What Successful Math Teachers Do, Grades 6–12: 80 ResearchBased Strategies for the Common Core–Aligned Classroom
Publication Year: 2013
DOI: http://dx.doi.org/10.4135/9781452299853
Subject: Mathematics, Common Core
 Chapters
 Front Matter
 Back Matter
 Subject Index

 Chapter 1: Make Sense of Problems and Persevere in Solving Them
 Aligning Chapter 1 to the Common Core State Standards
 Strategy 1: Help Students Develop SelfControl to Enhance Their Thinking and Independence as Well as to Ease Your Own Workload.
 Strategy 2: Encourage Students to Be Mentally Active While Reading Their Textbooks.
 Strategy 3: Praise Mistakes!
 Strategy 4: Make a Lesson More Stimulating and Interesting by Varying the Types of Questions You Ask Students.
 Strategy 5: Use a Variety of Strategies to Encourage Students to Ask Questions About Difficult Assignments.
 Strategy 6: Use a QuestionAsking Checklist and an Evaluation Notebook to Help Students Become Better Learners.
 Strategy 7: Find Out Why Students Rate a Mathematical Task as Difficult So You Can Increase the Difficulty of Exercises and Tests More Effectively.
 Strategy 8: Teach Students to Ask Themselves Questions About What They Already Know About a Problem or Task They Are Working on.
 Strategy 9: Structure Teaching of Mathematical Concepts and Skills Around Problems to Be Solved, Using a ProblemCentered or ProblemBased Approach to Learning.
 Notes
 Chapter 2: Reason Abstractly and Quantitatively
 Aligning Chapter 2 to the Common Core State Standards
 Strategy 10: Teach Students to Ask Themselves Questions About the Problems/Tasks They Are Working on.
 Strategy 11: Help Students Understand Their Own Thought Processes, and Guide Them in Learning to Think Like Mathematicians.
 Strategy 12: Select and Carefully Structure Homework Assignments So That They Require the Development of Mathematical Thinking and Reasoning. Anticipate Changes That Might Occur While Students Are Working at Home.
 Strategy 13: Emphasize HigherLevel Thinking Objectives in Regular Mathematics Classes So That All Students Incorporate the Features of Enriched Academic and Honors Classes.
 Notes
 Chapter 3: Construct Viable Arguments and Critique the Reasoning of Others
 Aligning Chapter 3 to the Common Core State Standards
 Strategy 14: Use Classwide Peer Tutoring to Help Your Students Learn Whether or Not They Have Learning Disabilities.
 Strategy 15: Carefully Select Problems for Use in Cooperative Learning Groups.
 Strategy 16: Encourage Students to Work Cooperatively with Other Students.
 Strategy 17: Use Group Problem Solving to Stimulate Students to Apply Mathematical Thinking Skills.
 Strategy 18: Don't Give Students Feedback on Their Performance Too Early.
 Strategy 19: Promptly Give Students Information or Feedback About Their Performance.
 Strategy 20: Increase Your Understanding of Factors That Affect Students' Attitudes Before and After Testing. You May Be Surprised!
 Strategy 21: Get Students to “Think Out Loud” When Solving Problems.
 Notes
 Chapter 4: Model with Mathematics
 Aligning Chapter 4 to the Common Core State Standards
 Strategy 22: Find Out About Your Students' Motivation Regarding Mathematics, and Use That Knowledge to Refine Your Instruction.
 Strategy 23: When Trying to Determine How to Motivate Students' Interest in Mathematics, Teachers Should Differentiate Between Personal and Situational Interest and Use Both Forms to Increase Students' Motivation to Learn Mathematics. Teachers Also Need to Both Stimulate and Maintain Their Students' Interest.
 Strategy 24: Use School FundRaising Projects, Such as Students' Selling Candy or Organizing a Walkathon, as the Basis of Mathematics Lessons.
 Strategy 25: When Doing Inquiry Lessons, Give Students Clearly Written Materials to Guide the Inquiry Process.
 Strategy 26: Use Graphic Representations or Illustrations to Enhance Students' Memory While They Are Listening to You. Abstract Representations Such as Flowcharts Are More Effective Than Colorful Pictures.
 Strategy 27: Playing Makes Understanding Mathematics Easier and More Fun.
 Strategy 28: Assign Homework and Other Projects Requiring Students to Write About Connections Between Mathematics and Other Subjects.
 Strategy 29: Encourage Students to Make Mental Pictures While Applying Rules to Solve Problems.
 Notes
 Chapter 5: Use Appropriate Tools Strategically
 Aligning Chapter 5 to the Common Core State Standards
 Strategy 30: Use the Jigsaw Technique of Cooperative Learning as an Interesting and Effective Way for Students to Learn.
