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- Herausgeber: John Wiley & Sons
- Kategorie: Bildung
- Serie: The Teacher's Toolbox Series
- Sprache: Englisch
Math teachers will find the classroom-tested lessons and strategies in this book to be accessible and easily implemented in the classroom
The Teacher’s Toolbox series is an innovative, research-based resource providing teachers with instructional strategies for students of all levels and abilities. Each book in the collection focuses on a specific content area. Clear, concise guidance enables teachers to quickly integrate low-prep, high-value lessons and strategies in their middle school and high school classrooms. Every strategy follows a practical, how-to format established by the series editors.
The Math Teacher's Toolbox contains hundreds of student-friendly classroom lessons and teaching strategies. Clear and concise chapters, fully aligned to Common Core math standards, cover the underlying research, required technology, practical classroom use, and modification of each high-value lesson and strategy.
This book employs a hands-on approach to help educators quickly learn and apply proven methods and techniques in their mathematics courses. Topics range from the planning of units, lessons, tests, and homework to conducting formative assessments, differentiating instruction, motivating students, dealing with “math anxiety,” and culturally responsive teaching. Easy-to-read content shows how and why math should be taught as a language and how to make connections across mathematical units. Designed to reduce instructor preparation time and increase student engagement and comprehension, this book:
- Explains the usefulness, application, and potential drawbacks of each instructional strategy
- Provides fresh activities for all classrooms
- Helps math teachers work with ELLs, advanced students, and students with learning differences
- Offers real-world guidance for working with parents, guardians, and co-teachers
The Math Teacher's Toolbox: Hundreds of Practical ideas to Support Your Students is an invaluable source of real-world lessons, strategies, and techniques for general education teachers and math specialists, as well as resource specialists/special education teachers, elementary and secondary educators, and teacher educators.
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Seitenzahl: 829
Veröffentlichungsjahr: 2020
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Table of Contents
Cover
List of Tables
About the Authors
About the Editors
Acknowledgments
Letter from the Editors
Introduction
Our Beliefs about Teaching Math
Structure of This Book
Why Good Math Teaching Matters
PART I: Basic Strategies
CHAPTER 1: Motivating Students
What Is It?
Why We Like It
Supporting Research
Common Core Connections
Application
Student Handouts and Examples
What Could Go Wrong
Technology Connections
Figures
CHAPTER 2: Culturally Responsive Teaching
What Is It?
Why We Like It
Supporting Research
Common Core Connections
Application
What Could Go Wrong
Technology Connections
CHAPTER 3: Teaching Math as a Language
What Is It?
Why We Like It
Supporting Research
Common Core Connections
Application
Student Handouts and Examples
What Could Go Wrong
Technology Connections
Figures
CHAPTER 4: Promoting Mathematical Communication
What Is It?
Why We Like It
Supporting Research
Common Core Connections
Application
What Could Go Wrong
Student Handouts and Examples
Technology Connections
Attribution
Figures
CHAPTER 5: Making Mathematical Connections
What Is It?
Why We Like It
Supporting Research
Common Core Connections
Application
Student Handouts and Examples
What Could Go Wrong
Technology Connections
Figures
PART II: How to Plan
CHAPTER 6: How to Plan Units
What Is It?
Why We Like It
Supporting Research
Common Core Connections
Application
Student Handouts and Examples
What Could Go Wrong
Technology Connections
Figures
CHAPTER 7: How to Plan Lessons
What Is It?
Why We Like It
Supporting Research
Common Core Connections
Application
Student Handouts and Examples
What Could Go Wrong
Technology Connections
Figures
CHAPTER 8: How to Plan Homework
What Is It?
Why We Like It
Supporting Research
Common Core Connections
Application
Student Handouts and Examples
What Could Go Wrong
Technology Connections
Figures
CHAPTER 9: How to Plan Tests and Quizzes
What Is It?
Why We Like It
Supporting Research
Common Core Connections
Application
Student Handouts and Examples
What Could Go Wrong
Technology Connections
Figures
CHAPTER 10: How to Develop an Effective Grading Policy
What Is It?
Why We Like It
Supporting Research
Common Core Connections
Application
What Could Go Wrong
Student Handouts and Examples
Technology Connections
Figures
PART III: Building Relationships
CHAPTER 11: Building a Productive Classroom Environment
What Is It?
Why We Like It
Supporting Research
Common Core Connections
Application
What Could Go Wrong
Student Handouts and Examples
Technology Connections
Figures
CHAPTER 12: Building Relationships with Parents
What Is It?
Why We Like It
Supporting Research
Common Core Connections
Application
What Could Go Wrong
Student Handouts and Examples
Technology Connections
Figures
CHAPTER 13: Collaborating with Other Teachers
What Is It?
Why We Like It
Supporting Research
Common Core Connections
Application
What Could Go Wrong
Technology Connections
PART IV: Enhancing Lessons
CHAPTER 14: Differentiating Instruction
What Is It?
Why We Like It
Supporting Research
Common Core Connections
Application
What Could Go Wrong
Student Handouts and Examples
Technology Connections
Figures
CHAPTER 15: Differentiating for Students with Unique Needs
What Is It?
Why We Like It
Supporting Research
Common Core Connections
Application
What Could Go Wrong
Student Handouts and Examples
Technology Connections
Figures
CHAPTER 16: Project‐Based Learning
What Is It?
Why We Like It
Supporting Research
Common Core Connections
Application
What Could Go Wrong
Student Handouts and Examples
Technology Connections
Figures
CHAPTER 17: Cooperative Learning
What Is It?
Why We Like It
Supporting Research
Common Core Connections
Application
What Could Go Wrong
Student Handouts and Examples
Technology Connections
Figures
CHAPTER 18: Formative Assessment
What Is It?
Why We Like It
Supporting Research
Common Core Connections
Application
What Could Go Wrong
Technology Connections
CHAPTER 19: Using Technology
What Is It?
Why We Like It
Supporting Research
Common Core Connections
Application
What Could Go Wrong
Student Handouts and Examples
Technology Connections
Figures
CHAPTER 20: Ending the School Year
What Is It?
