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- Herausgeber: Bentham Science Publishers
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Numberama: Recreational Number Theory in the School System presents number patterns and mathematical formulas that can be taught to children in schools. The number theories and problems are reinforced by enjoyable games that children can play to enhance their learning in a fun-loving way. Key features of the book include:
information about a number of well-known number theory problems such as Fibonaccci numbers, triangular numbers, perfect numbers, sums of squares, and Diophantine equations
organized presentation based on skill level for easy understanding
all basic mathematical operations for elementary school children
a range of algebraic formulae for middle school students
descriptions of positive feedback and testimonials where recreational number theory has been effective in schools and education programs
This book is a useful handbook for elementary and middle-school teachers, students, and parents who will be able to experience the inherent joys brought by teaching number theory to children in a recreational way.
Das E-Book können Sie in Legimi-Apps oder einer beliebigen App lesen, die das folgende Format unterstützen:
Seitenzahl: 273
Veröffentlichungsjahr: 2017
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Numberama:
Recreational Number Theory
In The School System
Authored by
Elliot Benjamin, Ph.D., Ph.D
Instructor of Mathematics at CAL Campus; Psychology Mentor,
Ph.D Committee Chair at Capella University, Minneapolis, USA
BENTHAM SCIENCE PUBLISHERS LTD.
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FOREWORD
With all the push toward applications of mathematics where some are at best artificial, it is refreshing to find a text that does not have the pretense of giving any real applications, but rather a book on number theory just for fun. The conception of the book, Numberama, could have been conceived by the first real number theorist, P. Fermat, through a bunch of problems (without any thought of applications).
This book is a text about problems in number theory intended to aid teachers from early elementary school to early high school in giving an appreciation of number theory to their students. The text is divided into four parts and an Appendix: Chapter 1 is devoted to several basic problems in number theory which can be appreciated using only elementary arithmetic: addition, subtraction, multiplication, and division. Chapters 2, 3, and 4 are devoted to board games based on the problems in Chapter 1, where each chapter requires successively higher level arithmetic skills to play the games in the chapter. For each of the problems given in Chapter 1, there is a code indicating the level of skills needed to work on at least parts of the problems. Teachers are given plenty of advice as to how to present the material to the students. The Appendix includes excerpts from various student and teacher participants in Dr. Benjamin's Numberama program, which describes both the benefits and the joy they received from participating in this program.
A couple of the problems included are the following, (some going back to antiquity): Finding perfect numbers. Here the students get to use their skills at multiplication and division, as well as being exposed to prime numbers. This problem leads, of course, to open questions such as the existence of infinitely many even perfect numbers and also the existence of at least one odd perfect number. There is also the problem of representing integers as the sum of two squares, a problem which Fermat himself worked on. Again the teachers are given hints as to how to proceed.
Chapters 2, 3, and 4 are devoted to 19 different board games concerning the problems in Chapter 1, which seek to hone the skills of the students, involving many of the properties of numbers given in the first chapter. The text is written well and seems to be accurate. I would certainly recommend it to teachers interested in enriching the mathematics content and honing basic arithmetic skills of the students.
PREFACE
It has been nearly 30 years since I began working on my Numberama book. However, in spite of the enormous developments in technology and social media over the past 30 years, the essential theme of my book remains intact. The essential theme is that mathematics can be a stimulating, challenging, and thoroughly enjoyable recreational mental activity to enhance and enrich substantial and creative thinking for children in our school system. As I describe in the Appendix, I have experienced a wide variety of appreciative and enthusiastic responses to my Numberama program over the years, ranging across teacher workshops, teacher education programs, children in gifted programs, children in regular classes, liberal arts college instruction, and family math workshops. I have also effectively utilized my Numberama program at a mental hospital for children, a senior citizen center, as a supplement in my algebra classes, and as an example of creative thinking in my psychology classes. Most recently, I found myself giving a “perfect number lesson” on a napkin at a restaurant at a spiritual development workshop I attended. I had the workshop presenter and some of the participants and staff enraptured, and I realized that Numberama is deeply ingrained in me, wherever I go and whatever I do. I am still a pure mathematician, and I practice what I preach. Doing mathematics for me is refreshing, stimulating, meditative, and enjoyable. I occasionally make use of technology to try to find examples of some of my pure mathematics algebraic number theory results, but this is always very secondary, as the priority is on my “thinking.” And this is the exact same philosophy I promote in my Numberama program in regard to the use of technology. Technology is a wonderful tool, but it is essential for the human being to be in control of the technology and not the other way around. Thus finding interesting, surprising, and enticing patterns of numbers, with the assistance of arithmetic calculators when the numbers invariable get very large, is a natural part of my Numberama problems. But what is most important here is the discovery of the patterns themselves, using technology to enhance the discovery.
