Proof and Proving in Mathematics Education

The 19th ICMI Study

New ICMI Study Series, Vol. 15

Hanna, Gila; de Villiers, Michael (Eds.)

Springer | 2012 | 475 páginas | PDF | 7,23 Mb

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• First comprehensive book covering a wide range of aspects related to the teaching and learning of proof
• Provides illustrative examples of proof practices and activities in different contexts and levels
• Extensive survey of available research findings, and indicating directions for future research

One of the most significant tasks facing mathematics educators is to understand the role of mathematical reasoning and proving in mathematics teaching, so that its presence in instruction can be enhanced. This challenge has been given even greater importance by the assignment to proof of a more prominent place in the mathematics curriculum at all levels.

Along with this renewed emphasis, there has been an upsurge in research on the teaching and learning of proof at all grade levels, leading to a re-examination of the role of proof in the curriculum and of its relation to other forms of explanation, illustration and justification.

This book, resulting from the 19^th ICMI Study, brings together a variety of viewpoints on issues such as:

• The potential role of reasoning and proof in deepening mathematical understanding in the classroom as it does in mathematical practice. 
• The developmental nature of mathematical reasoning and proof in teaching and learning from the earliest grades.
   
• The development of suitable curriculum materials and teacher education programs to support the teaching of proof and proving.

The book considers proof and proving as complex but foundational in mathematics. Through the systematic examination of recent research this volume offers new ideas aimed at enhancing the place of proof and proving in our classrooms.

Table of contents

1. Aspects of proof in mathematics education: Gila Hanna and Michael de Villiers.- Part I: Proof and cognition.- 2. Cognitive development of proof: David Tall, Oleksiy Yevdokimov, Boris Koichu, Walter Whiteley, Margo Kondratieva, and Ying-Hao Cheng .- 3. Theorems as constructive visions: Giuseppe Longo.- Part II: Experimentation: Challenges and opportunities.- 4. Exploratory experimentation: Digitally-assisted discovery and proof: Jonathan M. Borwein.- 5. Experimental approaches to theoretical thinking: Artefacts and proofs.- Ferdinando Arzarello, Maria Giuseppina Bartolini Bussi, Allen Leung, Maria Alessandra Mariotti, and Ian Stevenson (With response by J. Borwein and J. Osborn).- Part III: Historical and educational perspectives of proof.- 6. Why proof?  A historian’s perspective: Judith V. Grabiner.- 7. Conceptions of proof – in research and in teaching: Richard Cabassut, AnnaMarie Conner, Filyet Asli Ersoz, Fulvia Furinghetti, Hans Niels Jahnke, and Francesca Morselli.- 8. Forms of proof and proving in the classroom: Tommy Dreyfus, Elena Nardi, and Roza Leikin.- 9. The need for proof and proving: mathematical and pedagogical perspectives: Orit Zaslavsky, Susan D. Nickerson, Andreas Stylianides, Ivy Kidron, and Greisy Winicki.- 10. Contemporary proofs for mathematics education: Frank Quinn.- Part IV: Proof in the school curriculum.- 11. Proof, Proving, and teacher-student interaction: Theories and contexts: Keith Jones and Patricio Herbst.- 12. From exploration to proof production: Feng-Jui Hsieh, Wang-Shian Horng, and Haw-Yaw Shy.- 13. Principles of task design for conjecturing and proving: Fou-Lai Lin, Kyeong-Hwa Lee, Kai-Lin Yang, Michal Tabach, and Gabriel Stylianides.- 14. Teachers’ professional learning of teaching proof and proving: Fou-Lai Lin, Kai-Lin Yang, Jane-Jane Lo, Pessia Tsamir, Dina Tirosh, and Gabriel Stylianides.- Part V: Argumentation and transition to tertiary level.- 15. Argumentation and proof in the mathematics classroom: Viviane Durand-Guerrier, Paolo Boero, Nadia Douek, Susanna Epp, and Denis Tanguay.- 16. Examining the role of logic in teaching proof: Viviane Durand-Guerrier, Paolo Boero, Nadia Douek, Susanna Epp, and Denis Tanguay.- 17. Transitions and proof and proving at tertiary level: Annie Selden.- Part VI: Lessons from the Eastern cultural traditions.- 18. Using documents from ancient China to teach mathematical proof: Karine Chemla .- 19. Proof in the Western and Eastern traditions: Implications for mathematics education: Man Keung Siu.- Acknowledgements.- Appendix 1: Discussion Document.- Appendix 2: Conference Proceedings: Table of contents.- Author Index.- Subject Index.

