"Proof is central to all mathematical thinking. This book provides students with an excellent all round guide to proof including clear explanation and examples.

Author: Tom Bennison

Publisher: Tarquin Group

ISBN: 1913565327

Category:

Page: 248

View: 697

Proof is central to any mathematics curriculum and indeed, all mathematical thinking. Now we are delighted to provide an International Edition of our guide to proof for students...and for their teachers too. Contents: 1. Introduction to proof 2. Exploring Methods of Proof 3. Mathematical Language 4. Direct Proof 5. Indirect Proof 6. Proof by Induction 7. Proof and Applications of Pythagoras' Theorem 8. Proof in Calculus 9. Proving Trigonometric Identities 10. Proof in Statistics and Probability 11. Worked Solutions

We will also cover concepts such as mathematical conjecture, quantifiers, and
the concept of mathematical proof. Chapter 2 focuses squarely on reading,
writing, analyzing and understanding mathematical proofs. In this unit you will
learn how ...

Author: Connie M. Campbell

Publisher: Cengage Learning

ISBN: 9781133168782

Category: Mathematics

Page: 144

View: 982

This text offers a crucial primer on proofs and the language of mathematics. Brief and to the point, it lays out the fundamental ideas of abstract mathematics and proof techniques that students will need to master for other math courses. Campbell presents these concepts in plain English, with a focus on basic terminology and a conversational tone that draws natural parallels between the language of mathematics and the language students communicate in every day. The discussion highlights how symbols and expressions are the building blocks of statements and arguments, the meanings they convey, and why they are meaningful to mathematicians. In-class activities provide opportunities to practice mathematical reasoning in a live setting, and an ample number of homework exercises are included for self-study. This text is appropriate for a course in Foundations of Advanced Mathematics taken by students who've had a semester of calculus, and is designed to be accessible to students with a wide range of mathematical proficiency. It can also be used as a self-study reference, or as a supplement in other math courses where additional proofs practice is needed. Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version.

In particular, undergraduate mathematics students often experience difficulties in understanding and constructing proofs.Understanding Mathematical Proof describes the nature of mathematical proof, explores the various techn

Author: John Taylor

Publisher: CRC Press

ISBN: 9781466514911

Category: Mathematics

Page: 414

View: 111

The notion of proof is central to mathematics yet it is one of the most difficult aspects of the subject to teach and master. In particular, undergraduate mathematics students often experience difficulties in understanding and constructing proofs.Understanding Mathematical Proof describes the nature of mathematical proof, explores the various techn

When reading a proof , aim to understand the underlying idea . There are
multiple levels involved with reading and understanding proofs . At a basic level ,
one can read a proof by understanding every word in it and verifying that the
steps are ...

Author: Joel David Hamkins

Publisher:

ISBN: 9780262539791

Category: Mathematics

Page: 232

View: 390

"A textbook for students who are learning how to write a mathematical proof, a validation of the truth of a mathematical statement"--

Proof as bearers of mathematical knowledge. Bridging knowing and proving in mathematical reasoning. The role of mathematics in long-term cognitive development of reasoning. Proof as experiment in the work of Wittgenstein.

Author: Gila Hanna

Publisher: Springer Science & Business Media

ISBN: 1441905766

Category: Education

Page: 294

View: 944

In the four decades since Imre Lakatos declared mathematics a "quasi-empirical science," increasing attention has been paid to the process of proof and argumentation in the field -- a development paralleled by the rise of computer technology and the mounting interest in the logical underpinnings of mathematics. Explanantion and Proof in Mathematics assembles perspectives from mathematics education and from the philosophy and history of mathematics to strengthen mutual awareness and share recent findings and advances in their interrelated fields. With examples ranging from the geometrists of the 17th century and ancient Chinese algorithms to cognitive psychology and current educational practice, contributors explore the role of refutation in generating proofs, the varied links between experiment and deduction, the use of diagrammatic thinking in addition to pure logic, and the uses of proof in mathematics education (including a critique of "authoritative" versus "authoritarian" teaching styles). A sampling of the coverage: The conjoint origins of proof and theoretical physics in ancient Greece. Proof as bearers of mathematical knowledge. Bridging knowing and proving in mathematical reasoning. The role of mathematics in long-term cognitive development of reasoning. Proof as experiment in the work of Wittgenstein. Relationships between mathematical proof, problem-solving, and explanation. Explanation and Proof in Mathematics is certain to attract a wide range of readers, including mathematicians, mathematics education professionals, researchers, students, and philosophers and historians of mathematics.

But what does that mean? This book focuses on essential knowledge for teachers about proof and the process of proving. It is organised around five big ideas, supported by multiple smaller, interconnected ideas--essential understandings.

