Making and Breaking Mathematical Sense

Histories and Philosophies of Mathematical Practice

Author: Roi Wagner

Publisher: Princeton University Press

ISBN: 1400883784

Category: Mathematics

Page: 256

View: 4658

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In line with the emerging field of philosophy of mathematical practice, this book pushes the philosophy of mathematics away from questions about the reality and truth of mathematical entities and statements and toward a focus on what mathematicians actually do—and how that evolves and changes over time. How do new mathematical entities come to be? What internal, natural, cognitive, and social constraints shape mathematical cultures? How do mathematical signs form and reform their meanings? How can we model the cognitive processes at play in mathematical evolution? And how does mathematics tie together ideas, reality, and applications? Roi Wagner uniquely combines philosophical, historical, and cognitive studies to paint a fully rounded image of mathematics not as an absolute ideal but as a human endeavor that takes shape in specific social and institutional contexts. The book builds on ancient, medieval, and modern case studies to confront philosophical reconstructions and cutting-edge cognitive theories. It focuses on the contingent semiotic and interpretive dimensions of mathematical practice, rather than on mathematics' claim to universal or fundamental truths, in order to explore not only what mathematics is, but also what it could be. Along the way, Wagner challenges conventional views that mathematical signs represent fixed, ideal entities; that mathematical cognition is a rigid transfer of inferences between formal domains; and that mathematics’ exceptional consensus is due to the subject’s underlying reality. The result is a revisionist account of mathematical philosophy that will interest mathematicians, philosophers, and historians of science alike.

The Math of Money

Making Mathematical Sense of Your Personal Finances

Author: Morton D. Davis

Publisher: Springer Science & Business Media

ISBN: 1475743343

Category: Mathematics

Page: 199

View: 4191

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This lively and practical introduction to the mathematics of money invites us to take a fresh look at the numbers that underpin our financial decisions. Morton D. Davis talks about strategies to use when we are required to bet against the odds (purchasing auto insurance) or choose to bet against the odds (wagering in a casino or at the track). He considers the ways in which we can streamline and simplify the choices available to us in mortgages and other loans. And he helps us understand the real probabilities when we accept a tip on that "one in a thousand" stock, even when the tip comes from a successful day trader. With a wealth of entertaining and counterintuitive examples, The Math of Money delights as well as informs, and will help readers treat their financial resources more rationally.

Powerful Problem Solving

Activities for Sense Making with the Mathematical Practices

Author: Max Ray

Publisher: Heinemann Educational Books

ISBN: 9780325050904

Category: Education

Page: 192

View: 9634

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How can we break the cycle of frustrated students who "drop out of math" because the procedures just don't make sense to them? Or who memorize the procedures for the test but don't really understand the mathematics? Max Ray and his colleagues at the Math Forum @ Drexel University say "problem solved," by offering their collective wisdom about how students become proficient problem solvers, through the lens of the CCSS for Mathematical Practices. They unpack the process of problem solving in fresh new ways and turn the Practices into activities that teachers can use to foster habits of mind required by the Common Core: communicating ideas and listening to the reflections of others estimating and reasoning to see the "big picture" of a problem organizing information to promote problem solving using modeling and representations to visualize abstract concepts reflecting on, revising, justifying, and extending the work. Powerful Problem Solving shows what's possible when students become active doers rather than passive consumers of mathematics. Max argues that the process of sense-making truly begins when we create questioning, curious classrooms full of students' own thoughts and ideas. By asking "What do you notice? What do you wonder?" we give students opportunities to see problems in big-picture ways, and discover multiple strategies for tackling a problem. Self-confidence, reflective skills, and engagement soar, and students discover that the goal is not to be "over and done," but to realize the many different ways to approach problems. Read a sample chapter. Save 15% when you purchase 15 copies with a Book Study Bundle!

Mathematics as Sign

Writing, Imagining, Counting

Author: Brian Rotman

Publisher: Stanford University Press

ISBN: 9780804736848

Category: Mathematics

Page: 170

View: 8774

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In this book, Rotman argues that mathematics is a vast and unique man-made imagination machine controlled by writing. It addresses both aspects—mental and linguistic—of this machine. The essays in this volume offer an insight into Rotman's project, one that has been called "one of the most original and important recent contributions to the philosophy of mathematics."

