Differentiating Math Instruction, K–8: Common Core Mathematics in the 21st Century Classroom
Publication Year: 2013
Real-time strategies for real-life results!
It's all too familiar: A classroom of 25+ students—all with very different needs and learning styles. Do you sometimes feels more like a juggler than a math teacher, struggling to keep all those balls in air? William Bender is prepared to lend a hand with this new edition of what's certainly the most important book on differentiating math instruction written in recent years.
Just what makes Differentiating Math Instruction such a favorite among K-8 teachers? It's the only book that successfully blends the best of what technology has to offer with guidelines for meeting the objectives set forth by the Common Core. Every new innovation in math instruction is addressed:
Flipping math instruction; Project-based learning; Using Khan Academy in the classroom; Educational gaming; ...
- Front Matter
- Back Matter
- Subject Index
- Differentiated Mathematics Instruction
- Differentiated Instructional Models: Lesson Modification and Learning Centers for Mathematics
- Flipped Mathematics Classes and Project-Based Learning: New Differentiated Instructional Models in Mathematics
- Strategies for Differentiating Early Math Instruction
- Strategies for Differentiating Instruction in Grades 3 Through 6
- Strategies for Differentiating Instruction in Grades 6 Through 8
- Differentiated Assessments and Response to Intervention
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.
All trade names and trademarks recited, referenced, or reflected herein are the property of their respective owners who retain all rights thereto.
Printed in the United States of America
Library of Congress Cataloging-in-Publication Data
Bender, William N., author. [Differentiating math instruction] Differentiating math instruction, K-8 : common core mathematics in the 21st century classroom / William N.
Bender. —Third edition.
Revision of: Differentiating math instruction : strategies that work for K-8 classrooms! / William N. Bender.? 2009.
Includes bibliographical references and index.
ISBN 978-1-4522-5545-3 (alk. paper)
1. Individualized instruction. 2. Mathematics—Study and teaching (Preschool) 3. Mathematics—Study and teaching (Elementary) 4. Mathematics—Study and teaching (Middle school) I. Title.
This book is printed on acid-free paper.
13 14 15 16 17 10 9 8 7 6 5 4 3 2 1
A SAGE Company
2455 Teller Road
Thousand Oaks, California 91320
SAGE Publications Ltd.
1 Oliver's Yard
55 City Road
London EC1Y 1SP
SAGE Publications India Pvt. Ltd.
B 1/I 1 Mohan Cooperative Industrial Area
Mathura Road, New Delhi 110 044
SAGE Publications Asia-Pacific Pte. Ltd.
3 Church Street
#10-04 Samsung Hub
Acquisitions Editor: Jessica Allan
Associate Editor: Kimberly Greenberg
Editorial Assistant: Heidi Arndt
Permissions Editor: Karen Ehrmann
Marketing Manager: Maura Sullivan
Project Editor: Veronica Stapleton Hooper
Copy Editor: Terri Lee Paulsen
Typesetter: C&M Digitals (P) Ltd.
Proofreader: Dennis W. Webb
Indexer: Wendy Allex
Cover Designer: Karine Hovsepian
Corwin gratefully acknowledges the contributions of the following reviewers:
- Marcia Carlson
- Sixth-Grade Teacher
- Crestview Elementary School
- Clive, Iowa
- Julie Duford
- Fifth-Grade Math Teacher
- Polson Middle School
- Polson, Montana
- Esther Eacho
- Associate Faculty
- Johns Hopkins University
- Baltimore, Maryland
- Debi Gartland
- Professor of Special Education
- Towson University, Department of Special Education
- Towson, Maryland
- Edward C. Nolan
- Mathematics Supervisor, PreK to Grade 12
- Montgomery Public Schools
- Rockville, Maryland
- Rachel Spenner
- Sixth-Grade Teacher
- Westridge Elementary
- West Des Moines, Iowa
About the Author
Appendix A: Recently Developed or Widely Used Curricula in Mathematics[Page 268]
There are a plethora of innovative instructional, computer-based software programs and recently developed hard-copy mathematics curricula used in schools today. Many of these, like the Khan Academy, have been described in text, and some are free for teachers to use. This appendix presents several recently developed, broad-scale mathematics curricula that many mathematics teachers are currently using. Many of these tools allow for individual laptop or tablet-based instruction, and might well provide the option of flipping the classroom, as discussed in Chapter 3. Others have proven to be a great foundation for Tier 2 and 3 RTI interventions in mathematics as discussed in Chapter 7. While other curricula could have been included here, these were chosen for the reasons above, or because they are frequently used in schools today.Number Worlds
Griffin (2003a, 2003b, 2004a, 2005) has provided another research-based structure that represents children's early understanding of numbers, Number Worlds. This research-supported resource for teaching young students mathematical concepts is a mathematics readiness/math [Page 269]curriculum that teaches foundational mathematics concepts and skills on a conceptual level and an application level (Griffin, 2003b, 2004a, 2005). This curriculum is intended for children from prekindergarten through Grade 1, and includes software, manipulatives, problem-solving scenarios, games, lesson plans, and hard-copy workbooks, all of which focus on developing a hands-on understanding of numbers and how numbers are used in the real world (Griffin, 2004b; Griffin et al., 2003). While predating the Common Core State Standards for Mathematics, this curriculum is nevertheless congruent with those standards, in that students are taught mathematics readiness and number sense in a manner that stresses deep conceptual understanding. Number Worlds is now published by Scholastic Research Associates (http://www.sranumberworlds.com).
SRA Number Worlds is based on five instructional principles. The program was developed to
Build on the student's current knowledge using multilevel activities;
Use natural learning paths that children typically employ to develop number sense;
Present new knowledge in a way that supports the common progression children use when learning numbers;
Teach computational fluency; and
Stress hands-on exploration, problem solving, and communication using numbers.
