"One of the best critiques of current mathematics education I have ever seen."-Keith Devlin, math columnist on NPR's Morning Edition A brilliant research mathematician who has devoted his career to teaching kids reveals math to be creative and beautiful and rejects standard anxiety-producing teaching methods. Witty and accessible, Paul Lockhart's controversial approach will provoke spirited debate among educators and parents alike and it will alter the way we think about math forever. Paul Lockhart, has taught mathematics at Brown University and UC Santa Cruz. Since 2000, he has dedicated himself to K-12 level students at St. Ann's School in Brooklyn, New York.

Paul Lockhart’s Mathematician’s Lament outlined how we introduce math to students in the wrong way. Measurement explains how math should be done. With plain English and pictures, Lockhart makes complex ideas about shape and motion intuitive and graspable, and offers a solution to math phobia by introducing us to math as an artful way of thinking and living.In conversational prose that conveys his passion for the subject, Lockhart makes mathematics accessible without oversimplifying. He makes no more attempt to hide the challenge of mathematics than he does to shield us from its beautiful intensity. Favoring plain English and pictures over jargon and formulas, he succeeds in making complex ideas about the mathematics of shape and motion intuitive and graspable. His elegant discussion of mathematical reasoning and themes in classical geometry offers proof of his conviction that mathematics illuminates art as much as science.Lockhart leads us into a universe where beautiful designs and patterns float through our minds and do surprising, miraculous things. As we turn our thoughts to symmetry, circles, cylinders, and cones, we begin to see that almost anyone can “do the math” in a way that brings emotional and aesthetic rewards. Measurement is an invitation to summon curiosity, courage, and creativity in order to experience firsthand the playful excitement of mathematical work.

"A terrific biography. . . . The dramatic story of how the American army that beat the British was forged has never been better told." —Doris Kearns Goodwin, New York Times–bestselling author of Team of RivalsFrustrated with a stalled career in midlife, the Baron de Steuben uprooted himself from his native Europe to seek one last chance at glory and fame in the New World. Steeped in the traditions of the Prussian army of Frederick the Great—the most ruthlessly effective in Europe—he taught the ragged, demoralized soldiers of the Continental Army how to fight like Europeans. His guiding hand shaped the fighting force that triumphed over the British at Monmouth, Stony Point, and Yorktown. But his influence did not end with the Revolution. Steuben was instrumental in creating West Point and in writing the first official regulations of the American army, and his principles have guided the American armed forces to this day."Reveal[s] the deeds and character of a man whose life was full of surprises and frustrating failures but ultimately crowned with success . . . sheds light on the career of an important but relatively obscure figure." —Booklist"The author generally treats [Steuben] with balance, understanding and great good humor." —The Wall Street Journal"An archetypal American story: an immigrant, ambitious, blustering, insecure, who gives his talents and his passion to his new-found home." —Richard Brookhiser, author of George Washington on Leadership

A reassessment of one of the most famous battles in U.S. history combining military and political history by the author of The Drillmaster of Valley Forge.One hot June afternoon in 1775, on the gentle slopes of a hill near Boston, Massachusetts, a small band of ordinary Americans—frightened but fiercely determined—dared to stand up to a superior British force. The clash would be immortalized as the Battle of Bunker Hill: the first real engagement of the American Revolution and one of the most famous battles in our history. But Bunker Hill was not the battle that we have been taught to believe it was.Revisiting old evidence and drawing on new research, historian Paul Lockhart shows that Bunker Hill was a clumsy engagement pitting one inexperienced army against another. Lockhart tells the rest of the story, too: how a mob of armed civilians became America''s first army; how George Washington set aside his comfortable patrician life to take command of the veterans of Bunker Hill; and how the forgotten heroes of 1775—though overshadowed by the more famous Founding Fathers—kept the notion of American liberty alive, and thus made independence possible."[A] stimulating history. . . . Lockhart''s shrewd, well-judged interpretation corrects myths about the battle and the men who fought it while doing full justice to their achievement in creating an army—and a nation—out of chaos." —Publishers Weekly (starred review)

The history of Sweden in the seventeenth century is perhaps one of the most remarkable political success stories of early modern Europe. Little more than a century after achieving independence from Denmark, Sweden - an impoverished and sparsely-populated state - had defeated all of its most fearsome enemies and was ranked amongst the great powers of Europe. In this book, which incorporates the latest research on the subject, Paul Douglas Lockhart:- Surveys the political, diplomatic, economic, social and cultural history of the country, from the beginnings of its career as an empire to its decline at the end of the seventeenth century- Examines the mechanisms that helped Sweden to achieve the status of a great power, and the reasons for its eventual downfall- Emphasises the interplay between social structure, constitutional development, and military necessityClear and well-written, Lockhart''s text is essential reading for all those with an interest in the fascinating history of early modern Sweden.