Actas do Encontro

Fou-Lai Lin, Feng-Jui Hsieh
Gila Hanna, Michael de Villiers

The Department of Mathematics, National Taiwan Normal University | 2009 | PDF

vol 1

140.122.140.1 (link direto)

vol 2

140.122.140.1 (link direto)

Mathematics as an Educational Task

Hans Freudenthal

Springer | 1972 | 692 páginas | PDF | djvu | 5 Mb

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New Mathematics has been a slogan for a decade. Modern mathemtics, interpreted according to the letter, kills education; interpreted according to its spirit, it may give life. The author placesmathematics into its historical, developmental, and social context, and mathematics education into the context of general education, as it has developed and still continues to develop. At he summit of scientific production ot os gemera;;y acknowledged that mathematics is an activity rather than a well-established stock of knowledge. The author's teaching philosophy maintains that this applies to all levels of the learning process. These levels have been analyzed in numerous examples. This general theory is followed by an analysis of several crucial concepts and domains. In contrast to the modern tendency of overstressing the cardinal aspects, the number concepts is scrutinized from a wide spectrum of logical and developmental aspects. The author shows what the use of set theory are, while at the same time warning against its misuse. Geometry is no longer regarded as being summit of deductivity; its task is now seen as that of exploring the space we live in, and of offering a field of mathematical acitvity.

The Impact of Reform Instruction on Student Mathematics Achievement

An Example of a Summative Evaluation of a Standards-Based Curriculum 

(Studies in Mathematical Thinking and Learning Series)

Thomas Romberg,  Mary C. Shafer

Routledge | 2008 | 185 páginas | rar - PDF | 2 Mb

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Summarizing data derived from a four-year combined longitudinal/ cross-sectional comparative study of the implementation of one standards-based middle school curriculum program, Mathematics in Context, this book demonstrates the challenges of conducting comparative longitudinal research in the reality of school life.

The study was designed to answer three questions:

• What is the impact on student performance of the Mathematics in Context instructional approach, which differs from most conventional mathematics texts in both content and expected pedagogy?


• How is this impact different from that of traditional instruction on student performance?


• What variables associated with classroom instruction account for variation in student performance?

The researchers examined a range of variables that affected data collection. These variations highlight the need to study the effects of the culture in which student learning is situated when analyzing the impact of standards-based curricula on student achievement.

This book is directed to educational researchers interested in curriculum implementation, mathematics educators interested in the effects of using reform curriculum materials in classrooms, evaluators and research methodologists interested in structural modeling and scaling of instructional variables, and educational policy makers concerned about reform efforts.

Invitation to Number Theory

(New Mathematical Library)

Oystein Ore

Mathematical Assn of America | 1969 | 129 páginas | PDF | 3,16 Mb

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estoyanov.net (link direto)

Djvu | 1,1 Mb 
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Number theory has been instrumental in introducing many of the most distinguished mathematicians, past and present, to the charms and mysteries of mathematical research. The purpose of this simple little guide will have been achieved if it should lead some of its readers to appreciate why the properties of numbers can be so fascinating. It would be better still if it would induce you to try to find some number relations of your own; new curiosities devised by young people turn up every year. In any case, you will become familiar with some of the special mathematical concepts and methods used in number theory and will be prepared to embark upon the study of the more advanced books in its rich literature.

The Contest Problem Book IV

Annual High School Examinations, 1973-1982

Ralph A. Artino, Anthony M. Gaglione, Niel Shell

The Mathematical Association of America | 1982 | 198 páginas | PDF | 4,2 Mb

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This text contains the problems from the (American) Annual High School Mathematics Examinations (AHSME), from the years 1973 - 1982, their answers, and their solutions. The problems, which range from relatively straightforward calculations to fiendishly difficult problems that require considerable ingenuity to solve, draw upon algebra, geometry, trigonometry, number theory, combinatorics, probability, and logic. No calculus is assumed, but it can sometimes be employed to solve a problem. The problems, which are meant to be done without a calculator, increase in difficulty within each examination.