 Strategy 31: Use Homework as a Way of Delving More Deeply into Important Mathematical Concepts and Skills.
 Strategy 32: Help Students Learn without Relying on TeacherCentered Approaches. Give Them Carefully Chosen Sequences of WorkedOut Examples and Problems to Solve.
 Notes
 Chapter 6: Attend to Precision
 Aligning Chapter 6 to the Common Core State Standards
 Strategy 33: Treat Students in Ways That Reflect the Belief That You Have High Expectations for Their Performance.
 Strategy 34: Call on Students More Frequently to Promote Their Achievement.
 Strategy 35: Make Sure to Pause for at Least Four Seconds After Listening to a Student's Communication Before Responding.
 Strategy 36: Emphasize the General Principles That Underlie Solving Specific Types of Problems.
 Notes
 Chapter 7: Look for and Make Use of Structure
 Aligning Chapter 7 to the Common Core State Standards
 Strategy 37: Teachers Should Be Tactical in Their Use of Questions.
 Strategy 38: Use a Variety of Sequences to Ask Questions.
 Strategy 39: Adolescents Need Extended Support to Acquire the Ability to Visualize.
 Strategy 40: Give Students the Kind of Feedback That Will Most Help Them Improve Their Future Performance.
 Strategy 41: Complex Exercises That Give Students Freedom Tend to Fit the Way Older Students Learn.
 Strategy 42: Provide Hints or Clues or Ask Leading Questions When Students Need Help Solving Problems Instead of Giving Them the Answers. Gradually Phase Out This Support So as to Foster Independent Problem Solving.
 Strategy 43: Examine Your Students' Knowledge of Mathematics, and Use This Information to Write Challenging Word Problems That They Will Enjoy Solving.
 Strategy 44: Students Need Time to Practice Planning Their Solutions to Problems.
 Notes
 Chapter 8: Look for and Express Regularity in Repeated Reasoning
 Aligning Chapter 8 to the Common Core State Standards
 Strategy 45: Use Questions for Different and Versatile Functions in the Classroom.
 Strategy 46: Use InquiryBased Learning in Addition to ProblemBased Learning.
 Strategy 47: Teachers Can Help Students Learn to Ask Better Questions.
 Strategy 48: Use Homework Assignments as Opportunities for Students to Get Practice and Feedback on Applying Their Mathematical Knowledge and Skills.
 Strategy 49: Use Analogies to Help Students Develop More Valid Conceptions.
 Strategy 50: Have Students Study Written Model Solutions to Problems While Learning and Practicing Problem Solving.
 Notes
 Chapter 9: Manage Your Classroom
 Aligning Chapter 9 to the Common Core State Standards
 Strategy 51: Create Your Own Support Network as Soon as You Begin Your First Teaching Job.
 Strategy 52: Before Beginning a Lesson, Put an Outline of What You Are Going to Cover on the Blackboard.
 Strategy 53: Make Realistic Time Estimates When Planning Your Lessons.
 Strategy 54: Make Classroom Activities Flow Smoothly.
 Strategy 55: Have “Eyes in the Back of Your Head” So You Notice Misbehavior at an Early Stage.
 Strategy 56: Do More Than One Thing at a Time.
 Strategy 57: Work Directly with Individual Students as Often as Possible.
 Strategy 58: Avoid Reacting Emotionally When Evaluating Problematic Situations in the Classroom.
 Strategy 59: To Reduce Math Anxiety, Focus on Both the Thoughts and the Emotions of the Students.
 Strategy 60: Consider Whether a Student's Learning Weakness Might Involve a Deficiency in Auditory Perception.
 Notes
 Chapter 10: Assess Student Progress
 Aligning Chapter 10 to the Common Core State Standards
 Strategy 61: Feedback on Practice Is Essential for Improving Student Performance.
 Strategy 62: Make Sure Students Pay Attention to the Feedback You Give Them.
 Strategy 63: Systematically Incorporate Review into Your Instructional Plans, Especially Before Beginning a New Topic.
 Strategy 64: Provide All Students, Especially Students Lacking Confidence, with “Formative Assessments” to Allow Them Additional Opportunities to Succeed in Mathematics.
 Strategy 65: Be Aware of Students' Different Levels of Text Anxiety as It Relates to Different Subject Areas, and Use a Variety of Techniques to Help Them Overcome Their Test Anxiety.
 Strategy 66: Do Not Assume That Students Accept Responsibility for or Agree with Their Bad Grades on Tests.
 Strategy 67: If Students Do Not Follow Your Instructions and/or If Their Achievements Do Not Fulfill Your Expectations, the Cause May Not Be Students' Incompetence. It Could Be a Result of Your SelfOverestimation.