Each concept in the curriculum is discussed and utilized in various ways to assure flexibility in students' understandings of math constructs (Griffin, 2003b, 2004b). In addition to being recommended for average achievers, this program has been evaluated with children from low-income populations, as well as children with special needs, and has proven effective in enhancing number sense, computational fluency, and reasoning, as well as performance on traditional, standardized mathematics achievement tests (Griffin, 2004a, 2005). Positive results of this program were evident in some of these studies, up to one year after termination of the program (Griffin, 2004a, 2005).
Differentiated activities are relatively easy to develop using Number Worlds. Teachers can adjust the sequence of work to accommodate students with many different learning styles. Teachers will assess each student's level of ability and select specific activities for the individual children throughout the year (Griffin, 2004a, 2005).[Page 270]SAS Curriculum Pathways
One recently developed online curriculum for older students that is being used increasingly in schools is the SAS Curriculum Pathways (http://www.sascurriculumpathways.com). This curriculum is free for teachers worldwide and is used by more than 50,000 teachers in over 12,000 schools. This program provides an extensive, online curricula with materials and instructional activities in English, language arts, science, social studies, mathematics, and Spanish. The lesson activities are designed around the Common Core State Standards in math and are appropriate from grade levels 6–12. While the curriculum is free, teachers are required to login to access the material, and a brief tutorial video is available free of charge at the website above. The company behind this website designed the curriculum materials extremely carefully, with content experts in various subject areas consulting on all aspects of the curriculum. A wide variety of instructional activities are incorporated into this curriculum, as well as excellent graphics and short video demos that can be used in flipped classes. Every mathematics teacher in the middle grades should explore this free curriculum.Math in Focus: Singapore Math
This comprehensive math curriculum, by Marshall Cavendish, is based on instructional practices developed for mathematics in the nation of Singapore in recent decades (http://www.hmheducation.com/singaporemath/index.php). This curriculum has been developed for students from kindergarten through Grade 8 using a concrete, pictorial, and abstract framework to cover mathematics concepts in depth. In the Math in Focus curriculum, beginning in kindergarten, students are exposed to mathematical concepts, covering less content overall but at a greater depth, with a strong emphasis on conceptual learning as well as mathematical procedures. Rather than covering a concept in two or three days, students in this curriculum may spend two to three weeks exploring mathematical content much more fully, and students achieve mastery prior to moving to the next big idea. Because of this emphasis on deep understanding, this curriculum was one of 15 mathematics curricula that were reviewed in the development of the Common Core State Standards in mathematics. For that reason, this curriculum has and will continue to receive increasing attention.Accelerated math for Intervention
This newly developed commercially available curriculum from Renaissance Learning utilizes the widely popular Accelerated Math curriculum that has [Page 271]been around for a while, but adds an overlay of assessment and progress-monitoring tools to make this very appropriate for use in RTI interventions (http://www.renlearn.com/am/RTI.aspx). Accelerated Math for Interventions enables monitored, differentiated math practice, and is recognized as a mastery measure by the National Center on Response to Intervention. Reports from this curriculum provide daily information about student progress toward math mastery (TOPS report), skill by skill, and a variety of other reports (e.g., Status of the Class Report) are available to allow teachers to see student performance or compare students' performance. Because Accelerated Math has been widely used for years, and with the growing emphasis on specific targeted interventions in mathematics for students with mathematics deficits, teachers should expect to see this curriculum more frequently across the elementary grade levels.IXL
IXL is a subscription-based site that presents mathematics exercises for grades kindergarten through middle school (http://www.ixl.com). Students completed activities online, answering simple questions on various topics, and their progress is tracked online. IXL assesses students' understanding as they practice, and then generates detailed performance reports that can be used for instruction and/or progress monitoring in an RTI framework. The various reports include data on grade-level proficiency, trouble spots, and even progress toward mathematics mastery.VMATH
Vmath was developed by the Voyagers company, a company known for the Voyagers reading program (http://www.voyagerlearning.com/curriculum/math-solutions/vmath). Note: Be careful to not confuse Vmath with Vedic Math!). Vmath is a supplemental curriculum and benchmarking tool, supported by a number of case study applications (found at the website above) that is aimed at students who are struggling in math from Grade 2 through 8. Vmath presents a balance of teacher-led, explicit instruction, printed materials, assessment options, and student work online and is intended to fill gaps in student knowledge across the grade levels. In the daily lessons, teachers model the skills, facilitate both group and individual practice, and provide corrective feedback as students experience difficulty. Each lesson addresses conceptual understanding, skills practice, and problem solving. The program is founded on mathematics standard from the National Council of Teachers of Mathematics and uses CBM as [Page 272]one of the assessment tools. The program aligns well with RTI requirements, and can be used either in Tier 1 instruction or in Tiers 2 and 3 for specific targeted interventions. Also, intensive training and support are provided once a school undertakes implementation of this program.Study Island
Study Island, from Archipelago Learning, is a supplemental web-based instructional program offering instructional practice in almost every subject area, including math. This is tied specifically to each state's standards of learning (making this particularly useful for states that have not adopted the Common Core State Standards) as well as the Common Core. Study Island is also tied to each state's testing program (visit http://www.studyisland.com for more information). By basing this curriculum exclusively online, the Study Island developers have made these instructional and assessment materials available to licensed users, including teachers and students, both at school and at home, and this can be a significant advantage. Some students will undertake these instructional activities in the home environment, and software programs that are loaded exclusively on computers at school do not have that option.
Instructional activities are flexible, and teachers can either allow students to select topics on which to work, or they can assign specific topics that are based on the exact learning needs and styles of the individual student. The student will then be presented with various computer-based work or educational gaming activities.