The Philosopher and the Wolf

Lessons from the Wild on Love, Death, and Happiness

Author: Mark Rowlands

Publisher: Open Road Media

ISBN: 1453217762

Category: Nature

Page: 246

View: 1410

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The charming and poignant story of the relationship between a philosophy professor and his pet wolf Mark Rowlands was a young philosophy professor, rootless and searching for life’s greater meaning. Shortly after arriving at the University of Alabama, he noticed a classified ad in the local paper advertising wolf clubs for sale, and decided he had to investigate, if only out of curiosity. It was love at first sight, and the bond that grew between philosopher and wolf reaffirms for us the incredible relationships that exist between man and animal.More than just an exotic pet, Brenin exerted an immense influence on Rowlands both as a person, and strangely enough as a philosopher, leading him to reevaluate his attitude toward love, happiness, nature, death, and the true meaning of companionship.

Categories for the Working Philosopher

Author: Elaine Landry

Publisher: Oxford University Press

ISBN: 019874899X

Category: Mathematics

Page: 528

View: 7467

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This is the first volume on category theory for a broad philosophical readership. It is designed to show the interest and significance of category theory for a range of philosophical interests: mathematics, proof theory, computation, cognition, scientific modelling, physics, ontology, the structure of the world. Each chapter is written by either a category-theorist or a philosopher working in one of the represented areas, in an accessible waythat builds on the concepts that are already familiar to philosophers working in these areas.

Making Sense of Mathematics for Teaching Grades 6-8

(Unifying Topics for an Understanding of Functions, Statistics, and Probability)

Author: Edward C. Nolan,Juli K. Dixon

Publisher: Solution Tree Press

ISBN: 194249646X

Category: Education

Page: 176

View: 9746

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Develop a deep understanding of mathematics. This user-friendly resource presents grades 6–8 teachers with a logical progression of pedagogical actions, classroom norms, and collaborative teacher team efforts to increase their knowledge and improve mathematics instruction. Make connections between elementary fraction-based content to fraction operations taught in the middle grades. Explore strategies and techniques to effectively learn and teach significant mathematics concepts and provide all students with the precise, accurate information they need to achieve academic success. Benefits Dig deep into mathematical modeling and reasoning to improve as both a learner and teacher of mathematics. Explore how to develop, select, and modify mathematics tasks in order to balance cognitive demand and engage students. Discover the three important norms to uphold in all mathematics classrooms. Learn to apply the tasks, questioning, and evidence (TQE) process to grow as both learners and teachers of mathematics. Gain clarity about the most productive progression of mathematical teaching and learning for grades 6–8. Access short videos that show what classrooms that are developing mathematical understanding should look like. Contents Introduction 1 Fraction Operations and Integer Concepts and Operations 2 Ratios and Proportional Relationships 3 Equations, Expressions, and Inequalities 4 Functions 5 Measurement and Geometry 6 Statistics and Probability Epilogue: Next Steps References and Resources Index

How Not to be Wrong

The Power of Mathematical Thinking

Author: Jordan Ellenberg

Publisher: Penguin

ISBN: 0143127535

Category: Business & Economics

Page: 480

View: 9295

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"Using the mathematician's method of analyzing life and exposing the hard-won insights of the academic community to the layman, minus the jargon ... Ellenberg pulls from history as well as from the latest theoretical developments to provide those not trained in math with the knowledge they need"--

Sourcebook in the Mathematics of Medieval Europe and North Africa

Author: Victor J. Katz,Menso Folkerts,Barnabas Hughes,Roi Wagner,J. Lennart Berggren

Publisher: Princeton University Press

ISBN: 1400883202

Category: Mathematics

Page: 592

View: 3649

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Medieval Europe was a meeting place for the Christian, Jewish, and Islamic civilizations, and the fertile intellectual exchange of these cultures can be seen in the mathematical developments of the time. This sourcebook presents original Latin, Hebrew, and Arabic sources of medieval mathematics, and shows their cross-cultural influences. Most of the Hebrew and Arabic sources appear here in translation for the first time. Readers will discover key mathematical revelations, foundational texts, and sophisticated writings by Latin, Hebrew, and Arabic-speaking mathematicians, including Abner of Burgos's elegant arguments proving results on the conchoid—a curve previously unknown in medieval Europe; Levi ben Gershon’s use of mathematical induction in combinatorial proofs; Al-Mu’taman Ibn Hūd’s extensive survey of mathematics, which included proofs of Heron’s Theorem and Ceva’s Theorem; and Muhyī al-Dīn al-Maghribī’s interesting proof of Euclid’s parallel postulate. The book includes a general introduction, section introductions, footnotes, and references. The Sourcebook in the Mathematics of Medieval Europe and North Africa will be indispensable to anyone seeking out the important historical sources of premodern mathematics.

