teaching mathematics meaningfully

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Teaching Mathematics Meaningfully

Solutions for Reaching Struggling Learners

Second Edition

by

David H. Allsopp, Ph.D.University of South Florida

Tampa

LouAnn H. Lovin, Ph.D.James Madison University

Harrisonburg, Virginia

and

Sarah van Ingen, Ph.D.University of South Florida

Tampa

FOR MORE, go to http://www.brookespublishing.com/Teaching-Mathematics-Meaningfully

Excerpted from Teaching Mathematics Meaningfully: Solutions for Reaching Struggling Learners, Second Edition by David H. Allsopp, Ph.D.,LouAnn H. Lovin, Ph.D., & Sarah van Ingen, Ph.D.

Paul H. Brookes Publishing Co.Post Office Box 10624Baltimore, Maryland 21285-0624USA

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Copyright © 2018 by Paul H. Brookes Publishing Co., Inc.All rights reserved.Previous edition copyright © 2007.

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Typeset by Absolute Service, Inc., Towson, Maryland.Manufactured in the United States of America bySheridan Books, Inc., Chelsea, Michigan.

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Purchasers of Teaching Mathematics Meaningfully: Solutions for Reaching Struggling Learners, Second Edition, are granted permission to download, photocopy, and print the forms and activities found in Appendices A–D for educational and professional purposes. This material may not be reproduced to generate revenue for any program or individual. Photocopies may only be made from an original book. Unauthorized use beyond this privilege may be prosecutable under federal law. You will see the copyright protection notice at the bottom of each photocopiable page.

Library of Congress Cataloging-in-Publication Data

Names: Allsopp, David H., author. | Lovin, LouAnn H., author. | van Ingen, Sarah, author.Title: Teaching mathematics meaningfully: solutions for reaching struggling learners / by David H.

Allsopp, Ph.D., University of South Florida, Tampa, LouAnn H. Lovin, Ph.D., James Madison University, Harrisonburg, Virginia, and, Sarah van Ingen, Ph.D., University of South Florida, Tampa.

Description: Second edition. | Baltimore: Paul H. Brookes Publishing Co., [2018] | Includes bibliographical references and index.

Identifiers: LCCN 2017027635 (print) | LCCN 2017033234 (ebook) | ISBN 9781598575590 (epub) | ISBN 9781598575637 (pdf) | ISBN 9781598575583 (pbk.)

Subjects: LCSH: Mathematics—Study and teaching (Elementary) | Mathematics—Study and teaching (Middle school) | Mathematics—Study and teaching (Secondary) | Attention-deficit-disordered youth—Education. | Learning disabled teenagers—Education.

Classification: LCC QA13 (ebook) | LCC QA13 .A44 2018 (print) | DDC 371.9/0447—dc23LC record available at https://lccn.loc.gov/2017027635

British Library Cataloguing in Publication data are available from the British Library.

2021 2020 2019 2018 2017

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iii

About the Activities and Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

About the Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .vii

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

1 Critical Components of Meaningful and Effective Mathematics Instruction for Students with Disabilities and Other Struggling Learners . . . . . . . . . . . . . . . . . . . . . . . . 1

I Identify and Understand the Mathematics

2 The Big Ideas in Mathematics and Why They Are Important . . . . . . . . . . . . 15

3 Children’s Mathematics: Learning Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . 41

II Learning the Needs of Your Students and the Importance of Continuous Assessment

4 Barriers to Mathematical Success for Students with Disabilities and Other Struggling Learners . . . . . . . . . . . . . . . . . . . . . . . 69

5 Math Assessment and Struggling Learners . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

III Plan and Implement Responsive Instruction

6 Making Flexible Instructional Decisions: A Continuum of Instructional Choices for Struggling Learners . . . . . . . . . 137

7 Essential Instructional Approaches for Struggling Learners in Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

8 Changing Expectations for Struggling Learners: Integrating the Essential Instructional Approaches with the NCTM Mathematics Teaching Practices . . . . . . . . . . . . . . . . . . . . . . 217

9 Mathematics MTSS/RTI and Research on Mathematics Instruction for Struggling Learners . . . . . . . . . . . . . . . . . . . . . 239

Contents



10 How to Intensify Assessment and Essential Instructional Approaches within MTSS/RTI . . . . . . . . . . . . . . . . . 247

11 Intensifying Math Instruction Across Tiers within MTSS: Evaluating System-Wide Use of MTSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

IV Bringing It All Together

12 The Teaching Mathematics Meaningfully Process . . . . . . . . . . . . . . . . . . . . . 281

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

Appendices

A Take Action Activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

B ARC Assessment Planning Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

C Peer-Tutoring Practice Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

D Using a Think-Aloud . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343

E Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375

iv Contents



vii

David H. Allsopp, Ph.D., Professor of Special Education, College of Educa-tion, University of South Florida, 4202 East Fowler Avenue, EDU 105, Tampa, Florida 32620

Dr. Allsopp is Assistant Dean for Education and Partnerships in addition to being the David C. Anchin Center Endowed Chair and Director of the David C. Anchin Center at the College of Education at the University of South Florida. He is also Professor in the Department of Teaching and Learning—Special Education Programs. Dr. Allsopp holds degrees from Furman University (B.A., Psychology) and the University of Florida (M.Ed., Learning Disabilities; Ph.D., Special Education). Dr. Allsopp teaches at both the undergraduate and doctoral levels, and his scholarship revolves around effective instructional practices, with an emphasis on mathematics, for students with high-incidence disabili-ties (e.g., specific learning disabilities, attention-deficit/hyperactivity disorder, social-emotional/behavior disorders) and other struggling learners who have not been identified with disabilities. Dr. Allsopp also engages in teacher edu-cation research related to how teacher educators can most effectively prepare teachers to address the needs of students with disabilities and other struggling learners. Dr. Allsopp began his career in education as a middle school teacher for students with learning disabilities and emotional/behavioral difficulties in Ocala, Florida. After completing his doctoral studies at the University of Florida, Dr. Allsopp served on the faculty at James Madison University for 6 years. He has been a member of the faculty at University of South Florida since 2001.

LouAnn H. Lovin, Ph.D., Professor of Mathematics Education, Department of Mathematics and Statistics, James Madison University, 800 South Main Street, MSC 1911, Harrisonburg, Virginia 22807

Dr. Lovin began her career teaching mathematics to middle and high school stu-dents before making the transition to Pre-K through Grade 8. For over 20 years, she has worked in elementary and middle school classrooms. Then and now, Dr. Lovin engages with teachers in professional development as they implement a student-centered approach to teaching mathematics. At the time of this publica-tion, she focused her research concerning teachers’ mathematical knowledge for teaching on the developmental nature of prospective teachers’ fraction knowledge. She has published articles in Teaching Children Mathematics, Mathematics Teaching in the Middle School, Teaching Exceptional Children, and the Journal of Mathematics Teacher Education. She coauthored the Teaching Student-Centered Mathematics Professional Development Series with John A. van de Walle, Karen Karp, and Jenny

About the Authors



viii About the Authors

Bay-Williams ( Pearson, 2013). Dr. Lovin is an active member of the National Coun-cil of Teachers of Mathematics, the Association of Mathematics Teacher Education, and the Virginia Council of Teachers of Mathematics.

Sarah van Ingen, Ph.D., Assistant Professor of Mathematics Education, Depart-ment of Childhood Education and Literacy, College of Education, University of South Florida, 4202 East Fowler Avenue, EDU 202, Tampa, Florida 33620

Dr. van Ingen codirects the innovative and nationally recognized Urban Teacher Residency Partnership Program. In this role, she partners with Hillsborough County Public Schools’ teachers and administrators to improve the learning of both elementary students and prospective elementary teachers. She also teaches courses in mathematics education and teacher preparation at the undergraduate, masters, and doctoral levels. Dr. van Ingen holds a bachelor’s degree from St. Olaf College, a master of arts in teaching from the University of Tampa, and a doctoral degree from the University of South Florida. She was elected into membership in Phi Beta Kappa and was the recipient of the prestigious STaR fellowship in mathematics education. She taught mathematics for many years in urban, inclu-sive middle school classrooms before her work at the university level.

Dr. van Ingen’s research agenda lies at the intersection of equitable math-ematics education and clinically rich teacher preparation. Her research interests include teachers’ use of research to inform practice, the use of mathematics con-sultations to meet the mathematics learning needs of students with exceptionali-ties, and the implementation of integrated STEM lessons in K–5 classrooms. She regularly publishes and presents her research to audiences who work in math-ematics education, special education, and teacher preparation. She is the principal investigator and coprincipal investigator for federally funded research and is active in leadership in her professional organizations.



345

Case Study

E

This appendix provides a case study in which two teachers, Ms. Thompson and Mr. Hart, apply the Teaching Mathematics Meaningfully Process with their strug-gling learners. The teachers implement each step of the process. The contents are organized as follows:

• Purpose

• MeetMs.Thompson,Mr.Hart,andTheirStudents

• IdentifyandUnderstandtheMathematics

• ContinuouslyAssessStudents

• DetermineStudents’Math-SpecificLearningNeeds

• DetermineStrugglingLearners’SpecificLearningNeeds

• PlanandImplementResponsiveInstruction

• TakeAction

PURPOSE

The purpose of this case study is to provide you with a way to visualize how two teachers, an elementary general education math teacher and a special edu-cationteacher,mightworkcollaborativelytoutilizetheTeachingMathematicsMeaningfullyProcess.Wefirstintroduceyoutotheteachersandtheirstudents.Then,wedescribehowthetwoteachersimplementeachofthefivecomponentsoftheprocess.Ourgoalistoprovideyouwithanappliedcontextformakingsenseofthisprocessandtoillustratethetypesofdecisionmakingthatwillhelpyoudesigninstructionandinterventionsthatareresponsivetoyourstudents’needs.

Throughout the case study, marginal icons are included to indicate activities anddecisionsthatillustratespecificEssentialInstructionalApproaches(EIAs),National Council of Teachers of Mathematics (NCTM) Effective Mathemat-icsTeachingPractices (MTPs), andanchors for intensifying instructionwithinmulti-tiered systems of supports/response to intervention (MTSS/RTI). Eachicon includesanumber indicatingwhichEIA,MTP,oranchor for intensifyinginstructionthatitdenotes.Akeytotheseiconsisprovidednext.Usetheiconsas you read to see how various elements of instruction discussed throughout the book are applied and integratedwithin the teachers’ classroomplanning andinstruction.



346 Appendix E

Key

Essential Instructional Approaches (EIAs)

1. Teach systematically.2. Develop and explicitly share learning intentions.3. Make instructional decisions that are student-centered and based on

meaningful data.4. Teach mathematical fluency.5. Teach the language of mathematics through vocabulary

development and discourse.6. Provide many response opportunities with feedback.7. Emphasize use of mathematical practices.8. Utilize visuals.9. Use different appropriate grouping structures.

10. Teach students to be strategic in their approach to mathematics.11. Situate mathematics within meaningful contexts that help students

to develop abstract reasoning.

NCTM (2014b) Effective Mathematics Teaching Practices (MTPs)

1. Establish mathematics goals to focus learning2. Implement tasks that promote reasoning and problem solving3. Use and connect mathematical representations4. Facilitate meaningful mathematical discourse5. Pose purposeful questions6. Build procedural fluency from conceptual understanding7. Support productive struggle in learning mathematics8. Elicit and use evidence of student thinking

Anchors for Intensifying Instruction within MTSS/RTI (IAs)

1. Purposeful Content Focus2. Formative Assessment—Identifying What Students Know,

Don’t Know, and Why3. Explicitness and Teacher Direction4. Teach Math Metacognition5. Opportunities to Respond6. Amount of Time7. Teacher–Student Ratio

MEET MS. THOMPSON, MR. HART, AND THEIR STUDENTS

Ms.Thompsonisanelementarygeneraleducationteacherwhoteachesfifthgrade.Shehasbeenteachingelementaryschoolstudentsfor8years;shespent5ofthoseyearsteachingfourthandfifthgrade.Shehas22students(10boysand12girls)inherclass.Fifteenarewhite/Caucasian,fiveareAfricanAmerican,oneisMexicanAmerican,andoneisKoreanAmerican.Ms.Thompson’sclassincludesfivestu-dents identified as having disabilities. Four are identified as having learningdisabilitiesandreceivespecialeducationservicesthroughtheIndividualswithDisabilitiesEducationImprovementAct(IDEA)of2004(PL108-446).Oneisiden-tifiedashavingattention-deficit/hyperactivitydisorder(ADHD)andissupportedthroughaSection504accommodationplan.Ingeneral, thestudentsnot identi-fiedwithdisabilitiesperformatgradelevelorabove.Threeadditionalstudents,



Case Study 347

whoarenotidentifiedashavingdisabilities,sometimesstrugglewithmathandreading.TwoofthesestudentsareEnglishlanguagelearners.

Mr.HartisaspecialeducationteacherwhoworkswithMs.Thompsoninaconsultation and facilitation role, helping her support the needs of her students withdisabilities,particularly in reading andmathematics.Mr.Hart co-teacheswith Ms. Thompson during core instruction and is responsible for providingintensiveinstructionalsupportforstudentswhohavethemostdifficultyinmeet-ing core standards.Mr.Hart also provides input to the teams for Grades 3–5regarding supplemental instructional support for students receiving exceptional studenteducation(ESE)services.

InMs.Thompson’sclass,themorningbeginswitha60-minuteblockofcoremathematicsinstruction,followedbya120-minutereadingorEnglishlanguageartsblock.Duringanadditional50-minuteblockafterlunch,allstudentsengagein some type of supplemental or intensive instruction or enrichment for reading, mathematics,orboth.Mathstandards in thisstatearecloselyalignedwiththeCommonCoreStateStandards(CCSS).

IDENTIFY AND UNDERSTAND THE MATHEMATICS

For the Identify and Understand the Mathematics component, you will read how Ms. Thompson and Mr. Hart prepared themselves to teach the mathematical content.Inessence,wepullbackthecurtaintoshowyouthebehind-the-scenespreparation that equips teachers with the mathematical understanding necessary for this process. Ms. Thompson and Mr. Hart go through three stages for this firstcomponent.First,theyidentifytherelevantmathematicsstandards.Second,theylookforandlearnfromanavailabletrajectorythatdescribeshowstudentsprogress through various stages of learning related to the standards. Third, they consider the role mathematical practices have in the learning process with respect totheidentifiedcontent.