Based on adaptive assessment technology within the program, the curriculum adjusts itself according to the learning curve of particular students, either moving students through the reading content faster or moving students into a slower track with more practice on various content items. Once students have mastered a particular lesson and assessment, they receive a blue ribbon and are able to move to the next lesson. However, should a student receive a low score, the program may prompt the student to continue working on the same skills until he or she develops proficiency. Thus, students receive instructional feedback each time they answer a question. This can be implemented as either a stand-alone instructional program or as a supplemental program in conjunction with other instruction. Thus, this program can also be used as the basis for RTI procedures either as Tier 1 instruction or as a Tier 2 or Tier 3 intervention in mathematics across grade levels. The stand-alone nature makes this curriculum intervention and assessment program ideal for RTI implementation in elementary, middle, or even secondary schools. Like [Page 273]more modern computerized programs, reports can be generated either for individual students or the entire class.
Study Island has only limited anecdotal research support, and that research was prepared by an independent firm under contract with the company. The supporting research can be found on the Study Island website (http://www.studyisland.com). The Study Island website does present a variety of reports suggesting how Study Island can be implemented in the context of RTI, which will help schools considering RTI implementation in the future. A number of schools around the nation are using Study Island, with some success, in the RTI context.Transmath
TransMath (http://www.voyagerlearning.com/cs/Satellite/transmath) is a higher-level, hard-copy supplementary mathematics curriculum developed by John Woodward and Mary Stroh that focuses on moving students from elementary mathematics skill levels up to algebra readiness. This curriculum is intended for students functioning at or below the 40th percentile in Grades 5 through 9, but the content covers a wider range of skills, reaching from number sense to algebraic expression. This curriculum covers fewer overall topics than most core mathematics curricula but covers those topics in much more conceptual depth, making it consistent in intent with the Common Core Standards in Mathematics. This program progresses in three levels that focus on specific mathematics areas: number sense, rational numbers, and algebraic expressions.
Three placement assessments (one for each of the levels mentioned) come with the curriculum, and each instructional unit within each level also includes two performance assessments. Together, these assessments allow for frequent progress monitoring, making this a useful curriculum in the RTI context. There is limited research support for this curriculum. This curriculum has been implemented by many elementary, middle, and high schools in their RTI efforts, and given the transitional nature of this curriculum, a focus that is virtually unique in the mathematics area, this curriculum is likely to be implemented in many more schools.The Successmaker Math Curriculum
SuccessMaker Math is one component of the broader SuccessMaker curriculum from Pearson Learning. It is an instructional software core curriculum that is available for school or school district purchase. This curriculum [Page 274]provides individualized instruction for elementary and middle schools students in a variety of areas including mathematics (http://www.pearsonschool.com/index.cfm?locator=PSZk99). SuccessMaker is currently being used in many RTI intervention programs in math. Students take an initial assessment when they begin, and those data are used to place each student in specific levels in mathematics. As students complete the lessons, the levels and questions get increasingly complex to move students toward mastery. In addition, SuccessMaker Math generates a variety of progress-monitoring reports for teachers that make it possible to review student growth individually, in subgroups, or for the entire class. This is excellent data for RTI implementation, and the teachers can closely monitor students' growth. This program can be used with the entire class, or students needing a Tier 2 or Tier 3 intervention are able to work with SuccessMaker on an individual basis, at their own pace, with a customized program.
References[Page 275]1997). Using classwide peer tutoring to teach beginning algebra problem-solving skills in heterogeneous classrooms. Remedial and Special Education, 18(6), 367–379. http://dx.doi.org/10.1177/074193259701800606(1999). Using modeling, manipulatives, and mnemonics with eighth-grade math students. Teaching Exceptional Children, 32(2), 74–81.(2008). Mathematics dynamic assessment: Informal assessment that responds to the needs of struggling learners in mathematics. Teaching Exceptional Children, 40(3), 6–17., , , , , & (2003). New classroom rules to promote preservice elementary teachers' mathematics learning education. ChulaVista, 123(3), 609–615.(Annenberg Learner. (2013). Inductive and deductive reasoning. Retrieved from http://www.learner.org/courses/teachingmath/grades6_8/session_04/section_03_b.html2011). Games and simulations help children access science. Education Week, 30(27), 12.(2012, July 24). Many students now learning while having fun with video games. StarNews Online. Retrieved from http://www.starnewsonline.com/article/20120724/ARTICLES/120729863/-1/news300?p=3$tc=pg(2002). A synthesis of empirical research on teaching mathematics to low-achieving students. Elementary School Journal, 103, 51–73. http://dx.doi.org/10.1086/499715, , & (2002). Teaching reading in mathematics and science. Educational Leadership, 60(3), 24–28., , & (2003). Learning-disabled students make sense of mathematics. Teaching Children Mathematics, 9(5), 269–274.