 Notes
 Chapter 11: Consider Social Aspects in Teaching Mathematics
 Aligning Chapter 11 to the Common Core State Standards
 Strategy 68: Make Multicultural Connections in Mathematics.
 Strategy 69: Find Out About Your Students' Families and How Their Values and Practices Might Affect Students' Attitudes and Performance in Mathematics.
 Strategy 70: Reach Out to Parents to Form a Partnership for Educating Elementary and High School Students.
 Strategy 71: Inform Parents That They Should Not Let Media Reports About Studies of Other Children Change Their Views of Their Own Children's Abilities to Be Successful in Mathematics.
 Strategy 72: Some Students Do Not Think They Have Control Over Their Academic Successes and Failures. Help These Students Recognize That They Do Have Some Control.
 Strategy 73: Teach Students, Especially Girls, to Believe That Success in Mathematics Results from Their Efforts.
 Strategy 74: Give Girls the Same Quantity and Quality of Teacher Attention as Boys.
 Strategy 75: Make Special Efforts to Encourage Girls to Study Mathematics.
 Strategy 76: Use Different Motivational Strategies for Girls and Boys.
 Strategy 77: Take into Consideration How Students View Successful Teachers and How This Differs for Girls and Boys.
 Strategy 78: Praise, Encourage, and Help Your Older Students.
 Strategy 79: Does Grade Skipping Hurt Mathematically Talented Students Socially and Emotionally? Don't Worry About Accelerating Talented Students!
 Strategy 80: Use Technology Such as Dynamic Geometry Software to Enhance Student Understanding and Analysis.
 Notes

Dedication
To Barbara for her support, patience, and inspiration.
To my children and grandchildren, David, Lauren, Lisa, Danny, Max, Sam, and Jack, whose futures are unbounded.
And in memory of my dear parents, Alice and Ernest, who never lost faith in me.
—Alfred S. PosamentierTo my Mom and Dad, whose unceasing love and support, and examples of integrity and hard work have converged to allow me a life limited only by my imagination.
—Terri L. GermainWilliamsTo my wife, Tae Jin, who has made every sacrifice to ensure my happiness and success.
To my children, Jennifer and Rebecca, who are an eternal source of pride and joy, and in memory of my parents, Stanley and Beatrice, who were always there for me.
—Daniel JayeCopyright
Copyright © 2013 by Corwin
All rights reserved. When forms and sample documents are included, their use is authorized only by educators, local school sites, and/or noncommercial or nonprofit entities that have purchased the book. Except for that usage, no part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the publisher.
National Council of Teachers of Mathematics (NCTM) standards throughout the book are reprinted with permission by NCTM. They are quoted from Curriculum and Evaluation Standards for School Mathematics (1989), Professional Standards for Teaching Mathematics (1991), Handbook of Research on Mathematics Teaching and Learning (1992), Assessment Standards for School Mathematics (1995), Principles and Standards for School Mathematics (2002), and Research Companion to Principles and Standards for School Mathematics (2003). All rights reserved. Standards are listed with the permission of NCTM. NCTM does not endorse the content or validity of these alignments.
All trade names and trademarks recited, referenced, or reflected herein are the property of their respective owners who retain all rights thereto.
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Library of Congress CataloginginPublication Data
Posamentier, Alfred S.
What successful math teachers do, grades 6–12 : 80 researchbased strategies for the common core–aligned classroom. — Second edition / Alfred S. Posamentier, Terri L. GermainWilliams, Daniel Jaye.
pages cm
Includes bibliographical references and index.
ISBN 9781452259130 (pbk.)
1. Mathematics—Study and teaching (Secondary) —Standards—United States. I. GermainWilliams, Terri L. II. Jaye, Daniel. III. Title.
QA13.P67 2013
510.71′2—dc23 2013011072
This book is printed on acidfree paper.
13 14 15 16 17 10 9 8 7 6 5 4 3 2 1
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Associate Editor: Julie Nemer
Editorial Assistant: Ariel Price
Production Editor: Melanie Birdsall
Copy Editor: Amy Rosenstein
Typesetter: C&M Digitals (P) Ltd.
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Permissions Editor: Karen Ehrmann
Prologue
As a direct result of federal pressure on the states to continuously improve their instructional program and ensure that all students are being reached in the teaching process, teachers are being called on to meet professional standards and base their work on researchproven methods of teaching. Educational research, often conducted at universities or on educational sites by university researchers, is reported in educational journals and is most often read by other researchers. All too often, the style in which research reports or articles on research findings are reported is not friendly or appealing to the classroom teacher. The very community—classroom teachers—that could benefit enormously from the findings of many of these educational initiatives rarely learns about these endeavors. It is the objective of this book to bring some of the more useful research findings to the classroom teacher. In our quest for the most salient research findings, we were guided by the Common Core State Standards and the National Council of Teachers of Mathematics standards. Rather than merely presenting the research findings that support these standards, we have attempted to convert them into useful classroom strategies, thus capturing the essence of the findings and at the same time putting them into a meaningful context for the practicing mathematics teacher.
