Teaching with Tasks for Effective Mathematics Learning

(Mathematics Teacher Education, 9) 

 Peter Sullivan, Doug Clarke e Barbara Clarke 

Springer | 2013 | 214 páginas | pdf | 3,3 Mb

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Using classroom mathematics tasks to support student learning is the topic of this timely volume. Employing research-based data, the authors focus on teacher practice as well as teacher and student learning and knowledge creation to demonstrate the use of mathematics tasks which promote effective student understanding. Unique in the field, the book provides a thorough, comprehensive guide to the nature of tasks for researchers, teacher educators, curriculum designers, administrators and teachers. Chapters on the effective implementation of mathematics tasks in the classroom, distinct pedagogical concepts related to teaching with mathematics tasks, and sample lessons that clearly demonstrate successful uses for mathematics tasks in the classroom are included.  The book is designed to provide a mix of cutting-edge data on task use with concrete examples of successful tasks and implementation tactics.  All of the lesson plans and illustrative examples provided have been extensively evaluated and tested in actual learning situations and feature specific suggestions for combating student difficulties and promoting solution pathways. This is a book that is essential for anyone hoping to understand both the importance of mathematics tasks for enhancing student learning and ways in which mathematics tasks can be applied in the classroom to achieve learning goals and objectives.   

Contents
Researching Tasks in Mathematics Classrooms
Perspectives on Mathematics, Learning, and Teaching
Tasks and Mathematics Learning
Using Purposeful Representational Tasks
Using Mathematical Tasks Arising from Contexts
Using Content-Specific Open-Ended Tasks
Moving from the Task to the Lesson: Pedagogical Practices and Other Issues
Constructing a Sequence of Lessons
Students’ Preferences for Different Types of Mathematics Tasks
Students’ Perceptions of Characteristics of Desired Mathematics Lessons
Contrasting Types of Tasks: A Story of Three Lessons
Conclusion
A Selection of Mathematical Tasks

Computer Aided Assessment of Mathematics

Chris Sangwin

Oxford University Press | 2013 | páginas | rar - pdf | 1,5 Mb

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Assessment is a key driver in mathematics education. This book examines computer aided assessment (CAA) of mathematics in which computer algebra systems (CAS) are used to establish the mathematical properties of expressions provided by students in response to questions. In order to automate such assessment, the relevant criteria must be encoded and, in articulating precisely the desired criteria, the teacher needs to think very carefully about the goals of the task. Hence CAA acts as a vehicle to examine assessment and mathematics education in detail and from a fresh perspective. 
One example is how it is natural for busy teachers to set only those questions that can be marked by hand in a straightforward way, even though the constraints of paper-based formats restrict what they do and why. There are other kinds of questions, such as those with non-unique correct answers, or where assessing the properties requires the marker themselves to undertake a significant computation. It is simply not sensible for a person to set these to large groups of students when marking by hand. However, such questions have their place and value in provoking thought and learning. 
This book, aimed at teachers in both schools and universities, explores how, in certain cases, different question types can be automatically assessed. Case studies of existing systems have been included to illustrate this in a concrete and practical way.

Contents
1. Introduction
2. An Assessment Vignette
3. Learning and Assessing Mathematics
4. Mathematical Question Spaces
5. Notation and Syntax
6. Computer Algebra Systems for CAA
7. The STACK CAA System
8. Software Case Studies
9. The Future

Mathematics as sign : writing, imagining, counting

Brian Rotman

Stanford University Press | 2000 | 180 páginas | rar - pdf | 4,8 Mb

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Two features of mathematics stand out: its menagerie of seemingly eternal objects (numbers, spaces, patterns, functions, categories, morphisms, graphs, and so on), and the hieroglyphics of special notations, signs, symbols, and diagrams associated with them. The author challenges the widespread belief in the extra-human origins of these objects and the understanding of mathematics as either a purely mental activity about them or a formal game of manipulating symbols. Instead, he argues that mathematics is a vast and unique man-made imagination machine controlled by writing.