Math Standard

Ms.ThompsonandMr.Hart’sfirsttaskistoidentifythecontentthattheywillbeteaching.Basedonthecurriculummapforfifthgradersintheirdistrict,theyareplanningtoteachaunitonmultiplyingmulti-digitwholenumbersusingthestan-dardalgorithm.TherelevantCCSSstandardforfifthgradeisasfollows(NationalGovernorsAssociation[NGA]CenterforBestPractices&CouncilofChiefStateSchoolOfficers[CCSSO],2010).

CCSS.MATH.CONTENT.5.NBT.B.5

Fluentlymultiplymulti-digitwholenumbersusingthestandardalgorithm.

In thinking about teaching this standard,Ms. Thompson knows that sheandMr.Hart need tounpack the included content. She also knows theyneedto consider how this content connects to what students have previously beenexposedtoinfifthgradeaswellasinearliergradesinordertoensurestudentshavetheprerequisiteknowledgetoengagesuccessfullywiththiscontent.Keep-inginmindwheretheirstudentswillbeheadedinfuturemathematicslessons,Ms. Thompson and Mr. Hart also look to related standards, both within the



348 Appendix E

gradelevelandbeyond,tohelpthembepurposefulinmakingdecisionsaboutinstructional tasks and about how to leverage students’ current mathematicalideas.

Ms.Thompsonthinksabouthowthesestandardsconnecttootherfifth-gradestandards,includingwhatstudentshavealreadybeenexposedtothisyearandwhattheywillbeexpectedtolearnlaterintheyear,andshesharesherthoughtswithMr.Hart.With respect toNumberandOperations inBaseTen, their stu-dentshaveworkedonthefollowingstandardtofurthertheirunderstandingoftheplacevaluesystem(NGACenterforBestPractices&CCSSO,2010):

CCSS.MATH.CONTENT.5.NBT.A.1

Recognizethatinamulti-digitnumber,adigitinoneplacerepresents10timesasmuchasitrepresentsintheplacetoitsrightand1/10ofwhatitrepresentsinthe place to its left.

Ms.Thompsonnotes that the remainingfifth-gradestandards forNumberand Operations in Base Ten are related to two other mathematical ideas: 1)developingdivisionstrategieswithwholenumbersand2)developingthefouroperations(addition,subtraction,multiplication,anddivision)withdecimalstohundredths using place-value ideas, properties of operations, and relationships betweenthevariousoperations.

Ms. Thompson andMr.Hart look atwhat their studentswere exposed toin fourth grade related tomultiplication ofmulti-digit numbers. In particular,they consider the following fourth-gradeCCSS standard (NGACenter forBestPractices&CCSSO,2010):

CCSS.MATH.CONTENT.4.NBT.B.5

Multiplyawholenumberofuptofourdigitsbyaone-digitwholenumber,andmultiplytwotwo-digitnumbers,usingstrategiesbasedonplacevalueandthepropertiesofoperations.Illustrateandexplainthecalculationbyusingequa-tions, rectangular arrays, and/or area models.

Ms.Thompsonknowsthestandardalgorithmformulti-digitmultiplicationisoneofthemostdifficultalgorithms;studentsareveryerrorpronewhenusingthisalgorithm,especiallywhentheyhavenothadampleopportunitiestoworkwithmultiplicationstrategiesbasedonplace-valueconceptsandrepresentationssuchas area models. Both Ms. Thompson and Mr. Hart are aware of the importance ofdeveloping students’ procedural knowledge from conceptualunderstanding(NCTM,2014b).Thisleadsthemtorecognizethatbeforetheyworktohelpstu-dents develop the standard multi-digit multiplication algorithm, they will need to revisit this related fourth-grade standard to ensure students have developed a rigorous conceptual understanding of multiplication.

WiththeinsightsMs.ThompsonandMr.Harthavedevelopedbyconsideringnotonlythetargetstandardbutalsohowitrelatestootherstandardsfromthepreviousandcurrentgradelevels,theyareestablishingabettersenseabouthowto build on students’ prior knowledge to understand and apply the newfifth-grade multiplication standard. Mr. Hart appreciates the way Ms. Thompson goes deeperinthinkingaboutthecontentandrelatedlearningintentionstheyhavefortheirstudents,includingconnectingthemaththeirstudentswillbelearningto



Case Study 349

thatwhichtheyhavealreadyexperienced.ThishelpsMr.Hartthinkaboutwhatprerequisitecontent,bothatthegradelevelandbelowit,hewillneedtoempha-sizewhenheprovidesmoreintensiveinstructiontohisstudents.Heknowsthiscontent must connect to core math standards.

Related Learning Trajectory

Ms.Thompsonknowstheusefulnessoflearningtrajectoriesinprovidingaroadmap forhowstudent thinkingprogressesandhowtosequence learningexpe-riences tomaximize learning of amathematical concept or skill. She searchesonline formultiplication trajectoriesandfindsone thatdescribesaprogressionofstudents’multiplicativereasoningandstrategies.TableE.1includesthemoresophisticatedwaysofreasoningmultiplicativelyinthislearningtrajectory,morelikely tobeexhibitedbyolderelementary students. (For the full trajectory, seethe section on multiplicative reasoning in Chapter 3.) Ms. Thompson is alsoawareof several learning trajectories she canfindonlinewhen she isworkingon other mathematical concepts, such as https://www.turnonccmath.net andhttp://www.numeracycontinuum.com/continuum-chart.

Table E.1. The upper levels of multiplicative reasoning demonstrated by students

Level 3: Transitional multiplicative strategies (see Figure E.1a)

At this level, students demonstrate an increasingly robust capability of reasoning with multiples as their use of groups becomes more sophisticated. They no longer have to count each group by ones. Strategies such as using area models and open arrays are used to reason through multiplicative situations.

Level 3.3: Repeated abstract composite grouping

Students are aware that a number can be both composite and unitary at the same time, but at this level, students can only think of one of the numbers in a multiplication situation (one of the factors) in this way. For example, with 3 × 4, students are able to consider the 4 as both a composite unit and unitary at the same time, but they only think of the 3 as unitary—as a way to count the number of fours. They can see the 4 as consisting of 4 single units (unitary) but can also see (or make sense of ) the 4 as one “thing” (a composite unit). For 3 × 4, they would reason 4 + 4 + 4 (4 three times).

Level 4: Multiplicative strategies (see Figure E.1b)

At this point, students can reason about multiplication and division using the more sophisticated strategies that rely primarily on numerical representations such as partial products, the distributive property, and doubling and halving of quantities.

Level 4.1: Multiplication and division as operations

At this level, students can coordinate two composite units in the context of multiplication or division. For example, with a task such as six groups of four, the student is aware of both 6 and 4 as abstract composite units. The 6 can be used as a count of the groups of four but can also be considered its own composite unit. As a consequence, the commutative property of multiplication makes sense to the student. The student is able to immediately recall and quickly derive many of the basic facts for multiplication and division.



350 Appendix E

a

Later transitional strategies

Area model (less reliant on needing to see every square unit)

Open area model Open area model

4 × 6 = 24

6

4

7 × 12 = 70 + 14 = 84

10 2

7 70 14

23 × 45 = 800 + 120 + 100 + 15 = 1035

40 5

20 800 100

3 120 15

b

Multiplicative strategies

Known or derived facts

8 × 6 = 48

because 4 × 6 = 24 and we need to double that

Commutative property

6 × 8 = 8 × 6

Powers of 10

3 × 500 = 3 × 50 × 10 = 3 × 5 × 10 × 10 = 15 × 100 = 1500

Associative property

(4 × 6) × 5= 4 × (6 × 5)= 4 × 30= 120

Doubling and halving

half of 18

3 doubled

18 × 3 = 9 × 6 = 54

Distributive property

8 × 12 = 8(10 + 2) = 8(10) + 8(2) = 80 + 16 = 96

Partial products

23× 45

15100120

+ 8001035

(5 × 3)(5 × 20)(40 × 3)(40 × 20)

Standard algorithm

23× 45

115+ 920

1035

11

Figure E.1. Examples of strategies students use to engage in multiplicative reasoning: later transitional strategies (a) and multiplicative strategies (b).



Case Study 351

Fromstudyingthistrajectory,Ms.ThompsonandMr.Hartcometounder-stand that students who are ready to develop the standard algorithm for multi-digit multiplication should be using the later (more developed) multiplicativestrategiesseeninLevel4ofthelearningtrajectory,whicharebasedonthevariouspropertiesandonstrategiessuchasdoublingandhalving(seeFigureE.1B).Theyshouldalsoalreadybeproficientinusingthepartialproductsalgorithm.Studentswhoarenotthereyetmaybeusingthelatertransitionalstrategiesthatrelyonanareamodel,ortheymayjustbedevelopingproficiencywiththepartialproductsalgorithm.Somemayalsoexhibitevenlesssophisticatedreasoningthatreliesonskipcountingorinefficientadditivestrategies.(SeeChapter3forinformationper-tainingtotheselowerlevelsofreasoning.)Ms.ThompsonandMr.Hartdecidetousethislearningtrajectorytohelpthemidentifythelevelofsophisticationoftheirstudents’reasoning.Forstudentswhoseassessmentresultsindicatelesssophisti-catedreasoning,interventionswillneedtobeused.

Math Practices

Ms.ThompsonandMr.Hartrememberthattheirstudentsnotonlyneedtolearnthemathcontentwithinthetargetstandardsbutalsoneedtolearnhowtomean-ingfullyengagewith thiscontent indifferentways through theCommonCoreEight Standards forMathematical Practice (see Chapters 2 and 7). Textbox E.1shows these practices as a reference.

AsMs.Thompsonthinksabouteachofthemathpractices,sherealizesmanycouldbeappropriatelyutilizedinconjunctionwiththetargetstandard.Tomakethingsmoremanageable, she identifies two shewants to explicitly emphasizewithinherinstruction.(ThesepracticesareboldinTextboxE.1.)Ms.Thompsondetermines that one practice, Look for and make use of structure(NGACenterforBestPractices&CCSSO,2010),fitswellbecauseherstudentswillbelearningtomul-tiplymulti-digitwholenumbersbyrelatingtheirunderstandingofareamodels,thedistributiveproperty,andplacevaluetothestandardalgorithm.Shechoosesa second practice, Construct viable arguments and critique the reasoning of others(NGACenterforBestPractices&CCSSO,2010),becauseshewantstohelpherstudents

Textbox E.1. Common Core Eight Standards for Mathematical Practice

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

From Common Core State Standards Initiative. (2010). Common Core State Standards for mathematics. Retrieved from http://www.corestandards .org/assets/CCSSI_Math%20Standards.pdf; reprinted by permission. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.



352 Appendix E

becomemorecomfortableandproficientwithengaging indiscourse,appropri-atelylisteningtoandcritiquingtheirpeers’argumentsandreasoning.

Because the standard multiplication algorithm requires a level of precision in order to ensure the proper digits are multiplied together and recorded appropri-ately, Mr. Hart suggests that they also emphasize with their students the practice of Attend to precision (NGACenterforBestPractices&CCSSO,2010)whentheywritedown theprocedure.Mr.Hart knows thiswill be an importantpoint ofemphasisforseveralstudentswhotendtobemoreimpulsiveandstrugglewithself-regulation strategies related to organization and time management during independentmathwork.

CONTINUOUSLY ASSESS STUDENTS

For the Continuously Assess Studentscomponent,Ms.ThompsonandMr.Hartworkcollaborativelytoassessstudents’specificlearningneedsrelatedtotheidentifiedcontent standards and mathematical practices. First, Ms. Thompson reviews stu-dents’availablebenchmarkassessmentdata,usingthemtoprojectwhichareasrelatedtomultiplicationofwholenumbersmightrequirefurtherformativeinfor-mal assessment. Then, Ms. Thompson and Mr. Hart create several assessment taskstogetattheseareas.Last,theyreflectontheirstudentsandpossiblewaystoengagetheminrespondingtotheassessmenttaskssothattheycanbestdeter-minewhattheirstudentsknow,don’tknow,andwhy.

Review Available Benchmark Assessment Data

Ms.ThompsonandMr.Hart’sschooldistrictcollectsgrade-levelbenchmarkdatathreetimesduringtheschoolyear.Themultiple-choicebenchmarkassessmentsrelatedirectly to the state standardsand theend-of-yearhigh-stakes test.Eachmathbenchmarkassessment iscompletedonlineandevaluateswherestudentsareinrelationtothestandardstheyhavebeenexposedtowhenthebenchmarkassessmentisadministered(earlySeptember,earlyDecember,andmid-February).Eachtypicallytakesapproximately45–60minutestocomplete.SchoolpersonnelusethefirstbenchmarkassessmentinearlySeptembertomakeinitialdecisionsaboutsupplementalandintensivetieredinstruction.

Theschoolalsoutilizesacommercialonlinecurriculum-basedmeasurement(CBM)progressmonitoringtoolthattargetsparticulargrade-levelmathconceptsand skills. These measures are administered more often than the benchmarkassessments—every4weeks toall students.They targetamore specific subsetof grade-level,CCSSdomain–specificmath concepts and skills (determinedbygrade-levelteams).EachCBMassessmentincludesapproximately20–25multiple-choiceitemsandtypicallytakesstudents30minutestocomplete.Studentsreceiv-ing supplemental or intensive math instruction in addition to core instruction are administeredshorter,morefocused,andmorefrequentCBMprogressmonitor-ingprobesasneededduringtheirsupplementalandintensiveinstructionaltime.

GiventhatitislateOctober,Ms.ThompsonhasdatafromthebeginningyearbenchmarkassessmentandtwoCBMassessmentsforallstudentsinherclass.Theschool’sfirstbenchmarkassessmentfocusedprimarilyonessentialfourth-gradeconceptsandskillsthatareprerequisitesforsuccessinfifthgrade.Ms.Thompson



Case Study 353

reviewsbenchmarkresultsrelatedtoprerequisitesforthetargetmathcontent.Forher students currently receiving supplemental and intensive math support, she also has access to the progress monitoring data generated more frequently.