The history of Sweden in the seventeenth century is perhaps one of the most remarkable political success stories of early modern Europe. Little more than a century after achieving independence from Denmark, Sweden - an impoverished and sparsely-populated state - had defeated all of its most fearsome enemies and was ranked amongst the great powers of Europe. In this book, which incorporates the latest research on the subject, Paul Douglas Lockhart:- Surveys the political, diplomatic, economic, social and cultural history of the country, from the beginnings of its career as an empire to its decline at the end of the seventeenth century- Examines the mechanisms that helped Sweden to achieve the status of a great power, and the reasons for its eventual downfall- Emphasises the interplay between social structure, constitutional development, and military necessityClear and well-written, Lockhart's text is essential reading for all those with an interest in the fascinating history of early modern Sweden.

The history of Sweden in the seventeenth century is perhaps one of the most remarkable political success stories of early modern Europe. Little more than a century after achieving independence from Denmark, Sweden - an impoverished and sparsely-populated state - had defeated all of its most fearsome enemies and was ranked amongst the great powers of Europe. In this book, which incorporates the latest research on the subject, Paul Douglas Lockhart:- Surveys the political, diplomatic, economic, social and cultural history of the country, from the beginnings of its career as an empire to its decline at the end of the seventeenth century- Examines the mechanisms that helped Sweden to achieve the status of a great power, and the reasons for its eventual downfall- Emphasises the interplay between social structure, constitutional development, and military necessityClear and well-written, Lockhart's text is essential reading for all those with an interest in the fascinating history of early modern Sweden.

“Inspiring and informative…deserves to be widely read.”—Wall Street Journal“This fun book offers a philosophical take on number systems and revels in the beauty of math.”—Science NewsBecause we have ten fingers, grouping by ten seems natural, but twelve would be better for divisibility, and eight is well suited to repeated halving. Grouping by two, as in binary code, has turned out to have its own remarkable advantages.Paul Lockhart presents arithmetic not as rote manipulation of numbers—a practical if mundane branch of knowledge best suited for filling out tax forms—but as a fascinating, sometimes surprising intellectual craft that arises from our desire to add, divide, and multiply important things. Passionate and entertaining, Arithmetic invites us to experience the beauty of mathematics through the eyes of a beguiling teacher.“A nuanced understanding of working with numbers, gently connecting procedures that we once learned by rote with intuitions long since muddled by education…Lockhart presents arithmetic as a pleasurable pastime, and describes it as a craft like knitting.”—Jonathon Keats, New Scientist“What are numbers, how did they arise, why did our ancestors invent them, and how did they represent them? They are, after all, one of humankind’s most brilliant inventions, arguably having greater impact on our lives than the wheel. Lockhart recounts their fascinating story…A wonderful book.”—Keith Devlin, author of Finding Fibonacci

A joyful and intimate celebration of the beauty and creativity of algebra from one of the foremost math educators of our time.For many of us, algebra conjures up memories of dull classes spent wondering when we’d ever have to solve a system of equations or factor a polynomial. Indeed, most of the time, if we need to plan a budget or figure out how early to leave the house for work, common sense or a quick, seat-of-the-pants approximation is good enough.But as mathematician Paul Lockhart argues in The Mending of Broken Bones, once we stop thinking of algebra in terms of its practical applications, we can appreciate it for the beautiful and gratifying subject that it is. In his hands, algebra is the delicate craft of untangling numerical puzzles to reveal the hidden patterns and often surprising behaviors of the numbers themselves. As Lockhart traverses numerical systems, slips into and out of higher-dimensional space, and delights in the intimate connections between algebra and geometry, we come to see the discipline from his viewpoint: not as the mundane exercises of our school years, but instead as an art form whose beauty lies in its elegant simplicity.Written in Lockhart’s charming and conversational prose, The Mending of Broken Bones is an impassioned and deeply personal celebration of algebra that helps us experience the profound joys of mathematical discovery.

The history of Sweden in the seventeenth century is perhaps one of the most remarkable political success stories of early modern Europe. Little more than a century after achieving independence from Denmark, Sweden - an impoverished and sparsely-populated state - had defeated all of its most fearsome enemies and was ranked amongst the great powers of Europe. In this book, which incorporates the latest research on the subject, Paul Douglas Lockhart:- Surveys the political, diplomatic, economic, social and cultural history of the country, from the beginnings of its career as an empire to its decline at the end of the seventeenth century- Examines the mechanisms that helped Sweden to achieve the status of a great power, and the reasons for its eventual downfall- Emphasises the interplay between social structure, constitutional development, and military necessityClear and well-written, Lockhart''s text is essential reading for all those with an interest in the fascinating history of early modern Sweden.