Introduction to Mathematical Thinking

Author: Keith J. Devlin

Publisher: N.A

ISBN: 9780615653631

Category: Mathematics

Page: 92

View: 982

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In the twenty-first century, everyone can benefit from being able to think mathematically. This is not the same as "doing math." The latter usually involves the application of formulas, procedures, and symbolic manipulations; mathematical thinking is a powerful way of thinking about things in the world -- logically, analytically, quantitatively, and with precision. It is not a natural way of thinking, but it can be learned.Mathematicians, scientists, and engineers need to "do math," and it takes many years of college-level education to learn all that is required. Mathematical thinking is valuable to everyone, and can be mastered in about six weeks by anyone who has completed high school mathematics. Mathematical thinking does not have to be about mathematics at all, but parts of mathematics provide the ideal target domain to learn how to think that way, and that is the approach taken by this short but valuable book.The book is written primarily for first and second year students of science, technology, engineering, and mathematics (STEM) at colleges and universities, and for high school students intending to study a STEM subject at university. Many students encounter difficulty going from high school math to college-level mathematics. Even if they did well at math in school, most are knocked off course for a while by the shift in emphasis, from the K-12 focus on mastering procedures to the "mathematical thinking" characteristic of much university mathematics. Though the majority survive the transition, many do not. To help them make the shift, colleges and universities often have a "transition course." This book could serve as a textbook or a supplementary source for such a course.Because of the widespread applicability of mathematical thinking, however, the book has been kept short and written in an engaging style, to make it accessible to anyone who seeks to extend and improve their analytic thinking skills. Going beyond a basic grasp of analytic thinking that everyone can benefit from, the STEM student who truly masters mathematical thinking will find that college-level mathematics goes from being confusing, frustrating, and at times seemingly impossible, to making sense and being hard but doable.Dr. Keith Devlin is a professional mathematician at Stanford University and the author of 31 previous books and over 80 research papers. His books have earned him many awards, including the Pythagoras Prize, the Carl Sagan Award, and the Joint Policy Board for Mathematics Communications Award. He is known to millions of NPR listeners as "the Math Guy" on Weekend Edition with Scott Simon. He writes a popular monthly blog "Devlin's Angle" for the Mathematical Association of America, another blog under the name "profkeithdevlin", and also blogs on various topics for the Huffington Post.

What is Mathematics, Really?

Author: Reuben Hersh

Publisher: Oxford University Press, USA

ISBN: 9780195130874

Category: Mathematics

Page: 343

View: 1836

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Reflecting an insider's view of mathematical life, the author argues that mathematics must be historically evolved, and intelligible only in a social context.

Making Sense of Mathematics for Teaching Grades K-2

(Communicate the Context Behind High-Cognitive-Demand Tasks for Purposeful, Productive Learning)

Author: Juli K. Dixon,Edward C. Nolan

Publisher: Solution Tree Press

ISBN: 1942496400

Category: Education

Page: 168

View: 4579

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Develop a deep understanding of mathematics. This user-friendly resource presents grades K–2 teachers with a logical progression of pedagogical actions, classroom norms, and collaborative teacher team efforts to increase their knowledge and improve mathematics instruction. Explore strategies and techniques to effectively learn and teach significant mathematics concepts and provide all students with the precise, accurate information they need to achieve academic success. Clarify math essentials with figures and tables that facilitate understanding through visualization. Benefits Dig deep into mathematical modeling and reasoning to improve as both a learner and teacher of mathematics. Explore how to develop, select, and modify mathematics tasks in order to balance cognitive demand and engage students. Discover the three important norms to uphold in all mathematics classrooms. Learn to apply the tasks, questioning, and evidence (TQE) process to ensure mathematics instruction is focused, coherent, and rigorous. Use charts and diagrams for classifying shapes, which can engage students in important mathematical practices. Access short videos that show what classrooms that are developing mathematical understanding should look like. Contents Introduction 1 Number Concepts and Place Value 2 Word Problem Structures 3 Addition and Subtraction Using Counting Strategies 4 Addition and Subtraction Using Grouping Strategies 5 Geometry 6 Measurement Epilogue Next Steps Appendix A Completed Classification of Triangles Chart Appendix B Completed Diagram for Classifying Quadrilaterals