Basedonthesedata,Ms.Thompsoncreatesasimplechart(seeFigureE.2)tohelphervisualizewhatherstudentsappearbothtoknowandnotknowwithrespecttokeyprerequisiteconceptsandskillsrelatedtowholenumbermultiplication.

Assessment Tasks

Ms.ThompsonandMr.HartutilizetheinformationfromMs.Thompson’sbench-markandCBMsummarytabletothinkaboutwhichkindsofformativeassess-menttocreate.Theyaimtogetamorein-depthunderstandingoftheirstudents’thinkingand level ofunderstandingwith respect toprerequisite concepts andskills that are foundational to multi-digit multiplication. Based on the bench-markassessmentdata,Ms.ThompsonandMr.Hartdetermine that the forma-tiveassessmentshouldfocusonthreeprimaryareas:flexibleuseofmultiplicativestrategies suchasdoubling, appropriateuseof thepropertiesofmultiplication(commutative,associative,distributive),andaccurateuseofthepartialproductsalgorithm(numericaluseofpartialproducts).Theyagreeitwouldbemosteffi-cienttodevelopashortassessmentprobeaddressingtheseconceptsandskills,tobetakenbyallstudents.

Most Some Few

Appears to know or understand

• Hasaconceptualunderstandingofmultiplicationasequalgroups

• Canuserepeatedadditiontorepresentmultiplication

• Usesknownmultiplicationfactstoderiveunknownfacts

• Canuseanopenareamodeltorepresentdouble-digitbydouble-digitmultiplicationanddecomposesthetensandonesinthemodel

• Canexplainandappropriatelyutilizethecommutative,associative,anddistributiveproperties

• Usespartialproducts(withoutavisualmodel)tosolvedouble-digitmultiplicationproblems

• Usesthepowersof10whenappropriatetosolvedouble-digitmultiplicationproblems

• Flexiblyusesstrategiessuchasdoublingandhalvingtosolvedouble-digitmultiplicationproblems

• Usespartialproducts(withavisualmodel)tosolvedouble-digitmultiplicationproblems

• Usesanareamodelthatshowsallthesquareunitstorepresentdouble-digitbydouble-digitmultiplicationanddecomposesthetensandonesinthemodel

Figure E.2. Ms. Thompson’s summary of what her 22 students know, based on benchmark and curriculum-based measure-ment data through late October and organized by most students (80% +), some students (10%–15%), and few students (5% or less).



354 Appendix E

Textbox E.2. Formative assessment created by Ms. Thompson and Mr. Hart to appraise what students know and don’t know about prerequisite multiplication concepts and skills

1. Use drawings or manipulatives to demonstrate why 3 × 4 = 4 × 3.

2. Darran thinks 5 × (2 × 6) is not the same as (5 × 2) × 6. Please explain why you agree or disagree with Darran.

3. If you did not know how to multiply 5 × 14, which set of facts would help you find the product? Please explain why you selected a particular answer:

5 × 1 + 5 × 4

5 × 10 + 5 × 4

5 × 1 × 4

5 × 10 × 4

4. Xavier thinks that the product of 18 × 5 is the same as the product of 9 × 10. Do you agree or disagree with him? Explain why you agree or disagree.

5. Xiao and Maria both used partial products to solve 34 × 8. Look at their solutions. Explain why you think each solution is correct or incorrect:

Xiao: Maria:

34× 8

32+ 240

272

34× 8

32+ 24

56

6. Use the partial products algorithm to solve 23 × 16.

7. Jose’s father sells hotdogs at soccer games. There are 12 hotdogs in a pack, and Jose’s father goes through exactly 15 packs of hot dogs during one game. How many hot dogs did Jose’s father sell during that game? Please explain how you solved this prob-lem. (For students who finish early, ask them to solve using a different multiplication strategy.)

8. Niki collects stamps. She wants to buy an album to hold her stamps. One album holds 12 stamps on a page and contains 22 pages. Another album holds 15 stamps on a page and contains 18 pages. Niki wants to buy the album that holds more stamps. Which one should she buy? Please explain the reasoning behind your answer.

Thetwoteacherswanttobesurethattherearemultipletasksforeachareaof focus so that there are enough items to ensure they get an accurate appraisal ofwhatstudentsknowanddon’tknow.(SeeChapter5formoreaboutdevelop-inginformalformativeassessments.)Furthermore,theywanttobesurethetasksassessstudentengagementintheirtargetedmathematicalpractices:1)Construct viable arguments and critique the reasoning of others, 2) Look for and make use of structure,and3)Attend to precision(NGACenterforBestPractices&CCSSO,2010).Theydecide to includesome itemsthatcontainwordproblems(contextualizedandappliedproblems)andsomeitemsthatdonot(seeTextboxE.2).Tothisend,theycreatesixnoncontextualizedtasks:tworequiringknowledgeofmultiplica-tionproperties,twoaskingstudentstouseflexiblemultiplicationstrategies,and



Case Study 355

two requiring use of the partial products algorithm. They also include two word problems that require students to apply relevantmultiplication strategies. Thetwoteachersestimatethatitwilltakemoststudentsbetween20and30minutesto complete the assessment.

Student Responses

Asstudentsfinish,Ms.ThompsonandMr.Hartreviewstudents’responsestotheinformalassessmenttogetasenseofwhattheyknowandwhattheydon’tknow,andasenseofpossibleerrorpatternsthatmayindicatestudents’misconceptionsabout important underlyingmath concepts. Figure E.3 showsMs. Thompson’sandMr. Hart’s summary of their students’ responses, which theywill use todeterminetheirstudents’specificmathematicallearningneedsandthentoplanand implement responsive instruction.

DETERMINE STUDENTS’ MATH-SPECIFIC LEARNING NEEDS

For the Determine Students’ Math-Specific Learning Needs component, Ms. Thompson andMr.Hartusestudentresponsedatatodeterminetheirstudents’math-specificlearning needs. This involves three important activities, which all rely on the evalu-ationofstudentresponsestotheassessmenttasks.First,basedontheresponses,Ms. Thompson identifies students’ positions on the related learning trajectory(fromtheIdentify and Understand the Mathematicscomponent).Shethendeterminesstudents’misconceptions,aswellaswhatstudentsknow(knowledgestrengths)anddon’tknow(knowledgegaps)withrespecttotheidentifiedmathematics.Finally,Ms. Thompson and Mr. Hart determine together which mathematical ideas they needtotargetspecificallythatwillsupportstudentstofurtherdeveloptheirmath-ematicalknowledgeandskillsalongtheidentifiedlearningtrajectoryandtowardtheidentifiedstandard(s).

Identify Each Student’s Position on the Identified Learning Trajectory

Basedonstudents’responses,themajorityofstudentsappeartobefunctioningatLevel4ofthelearningtrajectory,meaningtheycandemonstratereasoningaboutmultiplication using the more sophisticated multiplicative reasoning strategies, which rely primarily on numerical representations that incorporate properties, suchastheassociativeanddistributiveproperties.Mostdonotrelyontheareamodel to use the partial products algorithm.

Somestudentshavedemonstratedalimitedunderstandingofkeyareas,suchas the prompteduse ofmultiplicative strategies such as doubling andhalvingand reliance on open area models to accurately complete the partial products algorithm. Based on this evidence gathered through assessment, Ms. Thompson judgesthesestudentstobefunctioningatLevel3ofthetrajectory.

Twostudents’assessmentperformanceindicatestheyarelikelyfunctioningatalowerlevelofthelearningtrajectory,possiblyLevel2,becausetheyneedtorelyonusinganareamodelshowingallthesquareunitstomakesenseofthepar-tial products algorithm. They demonstrated a solid understanding of the associa-tiveandcommutativepropertiesbutnotthedistributiveproperty,sotheycouldbemakingthetransitionfromLevel2toLevel3.



356 Appendix E

Teacher: Ms.ThompsonClass/period: 1stperiod

Areas for the assessment

On target (Write names.)

Limited (Write names.)

Insufficient (Write names.)

Flexibleuseofmultiplicativestrategiessuchasdoubling(Level4)

Task4

Moststudents TommyS.

JeromeB.

FelishaT.

NOTES:Thestudentsareabletousedoublingorhalvingwhenpromptedbutataveryslowpace.

SteveA.

TamikaW.

NOTES:Thestudentsusecalculation,notdoublingorhalving,todetermineequivalence(e.g.,uses18×5and9×10,seestheyareboth90).Thestudentsareunabletousethestrategy,evenwhenprompted.

Appropriateuseofthepropertiesofmultiplication(commutative,associative,distributive)(Level4)

Tasks1,2,and3

Commutativeproperty:Allstudents

Associativeproperty:Allstudents

Distributiveproperty:Moststudents

Distributiveproperty:SteveA.

TamikaW.

NOTES:Thestudentsusemultiplicationinsteadofaddition(e.g.,5×14=5×10×4).

Accurateuseofthepartialproductsalgorithm(numericaluseofpartialproducts)(Level4)

Tasks5,6,7,and8

Moststudents TommyS.

JeromeB.

FelishaT.

NOTES:Thestudentsareabletorecognizewhenpartialproductsareusedcorrectly(Task5)butnotabletousethestrategyaccuratelywithoutrelyingonanopenareamodel(withandwithoutcontexts)(Tasks6,7,and8).

SteveA.

TamikaW.

NOTES:Thestudentsareunabletoaccuratelyidentifyorusethepartialproductsalgorithm,evenwhenusinganopenareamodel.WhenMr.HartsketchedanopenareamodelforTask5,thesestudentsseemedconfusedastowhatthedimensionsoftheareamodelrepresented.Theywantedtoputthosenumbersinsidetheareamodel,notalongtheedges.Theymayneedtorelyonanareamodelthatshowseverysquareunit.

Figure E.3. Summary of students’ responses to assessment tasks that provides a sense of what they know, what they don’t know, and possible error patterns that may indicate students’ misconceptions.



Case Study 357

Evaluate Prior Knowledge, Strengths and Gaps, and Misconceptions

Most students indicated a readiness for learning the standard algorithm for multi-digitmultiplicationbecausetheycouldreasonusingthepartialproductsalgorithm in a purely numerical representation and explain their reasoning. Theycouldalsodemonstrateaflexibleuseofmultiplicativestrategies,includingdoublingandhalving,aswellastheassociative,commutative,anddistributiveproperties of multiplication.

Threestudentscouldappropriatelyusethepropertiesofmultiplicationbutstillrelied on open area models to support their reasoning with the partial products algo-rithm.Theyalsostruggledwithusingandknowingwhentouseadoublingandhalvingstrategy.Itappearsthesestudentsrelyheavilyondecomposingmulti-digitnumbersbasedonplacevalue,sotheyarenotlookingforalternativenumberrela-tionshipstoexploit.Althoughthisstrategywillhelpenhancetheirnumberandcom-putationsense,itshouldnothindertheirprogresstowardbecomingfluentwiththepartial products algorithm and the standard algorithm for multi-digit multiplication.

Two students demonstrated appropriate use of the associative and commutative properties,buttheydemonstratedissueswiththedistributivepropertyaswellasthepartial products algorithm, even when supported with open area models. Based on theirconfusionaboutwhatthenumbersrepresentedwhenMr.Hartusedanopenareamodeltoillustratethepartialproducts,itappearstheymayhavesomeknowl-edgegapsintheconceptofarea.Thesetwostudentsalsostruggledwithknowingwhenandhowtousedoublingandhalvingstrategies.Forallfiveofthesestudents,Ms.ThompsonandMr.Hartnotedthattheywanttothinkabouthowtoaddressthedifferentknowledgeandskillgapsastheyplanandimplementinstruction.

Target Math Ideas for Instruction

For most of the students, Ms. Thompson and Mr. Hart will target the mathemati-cal ideas closelyassociatedwith the identifiedmath standard:Fluently multiply multi-digit whole numbers using the standard algorithm(NGACenterforBestPractices&CCSSO, 2010). Inparticular, theywant to reinforce the relationshipbetweenthe partial products algorithm, the area model, and the standard algorithm. Embeddedinthesethreeconstructsistheimportantideaofthedistributiveprop-erty. Two students inparticular, SteveA. andTamikaW., demonstrated insuf-ficientevidenceofappropriateuseofthedistributiveproperty,soMs.ThompsonandMr.Hartmakenotethattheyneedtoincludethispropertyasatargetmathideaforthesestudents.Giventhesignificanceofthispropertytomultiplication,theydecidetoalsoincludeitasatargetideaforwhole-classinstruction.Inaddi-tion,Ms.ThompsonandMr.HartnotethattheywillneedtohelpSteveA.andTamikaW.enhancetheirunderstandingofareabeforetheycanbeexpectedtoworkproductivelyonmultiplicationingeneral.

DETERMINE STRUGGLING LEARNERS’ SPECIFIC LEARNING NEEDS

For the Determine Struggling Learners’ Specific Learning Needs component, Ms.Thompson andMr.Hartwork to determine the kinds of barriers that arelikelyaffectingtheirstrugglinglearnersandmakinglearningmathematicsdif-ficult.Theyfirstidentifythemathematicsperformancetraitstheyhaveobserved



358 Appendix E

withtheirstrugglinglearners, thenfocusondeterminingthepossible learningcharacteristicsandcurriculumfactorbarriersthatmaybecontributingtotheseperformance traits.

Identify Observed Performance Traits

Using the form illustrated inFigureE.4,Ms.ThompsonandMr.Hartnote theperformance traits theyhaveobservedwithmostor some students in thefirstperiod class or with individual students. They note that most students demon-strateknowledgeandskillsforsomemathdomainsandnotothersorforcertainstandards within particular math domains and not others. Therefore, for that per-formancetrait, theycheckofftheboxlabeledMost.Theyalsonotethatcertaingroups of students have consistently demonstrated two other performance traits, “Thestudentisabletocomputeorengagesinproblemsolvingaccuratelybutataveryslowpace”and“Thestudentavoidsengagingincertainmathematicaltasks.”IntheSomecolumn,theywritethesestudents’namesintheboxesnexttothesetwo performance traits. Finally, Ms. Thompson and Mr. Hart also note individ-ualstudents(twoorfewer)whodemonstratedstillotherperformancetraits(e.g.,“Thestudentdemonstratesfaultymathematicalthinkingorineffectivestrategies

Teacher: Ms.ThompsonandMr.HartClass/period: 1stperiod

Mathematics performance traitsMost (✓)

Some (Write names.)