(2009). Validity and problem-based learning research: A review of instruments used to assess intended learning outcomes. Interdisciplinary Journal of Problem-based Learning, 3(1), 59–89. http://dx.doi.org/10.7771/1541-5015.1059, , & (2009). Differentiating math instruction ((2nd ed.). Thousand Oaks, CA: Corwin2012a). Differentiated instruction for students with learning disabilities: New best practices for general and special educators (([Page 276]3rd ed.). Thousand Oaks, CA: Corwin2012b). Project-based learning: Differentiating instruction for the 21st century. Thousand Oaks, CA: Corwin.(2011). RTI in math. Bloomington, IN: Solution Tree Press., & (2007). Response to intervention: A practical guide for teachers. Thousand Oaks, CA: Corwin., & (2011a). The teaching revolution: RTI, technology, and differentiation transform teaching for the 21st century. Thousand Oaks, CA: Corwin., & (2011b). RTI and differentiated reading in the K–8 classroom. Bloomington, IN: Solution Tree Press., & (2012). The teaching revolution: RTI, technology, and differentiation transform teaching for the 21st century. Thousand Oaks, CA: Corwin., & (2013). Cool tech tools for lower tech teachers. Thousand Oaks, CA: Corwin., & (2012). Why flipped classrooms are here to stay. Education Week. Retrieved from http://www.edweek.org/tm/articles/2012/06/12/fp_bergmann_sams.html?tkn=WPCC1Rxu4%2FbCFsj3iEU3%2Bqk97aMS3xc0jkgq@cmp=cip-sb-ascd, & (2009). Implementation of responsiveness to intervention: A snapshot of progress. Journal of Learning Disabilities, 42(1), 85–95. http://dx.doi.org/10.1177/0022219408326214, , , & (2007). Reinventing project-based learning: Your field guide to real-world projects in the digital age. Washington, DC: International Society for Technology in Education., & (1993). A comparison of two approaches for teaching complex authentic mathematics problems to adolescents in remedial math classes. Exceptional Children, 59, 545–556., & (2001). Anchoring adolescents' understanding of math concepts in rich problem-solving environments. Remedial and Special Education, 22(5), 299–314. http://dx.doi.org/10.1177/074193250102200505, , , & (2002). Weighing the benefits of anchored math instruction for students with disabilities in general education classes. Journal of Special Education, 35(4), 186–200. http://dx.doi.org/10.1177/002246690203500401, , , & (2007). Integrating reform-oriented math instruction in special education settings. Learning Disabilities Research and Practice, 22(2), 96–109. http://dx.doi.org/10.1111/j.1540-5826.2007.00234.x, , , , & (1999, May). In search of … brain-based education. Phi Delta Kappan, 80, 645–657.(2006, Summer). Points of view: On the implications of neuroscience research for science teaching and learning: Are there any?CBE-Life Sciences Education, 5, 104–110. http://dx.doi.org/10.1187/cbe.06-03-0153(2010). Twitter. Teacher experience exchange. Retrieved from http://h30411.www.hp.com/discussions/68996?mcid=Twitter(2012, October 28). Wheaton H.S. to model project-based learning for Montgomery County schools. The Washington Post. Retrieved from http://www.washingtonpost.com/local/education/wheaton-high-to-model-project-based-learning-for-montgomery-county-schools/2012/10/28/b945602a-1a05-11e2-bd10-5ff056538b7c_story.html([Page 277]1996). Cognitively guided instruction: A knowledge base for reform in primary mathematics instruction. Elementary School Journal, 97(1), 3–20. http://dx.doi.org/10.1086/461846, , & (2002). Hands-on math projects (Vol. 2). Retrieved from http://www.edvantia.org/products/index.cfm?&t=products&c=math, , , , & (2005). Differentiated assessment strategies: One tool doesn't fit all. Thousand Oaks, CA: Corwin., & (1999). Math in the early grades: Laying a foundation for later learning. Association of School Curriculum Development. Available online at http://www.ascd.org/readingroom/cupcake/1999/1sum.html(2007). Attitude adjustments. Educational Leadership, 65(3), 72–77., , , & (2012). Project-based learning math projects. Retrieved from http://www.ehow.com/list_6498504_project-based-learning-math-projects.html#ixzz2BMhw6UHG(2011). From desktop to desk: A compelling way to teach math—“flipping” the classroom. Retrieved from http://www.boston.com/bostonglobe/editorial_opinion/oped/articles/2011/09/18/flipping_for_math/(1994). Assessment of cognitive processes: The PASS theory of intelligence. New York, NY: Allyn & Bacon., , & (2010). The calculating brain. In D. A.Sousa (ed.), Mind, brain, & education. Bloomington, IN: Solution Tree Press.(2003). Development in curriculum-based measurement. Journal of Special Education, 37(3), 184–192. http://dx.doi.org/10.1177/00224669030370030801(2010). The mathematical brain. In D. A.Sousa (ed.), Mind, brain, & education. Bloomington, IN: Solution Tree Press.(2012). Enhancing core mathematics instruction for students at risk for mathematics disabilities. Teaching Exceptional Children, 44(4), 48–57., , , , , , … (2007). The brain that changes itself. New York, NY: Penguin Books.(2013, February 28). Grant helps Idaho schools plug into online classes. Associated Press. Retrieved from http://www.kboi2.com/news/local/Grant-helps-Idaho-schools-plug-into-online-classes-194088691.html(2012). 