This book is to serve as a resource for mathematics teachers. It should provide a portal to access the many worthwhile findings resulting from educational, psychological, and sociological research studies done in Europe and in the United States. Heretofore, teachers have had very few proper vehicles for getting this information, short of combing through the tomes of research reports in the various disciplines. This book is designed to provide an easy way for the classroom teacher to benefit from the many ideas embedded in these academic exercises.
The book is designed to be an easy and ready reference for the mathematics teacher—both preservice and inservice. It consists of 11 chapters, each with a theme representing one aspect of the typical instructional program and each guided by the Common Core Standards. Each chapter presents a collection of teaching strategies, concisely presented in a friendly format:
The StrategyThis is a simple and crisp statement of the teaching strategy we recommend.
What the Research SaysThis offers a discussion of the research project that led to the strategy. This section should give the teacher some confidence in, and a deeper understanding of, the principle being discussed as a “teaching strategy.”
Teaching to the National Council of Teachers of Mathematics StandardsHere we present the salient National Council of Teachers of Mathematics standard that we are supporting with the strategy.
Aligning to the Common Core State StandardsThis section not only helps explain the standards but also shows how they can be met.
Classroom ApplicationsThis section tells the teacher how the teaching strategy can be used in the mathematics instructional program. Where appropriate, some illustrative examples of the teaching strategy in the mathematics classroom are provided.
Precautions and Possible PitfallsThis is the concluding section for each strategy and mentions some of the cautions that should be considered when using this teaching strategy so that the teacher can avoid common difficulties before they occur, thereby achieving a reasonably flawless implementation of the teaching strategy.
SourcesThese are provided so that the reader may refer to the complete research study to discover the process and findings in detail.
We see this book as a first step in bringing educational research findings to the practitioners: the classroom teachers. Perhaps teachers will see that there is much to be gained to enhance teaching by reviewing educational research with an eye toward implementing the findings in their instructional program. Furthermore, it would be highly desirable for researchers to make more of an effort to extend their publications/findings to the classroom teacher. To do otherwise would make the entire activity of educational research irrelevant.
As you read the many instructional suggestions offered in this book, we hope you will continuously think of yourself as the teacher who might implement them. Remember, your personality plays a large role in mapping out an overall instructional strategy. Each teacher brings to the classroom various strengths, and therefore, the research we bring to the reader should be viewed in that context. Nevertheless, the specific researchbased tips and strategies offered here will help you focus on certain aspects of your teaching. Teachers who continuously selfevaluate their instructional performance will, undoubtedly, become master teachers.
Acknowledgments
Amy Rosenstein is to be commended for her fine copy editing of the second edition.
Publisher's AcknowledgmentsCorwin thanks the following individuals for their contributions to this book:
 Kim Cottini, Math Learning Consultant
 Living Sky School Division
 North Battleford, Saskatchewan, Canada
 Diane S. Coupal, Adjunct Lecturer in Mathematics
 State University of New York at Plattsburgh
 Mathematics Department
 Plattsburgh, NY
 Nathan Herzog, Associate Professor
 William Jessup University
 Rocklin, CA
 Jami Stone, Assistant Professor of Education
 Black Hills State University
 College of Education
 Spearfish, SD
 Steven Willott, National Board Certified Mathematics Teacher
 Francis Howell North High School
 St. Charles, MO
About the Authors
Alfred S. Posamentier is Dean of the School of Education and Professor of Mathematics Education at Mercy College, New York. For the previous 40 years he held these same positions at The City College of The City University of New York. He is the author and coauthor of more than 55 mathematics books for teachers, students, and the general readership. He is also a frequent commentator in newspapers on topics relating to education.
After completing his BA degree in mathematics at Hunter College of The City University of New York, he took a position as a teacher of mathematics at Theodore Roosevelt High School in the Bronx (New York), where he focused his attention on improving the students' problemsolving skills and at the same time enriching their instruction far beyond what the traditional textbooks offered. He also developed the school's first mathematics teams (both at the junior and senior levels). He is currently involved in working with mathematics teachers and supervisors, nationally and internationally, to help them maximize their effectiveness.
Immediately upon joining the faculty of The City College (after having received his master's degree there), he began to develop inservice courses for secondary school mathematics teachers, including such special areas as recreational mathematics and problem solving in mathematics.