With this in mind, I am excited to now be offering my Numberama book as an e-book through Bentham publications. I welcome feedback from anyone who is using my Numberama problems and games, and I hope that I have succeeded in transmitting the joys of searching for captivating patterns of numbers in my book.
ACKNOWLEDGEMENTS
Elliot BenjaminI would like to take this opportunity to give special thanks to a few individuals who truly have made this book possible. First, I am greatly indebted to Dr. Chip Snyder—my mathematical mentor.
I have been working with Dr. Snyder in the pure mathematics discipline of Algebraic Number Theory since I moved to Maine in 1985, having earned my Ph.D. in this mathematical discipline in 1996. We have worked together all these years for essentially one reason—we both enjoyed it—and still do. I learned first-hand the joyful, challenging, frustrating, and transcendental experience of what it means to be a research mathematician. Thus I have been able to practice what I preach.
Next, I must give my heartfelt thanks to my son, Jeremy. As I describe in the Introduction to Chapter 2, it was Jeremy who inspired the games of recreational number theory. It was also my son Jeremy who lived through many of the recreational number theory problems—from ages 7 through 11. He has been wonderfully responsive and patient with his rather unusual father, and I love him dearly.
I must also express my appreciation to a student in my first teacher education in mathematics class at the University of Maine in 1990—Ethel Hill. Ethel thoroughly enjoyed my “special problems.” Ethel has also played an enormously important role in making this book a reality—she has done all of the typing! She learned how to put everything on the computer—including the games—in her spare time while working as administrative assistant to the Dean of the College of Education at the University of Maine. She has done this with a marvelous spirit and has encouraged me to persevere in making this book into a vehicle to help in the transformation of mathematics from drudgery to fun. I hope she continues to involve herself in the next stage of marketing this book.
The next individual I give my thanks to is no longer with us—her name is Stephanie Pall. My friend Stephanie was the person who inspired me to find a way to convey to others the joys I have experienced from mathematics. She enabled me to look deeply inside myself, to be who I truly am in my career of teaching mathematics. This occurred in 1988 as I began playing with many of the problems in David Wells’ (1986) book.
The Penguin Dictionary of Curious and Interesting Numbers, for the purpose of improving my mathematics teaching at Unity College. I subsequently found a number of additional resources that were helpful to me in formulating both my number theory problems and related teaching methods (Adams & Goldstein, 1976; Beiler, 1966; Brown, 1973, 1976, 1983; Dence, 1983; Dewey, 1933; King, 1993; Miller & Heeren, 1961; National Council of Mathematics, 1981, 1984, 1991; Neill, 1960; Postman & U Weingartner, 1969; Rogers, 1969; Walter & Brown, 1971; Walter & Brown, 1977).
Stephanie’s untimely death prevented her from seeing where her faith in me has led. But I give tribute to her now; I will forever be indebted to her for the gift she has given me.
I wish to thank all the students in my Finite Math classes at Unity College, my mathematics teacher education classes at the University of Maine, and the teachers and children who participated in my “Mathworks” program in Belfast, Maine, during the 1991−1992 school year.
I also would like to thank my Swanville, Maine, friends who so patiently indulged me in my mathematical entertainments on many a lazy Sunday afternoon in their home—Steve and Kate Webster.
Finally, I give my thanks to Thomas Hathaway Nason from the Word Shop in Orono, Maine, who patiently and effectively put finishing computer touches on the book, a task which turned out to be far more demanding than originally anticipated, and to Kay Retzlaff, my present book consultant who is responsible for the book’s new layout and design.
There are many more individuals I am not mentioning who have helped me to form my ideas about both mathematics and mathematics education. I give a final note of thanks to those many unnamed individuals.