8 crucial resources for flipped classrooms. Retrieved from http://edudemic.com/2012/03/8-crucial-resources-for-flipped-classrooms/(eSchool News. (2011, August 29). Press Release: Detroit schools choose movie maker to fuel creativity and boost test scores. eSchool News. Retrieved from http://www.eschoolnews.com/2011/08/29/detroit-schools-choose-movie-maker-to-fuel-creativity-and-boost-test-scores/eSchool News. (2012a). Georgia district implements virtual world technology: Forsyth County Schools will use ed tech to engage students with immersive experiences. Retrieved from http://www.eschoolnews.com/2012/03/28/Georgia-district-implements-virtual-world-technology/[Page 278]eSchool News. (2012b). Researchers debate gaming's effects on the brain: Scientists caution that more research is needed to prove benefits of video games in education. Retrieved from http://www.eschoolnews.com/2012/01/11/researchers-debate-gamings-effects-on-the-brain/2007). Mathematics accommodations for all students. Intervention in School and Clinic, 42(4), 190–203. http://dx.doi.org/10.1177/10534512070420040201(2011). Using Twitter in high school classrooms. Retrieved from http://teacherleaders.typepad.com/the_tempered_radical/2011/10/using-twitter-with-teens-.html(2010). Teaching the iGeneration: 5 easy ways to introduce essential skills with Web 2.0 tools. Bloomington, IN: Solution Tree Press., & (2008). Algebra progress monitoring and interventions for students with learning disabilities. Learning Disability Quarterly, 31(2), 65–78.(Frontline. (2010, February 8). Digital nation. A broadcast on Public Television. Also available online at http://www.pbs.org/wgbh/pages/frontline2005). Responsiveness to intervention: A blueprint for practitioners, policymakers, and parents. Teaching Exceptional Children, 18(1), 57–61., & (2007). Mathematics screening and progress monitoring in first grade: Implications for responsiveness to intervention. Exceptional Children, 73(3), 311–330. http://dx.doi.org/10.1177/001440290707300303, , , , , & (2008). Intensive interventions for students with mathematics disabilities: Seven principles of effective practice. Learning Disability Quarterly, 31(2), 79–92., , , , , & (1997). Children's conceptual structures for multidigit numbers and methods of multidigit addition and subtraction. Journal of Research in Mathematics Education, 28(2), 130–163. http://dx.doi.org/10.2307/749759, & (2001). Preparing students with disabilities in algebra. Teaching Exceptional Children, 34(1), 8–15., & (2007). Teaching students with LD to use diagrams to solve mathematics word problems. Journal of Learning Disabilities, 41(6), 341–353.(2012, November 20). A new kind of problem: The Common Core Math Standards. The Atlantic. Retrieved from http://www.theatlantic.com/national/archive/2012/11/a-new-kind-of-problem-the-common-core-mathe-standards/265444/(1983). Frames of mind. New York, NY: Basic Books.(1993). Multiple intelligences: The theory in practice. New York, NY: Basic Books.(2006). Multiple intelligences: New horizons. New York, NY: Basic Books.(2002, October). Improving the teaching of math from textbook concepts to real-world application. Paper presented at the annual meeting of the Council for Learning Disabilities, Denver, CO., & ([Page 279]1999). Number sense: Rethinking arithmetic instruction for students with learning disabilities. Journal of Special Education, 44, 18–28. http://dx.doi.org/10.1177/002246699903300102, & (2002, October). Instructional approaches for teaching mathematics to students with learning disabilities: Findings from a synthesis of experimental research. Paper presented at the annual meeting of the Council for Learning Disabilities, Denver, CO., , , & (1989). Strategy instruction in mathematics. Learning Disability Quarterly, 12, 43–55. http://dx.doi.org/10.2307/1510251(2006, September). The socially intelligent leader. Educational Leadership, 64, 76–81.(2012). My view: Flipped classrooms give every student a chance to succeed. Retrieved from http://schoolsofthought.blogs.cnn.com/2012/01/18/my-view-flipped-classrooms-give-every-student-a-chance-to-succeed/?htp=hp_bn1(1989). Longitudinal effects of classwide peer tutoring. Journal of Educational Psychology, 81, 371–383. http://dx.doi.org/10.1037/0022-06184.108.40.2061, , & (2003a). Laying the foundations for computational fluency in early childhood. Teaching Children Mathematics, February 2003, 306–309.(2003b). Number Worlds: A research-based mathematics program for young children. In D. H.Clements & A.DiBiase (Eds.), Engaging young children in mathematics: Findings of the 2000 national conference on standards for preschool and kindergarten mathematics education (pp. 325–342). Hillsdale, NJ: Erlbaum Associates, Inc.(2004a). Building number sense with number worlds. Early Childhood Research Quarterly, 19(1), 173–180. http://dx.doi.org/10.1016/j.ecresq.2004.01.012(2004b). Teaching number sense. Educational Leadership, 61(6), 39–42.(2005). Teaching mathematics in the primary grades: Fostering the development of whole number sense. In J.Bransford & S.Donovan (Eds.), How students learn: History, mathematics and science in the classroom (pp. 250–302). Washington, DC: National Academies Press (http://www.nap.edu).(2003). Laying the foundations for computational fluency in early childhood. Teaching Children Mathematics, 81, 371–383., , & (1999). Mathematics reform and learning differences. Learning Disability Quarterly, 22(1), 43–58. http://dx.doi.org/10.2307/1511151(2004). Promote number sense. Intervention in School and Clinic, 40(1), 55–58. http://dx.doi.org/10.1177/10534512040400010501(2002). Effective strategies for teaching mathematics. In E. J.Kame'enui, D. W.Carnine, R. C.Dixon, D. C.Simmons, & M. D.Coyne (Eds.), Effective teaching strategies that accommodate diverse learners. Upper Saddle River, NJ: Merrill/Prentice Hall., , , & (1995). Teaching initial multiplication skills to students with disabilities in general education classrooms. Learning Disabilities Research and Practice, 10(3), 180–195., , & (1995). Multiple intelligences and underachievement: Lessons from individuals with learning disabilities. Journal of Learning Disabilities, 28(7), 439–448. http://dx.doi.org/10.1177/002221949502800707, & ([Page 280]2013, January 7). Education and video games are no longer enemies: Educators say some games develop skills. Charlotte Observer. Retrieved from http://www.charlotteobserver.com/2013/01/07/3768358/education-and-video-games-are.html(2012). The fate of the Common Core: The view from 2022. Education Week. Retrieved from http://blogs.edweek.org/edweek/rick_hess_straight_up/2012/03/the_fate_of_the_common_core_the_view_from_2022.html?utm_source-twitterfeedutm_medium-twitter&utm_campaign=Walt+Gardner+Reality+Check(1987). Process mnemonics: Principles, prospects, and problems. In M. A.McDaniel & M.Pressley (Eds.), Imagery and related mnemonic processes: Theories, individual differences and applications (pp. 407–427). New York, NY: Springer.