Why We Like It
Supporting Research
Common Core Connections
Application
What Could Go Wrong
Technology Connections
Appendix A: The Math Teacher's Toolbox Technology Links
Chapter 1: Motivating Students
Chapter 2: Culturally Responsive Teaching
Chapter 3: Teaching Math as a Language
Chapter 4: Promoting Mathematical Communication
Chapter 5: Making Mathematical Connections
Chapter 6: How to Plan Units
Chapter 7: How to Plan Lessons
Chapter 8: How to Plan Homework
Chapter 9: How to Plan Tests and Quizzes
Chapter 10: How to Develop an Effective Grading Policy
Chapter 11: Building a Productive Classroom Environment
Chapter 12: Building Relationships with Parents
Chapter 13: Collaborating with Other Teachers
Chapter 14: Differentiating Instruction
Chapter 15: Differentiating for Students with Unique Needs
Chapter 16: Project‐Based Learning
Chapter 17: Cooperative Learning
Chapter 18: Formative Assessment
Chapter 19: Using Technology
Chapter 20: Ending the School Year
References
Index
End User License Agreement
List of Tables
Chapter 3
Table 3.1 Vocabulary Chart
Table 3.2 Geometry Vocabulary Chart
Table 3.3 Visual and Verbal Aids
Chapter 4
Table 4.1 Class Discussion
Table 4.2 Low‐Floor, High‐Ceiling Problems
Table 4.3 Problem‐Solving Chart
Table 4.4. Scaffolded Lesson Summary
Table 4.5 Scoring Guidelines for Lesson Summaries
Table 4.6 Scoring Guidelines for Quick Writes
Chapter 5
Table 5.1. Table for Vehicle Word Problem
Chapter 7
Table 7.1. Summary Questions
Chapter 9
Table 9.1 Confidence Level Scoring Guidelines
Table 9.2 Comparing Scoring Guidelines
Table 9.3 Two Test Questions with Unequal Difficulty
Table 9.4 Two Test Questions with Similar Difficulty
Chapter 10
Table 10.1 Standards‐Based Grading
Table 10.2 Sample Report Card Grades for a Student with 20% Content Mastery
Chapter 12
Table 12.1 Parent Communication Outline
Chapter 14
Table 14.1 Levels of Complexity for Tiered Lessons
Table 14.2 Ratio Word Problem Table
Table 14.3 Rubric
Chapter 15
Table 15.1 US and Latin American Prime Factorization Methods
Chapter 16
Table 16.1 Project Ideas
Table 16.2 Basic Project Rubric
Table 16.3 Oral Presentation Rubric
Chapter 17
Table 17.1 Self‐Assessment Rubric
Table 17.2
Table 17.3 Task Cards
Chapter 18
Table 18.1 Formative Assessment Questions
Chapter 20
Table 20.1 Using Technology for Multiple‐Choice Questions
List of Illustrations
Chapter 3
Figure 3.3 Why the Word “Height” Is Confusing
Figure 3.4 Draw a Picture
Figure 3.6 Polynomials Anchor Chart
Figure 3.7 Why the Formula
a
2
+
b
2
=
c
2
Is Confusing
Chapter 4
Figure 4.4 Lesson Summary
Chapter 5
Figure 5.4 Completing the Square
Figure 5.5 Determining the Center and Radius of a Circle
Figure 5.6 Why (
a
+
b
)
2
≠
a
2
+
b
2
Chapter 6
Figure 6.1 Unit Plan: List of Skills
Figure 6.2 Unit Plan: Concept Map
Figure 6.3 Unit Plan: Sequence of Lessons
Chapter 9
Figure 9.6 Completed Test Corrections Sheet
Chapter 10
Figure 10.1 Grade Calculation Sheet
Figure 10.2 Completed Grade Calculation Sheet
Chapter 11
Figure 11.5 Annotated Work
Chapter 12
Figure 12.1 Parent Communication Script
Figure 12.2 Parent Communication Log
Chapter 14
Figure 14.5 Review Sheet
Chapter 15
Figure 15.1 Frayer Model (Blank)
Figure 15.2 Frayer Model—Perpendicular Bisector
Figure 15.3 Concept Map
Chapter 17
Figure 17.4 Peer Editing
Chapter 19
Figure 19.1 Simulation of 1,000 Coin Flips
Figure 19.4 Two Views of a Graph Using Technology
Guide
Cover
Table of Contents
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“This magnificent book is a must for all mathematics teachers! Its practical value is derived from the fact that the authors are both seasoned and high‐quality mathematics teachers who have experienced and have explored every aspect of teaching they discuss in their book. It is comprehensive and challenges the reader to consider the pros and cons of the different strategies described, which go beyond a ‘toolbox’ of great ideas for teaching!”
Alice F. Artzt
Queens College of the City University of New York
“This resource is a must for all mathematics teachers! New and veteran teachers will find the practical strategies and explicit examples easy to implement in the classroom and helpful in enhancing one's own pedagogy. Authors Bobson Wong and Larisa Bukalov have crafted a fantastic student‐focused resource dedicated to ensuring high‐quality instruction. Highly recommended!”
Christine DeBono
K‐5 Math Instructional Coach, Higley Unified School District, Arizona
“This comprehensive book is an incredible resource for math teachers at any stage of their career. Master Teachers Bobson Wong and Larisa Bukalov do an excellent job describing practical strategies, justifying them with research, and bringing them to life with concrete examples. I highly recommend it.”
Michael Driskill
Chief Operating Officer, Math for America
“As someone who has been teaching for 30 years, I find The Math Teacher's Toolbox to be a very rich reference of teaching strategies and resources for practitioners, especially mentors and mentees. I plan to use this book in my lessons and my professional development.”
Irene Espiritu
Middle School Teacher, Math for America Master Teacher, New York State Master Teacher
“The Math Teacher's Toolbox provides the reader with a summation of research‐supported current best practices in mathematics teaching. The layout of this book masterfully helps move the reader from understanding through application of the central ideas most essential to teaching mathematics effectively. Practical ideas for the classroom, as well as discussion about what could possibly go wrong, combine to make this a useful guide for teachers of all experience levels.”
Tabetha Finchum
2014 Presidential Awardee for Excellence in Mathematics Teaching
“The Math Teacher's Toolbox provides concrete, innovative strategies for adapting often intangible pedagogical theories across a wide range of math content areas and grade levels. Larisa Bukalov and Bobson Wong draw extensively from current research as well as their own years of classroom experience to explore the benefits and possible limitations of each strategy. Having already sought to implement a number of their ideas into my own classroom, I cannot recommend this book enough.”