The history of warfare cannot be fully understood without considering the technology of killing. In Firepower, acclaimed historian Paul Lockhart tells the story of military technology from the Renaissance to the dawn of the atomic era -- five-hundred-year-long "age of firepower" during which the evolution of weaponry transformed the conduct of warfare in the West.Weapons technology had always influenced warfare. But the introduction of gunpowder weapons at the close of the Middle Ages made military technology the largest single factor shaping warfare's tactics, strategy, and logistics. Over the five centuries leading up to World War II, the art of war revolved around the ever-more-effective delivery of firepower, and the driving force of weapons development was the compulsion to make that possible. But for centuries, even as it became more effective, military weaponry remained simple and affordable enough that nearly any state could afford to equip a respectable army; weapons could be used and used again until they physically wore out. That all changed, very suddenly, around 1870. Widespread industrialization and rapid advances in metallurgy and chemistry meant that by the start of World War I, only a handful of great powers could afford to manufacture their own weapons. Revolutions in military technology, in short, triggered a revolution in the structure of power in the West, significantly reducing the number of nations that could act assertively in international politics -- and reducing the others to a condition of permanent subordination.Going beyond the battlefield to consider the profound political and social contexts of armed conflict, Firepower ultimately reveals how the evolution of weapons technology, and the uses to which it has been put, have together transformed human history.

This book considers the role played by Denmark’s King Frederik II (1559-88) in the international diplomacy of the 'age of religious wars'. As Europe’s leading Lutheran sovereign, Frederik commanded great influence; his conviction that an international Catholic 'conspiracy' threatened to destroy Protestantism led him to work towards the creation of a Protestant alliance that included both Calvinist and Lutheran states.Lockhart examines the role of religion in Frederik’s foreign policy, the motivations behind the king’s alliance-building projects, and the reasons behind the ultimate failure of Frederik’s policies.This volume will be of interest to students of early modern diplomacy, sixteenth-century Protestantism, and the Scandinavian monarchies in the early modern period.

"One of the best critiques of current mathematics education I have ever seen."-Keith Devlin, math columnist on NPR's Morning Edition A brilliant research mathematician who has devoted his career to teaching kids reveals math to be creative and beautiful and rejects standard anxiety-producing teaching methods. Witty and accessible, Paul Lockhart's controversial approach will provoke spirited debate among educators and parents alike and it will alter the way we think about math forever. Paul Lockhart, has taught mathematics at Brown University and UC Santa Cruz. Since 2000, he has dedicated himself to K-12 level students at St. Ann's School in Brooklyn, New York.

Paul Lockhart’s Mathematician’s Lament outlined how we introduce math to students in the wrong way. Measurement explains how math should be done. With plain English and pictures, Lockhart makes complex ideas about shape and motion intuitive and graspable, and offers a solution to math phobia by introducing us to math as an artful way of thinking and living.In conversational prose that conveys his passion for the subject, Lockhart makes mathematics accessible without oversimplifying. He makes no more attempt to hide the challenge of mathematics than he does to shield us from its beautiful intensity. Favoring plain English and pictures over jargon and formulas, he succeeds in making complex ideas about the mathematics of shape and motion intuitive and graspable. His elegant discussion of mathematical reasoning and themes in classical geometry offers proof of his conviction that mathematics illuminates art as much as science.Lockhart leads us into a universe where beautiful designs and patterns float through our minds and do surprising, miraculous things. As we turn our thoughts to symmetry, circles, cylinders, and cones, we begin to see that almost anyone can “do the math” in a way that brings emotional and aesthetic rewards. Measurement is an invitation to summon curiosity, courage, and creativity in order to experience firsthand the playful excitement of mathematical work.