(2012). The teacher report: 6 ways teachers are using video games in the classroom. Retrieved from http://www.weareteachers.com/community/weareteachers-blog/blog-wat/2012/11/06/the-teacher-report-6-ways-teachers-are-using-video-games-in-the-classroom(International Society for Technology in Education (ISTE). (2010). Topic: Student learning. Retrieved from http://caret.iste.org/index.cfm?fuseaction=evidence&answerID=12&words=Attention2002). Crossing content: A strategy for students with learning disabilities. Intervention in School and Clinic, 37(5), 279–283. http://dx.doi.org/10.1177/105345120203700503(2002). Teaching students math problem solving through graphic representations. Teaching Exceptional Children, 91, 345–356.(1999). Teaching middle school student with learning disabilities to solve word problems using a schema-based approach. Remedial and Special Education, 20(1), 50–64. http://dx.doi.org/10.1177/074193259902000108, , & (2001). CBAs that work: Assessing students' math content-reading levels. Teaching Exception Children, 34(1), 24–29.(1997). Mathematics instruction for secondary students with learning disabilities. Journal of Learning Disabilities, 30(2), 151–163. http://dx.doi.org/10.1177/002221949703000203, , & (2007). The need for number sense. Educational Leadership, 65(2), 63–68.(2007). Predicting first-grade math achievement from developmental number sense trajectories. Learning Disabilities Research and Practice, 22(1), 36–46. http://dx.doi.org/10.1111/j.1540-5826.2007.00229.x, , , & (1998). The effects of concrete to semi-concrete to abstract instruction in acquisition and retention of fraction concepts and skills. Learning Disabilities: A Multidisciplinary Journal, 9(3), 115–122., , & (2001). Differential application of cue card strategy for solving fraction problems: Exploring instructional utility of the cognitive assessment system. Child Study Journal, 31(2), 123–136., & (2000). Weaving mathematical instructional strategies into inclusive settings. Intervention in School and Clinic, 35(4), 206–215. http://dx.doi.org/10.1177/105345120003500402, & ([Page 281]2002). Instructional strategies and educational outcomes for students with developmental disabilities in inclusive “multiple intelligences” and typical inclusive classrooms. Research and Practice for Persons with Severe Disabilities, 27(4), 227–238. http://dx.doi.org/10.2511/rpsd.27.4.227, , & (2009). Gender differences in the functional and structural neuroanatomy of mathematical cognition. NeuroImage, 47, 342–352. http://dx.doi.org/10.1016/j.neuroimage.2009.04.042, & (2011). 5 best practices for educators on Facebook. Retrieved from http://mashable.com/2011/12/05/educators-on-facebook/(2006). Teaching to the minds of boys. Educational Leadership, 64(1), 56–61., & (2011). Multidimensional assessment: Guiding response to intervention in mathematics. Teaching Exceptional Children, 44(2), 48–57., , & (2005). Improving performance in high school algebra: What students with learning disabilities are saying. Learning Disability Quarterly, 28(3), 191–204. http://dx.doi.org/10.2307/1593658, , & (1993). Use of constant time delay in teaching multiplication facts to students with learning disabilities. Journal of Learning Disabilities, 26(8), 533–544, 567. http://dx.doi.org/10.1177/002221949302600807, & (2006). Using peer-assisted learning strategies to increase response to intervention in inclusive middle math settings. Teaching Exceptional Children, 38(5), 6–13., & (2007). The effects of peer-meditated instruction in mathematics for students with learning problems: A research synthesis. Learning Disabilities Research and Practice, 22(1), 1–12. http://dx.doi.org/10.1111/j.1540-5826.2007.00226.x, , & (2010). 7 essentials for project-based learning. Educational Leadership, 68(1), 34–37., & (2009). PBL starter kit: To-the-point advice, tools, and tips for your first project in middle or high school. San Rafael, CA: Unicorn Printing Specialists., , & (2010). Integrating interactive online content at an early college high school: An exploration of Moodle, Ning, and Twitter. Meridian Middle School Computer Technologies Journal, 12(1). Retrieved from http://www.ncsu.edu/meridian/winter2009/, & (1996). Adapting mathematics instruction in the general education classroom for students with mathematics disabilities. LD Forum, 21(2), 19–23. (ERIC Document Reproduction Service No. EJ529409).(2012). How well are American students learning? Retrieved from http://www.brookings.edu/~/media/Newsletters/0216_brown_education_loveless.PDF(2008). Computational skills, working memory, and conceptual knowledge in older children with mathematical learning disabilities. Journal of Learning Disabilities, 41(1), 5–28. http://dx.doi.org/10.1177/0022219407311003, & (2013, February 25). Revamping the “core” of education: New Common Core Standards will focus on critical thinking over memorization. Retrieved from http://www.utsandiego.com/news/2013/feb/25/revamping-the-core-of-education/([Page 282]2012). Using the concrete-representational-abstract sequence with integrated strategy instruction to teach subtraction with regrouping to students with learning disabilities. Learning Disabilities Research and Practice, 27(4), 152–166. http://dx.doi.org/10.1111/j.1540-5826.2012.00363.x, , & (1991). The incorporation of process mnemonic instruction in teaching computational skills: A case report on a mathematics learning disabled individual. Focus on Learning Problems in Mathematics, 13(4), 21–34.