Information and Communication Technologies, Modeling, Visualization and Experimentation
Author: Marcelo C. Borba,Monica E. Villarreal
Publisher: Springer Science & Business Media
This book offers a new conceptual framework for reflecting on the role of information and communication technology in mathematics education. Borba and Villarreal provide examples from research conducted at the level of basic and university-level education, developed by their research group based in Brazil, and discuss their findings in the light of the relevant literature. Arguing that different media reorganize mathematical thinking in different ways, they discuss how computers, writing and oral discourse transform education at an epistemological as well as a political level. Modeling and experimentation are seen as pedagogical approaches which are in harmony with changes brought about by the presence of information and communication technology in educational settings. Examples of research about on-line mathematics education courses, and Internet used in regular mathematics courses, are presented and discussed at a theoretical level. In this book, mathematical knowledge is seen as developed by collectives of humans-with-media. The authors propose that knowledge is never constructed solely by humans, but by collectives of humans and technologies of intelligence. Theoretical discussion developed in the book, together with new examples, shed new light on discussions regarding visualization, experimentation and multiple representations in mathematics education. Insightful examples from educational practice open up new paths for the reader.
During the last decade, argumentation has attracted growing attention as a means to elicit processes (linguistic, logical, dialogical, psychological, etc.) that can sustain or provoke reasoning and learning. Constituting an important dimension of daily life and of professional activities, argumentation plays a special role in democracies and is at the heart of philosophical reasoning and scientific inquiry. Argumentation, as such, requires specific intellectual and social skills. Hence, argumentation will have an increasing importance in education, both because it is a critical competence that has to be learned, and because argumentation can be used to foster learning in philosophy, history, sciences and in many other domains. Argumentation and Education answers these and other questions by providing both theoretical backgrounds, in psychology, education and theory of argumentation, and concrete examples of experiments and results in school contexts in a range of domains. It reports on existing innovative practices in education settings at various levels.
This radical, profoundly scholarly book explores the purposes and nature of proof in a range of historical settings. It overturns the view that the first mathematical proofs were in Greek geometry and rested on the logical insights of Aristotle by showing how much of that view is an artefact of nineteenth-century historical scholarship. It documents the existence of proofs in ancient mathematical writings about numbers and shows that practitioners of mathematics in Mesopotamian, Chinese and Indian cultures knew how to prove the correctness of algorithms, which are much more prominent outside the limited range of surviving classical Greek texts that historians have taken as the paradigm of ancient mathematics. It opens the way to providing the first comprehensive, textually based history of proof.
Although proving is core to mathematics as a sense-making activity, it currently has a marginal place in elementary classrooms internationally. Blending research with practical perspectives, this book addresses what it would take to elevate the place of proving at elementary school. The book uses classroom episodes from two countries to examine different kinds of proving tasks and the proving activity they can generate in the elementary classroom. It examines further the role of teachers in mediating the relationship between proving tasks and proving activity, including major mathematical and pedagogical issues that arise for teachers as they implement each kind of proving task. In addition to its contribution to research knowledge, the book has important implications for teaching, curricular resources, and teacher education.
13th International Conference, TPHOLs 2000 Portland, OR, USA, August 14-18, 2000 Proceedings
Author: Mark Aagaard
Publisher: Springer Science & Business Media
This book constitutes the refereed proceedings of the 13th International Conference on Theorem Proving in Higher Order Logics, TPHOLs 2000, held in Portland, Oregon, USA in August 2000. The 29 revised full papers presented together with three invited contributions were carefully reviewed and selected from 55 submissions. All current aspects of HOL theorem proving, formal verification of hardware and software systems, and formal verification are covered. Among the HOL theorem provers evaluated are COQ, HOL, Isabelle, HOL/SPIN, PVS, and Isabelle/HOL.
An Introduction to the World of Proofs and Pictures
Author: James Robert Brown
Publisher: Psychology Press
Philosophy of Mathematics is an excellent introductory text. This student friendly book discusses the great philosophers and the importance of mathematics to their thought. It includes the following topics: * the mathematical image * platonism * picture-proofs * applied mathematics * Hilbert and Godel * knots and nations * definitions * picture-proofs and Wittgenstein * computation, proof and conjecture. The book is ideal for courses on philosophy of mathematics and logic.
The Hundred-Year Quest to Solve One of Math's Greatest Puzzles
Author: George G. Szpiro
The amazing story of one of the greatest math problems of all time and the reclusive genius who solved it In the tradition of Fermat’s Enigma and Prime Obsession, George Szpiro brings to life the giants of mathematics who struggled to prove a theorem for a century and the mysterious man from St. Petersburg, Grigory Perelman, who fi nally accomplished the impossible. In 1904 Henri Poincaré developed the Poincaré Conjecture, an attempt to understand higher-dimensional space and possibly the shape of the universe. The problem was he couldn’t prove it. A century later it was named a Millennium Prize problem, one of the seven hardest problems we can imagine. Now this holy grail of mathematics has been found. Accessibly interweaving history and math, Szpiro captures the passion, frustration, and excitement of the hunt, and provides a fascinating portrait of a contemporary noble-genius.
Mit E. Husserls und H. Scholz' Anmerkungen herausgegeben von Ignacio Angelelli
Author: Gottlob Frege
Publisher: Georg Olms Verlag
Dieser Band enthält die vier Arbeiten Freges: Begriffsschrift, eine der arithmetischen nachgebildeten Formelsprache, 1879; Anwendungen der Begriffsschrift, 1879; Über den Briefwechsel Leibnizens und Huggens mit Papin, 1881; Über den Zweck der Begriffsschrift, 1883; Über die wissenschaftliche Berechtigung einer Begriffsschrift, 1882. Frege's research work in the field of mathematical logic is of great importance for the present-day analytic philosophy. We actually owe to Frege a great amount of basical insight and exemplary research, which set up a new standard also in other fields of knowledge. As the founder of mathematical logic he severely examindes the syllogisms on which arithmetic is built up. In doing so, Frege recognized that our colloquial language is inadequate to define logic structures. His notional language corresponded to the artaivicial logical language demandes by Leibniz. Frege's achievement in the field of logic were so important, that they radiated into the domain of philosophy and influenced the development of mathematical logic decisively.
Presents brief stories about the life and work of famous mathematicians, including Euler, Fermat, Fibonacci, Fourier, Gauss, Moebius, and Pythagoras, and introduces their theories with puzzles and tasks for students to solve.
Research on teaching and learning proof and proving has expanded in recent decades. This reflects the growth of mathematics education research in general, but also an increased emphasis on proof in mathematics education. This development is a welcome one for those interested in the topic, but also poses a challenge, especially to teachers and new scholars. It has become more and more difficult to get an overview of the field and to identify the key concepts used in research on proof and proving. This book is intended to help teachers, researchers and graduate students to overcome the difficulty of getting an overview of research on proof and proving. It reviews the key findings and concepts in research on proof and proving, and embeds them in a contextual frame that allows the reader to make sense of the sometimes contradictory statements found in the literature. It also provides examples from current research that explore how larger patterns in reasoning and argumentation provide insight into teaching and learning.
This book takes a theoretical perspective on the study of school algebra, in which both semiotics and history occur. The Methodological design allows for the interpretation of specific phenomena and the inclusion of evidence not addressed in more general treatments. The book gives priority to "meaning in use" over "formal meaning". These approaches and others of similar nature lead to a focus on competence rather than a user’s activity with mathematical language.