Author: Amy B. Ellis

Publisher: National

ISBN: 0873536754

Category: Effective teaching

Page: 102

View: 309

What is the difference between "proof" in mathematics and "proof" in science or a court of law? In mathematics, how does proof differ from other types of arguments? What forms can proof take besides the traditional two-column style? What activities constitute the process of proving? What roles do examples play in proving? Can examples ever prove a conjecture? Why does a single counterexample refute a conjecture? How much do you know, and how much do you need to know? Helping your students develop a robust understanding of mathematical proof and proving requires that you understand this aspect of mathematics deeply. But what does that mean? This book focuses on essential knowledge for teachers about proof and the process of proving. It is organized around five big ideas, supported by multiple smaller, interconnected ideas essential understandings. Taking you beyond a simple introduction to proof and the activities involved in proving, the book will broaden and deepen your mathematical understanding of one of the most challenging topics for students and teachers. It will help you engage your students, anticipate their perplexities, avoid pitfalls, and dispel misconceptions. You will also learn to develop appropriate tasks, techniques, and tools for assessing students' understanding of the topic. Focus on the ideas that you need to understand thoroughly to teach confidently. - Publisher.

Whatsoever we endeavour in obedience to reason is nothing further than to
understand; neither does the mind, in so far as it makes use of reason, judge
anything to be useful to it, save such things as are conducive to understanding. Proof.

Author: Benedict de Spinoza

Publisher: Courier Corporation

ISBN: 9780486121086

Category: Philosophy

Page: 420

View: 713

Also contains Ethics, Correspondence, all in excellent R. Elwes translation. Basic works on entry to philosophy, pantheism, exchange of ideas with great contemporaries.

Author: Frederick William RobertsonPublish On: 1875

There is an understanding in the beaver , and there is an understanding in the
bee , by which it builds its habitation . The fox has it as well ... For even the proof I
give , the impression my hand makes on his , is not that disputable ? May not that
...

Author: Frederick William RobertsonPublish On: 1878

There is an understanding in the beaver , and there is an understanding in the
bee , by which it builds its habitation . The fox has it as ... I can not prove the
being of a God ; if by proof I mean that addressed to the understanding . If I said I
could ...

But according to the principle here explained , this subtraction , with regard to all
popular religions , amounts to an entire annihilation ; and therefore we may
establish it as a maxim , that no human testimony can have such force as to prove
a ...

Author: Washington (State). Supreme CourtPublish On: 1893

As we understand the rule , any ambiguity appearing upon the face of the
contract may be explained by proof of the understanding of the parties at the time
such contract was made , or in the absence of any such understanding , by proof
of ...

Author: Washington (State). Supreme Court

Publisher:

ISBN: IOWA:31858019087992

Category: Law reports, digests, etc

Page:

View: 955

Vol. 1 includes the decisions of the Supreme Court of the Territory of Washington for 1889.

... not for proofs of new results , but for proofs that reflect logical and mathematical understanding , proofs that reveal their ... studies must include a structural theory
of proofs that extends proof theory through ( i ) articulating structural features of ...

Author: B. Jack Copeland

Publisher: Mit Press

ISBN: 9780262527484

Category: Computers

Page: 362

View: 108

Computer scientists, mathematicians, and philosophers discuss the conceptual foundations of the notion of computability as well as recent theoretical developments. In the 1930s a series of seminal works published by Alan Turing, Kurt Gödel, Alonzo Church, and others established the theoretical basis for computability. This work, advancing precise characterizations of effective, algorithmic computability, was the culmination of intensive investigations into the foundations of mathematics. In the decades since, the theory of computability has moved to the center of discussions in philosophy, computer science, and cognitive science. In this volume, distinguished computer scientists, mathematicians, logicians, and philosophers consider the conceptual foundations of computability in light of our modern understanding. Some chapters focus on the pioneering work by Turing, Gödel, and Church, including the Church-Turing thesis and Gödel's response to Church's and Turing's proposals. Other chapters cover more recent technical developments, including computability over the reals, Gödel's influence on mathematical logic and on recursion theory and the impact of work by Turing and Emil Post on our theoretical understanding of online and interactive computing; and others relate computability and complexity to issues in the philosophy of mind, the philosophy of science, and the philosophy of mathematics. Contributors Scott Aaronson, Dorit Aharonov, B. Jack Copeland, Martin Davis, Solomon Feferman, Saul Kripke, Carl J. Posy, Hilary Putnam, Oron Shagrir, Stewart Shapiro, Wilfried Sieg, Robert I. Soare, Umesh V. Vazirani

Want of Proofs . 2. Want of Ability to use them , 3. Want of Will to use them . 4.
Wrong Measures of Probability . S. 2. First , By Want of Proofs , I do not Pir , Want
mean only the Want of those Proofs which of Proofs . are no where extant , and
so ...

