This book collects the papers of the conference held in Berlin, Germany, 27-29 August 2012, on 'Space, Geometry and the Imagination from Antiquity to the Modern Age'. The conference was a joint effort by the Max Planck Institute for the History of Science (Berlin) and the Centro die Ricerca Matematica Ennio De Giorgi (Pisa).

The Ultimate Space Place presents information about the history of space flight, with emphasis on aviation, rocketry, Mercury, Gemini, Apollo, Skylab, and the Space Shuttle.

Investigators have moved back and forth between design efforts and basic studies in cognition to improve both application and fundamental knowledge. This volume's theme is this interaction between practice and science with the opportunity for reflecting on findings in order to understand them and suggesting improved forms of application and their underlying explanation. This is seen in various arenas including theory-based computer-assisted instruction for teaching mathematics, the design of communities of learning in elementary schools, teaching in the context of problem-solving situations and reasoning with models, self-explanation as a highly effective learning activity, conceptual change in medical training and health education, and workplace training in electronic troubleshooting. The results of extensive long-term experience and analysis in each of these areas are insightfully reported by the well-known contributors to this volume. Special features of this fifth edition include: * The work of eminent cognitive scientists in the design and evaluation of educational and training environments to increase current understanding of learning and development, as this understanding is applied to innovative instructional programs and teaching methods. * A description of learning theory and principles as well as implications and examples on research and development on educational application. * A presentation on the 10-year change in perspective on research and development in problem solving environments that invite inquiry about academic information and skills in the context of instruction of elementary school children. * An innovative approach to math and science instruction in which teaching is oriented around constructing, evaluating, and revising models. * An examination of the process of self-explaining, which involves explaining to one's self in an attempt to make sense of a new situation. * A description of a long-term program of cognitive task analysis and instructional design on problem solving in the operation of complex equipment. * An investigation on the acquisition of clinical reasoning skills and the understanding of biomedical concepts in both professional medicine and the health practices of the lay population.

Help students reveal the math behind the words "I don’t get what I’m supposed to do!" This is a common refrain from students when asked to solve word problems. Solving problems is about more than computation. Students must understand the mathematics of a situation to know what computation will lead to an appropriate solution. Many students often pluck numbers from the problem and plug them into an equation using the first operation they can think of (or the last one they practiced). Students also tend to choose an operation by solely relying on key words that they believe will help them arrive at an answer, without careful consideration of what the problem is actually asking of them. Mathematize It! Going Beyond Key Words to Make Sense of Word Problems, Grades 6–8 shares a reasoning approach that helps students dig into the problem to uncover the underlying mathematics, deeply consider the problem’s context, and employ strong operation sense to solve it. Through the process of mathematizing, the authors provide an explanation of a consistent method—and specific instructional strategies—to take the initial focus off specific numbers and computations and put it on the actions and relationships expressed in the problem. Sure to enhance teachers’ own operation sense, this user-friendly resource for Grades 6–8: · Offers a systematic mathematizing process for students to use when solving word problems · Gives practice opportunities and dozens of problems to leverage in the classroom · Provides specific examples of questions and explorations for multiplication and division, fractions and decimals, as well as operations with rational numbers · Demonstrates the use of visual representations to model problems with dozens of short videos · Includes end-of-chapter activities and reflection questions How can you help your students understand what is happening mathematically when solving word problems? Mathematize it!

While it is well known that the Delian problems are impossible to solve with a straightedge and compass – for example, it is impossible to construct a segment whose length is cube root of 2 with these instruments – the discovery of the Italian mathematician Margherita Beloch Piazzolla in 1934 that one can in fact construct a segment of length cube root of 2 with a single paper fold was completely ignored (till the end of the 1980s). This comes as no surprise, since with few exceptions paper folding was seldom considered as a mathematical practice, let alone as a mathematical procedure of inference or proof that could prompt novel mathematical discoveries. A few questions immediately arise: Why did paper folding become a non-instrument? What caused the marginalisation of this technique? And how was the mathematical knowledge, which was nevertheless transmitted and prompted by paper folding, later treated and conceptualised? Aiming to answer these questions, this volume provides, for the first time, an extensive historical study on the history of folding in mathematics, spanning from the 16th century to the 20th century, and offers a general study on the ways mathematical knowledge is marginalised, disappears, is ignored or becomes obsolete. In doing so, it makes a valuable contribution to the field of history and philosophy of science, particularly the history and philosophy of mathematics and is highly recommended for anyone interested in these topics.