Dr. Posamentier received his PhD from Fordham University (New York) in mathematics education and has since extended his reputation in mathematics education to Europe. He has been visiting professor at several European universities in Austria, England, Germany, and Poland, and at the University of Vienna, he was Fulbright Professor in 1990.
In 1989, he was awarded the title of Honorary Fellow at the South Bank University (London, England). In recognition of his outstanding teaching, The City College Alumni Association named him Educator of the Year in 1994, and New York City had May 1, 1994, named in his honor by the President of the New York City Council. In 1994, he was also awarded the Grand Medal of Honor by the Federal Republic of Austria. In 1999, upon approval of Parliament, the president of the Federal Republic of Austria awarded him the title of University Professor of Austria; in 2003, he was awarded the title of Ehrenbürger (Honorary Fellow) of the Vienna University of Technology, and he was awarded (June 2004) the Austrian Cross of Honor for Arts and Science, First Class by the President of the Federal Republic of Austria. In 2005, he was elected to the Hall of Fame of the Hunter College Alumni Association, and in 2006, he was awarded the Townsend Harris Medal from The City College of New York. Other honors bestowed upon Dr. Posamentier include Education Leader of the Year, Education Update newspaper, 2009; Educator of the Year, The City College of New York Education Alumni Association, 2009; New York State Mathematics Education Hall of Fame, New York State Association of Mathematics Supervisors, 2009; and the ChristianPeter Beuth Prize 2009, Beuth Society and University of Applied Science, Berlin, Germany, 2010.
He has taken on numerous important leadership positions in mathematics education locally. He was a member of the New York State Education Commissioner's Blue Ribbon Panel on the MathA Regents Exams. He served on the Commissioner's Mathematics Standards Committee, which was charged in 2004 with rewriting the Standards for New York State, and he is on the New York City Public Schools Chancellor's Math Advisory Panel.
After 40 years on the faculty of The City College of New York, and now three years as Dean of the School of Education at Mercy College, New York, he is still a leading commentator on educational issues and continues his longtime passion of seeking ways to make mathematics interesting to teachers (see Math Wonders: To Inspire Teachers and Students [2003] and The Art of Motivating Students for Mathematics Instruction [2011]), students, and the public—as can be seen from his latest books: Math Charmers: Tantalizing Tidbits for the Mind (2003); π, A Biography of the World's Most Mysterious Number (2004); 101+ Great Ideas for Introducing Key Concepts in Mathematics, Second Edition (2006); and The Fabulous Fibonacci Numbers (2006); Problem Solving Strategies for Efficient and Elegant Solutions (2008); Mathematical Amazements and Surprises: Fascinating Figures and Noteworthy Numbers (2009); The Pythagorean Theorem (2010); The Glorious Golden Ratio (2012); The Secrets of Triangles: A Mathematical Journey (2012); and Magnificent Mistakes in Mathematics (2013).
Terri L. GermainWilliams is Assistant Professor of Mathematics Education at Mercy College, teaching courses in methods of teaching mathematics as well as Assessment and Evaluation. She also works with a number of schools and organizations as an educational consultant, supporting schools in the areas of mathematics instruction, scheduling and programming, the implementation of small learning communities, data and accountability, strategic planning, and leadership.
GermainWilliams graduated with a bachelor's degree in mathematics from Adelphi University and began her career as an intern at Mepham High School in Bellmore, New York, during a fifthyear master's program. Upon graduating with her master's degree and completing her internship, GermainWilliams began her teaching career as an eighthgrade mathematics teacher in Jericho, New York. The lure to make a difference in an area of high need brought GermainWilliams to join the team of founding teachers of the Bushwick School for Social Justice housed in the Bushwick High School Campus in Brooklyn, New York. This group of dedicated educators brought the vision of the planning team to reality when, within four years, tripled the historically low graduation rate and her own personal vision: to have a class of students learn calculus before enrolling in college.
Upon graduating from Queens College's School Supervision and Administration program, GermainWilliams was fortunate to join the administrative team at the Academy of Urban Planning (AUP). At AUP, GermainWilliams focused on supporting student data practices and provided professional development and support to the mathematics and science departments. During her tenure as Assistant Principal, she was accepted into the PhD in Mathematics Education Program at Teachers College, Columbia University, where she is now in the final stages.
Prior to accepting the position with Mercy College Graduate School of Education, GermainWilliams worked as an Achievement Manager with the New York City Department of Education, supporting more than 25 K–12 schools in the areas of instruction, strategic planning, professional development, federal and state data and accountability, scheduling/programming, and student services.
Daniel Jaye is the Chief Academic Officer and Director of Academic Affairs at the Solomon Schechter Day School of Bergen County. He lectures frequently and enjoys presenting interesting techniques in problem solving as well as problems that provide enrichment for the mathematics classroom. He is also interested in comparing math standards throughout the nation and the world.