Nasriah Morrison
Math Teacher, Institute for Collaborative Education and Math for America Master Teacher
“This book is truly a ‘toolbox’ for math instruction. It offers great technology tools and resources for teachers and their students, free online resources for student learning, and practical ideas that every math teacher can use. I will use this book for years to come.”
Jendayi Nunn
Mathematics Virtual Instructional Specialist, Atlanta Public Schools
“The Math Teacher Toolbox provides a map that can guide new teachers as they begin their journey and help veterans navigate the shifting terrain. The authors summarize current research from many areas of teaching and connect it to structured classroom practices. The experienced writing team organizes complex parts of the profession into a structure that makes it easy for practitioners to put the ideas to use in their classroom.”
Carl Oliver
Assistant Principal, City‐As‐School, New York
“Both new and experienced teachers will have cause to reach into this box of tools and return time and again to dig deeper—and each time you return, you'll find the box just as organized as the last! You'll keep this book nearby throughout your career for its practical, detailed tips, copious references, and teacher‐to‐teacher tone.”
Ralph Pantozzi, Ed.D.
2014 MoMath Rosenthal Prize winner, 2017 Presidential Awardee for Excellence in Mathematics Teaching
A winning educational formula of engaging lessons and powerful strategies for math teachers in numerous classroom settings
The Teacher's Toolbox series is an innovative, research-based resource providing teachers with instructional strategies for students of all levels and abilities. Each book in the collection focuses on a specific content area. Clear, concise guidance enables teachers to quickly integrate low-prep, high-value lessons and strategies in their middle school and high school classrooms. Every strategy follows a practical, how-to format established by the series editors.
The Math Teacher's Toolbox is a classroom-tested resource offering hundreds of accessible, student-friendly lessons and strategies that can be implemented in a variety of educational settings. Concise chapters fully explain the research basis, necessary technology, standards correlation, and implementation of each lesson and strategy.
Favoring a hands-on approach, this book provides step-by-step instructions that help teachers to apply their new skills and knowledge in their classrooms immediately. Lessons cover topics such as setting up games, conducting group work, using graphs, incorporating technology, assessing student learning, teaching all-ability students, and much more. This book enables math teachers to:
Understand how each strategy works in the classroom and avoid commonmistakes
Promote culturally responsive classrooms
Activate and enhance prior knowledge
Bring fresh and engaging activities into the classroom
Written by respected authors and educators, The Math Teacher's Toolbox: Hundreds of Practical Ideas to Support Your Students is an invaluable aid for upper elementary, middle school, and high school math educators as well as those in teacher education programs and staff development professionals.
Books in the Teacher's Toolbox series, published by Jossey-Bass:
The ELL Teacher's Toolbox
, by Larry Ferlazzo and Katie Hull Sypnieski
The Math Teacher's Toolbox
, by BobsonWong, Larisa Bukalov, Larry Ferlazzo, and Katie Hull Sypnieski
The Science Teacher's Toolbox
, by Tara C. Dale, Mandi S. White, Larry Ferlazzo, and Katie Hull Sypnieski
The Social Studies Teacher's Toolbox
, by Elisabeth Johnson, Evelyn Ramos LaMarr, Larry Ferlazzo, and Katie Hull Sypnieski
The Math Teacher’s Toolbox
Hundreds of Practical Ideas to Support Your Students
BOBSON WONG
LARISA BUKALOV
LARRY FERLAZZO
KATIE HULL SYPNIESKI
The Teacher's Toolbox Series
Copyright © 2020 by John Wiley & Sons, Inc. All rights reserved.
Published by Jossey‐Bass
A Wiley Brand
111 River Street, Hoboken NJ 07030—www.josseybass.com
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, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the publisher, or authorization through payment of the appropriate per‐copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, 978‐750‐8400, fax 978‐646‐8600, or on the Web at www.copyright.com. Requests to the publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, 201‐748‐6011, fax 201‐748‐6008, or online at www.wiley.com/go/permissions.
Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. Readers should be aware that websites offered as citations and/or sources for further information may have changed or disappeared between the time this was written and when it is read.
Jossey‐Bass books and products are available through most bookstores. To contact Jossey‐Bass directly call our Customer Care Department within the US at 800‐956‐7739, outside the U.S. at 317‐572‐3986, or fax 317‐572‐4002.
Wiley also publishes its books in a variety of electronic formats and by print‐on‐demand. Some material included with standard print versions of this book may not be included in e‐books or in print‐on‐demand. For more information about Wiley products, visit www.wiley.com.
Library of Congress Cataloging‐in‐Publication Data:
Names: Wong, Bobson, 1971- author. | Bukalov, Larisa, 1973- author.
Title: The math teacher's toolbox : hundreds of practical ideas to support your students / Bobson Wong, Larisa Bukalov.
Description: Hoboken, NJ: Jossey-Bass, [2020] | Series: Teacher's toolbox | Includes bibliographical references and index.
Identifiers: LCCN 2019051973 (print) | LCCN 2019051974 (ebook) | ISBN 9781119573296 (paperback) | ISBN 9781119573203 (adobe pdf) | ISBN 9781119573241 (epub)
Subjects: LCSH: Mathematics—Study and teaching (Middle school)—Handbooks, manuals, etc. | Mathematics—Study and teaching (Secondary)—Handbooks, manuals, etc.
Classification: LCC QA11.2 .W663 2020 (print) | LCC QA11.2 (ebook) | DDC 510.71/2—dc23
LC record available at https://lccn.loc.gov/2019051973
LC ebook record available at https://lccn.loc.gov/2019051974
Cover Design: Wiley
Cover Image: © malerapaso/Getty Images
FIRST EDITION
List of Tables
Table 3.1
Vocabulary Chart
Table 3.2
Geometry Vocabulary Chart
Table 3.3
Visual and Verbal Aids
Table 4.1
Class Discussion
Table 4.2
Low‐Floor, High‐Ceiling Problems
Table 4.3
Problem‐Solving Chart
Table 4.4
Scaffolded Lesson Summary
Table 4.5
Scoring Guidelines for Lesson Summaries
Table 4.6
Scoring Guidelines for Quick Writes
Table 5.1
Table for Vehicle Word Problem
Table 7.1
Summary Questions
Table 9.1
Confidence Level Scoring Guidelines
Table 9.2
Comparing Scoring Guidelines
Table 9.3
Two Test Questions with Unequal Difficulty
Table 9.4
Two Test Questions with Similar Difficulty
Table 10.1
Standards‐Based Grading
Table 10.2
Sample Report Card Grades for a Student with 20% Content Mastery
Table 12.1
Parent Communication Outline
Table 14.1
Levels of Complexity for Tiered Lessons
Table 14.2
Ratio Word Problem Table
Table 14.3
Rubric
Table 15.1
US and Latin American Prime Factorization Methods
Table 16.1
Project Ideas
Table 16.2
Basic Project Rubric
Table 16.3
Oral Presentation Rubric
Table 17.1
Self‐Assessment Rubric
Table 17.2
Notice and Wonder
Table 17.3
Task Cards
Table 18.1
Formative Assessment Questions
Table 20.1
Using Technology for Multiple‐Choice Questions
About the Authors
Bobson Wong has taught math at New York City public high schools since 2005. He is a three‐time recipient of the Math for America Master Teacher Fellowship, a New York State Master Teacher, and a 2014–2015 recipient of the New York Educator Voice Fellowship. He is a member of the Advisory Council of the National Museum of Mathematics.