Individual (Write names.)

The student demonstrates knowledge and skill for some mathematical domains and not others, or for certain standards within a domain and not others.

✓

The student demonstrates faulty mathematical thinking or ineffective strategies when problem solving.

SteveA.TamikaW.

The student is able to compute or engages in problem solving accurately but at a very slow pace.

TommyS.JeromeB.FelishaT.

The student has difficulty with generalizing knowledge and skills to other mathematical concepts, skills, and contexts.

TommyS.JeromeB.

The student demonstrates mathematical abilities at one point in time but then is unable to demonstrate the same abilities later.

SteveA.JeromeB.

The student avoids engaging in certain mathematical tasks.

TommyS.JeromeB.SteveA.

TamikaW.

Figure E.4. One way to record students’ math performance by most, some, and individual.



Case Study 359

whenproblemsolving,”“Thestudenthasdifficultywithgeneralizingknowledgeand skills to other mathematical concepts, skills, and contexts,” “The studentdemonstratesmathematical abilities at one point in time but then is unable todemonstratethesameabilitieslater”).IntheIndividualcolumn,theywritethesestudents’namesintheboxesfortheselearningtraits.

Inreviewingtheirnotes,thetwoco-teachersdevelopapictureofthekindsofmathperformancetraitsdemonstratedbymostoftheirstudents,bysomestu-dents,andbyonlyoneortwostudents.Ms.ThompsonandMr.Hartcanalsoeas-ilyseewhichstudentsaredemonstratingmoremathperformancetraitdifficultiesthanothers.Thisinformationnowprovidesthemwithreferencepointstobeginthinkingaboutwhatpotentiallearningcharacteristicandcurriculumfactorbar-riers they will want to consider when planning and implementing instruction.

In considering potential learning characteristic barriers, Ms. ThompsonandMr.Hart think about their students as they review theirnotes onperfor-mance traits (seeFigureE.4).FourofMs.Thompson’s studentshave identifieddisabilities and thereforehave individualizededucationplans (IEPs): SteveA.,TamikaW., Tommy S., and Jerome B. In addition, one student identifiedwithADHD,FelishaT.,hasaSection504accommodationplan.EachofthesestudentsislistedineithertheSomeorIndividualcolumn(seeFigureE.4)asdemonstrat-ingmorethanoneperformancetrait.Ms.ThompsonandMr.Hartknowthat,forthesefivestudents,theywillneedtoconsiderallthetypicallearningcharacter-isticsofstrugglinglearnerswhenthinkingabouthowthesecharacteristicsmightbeassociatedwiththeirperformancetraits.

Ms.Thompsonfinds ithelpful tohaveMr.Hartasacollaboratorbecausehe helps her to better understand the disability-related needs of her studentswith identified disabilities: Steve A., Tamika W., Tommy S., and Jerome B.TamikaW.alsohasanidentifiedspeech-languageimpairment,relatedtodiffi-cultiesarticulatingparticularsoundswhenshespeaks.TommyS.andJeromeB.also are identified as having ADHD—Tommy S. with the primarily inatten-tive type (distractibility) and Jerome B. with the combined type (inattentionandimpulsivity/hyperactivity).FelishaT.,whodoesnothaveanIEPbuthasa504accommodationplan, is, likeTommyS., identifiedashavingtheprimarilyinattentivetypeofADHD.Ms.ThompsonandMr.Hartdecidetocreateatablethat shows the students with identified disabilities, their particular disabili-ties,andspecific informationaboutparticularcognitive,social-emotional,andbehavioralissues—asdocumentedinthestudents’IEPandcumulativefolders,andalsobasedon theconnectionsMr.Hartmadebasedonhisknowledgeofdisability-relatedlearningneeds.TableE.2showstheteachers’notes.

With this information at hand, Ms. Thompson and Mr. Hart can now start thinkingaboutwhat learningcharacteristicscouldbecontributingtotheirstu-dents’mathperformancetraits.Theygobacktotheir“most,some, individual”notes(seeFigureE.4)andconsidereachstudent,theperformancetrait,andtheinformationgatheredforeachstudentwithanidentifieddisability(seeTableE.2).For example, they note that Steve A. demonstrates three performance traits(inadditiontotheonemostofMs.Thompson’sstudentsdemonstrate).ForeachofSteveA.’sperformance traits, the twoteachersconsiderhiscognitive,social-emotional,andbehavioralneedsrelatedtohisdisabilityandwhichlearningchar-acteristics are likely contributing tohisperformance traitdifficulties.TableE.3showsanexampleoftheirthinking.



360 Appendix E

Table E.3. Example of Ms. Thompson’s thinking about Steve A., his performance traits, and potential learning characteristic barriers

Student Performance traitPotential learning

characteristic barriersPotential curriculum

factor barriers

SteveA. Thestudentdemonstratesfaultymathematicalthinkingorineffectivestrategieswhenproblemsolving.

MetacognitivethinkingdisabilitiesKnowledgeandskillgaps

Thestudentdemonstratesmathematicalabilitiesatonepointintimebutthenisunabletodemonstratethesameabilitieslater.

Memorydisabilities—memoryretrieval(andworkingmemory?)

Thestudentavoidsengagingincertainmathematicaltasks.

MathanxietyLearnedhelplessness

Table E.2. Ms. Thompson’s and Mr. Hart’s notes about their students with identified disabilities

Student Identified disabilitiesCognitive, social-emotional,

behavioral issues

SteveA. Learningdisabilities Cognitive—difficultyconnectingmorethanoneideatoanother,difficultyapplyinglearningstrategies,memoryretrievalimpairments

TamikaW. LearningdisabilitiesSpeech-languageimpairment—articulation

Cognitive—difficultyconnectingmorethanoneideatoanother,difficultyapplyinglearningstrategies,memoryretrievalimpairmentsSocial-emotional—feelsinferiortopeersbecauseofherspeecharticulationdifficulties

TommyS. LearningdisabilitiesAttention-deficit/hyperactivitydisorder(ADHD)–primarilyinattentivetype

Cognitive—difficultyconnectingmorethanoneideatoanother,difficultyapplyinglearningstrategies,auditoryprocessingdifficulties(e.g.,slowerprocessingratewithverbalinstructionsanddirections),distractibility

JeromeB. LearningdisabilitiesADHD–combinedtype(inattentionandhyperactivity/impulsivity)

Cognitive—difficultyconnectingmorethanoneideatoanother,difficultyapplyinglearningstrategies,memoryretrievalandworkingmemoryimpairments,sometimesrespondstoverbaldirectionsbeforeheunderstandswhatisbeingaskedofhimBehavioral—needstomovewhenengagedinseatwork

FelishaT. ADHD–primarilyinattentivetype Cognitive—distractibility



Case Study 361

Consider Possible Learning Characteristic Barriers

Forthefirstperformancetrait,Ms.ThompsonandMr.HartquicklyrealizethatSteve A.’s difficulties in making connections between certain math ideas andapplyingeffectivestrategiesaligncloselywithoneofthelearningcharacteristicbarriers,metacognitivethinkingdisabilities.TheyalsosuspectthatSteveA.hassomegapsinhismathematicalknowledgebase,confirmedbyhislowscoresonmathbenchmarktesting.Ms.Thompsonthinksthiscouldalsobeacontributingfactor.Withthesecondperformancetrait,Mr.HartfocusesonSteveA.’smemoryretrievaldifficulties,sohesuspectsthatmemorydisabilitieslikelycontributetohispatternofbeingable todemonstrateknowledgeofaconceptorskillatonepointintimebutnotatanotherpoint.Mr.Hartalsowonderswhetherworkingmemorycouldbeafactorinthis;duringinstruction,SteveA.appearstounder-standpartsoftheconceptbeingtaughtbutnotothers.Theteachersnotethiswithaquestionmark.Forthethirdperformancetrait,bothteachersrealizethat,giventhedifficultiesSteveA.has,heprobablyshutsdownwhenconfrontedwithmath-ematicstaskshedoesnotbelievehecancompletesuccessfully.SteveA.alsooftenraiseshishandforhelpwithmath,evenwhenhehastheknowledgeandskilltocompletethetask,sotheythinklearnedhelplessnessispotentiallyplayingaroleas well.

Ms. Thompson and Mr. Hart use the same process to identify the learning characteristicsthataremostlikelycontributingtothedifficultiesofTamikaW.,TommyS.,JeromeB.,andFelishaT.

Consider Possible Curriculum Factor Barriers

AsMs.ThompsonandMr.Hartcontinuetothinkabouteachoftheirstrugglinglearners, theyconsider thefivecurriculumfactorsandhowanyof thesemightbebarrierstotheirstudents’mathsuccessandmightcontributetothemathper-formancetraitstheydemonstrate.TableE.4showstheirthinkingforSteveA.inconnectiontotheirnotesaboutpotentialcurriculumbarriers.

AsMs.Thompson thinks about themath curriculumsheuses, she thinksabouthowcertaincharacteristicsthereofmightcontributetoherstudents’difficul-ties.Forthefirstperformancetrait,Ms.Thompsonreviewsafewlessonsfromtheteacher’seditionofthemathtextbook.Shenoticesthatalthougheachlessonhasasegmentrelatedtoconceptualunderstanding, littledirectconnectionismadebetweentheconceptandtheproceduresemphasizedduringtherestofthelesson.In someways, these twoaspectsof the lesson—conceptualunderstandingandreasoning,andproceduralunderstandingandproficiency—seemtobetreatedasseparate sections. Itmakes sense toher that thismight contribute toSteveA.’stendencytouseinefficientstrategieswhensolvingproblems;itmayalsoexplainwhyheissometimesoffbaseinconnectingwhatheisdoingtowhyheisdoingit.Therefore,Ms.Thompsonsuspectsthatthetextbook’slimitedemphasisoninte-grating conceptual understandingwithprocedural proficiency is a curriculumfactorbarrierforSteveA.

Inthinkingaboutthesecondperformancetrait,Ms.ThompsonconsidersthememorydifficultiesSteveA.canexperience.ThismakesherwonderwhetherthecurriculumallowsSteveA.tofullystorewhathelearnsaboutanewmathconceptorskillandhaveenoughopportunitiestoapplyorpracticeit.Thisinturnmakes



362 Appendix E

herconsiderwhethertheinstructionalpacingisappropriateforSteveA.Givenhismemorydisabilities,thepacingmightbetoorapidforSteveA.tofullylearnandbecomeproficientwithnewlyintroducedconcepts.Hemaybeabletodemon-stratewhatheunderstandsinthemoment,butwhenheisaskedtoapplyitlaterinthelessonoronanotherday,hecannoteffectivelyretrievethislearningfrommemorybecausehedidnothaveenoughopportunitiestoapplyhislearninginordertomakeretrievalautomatic.Itisalsopossiblethatthelesson’sinstructionalpacewasfasterthanSteveA.’sabilitytoprocesstheinformationefficiently,affect-inghisworkingmemory.

AsMs.ThompsonthinksmoredeeplyaboutSteveA.,hisperformancetraits,learning characteristic barriers, and potential curriculum factor barriers, shebeginstorealizetherearesomedisconnectsbetweentheinstructionemphasizedin themathtextbookandhis learningneeds.Ms.Thompsonhypothesizes thatthiscouldbeareasonforSteveA.’shesitationtoengageincertainmathematicsactivities:Hehasnotadequatelylearnedthem.So,Ms.Thompsonnotesutiliza-tionofeffectiveinstructionalpracticesforstrugglinglearnersasanotherimpor-tantpotentialcurriculumfactorbarrierforSteveA.

PLAN AND IMPLEMENT RESPONSIVE INSTRUCTION

Atthispoint,Ms.ThompsonandMr.Harthaveworkedfullythroughthefirstfour components of the Teaching Mathematics Meaningfully Process. They have integratedthetwoperspectivesillustratedinFigure12.1—thatofamathematics

Table E.4. Ms. Thompson’s and Mr. Hart’s thinking about Steve A., his performance traits, potential learning characteristic barriers, and potential curriculum factor barriers

Student Performance traitPotential learning

characteristic barrierPotential curriculum

factor barrier

SteveA. Thestudentdemonstratesfaultymathematicalthinkingorineffectivestrategieswhenproblemsolving.

MetacognitivethinkingdisabilitiesKnowledgeandskillgaps

Levelofemphasisplacedontheintegrationofconceptualunderstandingwithproceduralproficiency

Thestudentdemonstratesmathematicalabilitiesatonepointintimebutthenisunabletodemonstratethesameabilitieslater.

Memorydisabilities—memoryretrieval(andworkingmemory?)

Instructionalpacing

Thestudentavoidsengagingincertainmathematicaltasks.

MathanxietyLearnedhelplessness

Lackofutilizingeffectivemathematicspracticesforstrugglinglearnersacrossinstructionaltiersinmulti-tieredsystemsofsupports(MTSS)



Case Study 363

teacher and that of a special education teacher—as theyworked through eachaspect of the two related components, Determine Students’ Math-Specific Learning Needs and Determine Struggling Learners’ Specific Learning Needs.

For the Plan and Implement Responsive Instruction component, you will read howMs.ThompsonandMr.Harttakewhattheylearnedsofaranduseittoplanandimplementtheirinstructioninwaysthatrespondtotheirstrugglinglearners’needs.ThisfinalcomponentoftheTeachingMathematicsMeaningfullyProcessincludes the following steps:

• Developingamathinstructionalhypothesis(seeChapter5)toguideplanning

• Planning for and implementing effective instructional practices (seeChapters7–8)

• Reflectingandrevising instructionbasedonstudentperformancedata (seeChapter5)

AsyoureadabouthowMs.ThompsonandMr.Hartengageinthisprocess,you will learn how they identify instructional hypotheses to address the needs of thestudentswhoseformativeassessmentresultsindicatedknowledgeandskillgaps.Also,youwilllearnhowthetwoteachersplanandimplementinstructionthatisorganizedatthreelevelsbasedonhowMTSS/RTIisimplementedintheirschool:

• Lessintensive(whole-class,differentiatedcoreinstructionatTier1)

• Moreintensive(small-group,supplementaryinstructionatTier2)

• Evenmoreintensive(intensiveinstructionatTier3forafewstudents)

Asnotedpreviously,theschoolhasadaily50-minuteperioddevotedtopro-viding students with additional instructional time in reading and mathematics, usedeitherformoreintensiveinstructionorforextensionandenrichment.Dur-ing this time, general education teachers, such as Ms. Thompson, provide supple-mentaryinstruction(Tier2)forstudentswhoneedit;specialeducationteachers,such as Mr. Hart, and a math coach provide even more intensive instruction (Tier3).Teacherseitheralternatedaysforreadingandmathematicsinstructionorsplitthe50-minuteperiodinhalftoaddresseachcontentarea.