"A terrific biography. . . . The dramatic story of how the American army that beat the British was forged has never been better told." —Doris Kearns Goodwin, New York Times–bestselling author of Team of RivalsFrustrated with a stalled career in midlife, the Baron de Steuben uprooted himself from his native Europe to seek one last chance at glory and fame in the New World. Steeped in the traditions of the Prussian army of Frederick the Great—the most ruthlessly effective in Europe—he taught the ragged, demoralized soldiers of the Continental Army how to fight like Europeans. His guiding hand shaped the fighting force that triumphed over the British at Monmouth, Stony Point, and Yorktown. But his influence did not end with the Revolution. Steuben was instrumental in creating West Point and in writing the first official regulations of the American army, and his principles have guided the American armed forces to this day."Reveal[s] the deeds and character of a man whose life was full of surprises and frustrating failures but ultimately crowned with success . . . sheds light on the career of an important but relatively obscure figure." —Booklist"The author generally treats [Steuben] with balance, understanding and great good humor." —The Wall Street Journal"An archetypal American story: an immigrant, ambitious, blustering, insecure, who gives his talents and his passion to his new-found home." —Richard Brookhiser, author of George Washington on Leadership

A reassessment of one of the most famous battles in U.S. history combining military and political history by the author of The Drillmaster of Valley Forge.One hot June afternoon in 1775, on the gentle slopes of a hill near Boston, Massachusetts, a small band of ordinary Americans—frightened but fiercely determined—dared to stand up to a superior British force. The clash would be immortalized as the Battle of Bunker Hill: the first real engagement of the American Revolution and one of the most famous battles in our history. But Bunker Hill was not the battle that we have been taught to believe it was.Revisiting old evidence and drawing on new research, historian Paul Lockhart shows that Bunker Hill was a clumsy engagement pitting one inexperienced army against another. Lockhart tells the rest of the story, too: how a mob of armed civilians became America''s first army; how George Washington set aside his comfortable patrician life to take command of the veterans of Bunker Hill; and how the forgotten heroes of 1775—though overshadowed by the more famous Founding Fathers—kept the notion of American liberty alive, and thus made independence possible."[A] stimulating history. . . . Lockhart''s shrewd, well-judged interpretation corrects myths about the battle and the men who fought it while doing full justice to their achievement in creating an army—and a nation—out of chaos." —Publishers Weekly (starred review)

The history of Sweden in the seventeenth century is perhaps one of the most remarkable political success stories of early modern Europe. Little more than a century after achieving independence from Denmark, Sweden - an impoverished and sparsely-populated state - had defeated all of its most fearsome enemies and was ranked amongst the great powers of Europe. In this book, which incorporates the latest research on the subject, Paul Douglas Lockhart:- Surveys the political, diplomatic, economic, social and cultural history of the country, from the beginnings of its career as an empire to its decline at the end of the seventeenth century- Examines the mechanisms that helped Sweden to achieve the status of a great power, and the reasons for its eventual downfall- Emphasises the interplay between social structure, constitutional development, and military necessityClear and well-written, Lockhart''s text is essential reading for all those with an interest in the fascinating history of early modern Sweden.

The history of Sweden in the seventeenth century is perhaps one of the most remarkable political success stories of early modern Europe. Little more than a century after achieving independence from Denmark, Sweden - an impoverished and sparsely-populated state - had defeated all of its most fearsome enemies and was ranked amongst the great powers of Europe. In this book, which incorporates the latest research on the subject, Paul Douglas Lockhart:- Surveys the political, diplomatic, economic, social and cultural history of the country, from the beginnings of its career as an empire to its decline at the end of the seventeenth century- Examines the mechanisms that helped Sweden to achieve the status of a great power, and the reasons for its eventual downfall- Emphasises the interplay between social structure, constitutional development, and military necessityClear and well-written, Lockhart's text is essential reading for all those with an interest in the fascinating history of early modern Sweden.

The history of Sweden in the seventeenth century is perhaps one of the most remarkable political success stories of early modern Europe. Little more than a century after achieving independence from Denmark, Sweden - an impoverished and sparsely-populated state - had defeated all of its most fearsome enemies and was ranked amongst the great powers of Europe. In this book, which incorporates the latest research on the subject, Paul Douglas Lockhart:- Surveys the political, diplomatic, economic, social and cultural history of the country, from the beginnings of its career as an empire to its decline at the end of the seventeenth century- Examines the mechanisms that helped Sweden to achieve the status of a great power, and the reasons for its eventual downfall- Emphasises the interplay between social structure, constitutional development, and military necessityClear and well-written, Lockhart's text is essential reading for all those with an interest in the fascinating history of early modern Sweden.