He has also worked to improve the quality of high school mathematics standards and assessment in New York. He has served on several committees, including the state's Common Core standards review committee, the state's workgroup to reexamine teacher evaluations, and the United Federation of Teachers' Common Core Standards Task Force. As an educational specialist for the New York State Education Department, he writes and edits questions for high school math Regents exams.
He graduated from the Bronx High School of Science, Princeton University (B.A., history), the University of Wisconsin–Madison (M.A., history), and St. John's University (M.S.Ed., adolescent education, mathematics), where he received his teacher training through the New York City Teaching Fellows program.
He lives in New York City with his wife and children.
Larisa Bukalov has been teaching at Bayside High School since 1998. She has won several awards for excellence in classroom teaching. She is a four‐time recipient of the Math for America Master Teacher fellowship, a 2009 recipient of Queens College's Mary Fellicetti Memorial Award for excellence in mentoring and supervising student teachers, and a 2017 recipient of Queens College's Excellence in Mathematics Award for promoting mathematics teaching as a profession. A fourth‐generation math teacher, she simultaneously earned degrees from a specialized math high school in Ukraine and a distance learning high school at Moscow State University. After emigrating to the United States, she learned English while earning both her bachelor's degree in math and her master's degree in math education from Queens College, City University of New York.
Over the past 20 years at Bayside, Larisa has taught all levels of math from pre‐algebra to calculus, coached the school's math team, and created a math research program in which students wrote papers for the Greater New York City Math Fair, City College Engineering Expo, and the Intel Science and Talent Search.
Larisa has extensive experience providing professional development to pre‐service and in‐service teachers. She has mentored 16 student teachers. From 2007 to 2009 she provided professional development to early career teachers and math supervisors in New York City on Geometry, Probability, and Problem Solving. As part of her work with Math for America, Larisa has run several professional development sessions for teachers.
She lives in New York City with her husband and children.
About the Editors
Larry Ferlazzo and Katie Hull Sypnieski wrote The ELL Teacher's Toolbox and conceived of a series replicating the format of their popular book. They identified authors of all the books in the series and worked closely with them during their writing and publication.
Larry Ferlazzo teaches English, Social Studies, and International Baccalaureate classes to English Language Learners and others at Luther Burbank High School in Sacramento, California.
He's written nine books: The ELL Teacher's Toolbox (with co‐author Katie Hull Sypnieski); Navigating the Common Core with English Language Learners (with coauthor Katie Hull Sypnieski); The ESL/ELL Teacher's Survival Guide (with coauthor Katie Hull Sypnieski); Building a Community of Self‐Motivated Learners: Strategies to Help Students Thrive in School and Beyond; Classroom Management Q&As: Expert Strategies for Teaching; Self‐Driven Learning: Teaching Strategies for Student Motivation; Helping Students Motivate Themselves: Practical Answers to Classroom Challenges; English Language Learners: Teaching Strategies That Work; and Building Parent Engagement in Schools (with coauthor Lorie Hammond).
He has won several awards, including the Leadership for a Changing World Award from the Ford Foundation, and was the Grand Prize Winner of the International Reading Association Award for Technology and Reading.
He writes a popular education blog at http://larryferlazzo.edublogs.org/, a weekly teacher advice column for Education Week Teacher, and posts for The New York Times and The Washington Post.
He also hosts a weekly radio show on BAM! Education Radio.
He was a community organizer for 19 years prior to becoming a public school teacher.
Larry is married and has three children and two grandchildren.
A basketball team he played for came in last place every year from 2012 to 2017. He retired from league play after that year, and the team then played for the championship. These results might indicate that Larry made a wise career choice in not pursuing a basketball career.
Katie Hull Sypnieski has taught English Language Learners and others at the secondary level for over 20 years. She currently teaches middle school English Language Arts and Social Studies at Fern Bacon Middle School in Sacramento, California.
She leads professional development for educators as a teaching consultant with the Area 3 Writing Project at the University of California, Davis.
She is coauthor (with Larry Ferlazzo) of The ESL/ELL Teacher's Survival Guide, Navigating the Common Core with English Language Learners, and The ELL Teacher's Toolbox. She has written articles for the Washington Post, ASCD Educational Leadership, and Edutopia. She and Larry have developed two video series with Education Week on differentiation and student motivation.
Katie lives in Sacramento with her husband and their three children.
Acknowledgments
Teaching is a collaborative endeavor. We would never have accomplished everything that we've done – including writing this book – without the help of many individuals. We thank all of the people mentioned here. They've made us not just into better teachers but also into better people.
Several current and former administrators at Bayside High School have provided coaching and professional growth opportunities over the years: Michael Athy, Madeline Belfi‐Galvin, Harris Sarney, Susan Sladowski, and Judith Tarlo. Our colleagues in Bayside's Math Department have been a constant source of camaraderie, laughter, and valuable (if sometimes heated) pedagogical discussions. The thousands of students that we've taught at Bayside inspire and challenge us, giving us something to look forward to every day we go to work.
Math for America has created an active, supportive community that trusts and celebrates educators' expertise. Many of our ideas were refined in Math for America's professional development sessions.
Larry Ferlazzo and Katie Hull Sypnieski have been meticulous editors whose frequent questions about our thinking have improved our writing and teaching. Their timely and thought‐provoking edits have made this book much better than it would have been without them. David Powell has done an amazing job of formatting our manuscript. His attention to detail is amazing. Pete Gaughan at Wiley and Amy Fandrei at Jossey‐Bass patiently guided us through the stressful process of producing this book.
