Ernst Mach in the preceding chapter to point out how his oeuvre epitomizes all layers of visual cultures, especially drawing skills, an obsession with ... More

This chapter moves on from the discussion of Ernst Mach in the preceding chapter to point out how his oeuvre epitomizes all layers of visual cultures, especially drawing skills, an obsession with visual resources and visual thinking, which spread into his teaching and leisure. Mach’s life and oeuvre exemplify the 19th-century mania for making everything visible, a strand that also perfectly fits for Kekulé, Maxwell, Plateau and many other representatives of this a priori unlikely, yet highly characteristic, combination of unusual background, skills and visual inclinations. Avid Victorian photographers (Brewster, Herschel and Talbot), art-loving 20th-century American spectroscopists (W. W. Morgan) and highly skilled 17th-century draftsmen (Hooke, Wren and Huyghens) all became pioneers of new visual domains, be these spectroscopy, scientific photography, microscopy, planetary astronomy or magic lantern projection.Less

A Finished Task: Darwin’s Treatise on Coral Reefs

Alistair Sponsel

in Darwin's Evolving Identity: Adventure, Ambition, and the Sin of Speculation

This chapter analyzes the content and strategy of Darwin’s 1842 book, The Structure and Distribution of Coral Reefs. It was not the grand synthetic geological treatise he had originally envisioned ... More

This chapter analyzes the content and strategy of Darwin’s 1842 book, The Structure and Distribution of Coral Reefs. It was not the grand synthetic geological treatise he had originally envisioned writing, but it contained an elegant theory of reef formation supported by analysis of the structure and possible origin of every documented coral reef in the world. The first four chapters were ostensibly descriptive, but Darwin classified reefs into types that corresponded to developmental stages characterizing his theory, which emerged in chapter 5. The book concluded with extended discussion of the global distribution of different types of reefs, as illustrated and systematized on a fold-out thematic map (the only one of its sort Darwin ever published). Published reviews of the book emphasized (whether favorably or not) the ambitious scope of Darwin’s generalizing about reefs; he responded to some criticisms by heavily revising the chapter on coral reefs in a second (1845) edition of his Journal of Researches. Years hence he offered inconsistent and sometimes contradictory recollections about what he had accomplished with the book, reminding critics of his caution but privately reveling in the accuracy of his speculations when supporting evidence emerged from work by J.B. Jukes and J.D. Dana.Less