With this information in mind, we focus on how Ms. Thompson and Mr. Hart plan and implement instruction for struggling learners at each level or tier. Please notetheiconsintherightmarginthathighlighthowtheseteachers’instructionincorporatescertainpracticesdiscussedthroughoutthebook:EIAs(Chapter7),anchorsofinstructionthatcanbeintensifiedwithinMTSS/RTI(Chapter10),andtheMTPs(Chapter8).

Instructional Hypothesis

Reflecting on students’ responses from the formative assessment tasks,Ms.ThompsonandMr.Hartdeterminethatmosthavetheprerequisiteknowl-edgeandskillstoachievetheoveralllearningintention,multiplicationofmulti-digitnumbersusingthestandardalgorithm.However,fivestudentsstrugglewithdifferentprerequisites.AlthoughMs.ThompsonandMr.Hartdonotbelieveaninstructional hypothesis is needed to guide instruction for most of their students,

3



364 Appendix E

they decide instructional hypotheses would be helpful for these five. For thethreestudents(TommyS., JeromeB.,andFelishaT.)whocannotaccuratelyusethe partial products algorithm without relying on an area model, Ms. Thompson and Mr. Hart develop the following instructional hypothesis to support planning and instruction:

Giventwomulti-digitnumberstomultiply:

Students are able to recognize situations that involve multiplication and accurately use the partial products algorithm when using an open area model.

Students are unable to use the partial products algorithm without relying on an area model

. . . because theyhavedifficultykeepingtrackof thenumericalpartialproductswithout the visual cues provided with the area model.

Forthetwostudents(SteveA.andTamikaW.)whoarestrugglingwiththedistributivepropertyandpartialproductsalgorithm,Ms.ThompsonandMr.Hartdevelop the following instructional hypothesis:

Giventwomulti-digitnumberstomultiply:

Students are able to use the associative and commutative properties for multiplication.

Students are unable tousethedistributivepropertyorpartialproductsalgorithmeven with an area model

. . . becausetheydonotunderstandtherelationshipbetweenthenumbersinthepartialproductandtheirdistributionwithinanareamodel.

Ms. Thompson and Mr. Hart use these two instructional hypotheses to guide theirinstructionalplanningandteachingastheydifferentiateandintensifytheirinstructionacrosstiersforthesefivestudents.

Planning and Implementation

AsMs.ThompsonandMr.Hartstarttoplantheirinstructionbasedonthetwoinstructionalhypotheses,theyconsiderhowtheintensificationofinstructionforthosewhoneeditalignswithhowtheirschoolemploysMTSS/RTI(seetheintro-ductoryparagraphformoreaboutthiscomponentoftheTeachingMathematicsMeaningfullyProcess).Ms.ThompsonandMr.Hartbegintoplanasequenceofteachingandlearningactivitiesthatwillbeusedseveraldays.Theykeepinmindthat, because the students have alreadyhad lots of experiences using base-tenmaterialsandareamodelstothinkaboutmultiplicationandhaveconnectedtheseideastopartialproducts,theprimarygoalisforstudentstodevelopthewrittenrecord for the standard algorithm for multi-digit multiplication. Because most of thesestudentsareproficientwiththepartialproductsalgorithm,Ms.ThompsonandMr.Hartagreethatinstructionrelatedtothetargetstandardshouldbeginby connecting the two algorithms, partial products and standard, using con-creteorrepresentational(i.e.,semi-concrete,suchasanareamodel)modelsfirst.Theydecide to focus thefirst few lessonsonusingareamodels inconjunctionwith thewritten record. Once students are able to demonstrate and articulate

6

8

1



Case Study 365

an understanding of what is happening with the models, they will move to the writtenalgorithmwithoutrelyingontheareamodel.Theydecidetostartwithawordproblem to reinforcehowmultiplyingmulti-digitnumbersapplies to thereal world.

After working with both the standards and the learning trajectory,Ms. Thompson and Mr. Hart articulate the following overall and long-term learn-ing intentions for theirstudentsas follows:studentswillbeable to (1)conceptu-allyunderstandthemathematicalideasrelatedtomultiplyingmulti-digitnumbersusing the standard algorithm, (2)multiplymulti-digitwholenumberswithpro-ficiency,and(3)makesenseofandsolvewordproblemsthat involvemulti-digitmultiplicationofwholenumbers.

Differentiated Whole-Class Core Instruction (Less Intensive, Tier 1)

Both Ms. Thompson and Mr. Hart agree that using a systematic instruction approach is important forall theirstudents. Inparticular, itwillallowthemtoevaluate student learning after each lesson and determine what to emphasize in the next. Based on the assessment data gathered, Ms. Thompson and Mr. Hart decidetouseparalleltasks,onewithsmallernumbersandonewithlargernum-bers,duringtheinitialwhole-classlesson.Parallel teaching, in which two teachers teachtwodifferentgroupsthesamecontent,atthesametime,indifferentiatedways, is a co-teaching model that supports differentiating instruction withinwhole-classorlarge-groupcontexts(Friend&Cook,1996).Doingthiswilllowerthe teacher–student ratio for those students who are struggling with the math-ematicscontent.Forthosestrugglingwiththedistributivepropertyandtheareamodel,thetaskwithasingle-digitnumbermultipliedbyadouble-digitnumberwillreducethenumberofpartialproductsonwhichtheymustfocus.Forstudentswithworkingmemorydifficulties,suchasSteveA.,thisstrategywilllessenthecognitive loadnecessary to complete the task—making itmore likely that theywillbeabletoprocessthenumbersaccuratelyandcompletethenecessarycogni-tiveactions.Thisstrategycouldalsohelptoalleviatestudents’potentialanxietyaboutattemptingsomethingnew.Theteachersdecideonthesewordproblemsastheirparalleltasks:

• Inthefrontsectionoftheschoolauditoriumare6rowswhere45studentscansit in each row. How many students can sit in the front of the auditorium?

• Inthefrontsectionoftheschoolauditoriumare23rowswhere45studentscan sit in each row. How many students can sit in the front of the auditorium?

Studentsareaskedtodrawthecorrespondingrectangulararea,markoffthearea thatalignswitheachof thepartialproducts,and thencompleteawrittenrecord of the multiplication, using the partial products on a recording sheet that hasbase-tencolumnsidentified(seeFigureE.5).Ms. Thompson and Mr. Hart pro-videbase-tengridpaper(seeFigureE.6)toSteveA.andTamikaW.Thispapernotonlyexplicitlyshowstheindividualsquares(asopposedtoanopen-areamodelwherethesesquaresareimplied),butalsoorganizesthesquaresintogroupsof10rowsand10columns.Thisstructureisintendedtoeliminatetheneedtocountalltheindividualsquaresandalsosupportsstudents’partitioningthenumbersbeingmultipliedintotheirrespectiveplacevalues.

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366 Appendix E

40 5

20 800 100

7 280 35

Th H T O

2 7

× 4 5

1 3 5

1 0 0

2 8 0

+ 8 0 0

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Figure E.5. Place value recording sheet and an open area model showing the partial products for 27 × 45.

Atthispointinthelesson,itistimetointroducethewrittenrecordforthestandardalgorithm.Ms.ThompsonandMr.Hart recognize thatbasedon stu-dents’ learning needs, some students will need less guidance and others willneedmore.Forthosestudentsneedinglessguidance,theirtaskistoconsiderthewrittenrecordtheyhavecreatedusingpartialproductsandanotheronegiventothem that uses the standard algorithm along with the corresponding area model. Together,theywillworktofigureoutthewrittenrecordofthestandardalgorithm(i.e.,whatthenumbersmeanandwheretheycamefrom,andwhynumbersarerecording inparticularplaces).Ms.ThompsonandMr.Hartprovidethesestu-dentswithasequenceofwrittenrecords,asseeninFigureE.7,sothestudentscanseehowandwhennumbersareintroducedintothewrittenrecord.

Ms.ThompsonandMr.Hartknowtheyneedtooffermoresupporttosomestudentsbystartingwithasimplerproblemandbeingmoreexplicit.For these students, theyuse thefirstwordproblem that results in6 × 45.After studentshave completed the partial products algorithm and corresponding area model for thecomputation(seeFigureE.8),Mr.Hartworkswiththesmallgroup,askingaseries of focused questionstohelpthemrelatethetwowrittenrecords,such as the following:

Where is the 30 in the partial products and in the area model? What numbers were used to get 30? Where and how is the 30 recorded in the standard algorithm? Why is the 3 recorded above the 40 in the 45?

Heknowsheneeds touse strategies such asvisual cuing tohelp studentsmake these connections (see Figure E.9). Mr. Hart highlights the three tens inthe partial products algorithm and uses an arrow to show the connection to the regrouped three tens in the standard algorithm. The use of visuals to help students makeconnectionsbetweenmathematicalideasisaneffectiveinstructionalpracticefor struggling learnersbecause it helps students focuson important features ofmathematicalideasandtasksdespiteattentiondifficultiesorcognitiveprocessingimpairments.Mr.Hartworkswiththissmallgroupofstudentsonadditionalmul-tiplicationofsingle-digitnumbersbydouble-digitnumbers,eventuallyleadingtomultiplicationproblemsinvolvingtwodouble-digitnumbers.

Graduallyscaffoldingcontentinordertohelpstudentssucceed,withlessdiffi-cultcontentexpectationsinitiallyandthenwithmoredifficultexpectationslateron,supportsstrugglinglearnerstobewillingtotakerisks.Thisreducesthelikelihood

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Case Study 367

Figure E.6. Base-ten grid paper. (Various open source sheets can be found via an Internet browser search.)

that they will engage in learned helplessness. This practice also supports the needs of studentswhoprocess informationmore slowlyso theycanbecomeproficientwithlessdemandingmathematicstasksbeforemovingtomoredemandingtasks.

Ms.ThompsonandMr.Hartarealsosensitivetotheideathatwithboththepartial products and the standard algorithm, it is imperative that students still



368 Appendix E

Th H T O Th H T O Th H T O

32 3

2 3

2 7 2 7 2 7

× 4 5 × 4 5 × 4 5

1 3 5 1 3 5 11 3 5

1 0 8 0 + 1 0 8 0

1 2 1 5

Step 1 Step 2 Step 3

Figure E.7. Place value recording sheet showing the steps to the standard algorithm for 23 × 45.

Figure E.8. Place value recording sheet and an area model with the square units showing the partial products for 6 × 45.

6 240

40 5

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Th H T O

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× 6

3 0

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2 7 0

bepreciseinthelanguageused.Forexample,whenstudentswhoarefindingtheproductof23×45multiplythe2and5,theyshouldsay“20times5”sothatitismoreapparenttothestudentwhyheorshewritesthe1inthehundredsplace.

Supplemental Small-Group Instruction (More Intensive, Tier 2)

The three students who currently receive supplemental mathematics instruction—TommyS.,JeromeB,andFelishaT.—demonstratedtheywerereadytomovetowardunderstanding and becoming proficient with the standard algorithm formulti-plication. For this reason, Ms. Thompson and Mr. Hart agree that, although they willlikelyneedsomeadditionalinstructiononconceptsandskillscoveredduringwhole-classcoreinstruction,theymostlywillbenefitfromadditionalresponseandpracticeopportunitiesinordertobecomeproficientwiththeseconceptsandskills.Ms.Thompsondecidestobegineachsupplementalinstructionsessionatasmall-grouptablebyengagingstudentsinapre-instructional“check”activity,inwhichshe presents several prompts related to these core concepts and skills and asksstudentstoquicklyrespondonindividualdry-eraseboards.Asstudentsrespond,Ms. Thompson notes any error patterns or apparentmisconceptions andwritestheminasmalljournalsheusestoinformallytrackstudents’progressduringsup-plementalinstruction.(Shefindsusingasupplementalinstructionjournalthiswayhelpshertoefficientlyplanfromdaytodayandpinpointwheretotargetinstruc-tion.)Basedonherobservationsduringthispre-instructionalcheck,Ms.Thompsonthen determines which content or mathematical practices her students need more supportinunderstanding.Shecommunicatestostudentsherlearningintentionsforthesessionandhowtheserelatetowhatsheobservedduringthepre-instructionalcheck.

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Case Study 369

Knowingherstudentsandconsideringthenatureofthemathematicscontentstandard, Ms. Thompson decides to emphasize the three additional instructional intensificationanchors,Explicitness and Teacher Direction, Teach Math Metacognition, and Opportunities to Respond.Shebelievesthatherstudentswillbenefitfrommod-eratelevelsofintensificationforthefirsttwooftheseanchorsandahighlevelofintensificationforOpportunities to Respond.(Thisisanexampleofdifferentiatingintensityamong the seven instructionalanchors; seeChapter10 fordiscussionaboutdifferentiatingtheintensitylevelsofthesewithinMTSS/RTI.)Shedecidesthisbecause sheknows that tobuild appropriate levelsofproficiency,her stu-dentswilllikelyneedsomemodelingorreteachingandgreateropportunitiestorespondandpractice.SheunderstandsthatTommyS.,JeromeB.,andFelishaT.needmoreinstructionaltimedevotedtorespondingandpracticethanpossibleduring whole-class core instruction.

For times when her students need additional modeling or reteaching of con-ceptsorskillstobuildinitialandadvancedunderstanding,Ms.Thompsonuti-lizesEIA#8:Utilize visuals.Nexttohersmall-grouptable,shehasasmallstorageboxthatcontainsmanipulatives,variousgraphicorganizerideas,individualdry-eraseboardsandmarkers,folderinstructionalboardgames,andexamplesofdif-ferentwaystodrawsolutionstodifferenttypesofequations.Dependingonwhichconcept,skill,orpracticestudentsneedmoresupportinunderstanding,sheusesthese materials to help students visualize its important features, repeating the visual cuing practices she and Mr. Hart used during whole-group instruction.