(2000). The use of process mnemonics in teaching students with mathematics learning disabilities. Learning Disability Quarterly, 23(2), 137–156. http://dx.doi.org/10.2307/1511142, , & (1996). The effects of using manipulatives in teaching math problem solving to students with learning disabilities. Learning Disabilities Research and Practice, 11(1), 58–65., & (2011). Can an online game crack the code to language learning? Retrieved from http://mindshift.kqed.org/2011/11/can-an-online-game-crack-the-code-to-language-learning/(2012, December 20). Maine schools experimenting with web-based math homework. Bangor Daily News. Retrieved from http://bangordailynews.com/2012/12/20/education/maine-schools-experimenting-with-web-based-math-homework/(2009). Technical features of curriculum-based measures for beginning writers. Journal of Learning Disabilities, 42(1), 41–60. http://dx.doi.org/10.1177/0022219408326212, , & (2007). The effectiveness of problem based instruction: A comprehensive study of instructional methods and student characteristics. Interdisciplinary Journal of Problem-Based Learning1(2), 49–69., , & (2001). Cortical plasticity contributing to childhood development. In J. L.McClelland & R. S.Siegler (Eds.), Mechanisms of cognitive development: Behavioral and neural perspectives. Mahwah, NJ: Lawrence Erlbaum Associates.(1999). Some neurological principles relevant to the origins of—and the cortical plasticity-based remediation of—developmental language impairments. In J.Grafman & Y.Christen (Eds.), Neuronal plasticity: Building a bridge from the laboratory to the clinic. Berlin, Germany: Springer-Verlag., , , , & (2011a). Game-based learning units for the everyday teacher. Retrieved from http://www.edutopia.org/blog/video-game-model-unit-andrew-miller(2011b). Get your game on: How to build curricula units using the video game model. Retrieved from http://www.edutopia.org/blog/gamification-game-based-learning-unit-andrew-miller(2012). A new community and resources for games for learning. Retrieved from http://www.edutopia.org/blog/games-for-learning-community-resources-andrew-miller/([Page 283]2007). Helping students with disabilities understand what mathematics means. Teaching Exceptional Children, 39(1), 28–35., & (1997). Student perception, mathematical problem solving, and learning disabilities. Remedial and Special Education, 18(1), 46–53. http://dx.doi.org/10.1177/074193259701800108(1999). Classwide peer tutoring: Teaching students with mild mental retardation in inclusive classrooms. Exceptional Children, 65(4), 524–536., , , , , et al. (1997). Mathematics instruction and PASS cognitive processes: An intervention study. Journal of Learning Disabilities, 30(5), 513–520. http://dx.doi.org/10.1177/002221949703000507, & (2000). Effectiveness of a cognitive strategy intervention in improving arithmetic computation based on the PASS theory. Journal of Learning Disabilities, 33(6), 591–597. http://dx.doi.org/10.1177/002221940003300607, & (National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school mathematics. Available at http://standards.nctm.orgNational Mathematics Advisory Panel (NMAP). (2008). Foundations for success: The final report of the National Mathematics Advisory Panel:Washington, DC: U.S. Department of Education.2011). Digital portfolios and curriculum maps: Linking teacher and student work. In H. H.Jacobs (ed.), Curriculum 21: Essential education for a changing world. Alexandria, VA: Association for Supervision and Curriculum Development.(2010). Beyond differentiated instruction. Thousand Oaks, CA: Corwin.(2007). Private-speech and strategy use patters: Bidirectional comparisons of children with and without mathematical difficulties in a developmental perspective. Journal of Learning Disabilities, 40(1), 2–14. http://dx.doi.org/10.1177/00222194070400010101, & (2011, December 22). 5 tips for teachers to navigate Facebook's features and risks. Retrieved from http://www.schoolbook.org/2011/12/22/5-tips-for-teachers-to-navigate-facebooks-features-and-risks(2010). Blogs, wikis, podcasts, and other powerful tools for educators. Thousand Oaks, CA: Corwin.(2012). Preparing students to learn without us. Educational Leadership. Retrieved from http://www.ascd.org/publications/educational-leadership/feb12/vol69/num05/Preparing-Students-to-Learn-Without-Us.aspx(2011). Personal learning networks: Using the power of connections to transform education. Bloomington, IN: Solution Tree Press., & (2012; January 20). Blogs vs. term papers. The New York Times. Retrieved from http://www.nytimes.com/2012/01/22/education/edlife/musclinig-in-on-the-term-paper-tradition.html?_r=1(2012). Universities, districts to partner on Common-Core Secondary Math. Education Week. Retrieved from http://blogs.edweek.org/edweek/teacherbeat/2012/05/_there_has_been_quite.html([Page 284]2008). Teaching beyond the test: Differentiated project-based learning in a standards-based age. Minneapolis, MN: Free Spirit., & (1990). Using constant time delay to teach word definitions. Journal of Special Education, 24, 306–317. http://dx.doi.org/10.1177/002246699002400305, , & (2006). The cognitive correlates of computational estimation skill among third grade students. Learning Disabilities Research and Practice, 21(4), 233–243. http://dx.doi.org/10.1111/j.1540-5826.2006.00220.x, & (2005). Math games for adolescents. Teaching Exceptional Children, 37(3), 25–31., , & (2012). Special educators borrow from brain studies. Education Week, 31(17), 10.(2011). Assessment and the Common Core State Standards: Let's stay on top of it Retrieved from http://www.nctm.org/about/content.aspx?id=30169(2000). The effects of physical materials on kindergartners' learning of number concepts. Cognition & Instruction, 18(3), 32–64., , & (2011). High school teachers make gaming academic. Retrieved from http://www.usnews.com/education/high-schools/articles/2011/11/01/high-school-teachers-make-gaming-academic(2010). The eight Cs of engagement: How learning styles and instructional design increase student commitment to learning. In R.Marzano (ed.), On excellence in teaching. Bloomington, IN: Solution Tree Press., & (2000). So each may learn: Integrating learning styles and multiple intelligences. Alexandria, VA: Association for Supervision and Curriculum Development., , & (2001). How the special needs brain learns. Thousand Oaks, CA: Corwin.