Bayside High School students Safi Ansari, Stefany Flores, Emily Hermida, Hana Ho, Anna Ling, Anyu Loh, Camila Palmada, Hardeep Singh, Navneet Sohal, Jason Sun, Ariana Verbanac, Richard Xing, and Joy Zou contributed their work for this book. Susie Xu and Justin Zhuo helped us proofread the figures and tables in this book.
Finally, our spouses and children deserve special mention for tolerating our many early‐morning and late‐night conversations about this book and helping us keep things in perspective.
Bobson Wong: Larisa Bukalov has been a mentor, colleague, and friend for 13 years. Prof. Charles Cohen, Sherrill Mirsky, Dr. Carol Nash, Barbara Rockow, and Prof. Robert C.‐H. Shell were some of the many educators that I've met over the years that taught me the patience and attention to detail that I needed to write this book. The people that I've met online at #MTBoS and #ITeachMath have influenced and encouraged me. Robert Lebowitz has been a source of mathematical and philosophical conversation for decades; without him, this book would not be possible.
Larisa Bukalov: Bobson Wong has been a colleague and a friend. Thank you for putting my ideas about teaching mathematics in writing and always motivating me to do more. Mary Chiesi hasn't just been my co‐teacher but also a mentor and best friend. We developed and practiced many of the strategies in this book together. Dr. Alice Artzt and the faculty at the mathematics education program at Queens College provided countless hours of debates on constructivism, hands‐on approach, teaching mathematics as a language, group work, as well as advice on this book's structure and organization. Twenty years after graduation, I know that I can always count on her help and advice. Dr. Nick Metas at Queens College was famous for his attention to detail. A history buff, he shared stories about mathematicians and helped introduce to me the idea of culturally relevant mathematics. My husband, Boris Bukalov, a math and science teacher, checked the math in the book. My late grandfather, Izaya Vayzman, taught me to love mathematics and teaching. Many of the ideas described in this book came from my watching his everyday interactions with students back in the Soviet Union.
Letter from the Editors
We don't teach math, and we don't know much math, either.
We do, however, know pedagogy.
And there's more great pedagogy in Bobson's and Larisa's book than you can shake a stick at.
In fact, there's so much exceptional teaching advice in this book that any teacher – no matter what subject he/she teaches – can learn a great deal of information from this book about effective instructional strategies that can be used in any classroom.
We sure did!
And, because we have so much confidence in Bobson and Larisa, we're sure all the math is great, too.
If you don't believe us, just check out all the math people who have said so many terrific things about The Math Teacher's Toolbox – their endorsements can be found in the front of the book.
We're thrilled and honored that Bobson and Larisa's book follows our The ELL Teacher's Toolbox in the Toolbox series.
It was a pleasure working with them during the 12 months they spent writing it, and we'd wager this won't be the last book you see written by them.
Larry Ferlazzo and Katie Hull Sypnieski
Editors of the Toolbox series
Introduction
When people find out what we do for a living, they often admit to us that they hate math or they're not good at it. After repeated negative experiences, many develop math anxiety—feelings of fear and tension when doing math (Namkung, Peng, & Lin, 2019, p. 482; Shields, 2007, p. 56). Math anxiety is not simply a set of emotions but a physiological response that affects heart rate and neural activity (Ramirez, Shaw, & Maloney, 2018, p. 145). It can be even more problematic when teachers or parents have it, since they can pass it on to students, which can negatively affect academic achievement (Beilock, Gunderson, Ramirez, & Levine, 2010, p. 1,862; Maloney, Ramirez, Gunderson, Levine, & Beilock, 2015, p. 1,485; Ramirez, Hooper, Kersting, Ferguson, & Yeager, 2018, p. 8).
Unfortunately, math anxiety is common among students, their parents, and teachers. We wrote this book not just to help people overcome math anxiety but also to help them appreciate and use math in the real world. The strategies described in this book reflect techniques and methods that we've used during our combined 35 years of teaching 26,000 lessons to over 5,000 students (including English Language Learners and students with learning differences) from around the world. Many of these strategies rely on social‐emotional learning (SEL, sometimes called social and emotional learning), the process by which people develop the skills necessary to manage their emotions, show empathy for others, and maintain positive relationships with others (Collaborative for Academic, Social, and Emotional Learning [CASEL], n.d.). SEL is a critical part of effective teaching because students' mindsets can affect their cognitive processing. People who experience success in an activity may be motivated and able to learn, while those who experience failure may tend to withdraw, rendering even the most engaging and well‐planned lesson useless (Sousa, 2017, p. 61).
Our Beliefs about Teaching Math
What we write in this book reflects four of our core beliefs about math, pedagogy, and students:
Students need to feel safe before they can learn.
Research indicates that when the brain perceives a threat, it instinctively releases adrenaline, which inhibits cognitive functions and other activity viewed as unnecessary (Sousa,
2017
, p. 50). Students need to feel safe before they can be receptive to learning. As a result, teaching strategies that make students feel positively about learning can improve student motivation. As we explain in
Chapter 11
: Building a Productive Classroom Environment, feeling good or safe doesn't guarantee that students will absorb new information, but it is a necessary condition for learning.
Math should make sense to students.
We believe that math should be taught in a way that makes sense to students. In our opinion, part of the reason why math anxiety is so prevalent is that many see it as a collection of disconnected and confusing “tricks.” By the time students graduate, they should have the confidence and ability to apply mathematical and critical‐thinking skills to real‐world situations (Berry & Larson,
2019
, p. 40). As we say in
Chapter 3
: Teaching Math as a Language, using the language‐acquisition techniques commonly associated with teaching English Language Learners can help make math more accessible to all students. We also believe that
constructivism
—the idea that students should actively create knowledge by experiencing it and reflecting on it—should be a central part of math instruction.
All students need access to rigorous math.