The 1973 examination consisted of 35 questions. It had four parts of 10, 10, 10, and 5 questions, worth 3, 4, 5, and 6 points each. A wrong answer resulted in a deduction of 1/4 of the point value of the question. From 1974 - 1982, the examinations consisted of 30 questions. From 1974 - 1977, each question was worth five points, with one point deducted for wrong answers. From 1978 - 1982, the score was obtained by adding 30 points to four times the number of correct responses and then subtracting the number of wrong answers. Thus, starting in 1978, it was no longer possible to obtain a negative score. From 1973 - 1977, students were given 80 minutes to complete the examination. From 1978 - 1982, they had 90 minutes to complete the examination. Thus, speed was essential.

After working through a test, you can check your answers against the key, determine if you wish to retry a problem, then read the solutions that are provided. Reading the solutions given in the text is instructive since they are clearly written and tend to be elegant. Moreover, in some cases, alternate solutions are given or the solutions include comments about relevant theorems or related problems. However, you will not find the solutions useful if you are not familiar with the theorems used in solving the problems.

Working through this book is a great way to develop your problem-solving skills. However, it is no longer as useful for preparing for the American Mathematics Competition (AMC), as the AHSME is now called. The format and contents of the examination have changed. For instance, questions about non-decimal bases would not be familiar to current students given changes in the curriculum. If you are interested in preparing for the current examination, you would benefit from working through The Contest Problem: Book VIII (MAA Problem Book Series) if you are preparing for the AMC 10 or The Contest Problem Book IX (Maa Problem Books) (Bk. 9) if you are preparing for the AMC 12. Top scorers on those examinations qualify for the American Invitational Mathematics Examination (AIME). 

Contest Problem Book II: Annual High School Contests 1961-1965

Annual High School Contests of the Mathematical Association of America, 1961-1965

Charles T. Salkind

Mathematical Association of America | 1966 | 112 páginas | PDF | 2,56 Mb

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From the preface ...

The thesis that selective problem solving can be a vital factor in learning mathematics needs no extended defense. It is implicit in the suggestion by some curriculum experts that problems be made the central point of topical development. A good problem, like the acorn, has in it the potential for grand development.
...

Charles T. Salkind

This text compiles the (American) Annual High School Mathematics Examinations (AHSME), now called the American Mathematics Competition, that were administered from 1961-1965. It includes the questions, their answers, their solutions, and a listing of the problems classified by subject. This book is briefer than the others in this series. That may be because 1966 was the year in which a penalty was introduced for wrong answers in order to discourage random guessing.

Each examination consists of forty multiple choice problems based on the American high school curriculum at the time. The problems are meant to be done without a calculator. The principal topics are algebra and geometry, although there are questions on set theory and logic. The examination consists of three parts, which are successively more difficult. The first part, consisting of twenty questions, contains short problems that are meant to test for conceptual understanding. The next two parts each contain ten problems. These problems are designed to make you think more deeply about the subject matter. The problems range in difficulty from routine problems to ones that require considerable ingenuity to solve.

The format of the book enables you to work through the problems, check your answers, and correct any mistakes you make before examining the solutions that Salkind provides. Reading his solutions is instructive for a couple reasons. Salkind's solutions tend to be elegant, and he sometimes presents alternate methods of solving a problem. That said, if you are not familiar with the results that Salkind assumes, you may find his solutions cryptic.

Working through this text will help you develop your problem solving skills. If you are preparing for the American Mathematics Competition, you may want to use this as a source of practice problems. However, keep in mind that both the format of the examination and the curriculum have changed since these examinations were administered. There are problems dealing with non-decimal bases and logic which will be unfamiliar to current high school students.

Knowing and Learning Mathematics for Teaching: Proceedings of a Workshop

Mathematics Teacher Preparation Content Workshop Program Steering Committee, 
Center for Education , National Research Council 


National Academies Press | 2001| 236 páginas | PDF | 5,29 Mb


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online: nap.edu


There are many questions about the mathematical preparation teachers need. Recent recommendations from a variety of sources state that reforming teacher preparation in postsecondary institutions is central in providing quality mathematics education to all students. The Mathematics Teacher Preparation Content Workshop examined this problem by considering two central questions: what is the mathematical knowledge teachers need to know in order to teach well?, and how teachers can develop the mathematical knowledge they need to teach well? The Workshop activities focused on using actual acts of teaching such as examining student work, designing tasks, or posing questions, as a medium for teacher learning. The Workshop proceedings, "Knowing and Learning Mathematics for Teaching", is a collection of the papers presented, the activities, and plenary sessions that took place.