When students need support in recalling steps for completing the standard multiplicationalgorithmorsolvingmultiplicationwordproblems,Ms.Thompsonteaches theuse of explicit learning strategies (seeChapter 8 for examples) that supportstudents’memoryrecallandhelpthembuildmetacognitiveawareness.For example, she noted that when students were not provided the place value recording sheet (seeFiguresE.5–E.9), some forgotordidnot connect theplacevalueofdigitstheyregroupedusingthestandardalgorithm.So,shetaughtthesestudentstheFINDstrategy,whichhelpsthemidentifyandremembertheplacevalueofdigitsinmulti-digitnumbers(seeFigureE.10).TheFINDstrategyhelpsstudentsindependentlyrecreatetheplacevaluetemplatethatwasusedindiffer-entiated whole-class core instruction. This in turn allows them to respond inde-pendentlywhilereinforcingthinkingabouttheplacevalueofdigitswhenusingthestandardalgorithm.Overtime,Ms.Thompsonwillfadestudents’useoftheFINDstrategyasappropriate.

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Figure E.9. Place value recording sheet comparing the algorithms for partial products and the standard algorithm for 6 × 45.

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Partial Products

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370 Appendix E

Tohelpherstudentsbuildtheirlevelsofproficiency,Ms.Thompsonincorpo-ratesmultipleresponseopportunitiesduringboth instructionandpractice.Forexample, sheutilizes the individualwhiteboards to ensure all students in thegrouprespondtoherquestionsandprompts.Sheincorporatestheuseofinstruc-tionalboardgames(seeChapter7)forpractice.

Individualized Instruction (Even More Intensive, Tier 3)

Twostudents,SteveA.andTamikaW.,exhibitedanunderdevelopedunderstand-ingoftheconceptofareaanditsrelationshiptomultiplication.Bothhavebeenidentifiedasstudentswhoneedmore intensivemath instruction inadditiontocoreinstruction.AlthoughMs.ThompsonandMr.Harthaveattemptedtodiffer-entiatetheirpracticewithinwhole-classplanningandinstructiontobettersup-port struggling learners’needs, theyknowSteveA.andTamikaW.needevenmoreintensivesupportcomparedtothestudentsreceivingTier1instructionorsupplementalTier2support.

Based on the instructional hypothesis the two teachers developed from their informal assessment, Mr. Hart plans to organize his instruction according to three goals,whichhewillapplyineachdaily50-minuteintensivesessionwithSteveA.andTamikaW.

First,heknowstheyhavegapsintheirknowledgebaserelatedtothestandardtargeted in whole-class core instruction. They will need explicit systematic instruc-tioninrelatedfoundationalconceptsandskills:theareamodel,thedistributiveprop-erty,andthepartialproductsalgorithm.Second,heknowshisstudentswillneedmultipleresponseopportunitiesandpracticewiththeseconceptsandskillstobuildtheirproficiencyandfluency.Third,Mr.HartknowsthatSteveA.andTamikaW.willneedpre-instructionalsupportinordertobenefitfromwhole-classcoreinstruction.

Therefore, Mr. Hart organizes each intensive instructional session according to these three areas of focus as follows:

1. Duringthefirst20–25minutes,hefocusesonthefoundationalideas:area,distributive property, and the partial products algorithm. He decides tobeginwiththefoundationalconceptofareaandhowitrelatestomakingsense of multiplication, ideas with which these students are still struggling.

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Students use the FIND Strategy to monitor the value of digits as they complete the standard multiplication algorithm.

Find the columns (space between digits)

Insert the ts.

Name the place values.

Determine the value of each digit.

Th H T O

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4 2

× 2 9

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Figure E.10. Example of the FIND Strategy (Mercer & Mercer, 2005) being utilized to monitor the place value of digits when completing the standard multiplication algorithm.



Case Study 371

2. Duringthenext15–20minutes,heengageshisstudentsinpracticeopportu-nities related to one of these foundational concepts.

3. Duringthelast10minutesorso,hepreteachescontentheandMs.Thompsonwill cover during the next whole-class core instruction class period or peri-ods.HedoesthissoSteveA.andTamikaW.haveapreviewofwhattheywillbelearningthenextdayandhowitrelatestowhattheyarelearningduringTier3instruction,andtheycanbeginlearningthecontent.Mr.HartbelievesdoingthiswillbetterprepareSteveA.andTamikaW.forthenextday’scoreinstructionsothattheycanengageinthewhole-classlessonmoresuccessfully.

Area Model Instruction Mr. Hart decides to emphasize the same three instruc-tionalintensificationanchorsusedbyMs.ThompsonwithhersupplementalTier2group: Explicitness and Teacher Direction, Teach Math Metacognition, and Opportuni-ties to Respond. However, in contrast to Ms. Thompson, Mr. Hart greatly intensi-fiestheanchorsExplicitness and Teacher Direction and Teach Math Metacognition in addition to Opportunities to Respond(anotherexampleofdifferentiatingintensityamong the anchors).Hebeginswith single-digitmultiplication scenarios, suchas the following, and has the students create corresponding rectangular areas on grid paper to represent the scenarios.

Ms. Thompson wants to buy a rug for the classroom that is 5 feet by 6 feet. How much floor space (area) will the rug cover?

Mr.Hartbeginswithscenariosthatareeasytomodelconcretelyintheclass-room.Forthisscenario,studentscanutilizeatapemeasure,theclassroomfloortilesthateachmeasure1squarefoot,and1footby1footpapersquarestorepre-sent,thinkabout,andsolveareaproblems.Mr.Hartfirstmodelsthisusingthink-alouds thatshowhisthinkingabouthowmuchfloorspacetheruginthescenariowillcover.Then,heinvitesstudentstodothesameandtojustifywhytheareamodel they created is appropriate and how it accurately represents how much space the rug will cover.

Next,heposesdifferentareascenariosusingfeetandinchesandchallengesSteveA. and TamikaW. to determine the different areas using the tapemea-sureandsquarepaper“tiles.”Healsoasksthemtomarktheindividualfeetorincheson thepapersquaresand to justify their response.Aftereachresponse,the students then represent the same area using their grid paper, labeling thefeet or inches and identifying the total area represented on the grid paper inside. Mr.Hart checksboth students’ responseson thegridpaperandprovides spe-cificpositivereinforcementandcorrectivefeedback.Heaskseachstudenttoiden-tifytherelationshipsbetweentheareaonthegridandthescenario.Oncetheyareabletoidentifytheserelationships,heasksthemtowritethecorrespondingmultiplication equation.

Asthestudentsdemonstrateproficiencywithscenariosthatinvolvecontinu-ousquantities,Mr.Hartbeginstousescenariosthatinvolvediscretequantities(i.e.,onesstudentscount).Thisisdonetohelphisstudentsgeneralizemultiplica-tion to discrete quantities using an array structure. For example:

In a classroom, there are 4 rows of 5 desks. How many desks are in the classroom?

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372 Appendix E

Again, Mr. Hart models an example first, using think-alouds, with indexcards representing thedesks.Ashedidwith continuous quantities, he beginswith discrete-quantity scenarios that are easy to model concretely using an array. Forexample,SteveA.andTamikaW.canuseindexcardstocreatethe4×5array.Mr.Hartfirstmodelsthisusingthink-aloudsabouthowtoaligntheindexcardsin rows and columns.

TohelpSteveA.andTamikaW.trackthenumberofeach,hemarkscardswithredandbluehighlighterstoidentifythefourrowsandfivecolumns.Then,heasksthemtodothesameandtojustifywhytheirarraymodelisappropriateandhowitaccuratelyrepresentsthetotalnumberofdesksinthescenario.Next,heposesdifferentcombinationsofrowsandcolumns,challengesSteveA.andTamikaW.toworkinthesamewaytocreatetheappropriatearrayforeach,andpromptsthemtojustifytheirresponse.Aftereachresponse,thestudentsrepresentthesamearea,usingtheirgridpaperinmuchthesamewaytheydidearlierwhenworkingwithcontinuous quantities, including writing the multiplication equation underneath.

Mr.Hartchecksbothstudents’responsesonthegridpaperandprovidesspe-cificpositivereinforcementandcorrectivefeedback. Heaskseachstudenttoiden-tifytherelationshipsbetweenthearraymodelwiththeindexcards,theareaonthegrid,andtheequation(e.g.,the4intheequationisrepresentedbythefourrows,the5intheequationrepresentsthefivecolumns,the20intheequationrepresentsthetotalnumberofsquaresinthearea).HedoesthistomakesurethatSteveA.andTamikaW.aremakingtheconnectionbetweenareaandmultiplication.

Next,Mr.Hartmovestoproblemsthatinvolvethemultiplicationofasingle-digitnumberbyadouble-digitnumber.Headdresses these students’misunder-standingsaboutthedistributivepropertybyhelpingthempartitionintotensandonesthepartoftheareamodelcorrespondingtothedouble-digitnumber.Hefol-lowsaprocesssimilartothatusedforproblemsinvolvingsingle-digitnumbersonly.

AsMr.HartworkswithSteveA.andTamikaW.overtime,hetakesopportu-nities to explicitly connect what they are doing to the core content already covered duringwhole-classinstructionbecausehewantstocontinuouslyhelphisstudentsmaketheseconnections.Forexample,heshowsaproblemfromapriorwhole-classsessionasanexample(seeFigureE.8)anddemonstrateshowthesingle-digitbysingle-digitareamodelstheyhavebeenworkingwith(e.g.,4×5)relatetothepartialproductsareamodel(e.g.,6×45)bypointingouthowbothhaverowsandcolumns.Inotherwords,herelates4×5(fourrowsoffive)to6×45(sixrowsof45).

Practice Becausehehasonlytwostudentsandhewantstokeepthemmoti-vatedastheypractice,Mr.Hartdecidestoutilizebothinstructionalgamesandself-correctingmaterials(seeChapter7).Forexample,hethinkstheinstructionalgamePigwouldbegoodforpracticingsingle-digitmultiplicationofwholenum-bers(seeFigureE.11).

ForthePiggame,Mr.HartdecidestohaveSteveA.andTamikaW.rolltwodicetogeneratetwosingle-digitnumbers,representtheareaoftheresultingrect-angleusinggridpaper,labelitsrowsandcolumns,identifythearea(andproduct)bywriting thenumber inside the rectangle, andwrite the correspondingmul-tiplicationequationunderneath.Mr.Hartmonitorsandprovides feedbackandcoachingasneeded.Afterthesession,Mr.HartreviewsSteveA.’sandTamikaW.’sresponses to evaluate their progress and inform planning for the next session.

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Case Study 373

Pre-instruction for the Next Whole-Class Instruction Session During this por-tion of more intensive instructional time, Mr. Hart emphasizes the instructional intensificationanchorExplicitness and Teacher Direction, using an explicit instruc-tionprocess,LIPP,forconnectingwhatstudentswilllearn:

• Linkingthewhole-classlearningintentiontowhatSteveA.andTamikaW.arelearningaboutanareamodelintheirevenmoreintensive(Tier3)session

• Identifying the learning intention they will focus on during subsequentwhole-class instruction

• Providing a rationale for why the upcoming learning intention is important and relevant to their lives

• Previewing one or more foundational ideas related to it

Mr.HartlikesthemnemonicLIPPbecauseithelpshimrememberthefourareas to focus on for each pre-instruction session.

Reflect/Revise

Ms.ThompsonandMr.Hartmeet regularly tobothreflectonandrevise theirinstruction.Theydo thisbyevaluatingstudentperformancedataand throughtheirongoingobservations.Thisincludesstudents’levelsofengagementduringinstruction;at-the-momentdiagnosticinterviewsanderrorpatternanalysestheteacherscompletewithstudentsbasedontheirresponsesduringinstruction;andother informal formative assessments, such as weekly class mini-quizzes and

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Purpose: Provides students with computation practice

Materials: Grid paper and pencil; two dice, with a pig sticker on one face of one of the die. (If you do not have a pig sticker, purchase a pink dot sticker from an office supply store and draw a pig face on it.)

Directions: Students work in pairs and take turns. On each turn, a student rolls two dice at the same time. He or she uses the numbers rolled as dimensions of a rectangle. The student draws the rectangle on his or her grid paper. The student multiplies the two numbers and writes the product inside the area of the rectangle. The student keeps a running total of the areas he or she finds. When the pig is rolled, the student has to deduct 20 from his or her total area. Play continues until one student reaches a designated total area (e.g., 100).

Extension 1: Each student rolls three dice at a time to create a one-digit and a two-digit number. Use different colored dice to designate the one-digit number versus the two-digit number. For example, use one white die to generate the one-digit number and use two green dice for the two-digit number. Let the student determine how to use the digits from the dice to create a two-digit number (e.g., 32 or 23). The choice made will provide some insight into the student’s number sense and strategy awareness.

Extension 2: Each student rolls four dice at a time to create 2 two-digit numbers. Use two white dice for one two-digit number and two green dice for the other two-digit number. Again, let the student determine how to use the digits from the dice to create a two-digit number.

Figure E.11. Pig Math instructional game. (Source: Mercer & Mercer, 2005.)



374 Appendix E

school-wide continuous progress monitoring data from benchmark and CBMassessmentsasappropriate.(ThisisallpartoftheContinually Assess Students com-ponentoftheTeachingMathematicsMeaningfullyProcess.)

Ms.ThompsonandMr.Hartconcentrateonthreequestionsastheyreflectontheirinstructionacrosstiers:Whatisworking?Whatisnotworkingandwhy?Howcanwe improvewhatwearedoing?Because theycontinuallykeep thesethreequestionsinmindastheyteach,theyfindthattheydonotneedlotsoftimewhentheymeettoreflectandrevise;reflectionbecomesahabitofmind,sotheyalreadyhaveconcreteideasbeforetheymeet.Ms.ThompsonandMr.Hartagreeitiswonderfultobeabletocollaborateforseveralreasons:Theysharecommonper-spectives,buteachteacheralsohasanotherperspectivetopullfrom.Becausenei-thercanbepresentforinstructionacrossallthreeinstructionaltiers,collectivelytheyfeelthatcollaborationhelpsthemhaveamuchbetterhandleonallstudents’performance and related mathematics learning needs, particularly those who are strugglingwithmathematics.Theyalsofindthattheyaremuchbetterpreparedwhentheirgrade-levelteammeetstoreviewstudentperformancedatatomakeinstructionaldecisionsrelatedtoMTSS/RTIatthegradelevelandforindividualstudents.