We believe that
all
students should have access to rigorous math—mathematical learning that includes solving challenging problems and deeper thinking. It abandons outdated notions of the meaning of being good at math—today's powerful calculators have eliminated the need to equate speed and accuracy with mathematical mastery (Devlin,
2019
, p. 10; Ruef,
2018
). Rigorous math requires both procedural and conceptual understanding (Ben‐Hur,
2006
, pp. 7–8; Levin,
2018
, p. 273; McCormick,
1997
, p. 149; Rittle‐Johnson, Schneider, & Star,
2015
, p. 594). It helps students appreciate the beauty of mathematics and apply it to the world around them. Unfortunately, many barriers (such as low expectations and hidden biases among teachers) can restrict students' ability to experience math in a positive way (Berry & Larson,
2019
, p. 41). When trying to determine what type of work is appropriate for students, we keep two rules in mind:
What works for some students often works for others.
Specific strategies (such as using multiple representations or making mathematical connections) designed to help English Language Learners, students with learning differences, or advanced students can frequently benefit
all
students. We discuss this idea more in
Chapter 5
: Making Mathematical Connections and
Chapter 15
: Differentiating for Students with Unique Needs.
What works for some students often doesn't work for others.
We try to modify instruction to meet the diverse needs of all students (we discuss this more in
Chapter 14
: Differentiating Instruction). Periodically questioning our attitudes and constantly looking for ways to deepen students' understanding (which we explain more in
Chapter 2
: Culturally Responsive Teaching) can make us more effective teachers.
Teachers don't have to do everything to succeed.
Nobody (including ourselves!) could possibly implement all of the strategies described here at all times. This book should not be used as a checklist of everything that teachers must do to be effective. Instead, we view it as a collection from which teachers can pick what works for their classrooms. We feel that in teaching, as in life, selecting a few things and doing them well is more productive and sustainable than trying to do everything at once.
Structure of This Book
This book is divided into four parts. Part I: Basic Strategies expands on what we believe to be the central ideas necessary to teach math effectively: motivating students, culturally responsive teaching, teaching math as a language, promoting mathematical communication, and making mathematical connections. Part II: How to Plan discusses strategies for units, lessons, homework, tests and quizzes, and grades. Part III: Building Relationships talks about how to build relationships with students, parents, and co‐teachers. Finally, Part IV: Enhancing Lessons contains other important strategies—differentiation, project‐based learning, cooperative learning, formative assessment, and technology.
This book is part of a series in The ELL Teacher's Toolbox (2018a) by Larry Ferlazzo and Katie Hull Sypnieski. All chapters in the books in this series have the following sections:
What Is It?:
brief description of the strategy
Why We Like It:
explanation of why we like the strategy
Supporting Research:
research that supports the strategy
Common Core Standards:
relevant Common Core content and mathematical practice standards
Application:
description of ways that the strategy can be implemented
Student Handouts and Examples
: list of reproducibles and other figures
What Could Go Wrong:
explanation of what could go wrong with each strategy and what can be done in these situations
Technology Connections:
links to relevant websites (also available online)
Figures:
reproducibles, which are also available online at
www.wiley.com/go/mathteacherstoolbox
Why Good Math Teaching Matters
The bottom line is that in today's changing world, we need better thinkers and problem‐solvers. We believe that as teachers, we have to do more than just convey mathematical ideas. We also need to be role models for self‐confidence, self‐reflection, critical thinking, and conceptual understanding. What we say and do affects not only the way in which our students learn math but also their beliefs. We hope that this book can inspire you to do more for your students, your communities, and yourselves.
PART IBasic Strategies
CHAPTER 1Motivating Students
What Is It?
Motivation—why people do what they do—affects every aspect of schooling. Without motivation, student learning becomes difficult, if not impossible (Artzt, Armour‐Thomas, & Curcio, 2008, p. 48). Motivated students tend to have better performance, higher self‐esteem, and improved psychological well‐being (Fong, Patall, Vasquez, & Stautberg, 2019, p. 123; Gottfried, Marcoulides, Gottfried, & Oliver, 2013, p. 83; Liu & Hou, 2017, p. 49; Reeve, Deci, & Ryan, 2004, p. 22). Conversely, unmotivated students can become disengaged from academics and, in the worst cases, drop out of school (National Research Council, 2004, p. 24).
According to self‐determination theory, a theory of motivation developed by researchers Edward L. Deci and Richard M. Ryan, motivation can be intrinsic (doing something because it is inherently satisfying) or extrinsic (doing something because it leads to some other result) (Ryan & Deci, 2000, p. 55). Many times, motivation is difficult to characterize as purely intrinsic or extrinsic. A student may be drawn by an extrinsic reward but may eventually internalize the values and adapt a more intrinsic motivation (Usher & Kober, 2012b, p. 3).
In addition, motivation is not a fixed quantity (Ryan & Deci, 2000, p. 54). Factors like schools, parents, communities, teachers, and life experiences can positively or negatively affect motivation (Usher & Kober, 2012a, p. 7). Students' motivation can vary from class to class—a student who is highly motivated in one class may be completely disengaged in another (National Research Council, 2004, p. 33).
As a result, educators often need to foster both intrinsic and extrinsic motivation. Students who are intrinsically interested in a topic are more likely to seek challenging tasks, think more creatively, and learn at a conceptual level (National Research Council, 2004, p. 38). However, since many academic tasks may not be inherently interesting, teachers also need to learn how to promote different methods of extrinsic motivation (Ryan & Deci, 2000, p. 55).
To sustain motivation, educators often seek ways to encourage students to internalize values. When students do so, they become more persistent and have a more positive sense of themselves (Ryan & Deci, 2000, pp. 60–61).
Why We Like It
In our experience, keeping motivational strategies in mind can enhance student engagement, academic achievement, and confidence to do math. Boosting their confidence is particularly important since many of our students experience math anxiety (we discuss it more in the Introduction), which can hinder their academic growth.
Supporting Research
Many studies on motivation focus on ways to build inclusive communities that promote learning for all students (Kumar, Zusho, & Bondie, 2018, p. 78). Proponents of self‐determination theory argue that people are motivated to complete a task if doing so fulfills basic psychological needs, such as autonomy, relatedness, and competence (Ryan & Deci, 2000, p. 64).
However, some researchers have begun to challenge the idea of a universal theory of motivation, arguing that most of the existing work ignores the experiences and members of historically marginalized groups, such as people of color (Usher, 2018, p. 132). These researchers seek a more culturally responsive framework in which motivation is viewed not just as an individual characteristic but as the product of the social and historical context that shapes students' emotions and beliefs (King & McInerney, 2016, p. 2).