Common Sense Mathematics

Author: Ethan D. Bolker,Maura B. Mast

Publisher: The Mathematical Association of America

ISBN: 1939512107

Category: Mathematics

Page: 326

View: 3010

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Common Sense Mathematics is a text for a one semester college-level course in quantitative literacy. The text emphasizes common sense and common knowledge in approaching real problems through popular news items and finding useful mathematical tools and frames with which to address those questions. We asked ourselves what we hoped our students would remember about this course in ten year’s time. From that ten year perspective thoughts about syllabus–“what topics should we cover?"–seemed much too narrow. What matters more is our wish to change the way our students' minds work–the way they approach a problem, or, more generally, the way they approach the world. Most people “skip the numbers" in newspapers, magazines, on the web and (more importantly) even in financial information. We hope that in ten years our students will follow the news, confident in their ability to make sense of the numbers they find there and in their daily lives. Most quantitative reasoning texts are arranged by mathematical topics to be mastered. Since the mathematics is only a part of what we hope students learn, we've chosen another strategy. We look at real life stories that can be best understood with careful reading and a little mathematics.

Proofs from THE BOOK

Author: Martin Aigner,Günter M. Ziegler

Publisher: Springer

ISBN: 3662572656

Category: Mathematics

Page: 326

View: 9237

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This revised and enlarged sixth edition of Proofs from THE BOOK features an entirely new chapter on Van der Waerden’s permanent conjecture, as well as additional, highly original and delightful proofs in other chapters. From the citation on the occasion of the 2018 "Steele Prize for Mathematical Exposition" “... It is almost impossible to write a mathematics book that can be read and enjoyed by people of all levels and backgrounds, yet Aigner and Ziegler accomplish this feat of exposition with virtuoso style. [...] This book does an invaluable service to mathematics, by illustrating for non-mathematicians what it is that mathematicians mean when they speak about beauty.” From the Reviews "... Inside PFTB (Proofs from The Book) is indeed a glimpse of mathematical heaven, where clever insights and beautiful ideas combine in astonishing and glorious ways. There is vast wealth within its pages, one gem after another. ... Aigner and Ziegler... write: "... all we offer is the examples that we have selected, hoping that our readers will share our enthusiasm about brilliant ideas, clever insights and wonderful observations." I do. ... " Notices of the AMS, August 1999 "... This book is a pleasure to hold and to look at: ample margins, nice photos, instructive pictures and beautiful drawings ... It is a pleasure to read as well: the style is clear and entertaining, the level is close to elementary, the necessary background is given separately and the proofs are brilliant. ..." LMS Newsletter, January 1999 "Martin Aigner and Günter Ziegler succeeded admirably in putting together a broad collection of theorems and their proofs that would undoubtedly be in the Book of Erdös. The theorems are so fundamental, their proofs so elegant and the remaining open questions so intriguing that every mathematician, regardless of speciality, can benefit from reading this book. ... " SIGACT News, December 2011

Making Sense of Mathematics for Teaching High School

Understanding How to Use Functions

Author: Edward C. Nolan,Juli K. Dixon

Publisher: Solution Tree Press

ISBN: 1942496494

Category: Education

Page: 192

View: 6428

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Develop a deep understanding of mathematics by grasping the context and purpose behind various strategies. This user-friendly resource presents high school teachers with a logical progression of pedagogical actions, classroom norms, and collaborative teacher team efforts to increase their knowledge and improve mathematics instruction. Explore strategies and techniques to effectively learn and teach significant mathematics concepts and provide all students with the precise, accurate information they need to achieve academic success. Combine student understanding of functions and algebraic concepts so that they can better decipher the world. Benefits Dig deep into mathematical modeling and reasoning to improve as both a learner and teacher of mathematics. Explore how to develop, select, or modify mathematics tasks in order to balance cognitive demand and engage students. Discover the three important norms to uphold in all mathematics classrooms. Learn to apply the tasks, questioning, and evidence (TQE) process to ensure mathematics instruction is focused, coherent, and rigorous. Gain clarity about the most productive progression of mathematical teaching and learning for high school. Watch short videos that show what classrooms that are developing mathematical understanding should look like. Contents Introduction Equations and Functions Structure of Equations Geometry Types of Functions Function Modeling Statistics and Probability Epilogue: Next Steps Appendix: Weight Loss Study Data References Index