Mathematics as Sign addresses both aspects—mental and linguistic—of this machine. The opening essay, "Toward a Semiotics of Mathematics" (long acknowledged as a seminal contribution to its field), sets out the author's underlying model. According to this model, "doing" mathematics constitutes a kind of waking dream or thought experiment in which a proxy of the self is propelled around imagined worlds that are conjured into intersubjective being through signs.
Other essays explore the status of these signs and the nature of mathematical objects, how mathematical ideograms and diagrams differ from each other and from written words, the probable fate of the real number continuum and calculus in the digital era, the manner in which Platonic and Aristotelean metaphysics are enshrined in the contemporary mathematical infinitude of endless counting, and the possibility of creating a new conception of the sequence of whole numbers based on what the author calls non-Euclidean counting.
Reprising and going beyond the critique of number in Ad Infinitum, the essays in this volume offer an accessible insight into Rotman's project, one that has been called "one of the most original and important recent contributions to the philosophy of mathematics."

CONTENTS
PREFACE: WRITING, IMAGINING, COUNTING IX
1. Toward a Semiotics of Mathematics 1
2. Making Marks on Paper 44
3. How Ideal Are the Reals? 71
4. God Tricks; or, Numbers from the Bottom Up? 106
5. Counting on Non-Euclidean Fingers 125
NOTES 157
WORKS CITED 163
INDEX OF PERSONS 169

What's Math Got to Do with It?: Helping Children Learn to Love Their Least Favorite Subject-and Why It's Important for America

Jo Boaler

Viking Adult | 2008| 288 páginas | rar-epub | Mb

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A recent assessment of mathematics performance around the world ranked the United States twenty-eighth out of forty countries in the study. When the level of spending was taken into account, we sank to the very bottom of the list. According to Jo Boaler, who was a professor of mathematics education at Stanford University for nine years, statistics like these are becoming all too common—we have reached the point of crisis, and a new course of action is crucial.

In this straightforward and inspiring book, Boaler outlines the nature of the problem by following the progress of students in middle and high schools over a number of years, to find out which teaching methods are exciting students and getting results. Based on her research, she presents concrete solutions that will help reverse the trend, including classroom approaches, essential strategies for students, advice for parents on how to help children enjoy mathematics, and ways to work with teachers in schools.

The United States is continuing to fall rapidly behind the rest of the developed world when it comes to math education, and the future of our economy depends on the quality of teaching that our children receive today. In What’s Math Got to Do with It?, Jo Boaler offers us a new way forward, making this book in dispensable for all parents and educators, as well as anyone interested in the mathematical and scientific future of our society

Table of Contents
Title Page
Copyright Page
Dedication
Acknowledgements
1 / What Is Math? - And Why Do We All Need It?
2 / What’s Going Wrong in Classrooms? - Identifying the Problems
3 / A Vision for a Better Future - Effective Classroom Approaches
4 / Taming the Monster - New Forms of Testing That Encourage Learning
5 / Stuck in the Slow Lane - How American Grouping Systems Perpetuate Low Achievement
6 / Paying the Price for Sugar and Spice - How Girls and Women Are Kept Out of  ...
7 / Key Strategies and Ways of Working
8 / Giving Children the Best Mathematical Start - Activities and Advice
9 / Making a Difference through Work with Schools
Appendix A - Solutions to the Mathematics Problems
Appendix B - Mathematics Curricula
Appendix C - Recommended Math Puzzle Books (in order of difficulty)
Notes

Index

Livros da mesma autora, neste blog:

Experiencing school mathematics : traditional and reform approaches to teaching and their impact on student learning por Jo Boaler Idioma: Inglês   Editora: Mahwah, N.J. : L. Erlbaum, 2002.

Multiple perspectives on mathematics teaching and learning por Jo Boaler Idioma: Inglês   Editora: Westport, Conn. : Praeger, 2000.

God Created The Integers: The Mathematical Breakthroughs that Changed History

Stephen Hawking

Running Press | 2007 | 1375 páginas | rar - pdf | 6,9 Mb


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epub - 46 Mb
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Bestselling author and physicist Stephen Hawking explores the "masterpieces" of mathematics, 25 landmarks spanning 2,500 years and representing the work of 15 mathematicians, including Augustin Cauchy, Bernard Riemann, and Alan Turing. This extensive anthology allows readers to peer into the mind of genius by providing them with excerpts from the original mathematical proofs and results. It also helps them understand the progression of mathematical thought, and the very foundations of our present-day technologies. Each chapter begins with a biography of the featured mathematician, clearly explaining the significance of the result, followed by the full proof of the work, reproduced from the original publication.