Other studies have focused on the effect of emotions on student motivation (Hannula, 2019, p. 310). Students who feel more anxious about math often have decreased motivation and do more poorly in school (Gunderson, Park, Maloney, Beilock, & Levine, 2017, pp. 34–35; Mo, 2019, p. 2; Passolunghi, Cargnelutti, & Pellizzoni, 2018, p. 282). Discouragement from parents, inappropriate or overly difficult work, and lack of support from teachers can further erode students' self‐efficacy—the realistic expectation that making a good effort will lead to success (Usher, 2009, p. 308). In other words, social‐emotional learning is tied to motivation.
Research indicates that as students move through the K–12 school system, their attitudes toward math become less positive (Batchelor, Torbeyns, & Verschaffel, 2019, p. 204; Gottfried et al., 2013, p. 70). As a result, keeping middle and high school students motivated in math class can be particularly challenging.
Despite the different approaches and areas of emphasis in the literature, researchers agree on several ways to improve student motivation:
Teachers should have meaningful and challenging instruction (Kumar et al.,
2018
, p. 90).
Teachers should empathize with students and accept them unconditionally (Wormeli,
2014
).
Administrators should ensure that school personnel are culturally diverse (Usher,
2018
, p. 140).
In the Application section of this chapter, we discuss some strategies for improving motivation for all students.
Common Core Connections
Many of the motivational techniques that we describe in this chapter are related to Common Core standards. For example:
Showing the usefulness of a topic relates to standards such as modeling with mathematics (
Mathematical Practice
[
MP
].4), interpreting division of fractions by fractions (6‐NS.A.1), using a linear equation to model bivariate data (8‐SP.A.3), and interpreting equations as viable in a modeling context (A‐CED.A.3) (National Governors Association & Council of Chief State School Officers [NGA & CCSSO],
2010
, pp. 7, 42, 56, 65).
Finding a pattern connects to such standards as describing patterns in bivariate data (8‐SP.A.1), writing expressions in equivalent forms (A‐SSE.B.3), analyzing functions using different representations (F‐IF.C.7), and verifying the properties of dilations (G‐SRT.A.1) (NGA & CCSSO,
2010
, pp. 56, 64, 69, 77).
Promoting student autonomy relates to making sense of problems and persevering in solving them (MP.1) and using appropriate tools strategically (MP.5) (NGA & CCSSO,
2010
, p. 10). Problem solving and decision making are important elements of self‐determination (Heroux, Peters, & Randel,
2014
, p. 200).
Using technology is encouraged in such standards as drawing geometric shapes (7.G.A.2), interpreting scientific notation (8.EE.A.4), finding the solution to the equation
f
(
x
) =
g
(
x
) (A‐REI.D.11), showing key features of the graphs of functions (F‐IF.C.7), and describing transformations of functions (F‐BF.B.3) (NGA & CCSSO,
2010
, pp. 50, 54, 66, 69, 70).
Application
Researchers seeking to merge self‐determination theory with culturally responsive teaching (which we describe in more detail in Chapter 2: Culturally Responsive Teaching) have identified five characteristics of effective motivation:
Culture:
Students' learning is affected by the culture that surrounds them. Understanding it can shed light on how inequitable aspects of the dominant culture can negatively affect student learning (Usher,
2018
, p. 139).
Meaningfulness:
Students are most likely to see learning as valuable when teachers connect lessons to their lives in meaningful ways (Kumar et al.,
2018
, p. 83).
Competence:
Competence includes both
cultural competence
(understanding the cultural identities of oneself and others) and
academic competence
(the belief that one can complete a task) (Kumar et al.,
2018
, p. 83; Usher & Kober,
2012b
, p. 2).
Autonomy:
Teaching students how to set goals and make decisions can foster students' sense of
autonomy
(the extent to which they believe they can control their goals and actions) and lead to both individual growth and societal change (Kumar et al.,
2018
, p. 87).
Relatedness:
When educators develop authentic relationships with students (by learning about their home lives, culture, and values) and use that knowledge to communicate effectively with them, they are more likely to succeed academically (Bonner,
2014
, p. 397).
Here are some strategies that apply these characteristics in supporting student motivation.
NURTURING STUDENT CONFIDENCE
As we said in the Supporting Research section in this chapter, building students' self‐efficacy and autonomy can help alleviate their math anxiety and improve their motivation, which can in turn improve their academic performance.
One way that we nurture students' self‐confidence is to use language that supports their choice whenever possible. Saying, “I recommend that you rephrase this definition in your own words,” can often be more effective than simply commanding students to write it down. Explaining why completing a task is necessary can help students understand how they can benefit from doing so (Reeve & Halusic, 2009, p. 150).
Having private conversations with students about areas of concern can also help them feel more in control of their learning (“I've noticed that your work has slipped. Is everything OK? What can we do to improve it?”). Acknowledging and addressing their concerns can often turn complaints into more positive discussions (“Yes, I agree that this wording sounds confusing, but that's what they use on the state tests! Let's make it clearer.”) (Reeve & Halusic, 2009, p. 151).
In fact, building meaningful relationships with students—learning about their interests and activities, demonstrating authentic care, and genuinely trying to connect with students of all backgrounds—is the cornerstone of culturally responsive teaching (Heroux et al., 2014, p. 198; Kumar et al., 2018, p. 89). We believe that doing so can help not just students of color but all students.
While the techniques we describe above are often successful, they have limitations since many students face constraints that limit their autonomy (Kumar et al., 2018, p. 87). In order for students to become more self‐determined, they need more opportunities to set goals for themselves and regulate their learning (Heroux et al., 2014, p. 200). Some students may not complete online assignments because they lack Internet access at home, so simply telling them to “try harder” without figuring out what causes their behavior would accomplish little. Instead, they may need more time or an offline method to complete assignments. We discuss some helpful strategies in Chapter 11: Building a Productive Classroom Environment, Chapter 14: Differentiating Instruction, and Chapter 17: Cooperative Learning.
MOTIVATING THROUGH MATH
Many times, we use the math we teach as a motivational tool. These techniques work best when teachers use them to introduce the concept and elicit the lesson's goals (Posamentier, Smith, & Stepelman, 2010, p. 71).
When implementing these strategies, we also try to be sensitive to their emotional needs. We try to give them meaningful, scaffolded work that presents a moderate challenge. Giving students a reasonable chance to succeed can strengthen their self‐efficacy and ease their math anxiety (Margolis, 2014).