“Inspiring and informative…deserves to be widely read.”—Wall Street Journal“This fun book offers a philosophical take on number systems and revels in the beauty of math.”—Science NewsBecause we have ten fingers, grouping by ten seems natural, but twelve would be better for divisibility, and eight is well suited to repeated halving. Grouping by two, as in binary code, has turned out to have its own remarkable advantages.Paul Lockhart presents arithmetic not as rote manipulation of numbers—a practical if mundane branch of knowledge best suited for filling out tax forms—but as a fascinating, sometimes surprising intellectual craft that arises from our desire to add, divide, and multiply important things. Passionate and entertaining, Arithmetic invites us to experience the beauty of mathematics through the eyes of a beguiling teacher.“A nuanced understanding of working with numbers, gently connecting procedures that we once learned by rote with intuitions long since muddled by education…Lockhart presents arithmetic as a pleasurable pastime, and describes it as a craft like knitting.”—Jonathon Keats, New Scientist“What are numbers, how did they arise, why did our ancestors invent them, and how did they represent them? They are, after all, one of humankind’s most brilliant inventions, arguably having greater impact on our lives than the wheel. Lockhart recounts their fascinating story…A wonderful book.”—Keith Devlin, author of Finding Fibonacci

A joyful and intimate celebration of the beauty and creativity of algebra from one of the foremost math educators of our time.For many of us, algebra conjures up memories of dull classes spent wondering when we’d ever have to solve a system of equations or factor a polynomial. Indeed, most of the time, if we need to plan a budget or figure out how early to leave the house for work, common sense or a quick, seat-of-the-pants approximation is good enough.But as mathematician Paul Lockhart argues in The Mending of Broken Bones, once we stop thinking of algebra in terms of its practical applications, we can appreciate it for the beautiful and gratifying subject that it is. In his hands, algebra is the delicate craft of untangling numerical puzzles to reveal the hidden patterns and often surprising behaviors of the numbers themselves. As Lockhart traverses numerical systems, slips into and out of higher-dimensional space, and delights in the intimate connections between algebra and geometry, we come to see the discipline from his viewpoint: not as the mundane exercises of our school years, but instead as an art form whose beauty lies in its elegant simplicity.Written in Lockhart’s charming and conversational prose, The Mending of Broken Bones is an impassioned and deeply personal celebration of algebra that helps us experience the profound joys of mathematical discovery.

The history of Sweden in the seventeenth century is perhaps one of the most remarkable political success stories of early modern Europe. Little more than a century after achieving independence from Denmark, Sweden - an impoverished and sparsely-populated state - had defeated all of its most fearsome enemies and was ranked amongst the great powers of Europe. In this book, which incorporates the latest research on the subject, Paul Douglas Lockhart:- Surveys the political, diplomatic, economic, social and cultural history of the country, from the beginnings of its career as an empire to its decline at the end of the seventeenth century- Examines the mechanisms that helped Sweden to achieve the status of a great power, and the reasons for its eventual downfall- Emphasises the interplay between social structure, constitutional development, and military necessityClear and well-written, Lockhart''s text is essential reading for all those with an interest in the fascinating history of early modern Sweden.

The history of warfare cannot be fully understood without considering the technology of killing. In Firepower, acclaimed historian Paul Lockhart tells the story of military technology from the Renaissance to the dawn of the atomic era -- five-hundred-year-long "age of firepower" during which the evolution of weaponry transformed the conduct of warfare in the West.Weapons technology had always influenced warfare. But the introduction of gunpowder weapons at the close of the Middle Ages made military technology the largest single factor shaping warfare's tactics, strategy, and logistics. Over the five centuries leading up to World War II, the art of war revolved around the ever-more-effective delivery of firepower, and the driving force of weapons development was the compulsion to make that possible. But for centuries, even as it became more effective, military weaponry remained simple and affordable enough that nearly any state could afford to equip a respectable army; weapons could be used and used again until they physically wore out. That all changed, very suddenly, around 1870. Widespread industrialization and rapid advances in metallurgy and chemistry meant that by the start of World War I, only a handful of great powers could afford to manufacture their own weapons. Revolutions in military technology, in short, triggered a revolution in the structure of power in the West, significantly reducing the number of nations that could act assertively in international politics -- and reducing the others to a condition of permanent subordination.Going beyond the battlefield to consider the profound political and social contexts of armed conflict, Firepower ultimately reveals how the evolution of weapons technology, and the uses to which it has been put, have together transformed human history.

This book considers the role played by Denmark’s King Frederik II (1559-88) in the international diplomacy of the 'age of religious wars'. As Europe’s leading Lutheran sovereign, Frederik commanded great influence; his conviction that an international Catholic 'conspiracy' threatened to destroy Protestantism led him to work towards the creation of a Protestant alliance that included both Calvinist and Lutheran states.Lockhart examines the role of religion in Frederik’s foreign policy, the motivations behind the king’s alliance-building projects, and the reasons behind the ultimate failure of Frederik’s policies.This volume will be of interest to students of early modern diplomacy, sixteenth-century Protestantism, and the Scandinavian monarchies in the early modern period.