CONTENTS
Introduction xiii
EUCLID (C. 325BC–265BC)
His Life and Work 01
Selections from Euclid’s Elements
Book I: Basic Geometry—Definitions, Postulates, Common Notions; and Proposition 47, (leading up to the Pythagorean Theorem) 07
Book V: The Eudoxian Theory of Proportion—Definitions & Propositions 25
Book VII: Elementary Number Theory—Definitions & Propositions 63
Book IX: Proposition 20: The Infinitude of Prime Numbers 101
Book IX: Proposition 36: Even Perfect Numbers 102
Book X: Commensurable and Incommensurable Magnitudes 104
ARCHIMEDES (287BC–212BC)
His Life and Work 119
Selections from The Works of Archimedes
On the Sphere and Cylinder, Books I and II 126/168
Measurement of a Circle 194
The Sand Reckoner 200
The Methods 209
DIOPHANTUS (C. 200–284)
His Life and Work 241
Selections from Diophantus of Alexandria, A Study in the History of Greek Algebra
Book II Problems 8–35 246
Book III Problems 5–21 255
Book V Problems 1–29 265
RENÉ DESCARTES (1596–1650)
His Life and Work 285
The Geometry of Rene Descartes 292
ISAAC NEWTON (1642–1727)
His Life and Work 365
Selections from Principia
On First and Last Ratios of Quantities 374
LEONHARD EULER (1707–1783)
His Life and Work 383
On the sums of series of reciprocals (De summis serierum reciprocarum) 393
The Seven Bridges of Konigsberg 400
Proof that Every Integer is A Sum of Four Squares 407
PIERRE SIMON LAPLACE (1749–1827)
His Life and Work 411
A Philosophical Essay on Probabilities 418
JEAN BAPTISTE JOSEPH FOURIER (1768–1830)
His Life and Work 519
Selection from The Analytical Theory of Heat
Chapter III: Propagation of Heat in an Infinite Rectangular Solid (The Fourier series) 528
CARL FRIEDRICH GAUSS (1777–1855)
His Life and Work 591
Selections from Disquisitiones Arithmeticae (Arithmetic Disquisitions)
Section III Residues of Powers 599
Section IV Congruences of the Second Degree 625
AUGUSTIN-LOUIS CAUCHY (1789–1857)
His Life and Work 663
Selections from Oeuvres complètes d’Augustin Cauchy
Résumé des leçons données à l’École Royale Polytechnique sur le calcul infinitésimal (1823), series 2, vol. 4
Lessons 3–4 on differential calculus 671
Lessons 21–24 on the integral 679
NIKOLAI IVANOVICH LOBACHEVSKY (1792–1856)
His Life and Work 813
Geometrical Researches on the Theory of Parallels 820
JÁNOS BOLYAI (1802–1860)
His Life and Work 743
The Science of Absolute Space 750
ÉVARISTE GALOIS (1811–1832)
His Life and Work 797
On the conditions that an equation be soluble by radicals 807
Of the primitive equations which are soluble by radicals 820
On Groups and Equations and Abelian Integrals 828
GEORGE BOOLE (1815–1864)
His Life and Work 835
An Investigation of the Laws of Thought 842
BERNHARD RIEMANN (1826–1866)
His Life and Work 979
On the Representability of a Function by Means of a Trigonometric Series (Ueber die Darstellbarkeit eine Function durch einer trigonometrische Reihe) 992
On the Hypotheses which lie at the Bases of Geometry (Ueber die Hypothesen, welche der Geometrie zu Grunde liegen) 1031
On the Number of Prime Numbers Less than a Given Quantity (Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse) 1042
KARL WEIERSTRASS (1815–1897)
His Life and Work 1053
Selected Chapters on the Theory of Functions, Lecture Given in Berlin in 1886, with the Inaugural Academic Speech, Berlin 1857
§ 7 Gleichmässige Stetigkeit (Uniform Continuity) 1060
RICHARD DEDEKIND (1831–1916)
His Life and Work 1067
Essays on the Theory of Numbers 1072
GEORG CANTOR (1848–1918)
His Life and Work 1131
Selections from Contributions to the Founding of the Theory of Transfinite Numbers
Articles I and II 1137
HENRI LEBESGUE (1875–1941)
His Life and Work 1207
Selections from Integrale, Longeur, Aire (Intergral, Length, Area) Preliminaries and Integral 1212
KURT GÖDEL (1906–1978)
His Life and Work 1555
On Formally Undecidable Propositions of Principia Mathematica and Related Systems 1263
ALAN TURING (1912–1954)
His Life and Work 1285
On computable numbers with an application to the Entscheidungsproblem, Proceedings of the London Mathematical Society 1293