Jaye graduated from The City College of New York with a major in mathematics and began his career teaching mathematics at Seward Park High School (New York City). After one year, he was invited to teach at the prestigious Stuyvesant High School (New York City), where he distinguished himself by teaching the entire range of high school mathematics courses.
After receiving his master's degree in mathematics education from The City College of New York, Jaye took an interest in guiding student research projects in mathematics. Shortly thereafter, he served as the math research coordinator and coordinated the submission of thousands of studentgenerated research papers to local and national competitions, including the Westinghouse and Intel Science Talent Search Competitions. In 2001, he was awarded the Mathematical Association of America's Edyth Sliffe Award for Excellence in Teaching. He was also the recipient of Education Update's Outstanding Teacher of the Year award in 2004.
After 25 years of outstanding teaching, Jaye was selected as Assistant Principal for the Stuyvesant High School Department of Mathematics. He immediately began to put his energies into creating more opportunities for talented and gifted students to study advanced mathematics. He was chosen to be Executive Director of the New York City Math Team, where he coordinated the training activities of the 100 members of the team. In 2001, he created and directed the CCNY Summer Scholars Academy in Mathematics and Science. This program provides advanced courses in mathematics and science and is supplemented by a stellar guest lecture series featuring noted mathematicians, scientists, and educators.
In 2004, Jaye was chosen to serve on the New York State Math Standards Committee, which authored new state standards in mathematics. In 2004, he was elected President of the Association of Mathematics Assistant Principals for Supervision (New York City) and was awarded the Phi Delta Kappa Leadership in Education award. He has served on the New York City Public Schools Chancellor's Math Advisory Panel and the New York State Mathematics Curriculum Committee.
In 2006, Jaye was selected to lead the Bergen Academies in Hackensack, New Jersey, as Principal and Director, where he guided the institution to national acclaim as the sole recipient of the Intel School of Distinction Award for academic excellence. In 2010, he assumed the position of Chief Academic Officer and Director of Academic Affairs at the Solomon Schechter Day School of Bergen County.
Jaye's passion for teaching and interest in mathematics standards and problem solving were inspirational in creating this book.

Epilogue
Now that you have read through the many researchbased suggestions to improve your instruction, we want you to reflect on the larger picture of mathematics education today. Consider the following situation. A recent visit to a pictureframing shop highlighted a mathematical deficiency that seems to be common in our society. An inspection of the bill for framing two pictures, one 4 inches by 20 inches and the other 12 inches by 12 inches, revealed that they cost the same. When questioned, the proprietor responded that the same amount of framing was used for the two pictures and that the glass was figured on the basis of “united inches.”
He was immediately asked what this unit of measure meant. He indicated that it was the sum of the length and the width; in this case each had 24 united inches, and so the cost was the same for the two pieces of glass. The merchant was asked if he believed the two frames required the same amount of glass. He wasn't sure, but assumed they did, since the two had the same number of united inches. A math teacher listening to this discussion chimed in to give him a quick lesson on rectangle area. The proprietor was amazed to discover that he had been charging the same amount for the two pieces of glass, when, in fact one's area (144 sq. in.) was almost twice that of the other (80 sq. in.). This mathematical illiteracy is particularly alarming, especially in the context of our country's poor showing on a recent Organisation for Economic Cooperation and Development Programme for International Student Assessment study, where the United States came in 28th out of 40 countries being compared on the mathematical achievement of 15yearolds.
We have become complacent about achievement in mathematics. Adults more often than not take pride in their inability to have mastered school mathematics. Furthermore, when they are told that their children will need to master mathematics in school, they begin to question the reasons for such claims, especially when their children come home with math homework that looks unfamiliar to their parents. Over the years, we have tried to convince others that there is power and beauty in mathematics. This is no easy task. We are often confronted with responses like “I don't need to know arithmetic since I use the [ubiquitous] calculator.” Or, “I don't even have to calculate the ‘best buy’ in the supermarket, since every item has its unit price indicated.” Or “Even ‘miles per gallon’ need not be calculated, since my car's odometer does that for me.” Some even ask, “Why teach mathematics at all?” Why don't they ever ask, “Why teach poetry, literature, music, art, or even science if one is not planning to pursue a career in those fields?” When was the last time an adult needed to use any of these subjects in everyday life?
We need to convince the general populace of the importance of mathematics. Simply saying, as many do, that today's students are involved with realworld applications in the classroom just doesn't cut it. Unfortunately, there are at least two problems with the realworldmath claims: First, the real world of students is often not what adults have chosen as the real world, and to be truly of the real world—rather than artificial models—is generally far too complicated for a school audience. There are times when parents do a “project” at home that involves mathematics or reasoning skills. Often these skills were developed as a result of school math instruction. Parents should involve their children in these projects, which might include setting up a birthday party, buying flooring or carpeting, or calculating expenses (i.e., budget). These would be actual realworld activities.
