Writing Math Research Papers: A Guide for High School Students and Instructors PDF
Preview Writing Math Research Papers: A Guide for High School Students and Instructors
Writing Math Research Papers A Guide for High School Students and Instructors Fourth Edition Writing Math Research Papers A Guide for High School Students and Instructors Fourth Edition Robert Gerver, PhD Cover and illustrations by Michael Gerver Edited by Julianne Gerver, MA INFORMATION AGE PUBLISHING, INC. Charlotte, NC • www.infoagepub.com About the Cover A treasure map was chosen as the cover theme. Why? The math research journey involves exploration. It involves charting new territory and overcoming obstacles. It involves looking for clues in the mathematics, interpreting those clues, and doing lots of digging to uncover something very rewarding—a new mathematical result! Library of Congress Cataloging-in-Publication Data For the reasons I’ve got sunshine on a cloudy day; Linda, Julianne, A CIP record for this book is available from the Library of Congress http://www.loc.gov and Michael ISBN: 978-1-62396-863-2 (Paperback) 978-1-62396-864-9 (Hardcover) 978-1-62396-865-6 (ebook) Copyright © 2014 Information Age Publishing Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the publisher. Printed in the United States of America ii Writing Math Research Papers For the reasons I’ve got sunshine on a cloudy day; Linda, Julianne, and Michael ii Writing Math Research Papers Acknowledgments Without the help, encouragement, and support of the following people, this book would not have been possible: Elaine Boyrer Joan Fowler Regina Newman William Wagner Everyone who benefits from Writing Math Research Papers can join me in thanking the North Shore School District and these fine young scholars from North Shore High School, Cold Spring Harbor Junior-Senior High School, and Smithtown High School West: Sara Abbondondolo Logan Kupferman Simran Bagga Stephanie Bazan Aidan Lewis Dalita Balassanian Allison Black Liz Lucas Tim Bramfeld Andrew Carnevale Max Moore Jeff Chew Claire Castellano Aidan Murray Jocelyn Chua Cecilia Castellano Alexandra Osman Gabrielle De Rosa Nicole Ciotoli Pam Otis Nancy Friedlander Stephen Davis Talia Pelts Michael Gerbush Jill DeDona Miranda Purcell Blake Gilson Jeanne DiSimone Julia Reilly Parisa Golestaneh Alexia Efstathiadis Molly Reilly Robin Hadley Andrew Fierstein Will Rosencrans Laura Henning Fossett Mike Rovner Tiffany Indence Farb Danielle Sacco Kim Martini Raquel Fossett Charlie Schubach Vandana Minnal Gabrielle Ganasoulis Erin Sheerin Nicole Miritello AMyiachnaneal Gerver Dylan Smith Dan Moadel KJualtiaenrinnea G erver Christian Van Pak Sharon Poczter Stephen Grudman Diana Vizza Daniele Rottkamp Meghan Henneberger Sunaina Vohra Shikha Singhvi Sarah Ivanovic Stephanie Zelenetz Alexis Soterakis Olivia Jacome Lauren Adelman Stephen Johansen Special thanks also to Joseph Quarteraro and family, Jeff Lesser, and the entire Al Kalfus Long Island Math Fair organization, for giving tens of thousands of Long Island students the chance to write math papers, exchange presentations, and meet like-minded students from other high schools. This book would not exist without this forum for exploration. A Guide for Students and Instructors iii Table of Contents Introduction ...........................................................................................ix Chapter 1 Mathematics: Shouting Questions and Whispering Answers ..................1 Chapter 2 Finding a Topic.......................................................................................17 Chapter 3 Problem Solving: A Prerequisite for Research……………………...……31 Chapter 4 The Importance of Clear Communication..............................................56 Chapter 5 The Notes You Take in Math Class ......................................................63 Chapter 6 Technical Writing Techniques................................................................84 Chapter 7 Conjectures, Theorems, and Proofs……………..................................118 Chapter 8 Reading and Keeping a Research Journal .........................................146 Chapter 9 Components of Your Research Paper ................................................160 Chapter 10 Oral Presentations ..............................................................................171 Chapter 11 Sample Pages from Actual Papers......................................................194 Chapter 12 Some Suggested Resources ..............................................................229 Chapter 13 A Guide for Instructors and Administrators .........................................243 References .........................................................................................277 vii iv Writing Math Research Papers Introduction As an avid fan of writing and mathematics, I began giving my students writing assignments as a new teacher in 1977. The initial reaction was “This is math— why are we doing English?” As the students completed their math writing projects on varied topics such as automobile insurance, income taxes, stocks and bonds, consumer credit, and more, they began to realize the value of being able to communicate the powerful results mathematics can provide. Students who wrote about surface area, parabolas, the Pythagorean theorem, and other mathematics topics were quick to point out that they internalized the material so much better because in order to explain it they really had to understand it. Students concluded that writing is an essential component of any discipline, not just humanities courses. Based on the successes of these early writing experiences, I began to incorporate writing in my mathematics classes on a daily basis. Needless to say, I was very excited when the National Council of Teachers of Mathematics (NCTM), an organization of educators dedicated to the improvement of mathematics instruction, recommended the integration of writing across the mathematics curriculum. Mathematics and writing, to me, have always complemented each other naturally. Since problem solving is a central focus of math classes, it is clear that writing must be integrated with mathematics courses. Problem solvers have to be able to explain the procedures they used and the solutions they found. Have you ever been absent from a class and asked a fellow student, “What did we do in math yesterday?” only to hear the response “I know it but I can’t explain it”? How did you react? Imagine a detective telling a superior, “I’ve solved the case but I can’t explain my solution”! Mathematicians must be able to communicate their findings to others so their important results can be used to solve future problems. This is not the only value of writing in mathematics. Writing will also help you understand mathematical concepts. As you construct written explanations, you will need to explore and review mathematical concepts in your mind. Writing Math Research Papers will help you expand your ability to read about mathematics, explore mathematics, and write about your mathematical experiences. Writing in mathematics also aligns beautifully with the recently implemented Common Core State Standards for Mathematical Practice. ix A Guide for Students and Instructors A Guide for Students and Instructors v
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