Imagination occupies a central place in philosophy, going back to Aristotle. However, following a period of relative neglect there has been an explosion of interest in imagination in the past two decades as philosophers examine the role of imagination in debates about the mind and cognition, aesthetics and ethics, as well as epistemology, science and mathematics. This outstanding Handbook contains over thirty specially commissioned chapters by leading philosophers organised into six clear sections examining the most important aspects of the philosophy of imagination, including: Imagination in historical context: Aristotle, Descartes, Hume, Kant, Husserl, and Sartre What is imagination? The relation between imagination and mental imagery; imagination contrasted with perception, memory, and dreaming Imagination in aesthetics: imagination and our engagement with music, art, and fiction; the problems of fictional emotions and ‘imaginative resistance’ Imagination in philosophy of mind and cognitive science: imagination and creativity, the self, action, child development, and animal cognition Imagination in ethics and political philosophy, including the concept of 'moral imagination' and empathy Imagination in epistemology and philosophy of science, including learning, thought experiments, scientific modelling, and mathematics. The Routledge Handbook of Philosophy of Imagination is essential reading for students and researchers in philosophy of mind and psychology, aesthetics, and ethics. It will also be a valuable resource for those in related disciplines such as psychology and art.

This collection of essays examines how spatial mobilities of people and practices, technologies and objects, knowledge and ideas have shaped the production, circulation, and transfer of knowledge in different historical and geographical contexts. Targeting an interdisciplinary audience, Mobilities of Knowledge combines detailed empirical analyses with innovative conceptual approaches. The first part scrutinizes knowledge circulation, transfer, and adaption, focussing on the interpersonal communication process, early techniques of papermaking, a geographical text, indigenous knowledge in exploration, the genealogy of spatial analysis, and different disciplinary knowledges about the formation of cities, states, and agriculture. The second part analyses the interplay of mediators, networks, and learning by studying academic careers, travels, and collaborations within the British Empire, public internationalism in Geneva, the global transfer of corporate knowledge through expatriation, graduate mobility from the global south to the global north, and the international mobility of degree programs in higher education.This book is open access under a CC BY 4.0 license.

"The list of math books to truly synthesize what we know so far and what we need to know is a very short and exclusive list. Well, you can confidently add Mathematize It to this collection. Written by three of the most respected math educators today, the book zeros in on that often poorly traveled journey between the question and answer in problem solving. Mathematize It will be your go-to resource to install the mathematical play revolution in elementary classes everywhere!" Suni Singh Author of Pi of Life: the Hidden Happiness of Mathematics and Math Recess: Playful Learning in an Age of Disruption Help students reveal the math behind the words "I don’t get what I’m supposed to do!" This is a common refrain from students when asked to solve word problems. Solving problems is about more than computation. Students must understand the mathematics of a situation to know what computation will lead to an appropriate solution. Many students often pluck numbers from the problem and plug them into an equation using the first operation they can think of (or the last one they practiced). Students also tend to choose an operation by solely relying on key words that they believe will help them arrive at an answer, which without careful consideration of what the problem is actually asking of them. Mathematize It! Going Beyond Key Words to Make Sense of Word Problems, Grades 3-5 shares a reasoning approach that helps students dig into the problem to uncover the underlying mathematics, deeply consider the problem’s context, and employ strong operation sense to solve it. Through the process of mathematizing, the authors provide an explanation of a consistent method—and specific instructional strategies—to take the initial focus off specific numbers and computations and put it on the actions and relationships expressed in the problem. Sure to enhance teachers’ own operation sense, this user-friendly resource for Grades 3–5 • Offers a systematic mathematizing process for students to use when solving word problems • Gives practice opportunities and dozens of problems to leverage in the classroom • Provides specific examples of questions and explorations for all four operations (addition, subtraction, multiplication, and division) with whole numbers, fractions, and decimals • Demonstrates the use of concrete manipulatives to model problems with dozens of short videos • Includes end-of-chapter activities and reflection questions How can you help your students understand what is happening mathematically when solving word problems? Mathematize it!

The book offers a collection of essays on various aspects of Leibniz’s scientific thought, written by historians of science and world-leading experts on Leibniz. The essays deal with a vast array of topics on the exact sciences: Leibniz’s logic, mereology, the notion of infinity and cardinality, the foundations of geometry, the theory of curves and differential geometry, and finally dynamics and general epistemology. Several chapters attempt a reading of Leibniz’s scientific works through modern mathematical tools, and compare Leibniz’s results in these fields with 19th- and 20th-Century conceptions of them. All of them have special care in framing Leibniz’s work in historical context, and sometimes offer wider historical perspectives that go much beyond Leibniz’s researches. A special emphasis is given to effective mathematical practice rather than purely epistemological thought. The book is addressed to all scholars of the exact sciences who have an interest in historical research and Leibniz in particular, and may be useful to historians of mathematics, physics, and epistemology, mathematicians with historical interests, and philosophers of science at large.