So where does that leave us? Do we merely stop teaching mathematics for the abovementioned reasons, or do we try to demonstrate its purpose in other ways? Mathematics has manifested itself in the school curriculum in different ways at various times in our history. In the 18th century, students learned to “reckon,” or do arithmetic, so that they would be able to do the necessary calculations required in trade or farming, for which they also needed some geometry. As time went on, the time available for math instruction increased, as did the material taught: algebra and trigonometry. More advanced instruction began to include some probability and statistics, although this was limited until the “number crunching” machines (i.e., calculators and computers) appeared. Today, we find ourselves with the dilemma of deciding exactly what and how mathematics should be taught. The advent of the computer has had a marked effect on the curriculum. Topics that used to be commonly taught are no longer needed, such as extracting the square root of a number or using logarithms to simplify complex calculations. In short, learning mathematics entails much more than obtaining tools to use in other fields. It is the subconscious acquisition of thinking and reasoning skills coupled with the more sophisticated way we view the physical world that leads the list of the many life enhancements that come with learning mathematics.
We, as members of the select New York State Math Standards Committee, were charged with preparing standards that would provide the necessary understandings and applications of arithmetic and geometry so that a proper transition could be made to a solid understanding and use of algebra, trigonometry, probability, and statistics. Today's youth need a different facility with, and understanding of, numerical concepts than previous generations. The calculator and computer have reduced the need for one to be a lightningfast calculator, yet despite today's technology, the need to understand number concepts and relationships has not diminished. Calculating a tip in a restaurant or balancing a checkbook still ought to be done easily.
For us to compete favorably in mathematics achievement in the world arena, we need to spend more time doing those things that school mathematics has done well for decades: provide appropriate reckoning skills, enable a reasonably sophisticated view of geometry for use in academics and beyond, and fortify students with the necessary tools of mathematics to pursue whatever academic endeavor they choose to study. The byproduct of all of this will be a wellreasoning and able problemsolver. Only if we do this in the context of motivating students with the beauty of mathematics (rather than telling them that what they are doing is for their “realworld” experiences, when they have difficulty accepting this) will our efforts bear fruit.
Resource: What the Authors Say: Enriching Instruction
Strategy: Enrich Your Instruction Before You Look to Accelerate the CurriculumWhat the Authors SayRecently, there has been a surge to move students along in the standard high school mathematics curriculum at a faster pace. In the course of this acceleration, a number of topics have been relegated to the back burner. In short, they were cut. Years ago, there were many useful techniques in algebra that were regularly taught but are not even mentioned today. One example is the teaching of factoring. Students were shown how to factor the sum and difference of two cubes, but today the practice is not mentioned. Every teacher owes it to his or her students to enrich their instruction. This can be done by expanding on a topic being taught, by extending the students' knowledge by showing how the topic being taught can relate to another topic out of the curriculum, or by bringing in historical aspects related to the topic.
Teaching to the National Council of Teachers of Mathematics StandardsThe National Council of Teachers of Mathematics says enrichment is part of any good teaching performance. The Principles and Standards for School Mathematics state that “mathematics teachers generally are responsible for what happens in their own classrooms and can try to ensure that their classrooms support learning by all students … teachers must challenge and hold high expectations for all their students, not just those they believe are gifted.”1 It is a good professional practice to provide opportunities for all students to enrich their learning of mathematics by encouraging them to explore topics beyond the scope of the syllabus. The following classroom application contains some suggestions for student projects in mathematics.
Aligning to the Common Core State StandardsThe Common Core State Standards have released documents discussing the application of the standards (both literacy and mathematics) for students with disabilities as well as English language learners. However, the website and published materials lack explicit discussion or documentation regarding enrichment. This may be a result of the “higher” expectations of the standards, as claimed in their introduction to the mathematics standards. This may be a result of the novelty of the movement as a whole. In either case, mathematics educators must consider how to meet the needs of their students. Consider the habits of athletes. They push themselves beyond the expectations of a meet or game during practices so that the requirements of the contest seem easier when compared to their preparation. In mathematics, we can challenge students with solving more intricate or difficult problems (including problems with multiple answers and those with noninteger solutions), as well as the use of proof to enhance understanding. The CCSS provide high expectations that are internationally benchmarked, and we can use them as a guide to help our students become even more skillful mathematicians through meeting and exceeding their standards.
Classroom ApplicationsThere are a multitude of topics that can be used to enrich instruction. You can assign individual students to do a small report on a topic that relates to the material being taught. Listed next are some possible topics that students might use for such a project. This list is merely intended as a guide for generating additional topics.