Enlightening Symbols A Short History of Mathematical Notation and Its Hidden Powers

Joseph Mazur

Princeton University Press | 2014 | 312 páginas | rar - pdf (from html) | 4,15 Mb

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While all of us regularly use basic math symbols such as those for plus, minus, and equals, few of us know that many of these symbols weren't available before the sixteenth century. What did mathematicians rely on for their work before then? And how did mathematical notations evolve into what we know today? In Enlightening Symbols, popular math writer Joseph Mazur explains the fascinating history behind the development of our mathematical notation system. He shows how symbols were used initially, how one symbol replaced another over time, and how written math was conveyed before and after symbols became widely adopted.

Traversing mathematical history and the foundations of numerals in different cultures, Mazur looks at how historians have disagreed over the origins of the numerical system for the past two centuries. He follows the transfigurations of algebra from a rhetorical style to a symbolic one, demonstrating that most algebra before the sixteenth century was written in prose or in verse employing the written names of numerals. Mazur also investigates the subconscious and psychological effects that mathematical symbols have had on mathematical thought, moods, meaning, communication, and comprehension. He considers how these symbols influence us (through similarity, association, identity, resemblance, and repeated imagery), how they lead to new ideas by subconscious associations, how they make connections between experience and the unknown, and how they contribute to the communication of basic mathematics.

From words to abbreviations to symbols, this book shows how math evolved to the familiar forms we use today.

Contents

Introduction ix

Definitions xxi

Note on the Illustrations xxiii

Part 1 Numerals 1

1. Curious Beginnings 3

2. Certain Ancient Number Systems 10

3. Silk and Royal Roads 26

4. The Indian Gift 35

5. Arrival in Europe 51

6. The Arab Gift 60

7. Liber Abbaci 64

8. Refuting Origins 73

Part 2 Algebra 81

9. Sans Symbols 85

10. Diophantus’s Arithmetica 93

11. The Great Art 109

12. Symbol Infancy 116

13. The Timid Symbol 127

14. Hierarchies of Dignity 133

15. Vowels and Consonants 141

16. The Explosion 150

17. A Catalogue of Symbols 160

18. The Symbol Master 165

19. The Last of the Magicians 169

Part 3 The Power of Symbols 177

20. Rendezvous in the Mind 179

21. The Good Symbol 189

22. Invisible Gorillas 192

23. Mental Pictures 210

24. Conclusion 216

Appendix A Leibniz’s Notation 221

Appendix B Newton’s Fluxion of xn 223

Appendix C Experiment 224

Appendix D Visualizing Complex Numbers 228

Appendix E Quaternions 230

Acknowledgments 233

Notes 235

Index 269

Doing simple math in your head

W.J. Howard

Chicago Review Press | 2001 | 140 páginas | rar - pdf |3,7 Mb

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Almost all adults suffer a little math anxiety, especially when it comes to everyday problems they think they should be able to figure out in their heads. Want to figure the six percent sales tax on a $34.50 item? A 15 percent tip for a $13.75 check? The carpeting needed for a 12½-by-17-foot room? No one learns how to do these mental calculations in school, where the emphasis is on paper-and-pencil techniques. With no math background required and no long list of rules to memorize, this book teaches average adults how to simplify their math problems, provides ample real-life practice problems and solutions, and gives grown-ups the necessary background in basic arithmetic to handle everyday problems quickly.

CONTENTS
Introduction 1
1 Making Things Easier 7
2 Problems and Solutions 25
3 Background: Basic Arithmetic 79
Glossary 119
Index 12