(2006). How the special needs brain learns ((3rd ed.). Thousand Oaks, CA: Corwin2008). How the brain learns mathematics. Thousand Oaks, CA: Corwin.(Sousa, D. A. (Ed.). (2010). Mind, brain, & education. Bloomington, IN: Solution Tree Press.2011). Differentiation and the brain: How neuroscience supports the learner-friendly classroom. Bloomington, IN: Solution Tree Press., & (2011). Schools “flip” for lesson model promoted by Khan Academy. Education Today, 31(5), 1–14. http://dx.doi.org/10.1016/j.nedt.2010.11.003(1996). Effects of a copy, cover, and compare procedure on multiplication facts mastery with a third grade girl with learning disabilities in a home setting. Education and Treatment of Children, 19, 425–434., , & (2012a). A first-hand look inside a flipped classroom. eSchool News. Retrieved from http://www.eschoolnews.com/2012/02/09/a-first-hand-look-inside-a-flipped-classroom/([Page 285]2012b). Six ed-tech resources for ELL/ESL instruction. eSchool News. Retrieved from http://www.eschoolnews.com/2012/02/10/six-ed-tech-resources-for-ellesl-instruction/2/?(2003, December 2). Trying to figure out why math is so hard for some; theories abound: Genetics, gender, how it's taught. The Washington Post, p. A13.(1949). Children discover arithmetic. New York, NY: Harper.(1985). Beyond IQ: A triarchic theory of human intelligence. New York, NY: Cambridge University Press.(2006). Recognizing neglected strengths. Educational Leadership, 64(1), 30–35.(2005, December). From formative assessment to assessment for learning: A path to success in standards-based schools. Phi Delta Kappan, 87(4), 324–328. http://dx.doi.org/10.1177/003172170508700414(2012, February 8). Schools seeing improvement in math scores as students play video game. Las Vegas Sun. Retrieved from http://m.lasvegassun.com/news/2012/feb/08/school-district-seeing-improvement-math-scores-stu/(2005). Reading and language arts worksheets don't grow dendrites. Thousand Oaks, CA: Corwin.(1999). The differentiated classroom: Responding to the needs of all learners. Alexandria, VA: Association for Supervision and Curriculum Development.(2001). How to differentiate instruction in mixed-ability classrooms ((2nd ed.). Alexandria, VA: Association for Supervision and Curriculum Development2003). Differentiation in practice: A resource guide for differentiating curriculum: Grades K–5. Alexandria, VA: Association for Supervision and Curriculum Development.(2010). Differentiating instruction in response to academically diverse student populations. In R.Marzano (ed.), On excellence in teaching. Bloomington, IN: Solution Tree Press., (2008). The differentiated school: Making revolution changes in teaching and learning. Alexandria, VA: Association for Supervision and Curriculum Development., , & (2011, October 6). “Flipped” classrooms take advantage of technology. USA Today. Retrieved from http://usatoday30.usatoday.com/news/education/story/2011-10-06/flipped-classrooms-virtual-teaching/50681482/1(2012, May 2). Common Core Standards drive wedge in education circles. USA Today. Retrieved from http://www.usatoday.com/news/education/story/2012-04-28/common-core-education/54583192/1(2012). ALEC's Common Core vote now under public microscope. Education Week. Retrieved from http://blogs.edweek.org/edweek/state_edwatch/2012/05/alec_common_core_vote_now_under_public_microscope.html(2010). Responding to the research: Harvey Silver and Matthew Perini address learning styles. Education Update, 52(5). Retrieved from http://www.ascd.org/publications/newsletters/education-update/may10/vol52/num05/Responding-to-the-Research.aspx([Page 286]2011a). Khan Academy expands to art history, Sal Khan no longer its only faculty member. Retrieved from http://www.hackeducation.com/2011/10/19/khan-academy-expands-to-art-history-sal-khan-no-longer-its-only-faculty-member/(2011b). Why wikis still matter. Retrieved from http://www.edutopia.org/blog/wiki-classroom-audrey-watters(2011c). Distractions begone! Facebook as a study tool. Retrieved from http://mindshift.kqed.org/2011/09/distractions-set-aside-facebook-as-a-study-tool/(2009). Project based learning in mathematics. Retrieved from http://suite101.com/article/project-based-learning-in-mathematics-a142678(2012). Using wikis in math classes. Retrieved from http://suite101.com/article/using-wikis-in-math-classes-a67900(2002). Supporting first graders' development of number sense. Teaching Children Mathematics, 9(1), 26–33., , , , & (2008). Implementing CRA with secondary students with learning disabilities in math. Intervention in School and Clinic, 43(5), 270–276. http://dx.doi.org/10.1177/1053451208314734, , & (1988). Effective teaching: Principles and procedures of applied behavior analysis with exceptional students. Boston, MA: Allyn & Bacon., , & (1991). Use of constant time delay and attentional responses with adolescents. Exceptional Children, 57, 462–474., , , & (2001). Constructivism and the role of skills in mathematics instruction for academically at-risk secondary students. Special Services in the Schools, 17(1), 15–32. http://dx.doi.org/10.1300/J008v17n01_02(2006). Developing automaticity in multiplication facts: Integrating strategy instruction with timed practice drills. Learning Disability Quarterly29 (4), 269–290. http://dx.doi.org/10.2307/30035554(2002). Meeting the challenge of mathematics reform for students with LD. Journal of Special Education, 36(2), 89–102. http://dx.doi.org/10.1177/00224669020360020401, & (2012). The Common Core Math Standards: Are they a step forward or backward?Education Next, 12(3). Retrieved from http://educationnext.org/the-common-core-math-standards/, & (2012, December 14). How I use Twitter in my classroom. Retrieved from http://edudemic.com/2012/12/how-i-use-twitter-in-my-classroom/(