Connecting to the Real World
Many times, we try to connect what students are learning to the real world. Practical applications should be brief and accessible enough to advance the lesson, not detract from it (Posamentier et al., 2010, p. 66). For example, students can analyze college loan payments when discussing compound interest or determine which mobile phone plan is most economical when learning about piecewise functions. Real‐world applications often provide a strong motivation for mathematical learning (Walkington, Sherman, & Howell, 2014, p. 277).
Personalizing problems based on student interests or cultures can make mathematical tasks more relevant and improve their persistence and learning. Personalizing problems can connect what students already know to abstract mathematical concepts (a phenomenon known as grounding) and help them determine if their work is reasonable (Walkington et al., 2014, p. 275). Teachers can get information from students by asking them how they use numbers in their lives or what they are interested in. We talk about specific strategies for learning more about students in Chapter 11: Building a Productive Classroom Environment.
Here are some examples of problems that connect to students' lives and experiences:
Showing how one unusually low score can negatively affect an average can lead to a meaningful discussion of the harmful effects of outliers and the value of using other statistical measures like the median.
Students can use pattern blocks or other models of regular polygons to explore why honeycombs are shaped like hexagons. This can introduce a lesson on rotational symmetry, perimeter, and area. Students can relate this topic to chemistry by examining why hexagons appear in molecular structures.
Precalculus students can discover the need for a polar coordinate system by examining the maps of ancient cities like Paris, Moscow, or Beijing. We first ask students to give directions in a familiar city or town with a rectangular street grid, such as Manhattan. Students are then asked to give directions in a city in which streets radiate outwards from a center. This lesson can lead to a discussion of how math is used in other fields like city planning and navigation.
Students can even create their word problems (we discuss this more in Chapter 4: Promoting Mathematical Communication).
Finding Patterns
Many students learn how to recognize patterns as early as kindergarten. This skill helps them make connections with more complicated ideas in middle and high school (Markworth, 2016, p. 23). Pattern recognition boosts students' feelings of competence since they usually need little prior knowledge (Smith, Hillen, & Catania, 2007, p. 39). Research indicates that the ability to process complex patterns is one of the brain's most important features since it underlies many other cognitive functions, including thought, imagination, invention, and reasoning (Mattson, 2014, p. 13). Pattern recognition is used to make many real‐life decisions, including finding the fastest route home, predicting what people are thinking based on their body language, and estimating how much money to spend on a construction project (Barkman, 2018; Miemis, 2010). In our experience, pattern recognition can be incorporated into almost any lesson, often as an introductory exercise (which we discuss in Chapter 7: How to Plan Lessons).
Here are some examples in which we find patterns in lessons:
Figure 1.1
: Pattern Blocks shows an example using triangular numbers that can introduce sequences for middle school or Algebra I students.
Figure 1.2
: Rotational Symmetry has an activity that middle school or Geometry students can use to discover the formula for the minimum angle of rotation that maps a regular polygon back onto itself.
Figure 1.3
: Exponential Growth contains an activity in which students explore exponential growth by looking at the number of layers created when papers are folded in half.
Identifying a Void in Knowledge
Students often want to complete their understanding of a topic, so making them aware of a gap in that knowledge may motivate them to learn more (Posamentier et al., 2010, p. 62). We like this strategy because it allows us to make connections to knowledge that is familiar to students. This helps us promote retrieval practice, which we also discuss in Chapter 4: Promoting Mathematical Communication and Chapter 7: How to Plan Lessons.
Here is an example from Algebra I involving solving systems of linear equations algebraically:
1.
Solve for
x
and
y
: 2
x
+ 5
y
= 34 −2
x
− 4
y
= −28
2.
Solve for
x
and
y
: 2
x
+
y
= 7 3
x
−
y
= 3
3.
Solve for
x
and
y
: 2
x
+ 5
y
= 34
x
+ 2
y
= 14
In the first two systems, students should be able to add the equations to eliminate one variable and solve for both. However, adding the equations in the third system will not eliminate the variable. Students will then realize that their prior knowledge (in this case, solving by adding equations) will not work for the third system. They could then be motivated to learn what they can do to solve the third system—in this case, multiplying the second equation by −2 so that they can eliminate a variable by adding the equations.
Figure 1.4
: Identify a Void shows an example from Geometry. The first two examples can be solved using the Pythagorean theorem since two side lengths from right triangles are given and students must find the third side. However, the third example can't be solved using the Pythagorean theorem since only one side length is given. This prepares students for seeing the value of trigonometry, which deals with angle measures and the ratios of side lengths in right triangles.
REWARDS
The use of rewards—extrinsic incentives like prizes or points—to motivate students is controversial. Many researchers agree that rewards can work when individuals are not initially motivated. However, the effect of incentives on individuals who are already motivated is less clear (Hidi, 2015, p. 87). Some studies conclude that rewards generally undermine intrinsic motivation (Kohn, 1994; Deci, Ryan, & Koestner, 2001, p. 50). Others find that the detrimental effect of rewards on motivation is overstated and that rewards may sometimes have a positive effect (Cameron, Banko, & Pierce, 2001, p. 21; Eisenberger & Shanock, 2003, p. 128).
We believe that rewards, when used with the other strategies that we describe in this chapter, can help keep students motivated to learn. In reality, rewards are necessary for most tasks. After all, almost everyone works for some kind of compensation, such as money, awards, or high grades.
When possible, we make rewards meaningful and immediate. Incentives work best when students see an immediate benefit—offering extra computer time now is usually more effective than offering an end‐of‐year party. Minimizing the gap between the effort and the reward reduces delay discounting, in which people assign less value to future rewards (Cheng, 2016).
We use a variety of incentives. Rewards like extra credit, stars, or “good work” tickets are cheap, easy, and accessible to all students (we mention several apps that can keep track of points in the Technology Connections section of this chapter). Students could even be involved in distributing stars or “good work” tickets (Lewis, 2017). To encourage autonomy, teachers can allow students to convert such tickets into more tangible rewards, like extra credit (Chapter 10: How to Develop an Effective Grading Policy describes how we incorporate extra credit into our grades). Teachers may also consider awarding classroom privileges, such as extra computer time (we discuss rules and procedures more in Chapter 11: Building a Productive Classroom Environment).
We sometimes give candy or some other inexpensive prize, such as pens or erasers. However, we find that regularly offering these prizes can be problematic. They obviously require time and money, and they may put students in awkward