TAKE ACTION

We designed this case study to illustrate how teachers can integrate the compo-nents of the Teaching Mathematics Meaningfully Process. Our illustration is simply onewaytheprocesscanbecarriedout; thepotentialvariationsandadaptationsarelimitlessanddependontheneedsandcharacteristicsofspecificstudentsandteachers.Throughoutthisbook,wechallengedyoutoputeachcomponentofthedecision-makingprocessintoaction.Ourfinalchallengeforyouistointegratethecomponentsbyputtingtheentireprocessintoactioninyourownclassroom.

Becauseofthecomplexityofthistask,youmayfinditusefultoworkwithapartnerandtalkthrough,orwritedown,howyouenvisioneachstepplayingoutinyourclassroom.Then,goforit!Itmayfeeltimeconsumingandlaborinten-sive,butweencourageyoutoseetheentireprocessthroughfrombeginningtoend.Paycloseattentiontohowyouandyourstudentsrespond.Weretheresultsuseful?Whatweretheeffectsonstudentlearningandengagement?Whichpartsof theprocessworkedwell andwhichneed tobe tweaked? If youwere fortu-nateenoughtohaveanotherteachercollaboratewithyouthroughouttheprocess,whatwereyourinteractionslike?Howcouldtheybestrengthenedinthefuture?Werethereanypointsoftensionormisunderstandingsthatcouldbeaddressed?

Finally,wewanttoconcludethisbookbyhonoringyouforthetimeyouhavetakentoimproveyourmathematicsinstruction.Strugglingstudentsandstudentswithspecialeducationneedshistoricallyhavenotreceivedequitablemathematicsinstruction.Theeffortsyouhavemadetoreadandapplytheresearch-supportedstrategies in thisbookare clear indicatorsofyour commitment tomeeting themathematicslearningneedsofallstudents.Thankyouforthiscommitment.Theoutcome—improvements in mathematics teaching and learning for struggling students—issurelyworththeeffort!



375

Index

Page numbers followed by f and t indicate figures and tables, respectively.

Abstract-level understanding, assessment centers and, 125

Abstract-Representational-Concrete (ARC) assessmentassessment and, 103–104examples of, 105f, 106fMDA and, 103, 127Struggling learners and, 104–108

Academic self-concept, 105Access

assessment and, 122–124core instructions as, 275equity and, 10providing for, 162UDL and, 249

AccommodationsIDEA 2004 and, 251learning barriers and, 95for testing, 289

Accuracy in understandingadvanced acquisition stage of learning and, 117initial acquisition stage of learning and, 117scaffolding instruction for, 201

Acquisition stages, see Advanced acquisition stage of learning; Initial acquisition stage of learning

Acronyms, see MnemonicsActions That Will Help You Implement Each Step of

MDA activity, 133Activities

Bifocal Vision for Math Teaching as, 38–39explicitness instruction and, 145finding the area as, 119fIncorporating Effective Teaching Practice into a

Lesson Plan as, 237learning trajectory as, 64–65MTSS/RTI Instructional Tiers as, 246as peer-mediated, 143t, 206, 341–342Take Action Activities as, 96, 133, 153, 216, 245,

266–267, 278, 374see also Instructional strategies and practices

Adaptationsillustration of, 103tfor nonresponders, 276–277

Adaption stage of learningoverview of, 116teaching strategies for, 120understanding and, 118–119

Adaptive reasoning, 31procedural fluency and, 163see also Reasoning and proof skills

Addends, 230fAdding It Up, National Research Council (2001) report,

31, 162Addition

fluency in, 164fplace values and, 167

Additive strategies, 51–52, 56tAdd-on strategy, 77Advance organizers, 158

Advanced acquisition stage of learning, 60, 63taccuracy in understanding in, 117overview of, 116

Affective learning network, 250Algebra

building foundation for, 24–26, 25tinstructional games for, 183foperations and, 17–19solution drawings of, 198fstruggling learners and, 26, 60–61

Algebraic thinking, 78development of, 184instructional games for, 183foperations and, 17–19

Algorithmic fluencyexamples of, 165–171, 167f, 169fprocedural fluency and, 228, 229–230

Alternative proceduresfractions and, 21–22for multiplication, 20–21, 112

Amount of Time anchor, 263–264Answers-only versus reasoning and answers, 29fApplication fluency, procedural fluency and, 171–172,

228, 230ARC, see Abstract-Representational-Concrete

assessmentARC Assessment Planning Form, 107ARC assessment response sheet, 126f, 127fArea model instruction, 371Area problems, 119fArithmetic properties, 18Assessment, student, 2f

access and, 122–124CRA instruction and, 125definition of, 97diagnostic interview and, 112–113, 128error pattern analysis as, 108, 255fractions and, 60, 62tinformation from, 107instructional accommodations and, 251instructional decisions and, 288literature support and, 6methods for, 276MTSS/RTI and intensifying of, 247–267of prior knowledge, 227purpose and process of, 107the SOLO taxonomy and, 173struggling learners and, 97–133students response to, 356fsummary of, 290types of, 98tword problems for, 125see also Mathematics Dynamic Assessment (MDA);

Monitoring and charting performanceAssessment-related constructs, struggling learners and,

115–124Associative property, multiplicative strategy, 55f, 350fAttention disabilities, struggling learners and, 33, 83–84



376 Index

Authentic contextsCRA instruction, 195explicit instruction and, 215identifying, 125instructional practices and, 78, 161, 214students interests and, 213f, 215

Barrierscurriculum factors and, 9, 71f, 95, 296–298information on, 292learning characteristics as, 293, 361special education and, 8struggling learners and, 36, 69–96understanding and, 60–61, 77–78

Base-10 system, 17CCSS domain, 253–254conceptual understanding with, 227fconcrete materials for, 197fgrid paper for, 367foperations and, 19–21regrouping with, 228f

Behaviors of struggling learners, 70t, 72t, 74, 76–77, 289, 359

Benchmark Assessments, 101Bifocal Vision for Math Teaching activity, 38–39Big ideas

CCSS and, 17content strands and, 17, 38–39importance of, 15–39teacher self-examination and, 38–39

Case study, 345–374CBM, see Curriculum-based measurementCCSS, see Common Core State StandardsCGI, see Cognitively Guided InstructionCharting performance, see Monitoring and charting

performanceChild versus adult views, 7, 41Children’s mathematics, learning trajectories for, 41–65Choices, 121–122Class Mathematics Student Interest Inventory Form, 213fClassroom instruction, see Whole-class instructionClassroom-Based Formative Assessments, 102CLD students, see Culturally and linguistically diverse

studentsCognition, see MetacognitionCognitive interview, 226tCognitively Guided Instruction (CGI), 46–49Collaborative approach, 271color-coding, 222, 256Common Core State Standards (CCSS), 4, 5–6

adaption illustration of, 103tassociated skills cluster as, 284tbig ideas and, 17, 24Eight Standards for Mathematical Practice in, 102,

159, 171, 191tNCTM process standards and, 31–32, 32t, 37tNumber Operations in Base Ten as, 253–254Common error patterns, 110–111, 356f

Communicationin classroom, 1disconnect in, 89impact of learning characteristics on, 84process standards and, 29–30, 115

Commutative property, as multiplicative strategy, 55f, 350f

Compensation strategy, 229tComputational fluency

error pattern analysis, 109as fundamental skill, 18–19, 165procedural fluency and, 228–229

Concepts, 16, 42fractions and, 58–61, 63learning objectives and, 212skills and, 143t, 207, 287

Conceptual knowledge, 76lack of, 105symbols and, 198

Conceptual understanding, 31instructional choices and, 226instructional decisions and, 226procedural fluency and, 93, 163, 172–174, 218t, 223–230vocabulary knowledge and, 176

Concrete-level understandingassessment and, 105modeling and, 195–196

Concrete-Representational-Abstract (CRA) instructionassessment and, 104Explicitness, instructional levels of and, 195, 202,

203–204tinstructional programs, 128Scaffolding and, 243tsequence of, 192–195, 201fstudies on, 241understanding and, 199visual cues and, 194f

Concrete-semiconcrete-abstract (CSA), 104Connections between ideas

graphic organizers and, 211fimpact of learning characteristics on, 231process standard as, 30–31, 285representations and, 218–223, 236scaffolding and, 210

Construct viable arguments and critique the reasoning of others, 286, 354

Contentbig ideas and, 17, 38–39complexity levels, 144texpectations for, 159geometry as, 24learning intentions and, 191learning needs and, 239learning trajectory and, 65measurement and data as, 22–23overview of, 5proficiency stage of learning and, 31, 94

Continuous assessment, 2–3, 6–7case study and, 352–355instructional decisions and, 287

Continuum of instructional choices, 138–142

application of, 145–153making choices across, 142–145scaffolding across, 149–150

Continuum of learning, 61, 121, 117fsee also Learning, stages of

Cooperative learning groupsCLD students in, 91as group instruction, 83teacher directed instruction and, 152, 205use of, 140f

Core beliefs, 76Core instruction, 240f, 249f

for all students, 272–273, 275differentiated instruction and, 273–274flexible grouping with, 204–208

Counting, 44–45manipulatives and, 45, 51opportunities for, 233strategies for, 48–50types of, 47t

CRA, see Concrete-Representational-Abstract instruction

CSA, see Concrete-semiconcrete-abstract



Index 377

Cuingattention and memory disabilities and, 82choices as, 121–122explicitness and, 152multisensory, processing disabilities and, 87–88tools for, 207see also Mnemonics

Culturally and linguistically diverse (CLD) students, 89–91

cultural differences among, 90–91funds of knowledge and experiences from, 250–251

Culturally responsive materials, 214Curriculum considerations and instructional reforms,

92Curriculum factors

as barriers, 9, 71f, 95 297–298success and, 91–93

Curriculum-based measurement (CBM), 6, 101

Datafrom benchmark assessments, 287, 352–353methods for use of, 276from summative assessments, 271

Data analysiscontent strands as, 22–23database from, 212decision making and, 239, 253

Decision making, 34data for, 239, 253information and, 97instructional process of, 281

Describing wheel, graphic organizer, 177fDiagnostic Assessments of Achievement, 101Diagnostic interview, 112–113Diagnostic interviews, information from, 128–129Diagrams for problem solving, 219Differentiated instruction

core instruction and, 273–274determining need for, 263instructional intensity and, 265–266planning and, 247–251whole class instruction as, 365–368

Disability-related characteristics, 80–88Disconnections, communication, 89Discrete independent variable, 241Discrete learning, see Concrete-level understandingDiverse learners, see Struggling learnersDivision

explicit trade algorithm for, 170fas operation, 54for struggling learners, 58

Documents, on instructional strategies and practices, 37t

Doing mathematics, see Process standardsDomains, as standards, 5, 285

see also ContentDouble-digit addition, strategies for, 225f, 227 229fDrawings

Algebra solutions in, 198fas assessment tool, 30fconcrete materials and, 201kinesthetic cues and, 197representational-level learning with, 129, 219strategies for, 199f, 200f

Dynamic assessment, see Mathematics Dynamic Assessment (MDA)

Educational contexts, 241Educational materials, see MaterialsEffective Teaching Practices for General and Struggling

Students activity, 246

Efficiency, developing of, 118EIAs, see Essential instructional approachesEmergent stage of learning, 59, 62tEngaged dialogue assessment strategy, 113Engagement, 70t

application and, 171expectations for, 89in mathematical discourse, 179–180response opportunities and, 182–185, 259struggling learners and, 151–153student practice and, 148, 285–286word walls for, 176

English language learners, 1linguistic differences for, 90mathematical discourse, 179–180, 224fnative language use for, 176RD/MD, as related for, 88

Equity, access and, 10Error pattern analysis

assessment and, 108, 255computational fluency, 109–111mistakes for, 233fobservations and, 129, 164

Essential instructional approaches (EIAs)case study and, 345–346Language and, 174–180MTPs integration with, 217–237, 220f, 224f, 232fresearch support for, 241–245

Essential Instructional Approaches (EIAs)struggling learners and, 155–216, 156t, 270ttiered instruction and, 248

Evaluating an Assessment Against NCTM Standards activity, 133

Evaluationactivity for, 278model for, 277MTSS and, 269–278of performance, 157of prior knowledge, 291see also Monitoring and charting performance

Expectationsof content, 159for engagement, 89for productive struggle, 231for representations work, 219for struggling learners, 217–237

Experiences of studentsCLD students and, 90generalization of, 18–19see also Prior knowledge

Explicitness, instructional levels ofactivities and, 145authentic context and, 215characteristics of, 140fCRA instruction and, 195, 202, 203–204tcuing and, 152examples of, 139fto implicitness as continuum, 141–142instructional practices and, 143t, 151, 152fteacher direction and, 255–257in think-aloud strategies, 221vocabulary practices and, 175–178

Expressive responseassessment and, 120–122examples of, 123f, 124fin practice activities, 190f

Extension, see Adaption; Generalization

Factsfluency and, 43, 50t, 63as known or derived, 55fmastery of, 49–51, 50t



378 Index

FASTDRAW strategypeer-tutoring activity with, 342problem solving and, 208word problems and, 260t, 262

Feedbackas corrective, 202as effective, 235instruction and, 158during modeling, 35response opportunities with, 180–189, 232f, 259during scaffolding, 152see also Monitoring and charting performance

Figurative composite units, 54, 56t, 59fFIND strategy, 258, 370fFluency

with algorithms, 165–171, 167f, 169fin application, 171–172of computation, 165, 166fEIAs and, 162–164facts and, 43, 50t, 63in proficiency and maintenance stages of

learning, 118with whole numbers, 25t

Formative assessmentsin case study, 354tdecision making and, 253hypothesis and, 131–132identifying for, 254–255rubrics as, 114Summative versus, 101–102use of, 226