Math with Bad Drawings

Illuminating the Ideas That Shape Our Reality

Author: Ben Orlin

Publisher: Black Dog & Leventhal

ISBN: 0316509027

Category: Mathematics

Page: 376

View: 6924

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A hilarious reeducation in mathematics-full of joy, jokes, and stick figures-that sheds light on the countless practical and wonderful ways that math structures and shapes our world. In Math With Bad Drawings, Ben Orlin reveals to us what math actually is; its myriad uses, its strange symbols, and the wild leaps of logic and faith that define the usually impenetrable work of the mathematician. Truth and knowledge come in multiple forms: colorful drawings, encouraging jokes, and the stories and insights of an empathetic teacher who believes that math should belong to everyone. Orlin shows us how to think like a mathematician by teaching us a brand-new game of tic-tac-toe, how to understand an economic crises by rolling a pair of dice, and the mathematical headache that ensues when attempting to build a spherical Death Star. Every discussion in the book is illustrated with Orlin's trademark "bad drawings," which convey his message and insights with perfect pitch and clarity. With 24 chapters covering topics from the electoral college to human genetics to the reasons not to trust statistics, Math with Bad Drawings is a life-changing book for the math-estranged and math-enamored alike.

How Math Can Save Your Life

(And Make You Rich, Help You Find The One, and Avert Catastrophes)

Author: James D. Stein

Publisher: Wiley

ISBN: 9780470437759

Category: Mathematics

Page: 256

View: 9442

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How to make lots of money, keep yourself safe, and even save the world-all by using a little simple math Forget the dull, boring math you learned in school. This book shows you the powerful things math can do for you, with applications no teacher ever taught you in algebra class. How can you make money off credit card companies? Will driving a hybrid save you money in the long run? How do you know when he or she is "the one"? From financial decisions to your education, job, health, and love life, you'll learn how the math you already know can help you get a lot more out of life. Gives you fun, practical advice for using math to improve virtually every area of daily life Includes straightforward explanations and easy-to-follow examples Written by the author of the successful guide, How Math Explains the World Filled with practical, indispensable guidance you can put to work every day, this book will safeguard your wallet and enrich every aspect of your life. You can count on it!

Making sense

teaching and learning mathematics with understanding

Author: James Hiebert

Publisher: Heinemann Educational Books

ISBN: 9780435071325

Category: Education

Page: 184

View: 1374

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This book presents several key principles for teaching mathematics for understanding that you can use to reflect on your own teaching, make more informed decisions, and develop more effective systems of instruction.

Making Sense of Mathematics for Teaching Grades 3-5

(Learn and Teach Concepts and Operations with Depth: How Mathematics Progresses Within and Across Grades)

Author: Juli K. Dixon,Edward C. Nolan

Publisher: Solution Tree Press

ISBN: 1942496435

Category: Education

Page: 200

View: 5174

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Develop a deep understanding of mathematics. This user-friendly resource presents grades 3–5 teachers with a logical progression of pedagogical actions, classroom norms, and collaborative teacher team efforts to increase their knowledge and improve mathematics instruction. Focus on an understanding of and procedural fluency with multiplication and division. Address how to learn and teach fraction concepts and operations with depth. Thoroughly teach plane and solid geometry. Explore strategies and techniques to effectively learn and teach significant mathematics concepts and provide all students with the precise, accurate information they need to achieve academic success. Benefits Dig deep into mathematical modeling and reasoning to improve as both a learner and teacher of mathematics. Explore how to develop, select, and modify mathematics tasks in order to balance cognitive demand and engage students. Discover the three important norms to uphold in all mathematics classrooms. Learn to apply the tasks, questioning, and evidence (TQE) process to ensure mathematics instruction is focused, coherent, and rigorous. Use charts and diagrams for classifying shapes, which can engage students in important mathematical practices. Access short videos that show what classrooms that are developing mathematical understanding should look like. Contents Introduction 1 Place Value, Addition, and Subtraction 2 Multiplication and Division 3 Fraction Concepts 4 Fraction Operations 5 Geometry 6 Measurement Epilogue Next Steps Appendix A Completed Classification of Triangles Chart Appendix B Completed Diagram for Classifying Quadrilaterals

Kiss My Math

Showing Pre-algebra Who's Boss

Author: Danica McKellar

Publisher: Penguin

ISBN: 9781594630491

Category: Juvenile Nonfiction

Page: 335

View: 1947

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The television actress and mathematics guru author of Math Doesn't Suck presents a pre-algebra primer for seventh- to ninth-graders, in an accessible reference that shares time-saving tricks, real-world examples, and detailed practice problems. 100,000 first printing.