Number Theory Through Inquiry; is an innovative textbook that leads students on a carefully guided discovery of introductory number theory. The book has two equally significant goals. One goal is to help students develop mathematical thinking skills, particularly, theorem-proving skills. The other goal is to help students understand some of the wonderfully rich ideas in the mathematical study of numbers. This book is appropriate for a proof transitions course, for an independent study experience, or for a course designed as an introduction to abstract mathematics. Math or related majors, future teachers, and students or adults interested in exploring mathematical ideas on their own will enjoy ;Number Theory Through Inquiry.; Number theory is the perfect topic for an introduction-to-proofs course. Every college student is familiar with basic properties of numbers, and yet the exploration of those familiar numbers leads us to a rich landscape of ideas. Number Theory Through Inquiry contains a carefully arranged sequence of challenges that lead students to discover ideas about numbers and to discover methods of proof on their own. It is designed to be used with an instructional technique variously called guided discovery or Modified Moore Method or Inquiry Based Learning (IBL). Instructors materials explain the instructional method. This style of instruction gives students a totally different experience compared to a standard lecture course. Here is the effect of this experience: Students learn to think independently: they learn to depend on their own reasoning to determine right from wrong; and theydevelop the central, important ideas of introductory number theory on their own. From that experience, they learn that they can personally create important ideas. They develop an attitude of personal reliance and a sense that they can think effectively about difficult problems. These goals are fundamental to the educational enterprise within and beyond mathematics.
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Book Description
Editorial Reviews
Book Description
This innovative textbook leads students on a carefully guided discovery of introductory number theory. The book is designed to develop students' mathematical thinking skills, particularly theorem-proving skills, whilst helping them understand some of the wonderfully rich ideas in the mathematical study of numbers.
About the Author
David Marshall received his Ph.D. from the University of Arizona, specializing in the field of algebraic number theory.Currently he teaches mathematics at Monmouth University. Edward Odell received his Ph.D from MIT. Currently he teaches at the University of Texas, Austin where he is the John T. Stuart III Centennial Professor of Mathematics. Michael Starbird received his Ph.D. from the University of Wisconsin, Madison. Currently he teaches at the University of Texas, Austin.
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Most Helpful Customer Reviews
7 of 8 people found the following review helpful:
4.0 out of 5 stars
Can be used as a text, would require significant instructor supplements,
By Charles Ashbacher (Marion, Iowa United States) - See all my reviews (TOP 500 REVIEWER) (VINE VOICE) (HALL OF FAME REVIEWER)
This review is from: Number Theory Through Inquiry (Maa Textbooks) (Mathematical Association of America Textbooks) (Hardcover)
If you were to use this book as a text in a number theory course, you first must have made the decision to teach it in a nonstandard manner. The approach used in the presentation of number theory is not the traditional listing of the fundamental theorems with their proofs. Concepts are stated as theorems but in no case is a proof offered.
There are several groups of specific exercises such as Illustrate the division algorithm m = nq + r for m = 25, n = 7; m = 277, n = 4; m = 33, n = 22; m = 33, n = 45. Questions such as Do every two integers have at least one common divisor? What other numbers can you show to be irrational? Make and prove the most general conjecture you can. Which natural numbers can be written as the sum of two squares of natural numbers? State and prove the most general theorem possible about which natural numbers can be written as the sum of two squares of natural numbers, and prove it. At several points in the text, there are exercises called "Blank paper exercises" which have the following structure. After not looking at the material in this chapter for a day or two, take a blank piece of paper and outline the development of that material in as much detail as you can without referring to the text or to notes. Places where you get stuck or can't remember highlight areas that may call for further study. The coverage is generally what is found in an introductory course in number theory. However, the lack of proofs means that either the students must derive them on their own, look them up in another reference or have the instructor provide them. While this does not preclude the use of this book, it will require extra effort on the part of the student and/or the instructor. Published in Journal of Recreational Mathematics, reprinted with permission
1 of 1 people found the following review helpful:
5.0 out of 5 stars
Great book,
By
This review is from: Number Theory Through Inquiry (Maa Textbooks) (Mathematical Association of America Textbooks) (Hardcover)
I used this book to teach a semester course to talented high school students all of whom had at least two semesters of calculus before entering the course, so they were approximately at the college sophomore level. The students really learned how to prove things in a way that a standard approach did not. Because the students had to prove all the theorems, we covered less material than a standard lecture approach. My purpose, however, was not a huge amount of material, but rather to get the students to be able to write proofs, read proofs, and be critical of what they wrote themselves and what others wrote. It was a great experience for me and for them. At the end of the semester the students spontaneously broke into applause for the course. There were a few times when I helped them do proofs or gave some of the proofs as extra credit (if they were not central to the development of the material). I collected homework for almost every class and read it sometimes grading sometimes commenting. The book is well organized and has standard topics for such a course. I added the explicit statement of the division algorithm at the beginning of the course which helped in several instances later. I am using this book and this approach again.
0 of 1 people found the following review helpful:
5.0 out of 5 stars
Good information,
By BEAUTYCHICA "QUALITY BOOKS" (United States) - See all my reviews
This review is from: Number Theory Through Inquiry (Maa Textbooks) (Mathematical Association of America Textbooks) (Hardcover)
The information in this book was clear and very helpful for my coursework. Expensive, but useful!
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Inside This Book
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First Sentence:
How can one natural number be expressed as the product of smaller natural numbers? Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
mathematical induction, probable prime test, perfect squares modulo, next natural number, complete residue system, polynomial congruences, prime natural numbers, primality test, natural number greater, largest known primes, linear congruences, quadratic reciprocity, positive integer solutions, modular arithmetic, number congruent Key Phrases - Capitalized Phrases (CAPs): (learn more)
Number Theory Through Inquiry, Fermat's Little Theorem, Sophie Germain, Fermat's Last Theorem, Blank Paper Exercise, Fundamental Theorem of Arithmetic, Euclidean Algorithm, Division Algorithm, Wilson's Theorem, Prime Time, Dirichlet's Rational Approximation Theorem, Rationals Close, Public Key Coding System, Euler's Theorem, The Golden Rule, Law of Quadratic Reciprocity, Sums of Squares, Miller's Theorem, Binomial Theorem, Euler's Criterion, Chinese Remainder Theorem, Modular World, Primes Theorem, Fundamental Theorem of Algebra, Public Key Cryptography Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
How can one natural number be expressed as the product of smaller natural numbers? Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
mathematical induction, probable prime test, perfect squares modulo, next natural number, complete residue system, polynomial congruences, prime natural numbers, primality test, natural number greater, largest known primes, linear congruences, quadratic reciprocity, positive integer solutions, modular arithmetic, number congruent Key Phrases - Capitalized Phrases (CAPs): (learn more)
Number Theory Through Inquiry, Fermat's Little Theorem, Sophie Germain, Fermat's Last Theorem, Blank Paper Exercise, Fundamental Theorem of Arithmetic, Euclidean Algorithm, Division Algorithm, Wilson's Theorem, Prime Time, Dirichlet's Rational Approximation Theorem, Rationals Close, Public Key Coding System, Euler's Theorem, The Golden Rule, Law of Quadratic Reciprocity, Sums of Squares, Miller's Theorem, Binomial Theorem, Euler's Criterion, Chinese Remainder Theorem, Modular World, Primes Theorem, Fundamental Theorem of Algebra, Public Key Cryptography Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
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