 Advanced Euclidean Geometry
 Algebraic Fallacies
 Algebraic Models
 Algebraic Recreations
 Analog Computer
 Ancient Number Systems and Algorithms
 Arithmetic Fallacies
 Arithmetic Recreations
 Bases Other Than Ten
 Binary Computer
 Boolean Algebra
 Brocard Points
 Calculating Shortcuts
 Cavalieri's Theorem
 Checking Arithmetic Operations
 Conic Sections
 Continued Fractions
 Cryptography
 Crystallography
 Curves of Constant Breadth
 Cylindrical Projections
 Desargues' Theorem
 Determinants
 Diophantine Equations
 Divisibility of Numbers
 Duality
 Dynamic Symmetry
 Elementary Number Theory Applications
 The Euler Line
 Extension of Euler's Formula to N Dimensions
 Extension of Pappus's Theorem
 Fermat's Last Theorem
 Fibonacci Numbers
 Fields
 Finite Differences
 Finite Geometry
 The Five Regular Polyhedra
 Flexagons
 The FourColor Problem
 The Fourth Dimension
 Fractals
 Game Theory
 Gaussian Primes
 Geodesics
 Geometric Dissections: Tangrams
 Geometric Fallacies
 Geometric Models
 Geometric Stereograms
 Geometric Transformations
 Geometry of Bubbles and Liquid Film
 Geometry of Catenary
 Geometry Constructions (Euclid)
 Gergonne Problem
 The Golden Section
 Graphical Representation of Complex Roots of Quadratic and Cubic
 Equations
 Groups
 Higher Algebra
 HigherOrder Curves
 Hyperbolic Functions
 The Hyperbolic Paraboloid
 Hypercomplex Numbers
 Intuitive Geometric
 Recreations
 Investigating the Cycloid
 The Law of Growth
 Linear Programming
 Linkages
 Lissajous Figures
 Lobachevskian Geometry
 Logarithms of Negative and Complex Numbers
 Logic
 Magic Square Construction
 Map Projections
 Mascheroni Constructions
 Mathematics and Art
 Mathematics and Music
 Mathematics of Life Insurance
 Matrices
 MaximumMinimum in Geometry
 Means
 Methods of Least Squares
 The Metric System
 Minimal Surfaces
 Modulo Arithmetic in Algebra
 Monte Carlo Method of Number Approximation
 Multinomial Theorem
 Napier's Rods
 Networks
 The NinePoint Circle
 Nomographs
 The Number Pi, Phi, or e
 Number Theory Proofs
 Paper Folding
 Partial Fractions
 Pascal's Theorem
 Perfect Numbers
 Polygonal Numbers
 Prime Numbers
 Probability
 Problem Solving in Algebra
 Projective Geometry
 Proofs of Algebraic Theorems
 Properties of Pascal's Triangle
 Pythagorean Theorem: Triples
 Regular Polygons
 The Regular SeventeenSided Polygon
 Relativity and Mathematics
 Riemannian Geometry
 Solving Cubics and Quartics
 Special Factoring
 Spherical Triangles
 The Spiral
 Statistics
 Steiner Constructions
 Tesselations
 Theory of Braids
 Theory of Equations
 Theory of Perspectives
 ThreeDimensional Curves
 The Three Famous
 Problems of Antiquity
 Topology
 Unsolved Problems
 Vectors
You might also merely digress from a topic being taught. For example, when teaching probability, you might mention the famous “birthday problem.” When discussing concurrency in geometry, you might want to introduce Ceva's theorem, which makes many difficult theorems almost trivial to prove. We suggest you consider any of the following books, each of which will give you lots of ideas for enrichment:
2003). Math wonders to inspire students and teachers. Alexandria, VA: ASCD.(2006). 101+great ideas for introducing key concepts in mathematics. Thousand Oaks, CA: Corwin., & (2010). Teaching secondary mathematics: Techniques and enrichment units. Columbus, OH: Allyn & Bacon/Prentice Hall., , & (Precautions and Possible PitfallsWhenever you embark on material that is not part of the curriculum and where students will essentially not be held responsible for learning the material, there is a tendency on the part of the students to stretch this activity out as long as possible. By keeping the teacher from moving along, figuring that if the teacher runs out of time and cannot cover a required topic, the students hope to have less material to study for when the next test comes along. So we urge you to do the enrichment with an eye toward a properly balanced time schedule. Don't let the enrichment material dominate the class activities. Remember that you are still responsible for covering the prescribed course work. Experience will help you appropriately estimate the proper time allotment for such enrichment activities.
Note1. The National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics (p. 375). Reston, VA: Author.
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