Description of Skill Levels
Elliot BenjaminThe following letters will be used to denote the designated math skill levels. All problems and games are followed by the appropriate skill level; problems followed by two or more skill levels imply that they can be used in various degrees of skill complexity. It should be noted that all students can gain value from working on problems from previous skill levels:
addition of two-digit numbersgeneral addition and subtractionone-digit multiplicationgeneral multiplicationmultiplication division by 2multiplication division by one-digit numbermultiplication division in generalfractionssigned numbersalgebraINTRODUCTION TO THE BOOK
It is now over 20 years since I wrote the above acknowledgments for this book, as well as the basis of this introduction. However, it is a tribute to the timeless nature of these Recreational Number Theory problems and games that I have designated with the title of Numberama, that there is little I feel the need of adding to at this time. My philosophy of “math for fun” has not changed, and I am still collaborating with my ex-Ph.D mathematics mentor Dr. Chip Snyder as we continue to work together, publishing papers in the field of algebraic number theory. I have utilized my Numberama problems and games in diverse educational settings, inclusive of various elementary school classrooms, gifted and talented school programs, developmental mathematics classes at colleges and universities, teacher workshops, and even at a senior retirement home. The past few years I have utilized the Subsets & Circles problem (see Problem #1 in Chapter 1) in my Introductory Psychology classes to illustrate the experience of creative thinking. The results of all my Numberama explorations with both students and teachers have been overwhelmingly positive, and I have received many written descriptions of the benefits that participants have received from their experiences in my Numberama program, a sample of which I have included in the Appendix.
I believe that today, more than ever, it is so very important to not let our children lose (or never experience) the intrinsic joy of doing mathematics. Our technology is so sophisticated that it is all too easy for both our children and ourselves to discontinue our “thinking” and let our computer gadgets “think” for us. But there is an inherent potential joy in thinking, and I am thankful that I continue to experience this inherent joy of mathematical thinking in my pure mathematics field of algebraic number theory. And it continues to be part of my mission in life to convey the inherent joy of mathematical thinking to children in the context of Numberama Recreational Number Theory problems and games in the school system, and to people of all ages, through my Numberama book.
As a child I enjoyed adding numbers in my head. People were amazed at how quickly I could do so without using pencil or paper. Throughout school I enjoyed math and, as a result, I was good at it. It was no surprise to anyone when I decided to become a math teacher; however, I soon realized that the intrinsic rewards I received from studying mathematics were by no means a common experience for other students. After teaching elementary and high school, college, and in various adult education programs, I came to the conclusion that the vast majority of our population has a very limited perspective of what mathematics is truly all about.
Mathematics can certainly be an extremely pragmatic science, chock full of useful applications in virtually every field studied; however, there is another side to mathematics. Pure mathematics can be described as an art form, in the same way music, art, dance and theater are arts. Nearly every professor of mathematics knows this deep down in his/her heart. Mathematics is truth and beauty within the spirit of the mind. The natural process of thinking is inherently pleasurable. Pressures, grades, competition, etc., can destroy this potential intrinsic pleasure. What I refer to as a “natural dimension of mathematics” is doing math for the pure enjoyment of learning and discovering. Math can be fun.
This book attempts to impart the enjoyment of mathematics to the children in our schools, whether these schools are at home or part of a public or private system. The branch of mathematics that literally plays with numbers is known as number theory. Topics in number theory range from the highly theoretical, employing deep layers of abstract mathematical proof, to questions about numbers that any school child learning arithmetic can understand. These questions are enticing, adventuresome, challenging, and most important of all—fun.
I call this form of number theory, recreational number theory. Most of the problems described in Chapter 1 in this book can be worked by children who know how to add, subtract, multiply, and divide. A number of the problems do not even require division; some of the problems only require addition. There are also problems for children first learning fractions, and in many of the problems I have given suggestions on how they can be formulated into algebra problems for older students, in junior and senior high school. For each problem, the exact prerequisite skills are indicated. The general format is described at the end of the Table of Contents. The problems I have chosen to describe are by no means exhaustive. An examination of the bibliography I have included will give the interested reader some supplementary material. There is a place in our schools for “math for fun” problems. The earlier such problems are introduced, the easier it will be for a child to learn the basics of arithmetic. Working on these problems requires a lot of practice in nearly all of the arithmetic skills that are now being taught in the elementary schools. But the practice and drill are made fun through the discovery of patterns, formulas, unusual numbers, etc. The approach I am recommending is very much like playing a game.
Chapters 2, 3, and 4 consist of a series of 19 games based upon the ideas from recreational number theory introduced in Chapter 1, with each chapter requiring successively higher arithmetic skills for children to play the games included in the chapter. These games hones the skills of the students, involving many of the properties of numbers given in the first chapter. Once again, the games are by no means exhaustive, but merely serve as a rough.