Author: Paul E. LingenfelterPublish On: 2015-12-16

How can we “fix” our schools? Improve graduation rates in college? What works? These are questions that make the headlines and vex policy makers, practitioners, and educational researchers.

Author: Paul E. Lingenfelter

Publisher: Stylus Publishing, LLC

ISBN: 9781579227531

Category: Education

Page: 264

View: 788

How can we “fix” our schools? Improve graduation rates in college? What works? These are questions that make the headlines and vex policy makers, practitioners, and educational researchers. While they strive to improve society, there are frequently gulfs of mutual incomprehension among them. Academics, longing for more influence, may wrongly fault irrationality, ideology, or ignorance for the failure of research to inform policy and practice more powerfully. Policy makers and practitioners may doubt that academics can deliver ideas that will reliably yield desirable results. This book bridges the divide. It argues that unrealistic expectations lead to both unproductive research and impossible standards for “evidence-based” policy and practice, and it offers promising ways for evidence to contribute to improvement. It analyzes the utility and limitations of the different research methods that have been applied to policy and practice, as well as the strengths and weaknesses of educational reform strategies. It explains why using evidence for “accountability” often makes things worse rather than better. Paul Lingenfelter offers educational researchers and policy makers a framework for considering such questions as: What problems are important and accessible? What methods will be fruitful? Which help policy makers and practitioners make choices and learn how to improve? What information is relevant? What knowledge is valid and useful? How can policy makers and practitioners establish a more productive division of labor based on their respective capabilities and limitations? He cautions against the illusion that straight-forward scientific approaches and data can be successfully applied to society’s most complex problems. While explaining why no single policy or intervention can solve complex problems, he concludes that determination, measurement, analysis, and adaptation based on evidence in specific situations can lead to significant improvement. This positive, even-handed introduction to the use of research for problem-solving concludes by suggesting emerging practices and approaches that can help scholars, practitioners, and policy leaders become more successful in reaching their fundamental goals.

The book provides detailed analysis of the legal requirements of international sex crimes and types of fact that can be used to meet these requirements. It includes a unique knowledge-base that digests international case law on such crimes.

Author: Morten Bergsmo

Publisher: Torkel Opsahl Academic EPublisher

ISBN: 9788293081296

Category: Law

Page: 894

View: 401

"[This anthology] addresses the gap betwen international standard-setting prohibiting international sex crimes and actual accountability for individuals who are responsible for such crimes. The book provides detailed analysis of the legal requirements of international sex crimes and types of fact that can be used to meet these requirements. It includes a unique knowledge-base that digests international case law on such crimes. The anthology also contains several studies of institutional and evidentiary challenges in the prosecution of international sex crimes"--Series pref.

First , By want of proofs , I do not mean only the waņt of those proofs which are
nowhere extant , and so are nowhere to be ... and their understandings are but
little instructed , when all their whole time and pains is laid out to still the croaking
of ...

Author: Great Britain. Council on EducationPublish On: 1891

The most important question in the paper is the first , requiring explanation , proof
, and definition . ... I am bound to state , however , that there are some evidences
of training 3 15 in the proofs given to show that = Many appeal to first principles ...

If I understand you correctly , you see the key to appreciating the beauty of
mathematics , and to understanding proof , as doing mathematics in a legitimate
way . Analogously , to appreciate paintings you would suggest a course in
painting ...

That they “ prove ” anything , understanding “ proof ” as science usually
understands the term , can hardly be seriously maintained : till scientific proof be
produced , to suspend our judgment should not be unscientific . But , to retrace
our course ...

Once you have learned these rules , you will find that your proofs go much more
quickly and are much easier and more ... Learn the rules of Conditional Proof ( C
. P . ) and Indirect Proof ( I . P . ) in their symbolic forms , and understand the ...

Author: Virginia Klenk

Publisher:

ISBN: 0139427643

Category: Logic, Symbolic and mathematical.

Page: 433

View: 279

This comprehensive introduction presents the fundamentals of symbolic logic clearly, systematically, and in a straightforward style accessible to readers. Each chapter, or unit, is divided into easily comprehended small "bites" that enable learners to master the material step-by-step, rather than being overwhelmed by masses of information covered too quickly. The book provides extremely detailed explanations of procedures and techniques, and was written in the conviction that anyone can thoroughly master its content. A four-part organization covers sentential logic, monadic predicate logic, relational predicate logic, and extra credit units that glimpse into alternative methods of logic and more advanced topics. For individuals interested in the formal study of logic.