Mathematics can certainly be an extremely pragmatic science, chock full of useful applications in virtually every model of how many math ideas can be made into games where children are joyfully practicing their arithmetic skills while playing the game. The prerequisite skills necessary to play the games are listed for each game, in the same format described for the problems in Chapter 1. These games serve to reinforce ideas encountered in the problems. Although a major emphasis of recreational number theory is the elementary school, this is by no means the only place where it can be used. I have purposely included many generalizations to algebraic formulas in order to make the point that recreational number theory can be used throughout the school years.
Junior high and high school students can be taught to use their newly acquired algebra skills to generate their own algebraic formulas that describe experimental facts about numbers that they have gathered together. This approach to teaching algebra is a radical change from the often tedious, monotonous, and overly pragmatic way that algebra is generally taught in our school system. I am by no means recommending that all of the traditional material in arithmetic or algebra be deleted from our schools; rather I am advocating an exciting new tool and method of education that can be used to help our children learn many of these skills. The key word is “balance.” There is a place for lecture, a place for tradition, and also a place for process, adventure, and discovery.
Another challenge is to successfully use the discovery approach of recreational number theory with college students in the area known as developmental mathematics, which is little more than arithmetic and algebra for college students and adults going back to school. Community colleges and continuing education departments are teaching more of arithmetic and algebra to their students than any other kind of math. For much of my career as a mathematics professor, this was my own specialized field, and math anxiety, resistance, built-up failures, etc., are painfully high in this student population. To enable these students to view mathematics as a pleasurable pastime is indeed challenging.
This is the challenge this book is intended to meet. I have seen extremely dramatic results with students who hated math all of their lives. The prospect that they could now play with numbers for the purpose of making joyful discoveries was a welcome change of pace for them; however, the results were best when I was able to use the discovery approach exclusively without having to worry about required topics, exams, and follow-up courses. I realize this is not the typical situation our students are in, and throughout my mathematics college teaching years.
I continued to search for an effective way of balancing the old and the new; i.e., to incorporate the ideas and processes of recreational number theory within the traditional format of our developmental mathematics courses.
I hope that you will find value in the following problems and games, whether you are a math teacher, prospective math teacher, math student, parent, or interested reader in general. I welcome any feedback you have, and look forward to hearing from you.
Introduction to the Games
When my son Jeremy was 7 years old, he made me a little math game for Father’s Day. He seemed to think that it would be fun to play games based upon some of the math ideas I had been trying out on him, and he made me a cute little precursor of the Syracuse Algorithm Game. I took my son’s idea seriously, as you can see from the games in Chapters 2, 3, and 4 of this book. The 19 games I am including are only a sample of the kinds of games you can make out of the Recreational Number Theory problems introduced in Chapter I. Many of the games described in Chapters 2, 3, and 4 have been played by elementary school teachers in some of my Numberama teacher workshops. Teachers generally found them to be a productive, fun-loving way of helping children learn their arithmetic skills. They also had some excellent suggestions in terms of modifying various aspects of the games (see the section below for their suggestions). However, please keep in mind that my purpose in describing these games is only to offer you a basic framework. It is my hope that you will develop the game ideas for yourself, according to your own unique needs, interests, imagination, and artistic capabilities. Lastly, it is important to keep in mind that the theme for all of the games in Chapters 2, 3, and 4 and all of the problems in Chapter 1, is that the children should be having fun while they are learning mathematics. The game equipment that I have used are gameboards, dice, number cards, play money—in all denominations from $1 to $1,000, Lego® pieces for the players, and game rules. Chapters 2, 3, and 4 are divided according to the required arithmetic skill levels for children to play these games.
Game Ideas from Teachers at Numberama Workshops
Some ideas that came out of my Numberama teacher workshops in regard to the games are as follows:
Form teams of two or more children.Put a time limit on how long a child can take to give an answer.Use an attachment to the gameboard instead of separate number cards.Give a child at least some money even when an answer is incorrect.Make the games colorful and artistically attractive.Include paper with numbered items for a child to keep a record of.Instruct all children to work on the problem while a child has a turn.Encourage children to make exchanges of money in the game.I find all of these ideas to be excellent suggestions, and I’m sure you will have more of your own as you read through Chapters 2, 3, and 4. My own suggestion is to play the game after the children have had a chance to explore the ideas that the game is based upon. In other words, I believe the games will be most effective when the children have already experienced the discovery processes described in Chapter 1.