Foundational knowledgecognitive interview and, 226tidentifying and understanding as, 1–2, 3, 4, 64, 282standards relation to, 253

Fractionsalternative procedures and, 21–22assessment and, 60, 62tconcepts and, 58–61, 63fluency with, 25tinstructional program recommendations and, 61representation of, 59–60SOLO taxonomy, 173

Frayer Model graphic organizer, 177, 178

Generalization, experience of, 18–19Generalization stage of learning

overview of, 116teaching strategies and, 120understanding and, 118–119

Geometrycontent strands and, 24measurement and, 25t

Goalsof instruction, 179instructional decisions and, 288learning objectives and, 218tof modeling, 156see also Learning, stages of

Graphic organizer visuals, 193fGraphic organizers, 176–178

connections between ideas and, 211fGroup instruction

anchors for, 266cooperative learning as, 83differentiated instruction and, 204instructional decisions for, 148–149prior knowledge and, 150scaffolding and, 148fas supplemental, 368whole class instruction as, 127

Grouping, structures and, 204–208, 265

High-Stakes Tests, standardized and, 100Hypotheses, instructional, 67f

formative assessment information for, 131–132information for, 283reasoning and, 42–43revision of, 138f

IDEA, see Individuals with Disabilities Education Improvement Act of 2004 (PL 108-446)

Identifying and understandingappropriate procedures, 224authentic contexts, 125concepts or skills for practice, 207content preparation for, 347with disabilities, 234, 360tformative assessments and, 254–255as foundational, 1–2, 3, 4, 64, 282information structure, 85instructional decisions and, 283learning intentions, 156mathematical areas for, 284performance traits, 292–293t, 358–359place on learning trajectory, 291, 355–356tasks, 233

IEP, see Individualized education programIES, see Institute of Education SciencesIllustration

adaptations in, 103tFIND strategy in, 258f, 370finstruction with visuals, 256procedural fluency in, 164fproficiency in, 163ftiered instruction in, 240f, 249f, 272f

Implementing the 11 Essential Instructional approaches activity, 216

Implementing the Essential Instructional Approaches activities, 216

Implicitness, levels of instructioncharacteristics of, 140f, 142fcontinuum as explicitness to, 141–142examples of, 139finstructional practices and, 143t, 151, 152fstudent directed instruction and, 151–153

Incorporating Effective Teaching Practice into a Lesson Plan activity, 237

Independent practice, 157, 182see also Practice opportunities

Individual Mathematics Student Interest Inventory Form, 211–213

examples of, 212fIndividualized education program (IEP), 123Individualized instruction, 370–371Individuals with Disabilities Education Improvement

Act (IDEA) of 2004 (PL 108-446), 242accommodations and, 251

Informal data collection form, 186fInformation, 3

from assessment, 107about barriers, 292decision-making and, 97diagnostic interviews as, 128–129for hypothesis, 283identifying structure of, 85for instruction, 108, 112, 132passive learning and, 77about student’s stages, 46about student’s understanding, 122

Initial acquisition stage of learningaccuracy in understanding in, 117authentic contexts and, 214overview of, 116

Initial grouping, 51, 56t



Index 379

Institute of Education Sciences (IES), 61Instruction

feedback loop for, 158focus for, 274–275goals of, 179ideas for, 244t, 292, 357information for, 108, 112, 132intervention and, 276language and, 250–251scaffolding and, 146f, 148f, 150fseven anchors for, 252f, 265fspecial education and, 92student’s ideas inform, 64with visuals illustration as, 256see also Instructional strategies and practices

Instructional approachto mathematics, 208–211positive outcomes with, 243tsee also Essential instructional approaches (EIAs)

Instructional Approaches activity, 153–154Instructional decisions

conceptual understanding and, 226along continuum, 153–154flexibility in, 137–154group instruction and, 148–149MDA and, 130–131, 291performance data and, 271, 287process for, 281real time making of, 182as student-centered, 162

Instructional gamespractice opportunities with, 183ftips for making, 184f

Instructional hypotheses, see Hypotheses, instructionalInstructional intensity, 240f, 249f

differentiating, 265–266increasing levels of, 252f, 272fMTSS and, 269–278, 275t

Instructional materials, see MaterialsInstructional Pacing, 92–93Instructional program recommendations

CRA instruction and, 128fractions and, 61number and operation sense, 159practices for, 182problem solving and, 28research and, 242SOLO taxonomy and, 173substandards and, 158

Instructional reforms and curriculum considerations and, 92

Instructional strategies and practicesauthentic contexts and, 78, 161, 214cooperative learning groups/peer tutoring and, 205documents on, 37tEIAs and, 155explicitness or implicitness and, 143t, 151, 152fgames/self-correcting materials and, 183f, 184f, 185fgeneral strategies as, 42learning stages and, 44–45literature support for, 9meaningful contexts for, 212modeling and, 34, 48, 217as multidimensional, 241National Mathematics Advisory Panel on, 137problem-solving, 27–28, 32–33receptive and expressive response formats, 131research base to improve, 244–245scaffolding and, 145–147, 201school-wide, 269–270for struggling learners, 1–11, 32–33, 91–94, 239–246see also Assessment; Monitoring and charting

performance

Instructional time, 243t, 244tInstrumental understanding, 26Interactive learning, 202Interest inventories, 216Intermediate stage of learning, 60, 62tIntervention

instruction and, 276research on, 264

Inverse operations, 168f

Kinesthetic cuesdrawings and, 197for processing disabilities, 87

Knowledge and skill gapsinstructional intensity and, 254for struggling learners, 78–80

LanguageEIAs and, 174–180, 224finstruction and, 250–251symbols and, 86, 89–90understanding and, 199

Language development, mathematics achievement and, 79, 81

Language-based processing difficulties, 87Learned helplessness, learning characteristic as, 8, 74–77,

231Learning

deep, teaching for 4determing of, 158–159memory and, 81mistakes for, 233through practices, 27stages of, 44–46, 59–60, 115–119, 116t

Learning characteristicsas barriers, 293, 361performance traits and, 95f, 297fstruggling learners, 8–9, 71–88, 145

Learning disabilities, 214see also specific disabilities

Learning intentions, 156content and, 191EIAs and, 158–161examples of, 160fsharing, 156, 160–161

Learning needscontent and, 239EIAs and, 155grouping based on, 205of students’ with disabilities, 92, 251

Learning objectives, concepts and, 212Learning standards, see Instructional program

recommendationsLearning trajectory, 7–8, 281

capabilities during, 50, 233identifying place on, 291, 355–356mathematics and, 41–65relation of, 34standards and, 286

Lesson plans, see PlanningLine segments, 178fLine symmetry, 24Linear equations, algorithm for, 168fLinguistic differences, 89–90Literature support

for assessment, 6for differentiated instruction, 204for instructional practices, 9–10for peer-mediated learning, 206for representations, 220for visuals in mathematics instruction, 192



380 Index

Maintenance stage of learningfluency in, 118overview of, 116

Manipulativesconcrete-level understanding, 195–196counting and, 45, 51materials as, 23

Masteryof basic facts, 49–51, 50t, 165demonstration of, 108

Materialsas concrete, 196f, 197finstructional games/self-correcting materials, 183f,

184f, 185fmanipulatives as, 23ten frames as, 20f

Math anxiety, 80, 289Math practices, 285–286, 372–373Math standards, 284–285

see also Common Core State Standards (CCSS)Mathematical discourse, 89

engagement in, 179–180, 218t, 243tfor English language learners, 179–180, 224f

Mathematical practicesdevelopment of, 27emphasis on use of, 189–192

Mathematics achievementDiagnostic Assessments of Achievement and, 101language development and, 79, 81

Mathematics difficulties, see Reading difficulties and Mathematics difficulties (RD/MD), as related

Mathematics Dynamic Assessment (MDA)ARC assessment and, 103assessment through, 125–128conducting, 128–130instructional decisions, 130–131, 291overview of, 125results of, 130f

Mathematics learningproductive struggle in, 231–235, 236writing and, 180

Mathematics processesadult versus child views in, 7, 41see also Process standards

Mathematics vocabularycategories of, 175EIAs and, 174–180visuals cues and, 194f

Math-specific learning needs, determining of, 3, 7assessment tasks and, 288, 290–291instructional decisions and, 283

MDA, see Mathematics Dynamic AssessmentMeaningful connections, metacognitive disabilities and,

84–85Meaningful contexts, abstract reasoning development

in, 211–215Measurement

geometry and, 25tunits of, 58–59

Measurement and Data, content as, 22–23Memory, working and learning with, 81–82Memory disabilities

retrieval and, 81summary of, 72t, 80

Metacognitionstrategy examples and, 28–29, 35–36, 85supported practice in, 207teaching math and, 257–258

Metacognitive thinking disabilitiesmeaningful connections and, 84–85struggling learners and, 33, 69

Misconceptions, 291, 357error patterns and, 356f

Mistakes, see Error pattern analysisMnemonics

strategy instruction and, 82–83struggling learners use of, 262visual strategies for, 209f

Model with mathematics (NGA Center for Best Practices & CCSSO, 2010), 171

Modelingat abstract level, 198–204CCSS and, 191tconcrete-level understanding and, 195–196feedback during, 35goal of, 156instructional strategies and practices, 34, 48, 217overview of, 34process standards and, 34at representational level, 196–198of thinking, 208

Modelsfor evaluation, 277problem solving with, 222self-monitoring and, 206

Modes of input, 85–86Modifications, see AdaptationsMonitoring and charting performance

case study example of, 358tstrategies for, 128systemic teaching and, 157techniques for, 188f, 189f

Motor integration disabilities, 87, 110MTSS, see Multi-tiered systems of supportsMultiplication

add-on strategy in, 77alternative procedures for, 20–21, 112as operation, 54repeated addition process for, 256ffor struggling learners, 58

Multiplicative reasoning, 43, 51–58, 63strategy examples of, 52f, 53f, 54f, 56t, 350fsee also Reasoning and proof skills

Multi-tiered systems of supports (MTSS), 9assessment and, 98t, 100–101characteristics of, 269–274, 270tflexible grouping and, 205instructional intensity and, 269–278, 275tnumber sense and, 93–94RTI and, 239–246

Multi-tiered systems of supports (MTSS)/Response to intervention (RTI)

case study and, 345–346intensifying assessment and, 247–267

Multi-tiered systems of supports (MTSS)/Response to intervention (RTI) instructional Tiers activity, 246

National Council of Teachers of Mathematics (NCTM)on access and equity, 10–11on assessment, 97curriculum content strands and, 37tdefinition of standards and prompts by, 99tEffective Mathematics Teaching Practices by, 218tprocess standards and, 5–6, 31–32, 114tiered instruction and, 248see also Instructional program recommendations

National Mathematics Advisory Panelfinal report (2008) by, 24–26, 242on instructional practices, 137

National Research Council (2001), 242NCTM, see National Council of Teachers of

MathematicsNCTM Mathematics Teaching Practice (MTPs), 345–346

EIAs integration with, 217–237, 220f, 224f, 232fNonmultiplicative strategies, 51, 56t



Index 381

Nonresponders, adaptations for, 276–277Number sense

common errors and, 111definition of, 15development of, 184, 225MTSS and, 93–94

Number sequences, 45–46student capabilities and, 47–48t

Numbers and Operationsalgebraic thinking and, 17–19base-10 system and, 19–21connections between, 18as content strand, 15division as, 54error patterns and, 110fractions and, 21–22instructional program recommendations for, 159multiplication as, 54MTSS and, 93–94order of, 159place value and, 20see also Number sense

Numeracy, 63research on, 184

Objects, see Concrete-level understanding; ManipulativesObservations

from diagnostic interviews, 130error pattern analysis and, 129, 164

OGAP, see Ongoing Assessment Project Multiplicative Framework

Ongoing Assessment Project (OGAP) Multiplicative Framework, 51–55

OperationsAlgebra and, 17–19in Base Ten as CCSS, 253–254Base-10 system and, 19–21division as, 54instructional program recommendations for, 159multiplication as, 54place values and, 20see also Numbers and Operations

Opportunities for improvement, 11Oral directions, 123

Parallel, 178Part-part-whole strategies, 50Passive learning

productive struggle and, 231struggling learners and, 77–78

Patternsfluency and, 164of performance, 107problem solving and, 109search for and use of, 18–19, 36see also Common error patterns

Peer-mediated learningactivities for, 143t, 341–342grouping structures for, 206–208

Peer-tutoring, 206, 341–342Perceptual multiples, 52, 56tPerformance

evaluation of, 157level of understanding and, 108pattern of, 107

Performance charting, see Monitoring and charting performance

Performance datainstructional decisions and, 271, 287struggling learners and, 244tstudent responses and, 185–187

Performance Rubric, 186fPerformance traits, 8

identifying observance of, 292–293t, 358–359impact of learning characteristics and barriers, 95f, 297frecord of, 294fstruggling students and, 69, 70t

Perseverance, 235Phonological processing, effect on, mathematics

learning, 79Pictorial representations, 220, 221fPlace values

algorithms and, 167, 169multi-digit numbers, 258foperations and, 20problem solving and, 110–111recording sheet for, 368f, 369f

Planningdifferentiated instruction and, 247–251instructional intensity, 266for success, 248whole-class instruction and, 128

Planning Intensive Instruction Using Instructional Anchors activity, 267

Polygon, 178fPositive reinforcement, 157, 290Practice opportunities

for counting, 233engagement and, 148instructional games for, 183fproviding for, 35, 120, 175structured language experiences and, 1teacher support levels and, 120, 229visual diagrams for, 222

Pre instruction/anticipatory set, 158Precision, 35, 222, 354Preservice teachers, see TeachersPrinciples and Standards for School Mathematics (NCTM), 4

problem solving and, 27–28process standards and, 5–6, 27–28representation and, 220f

Principles to Actions (NCTM)effective formative assessment definition by, 102MTPs, 217, 270

Prior knowledgeactivation of, 33, 89, 121, 201, 214assessment of, 227gaps and strengths in, 288, 291, 357group instruction and, 150lack of, 274response cards and, 182f, 183f

Problem solving, 5drawings and, 197FASTDRAW strategy and, 208impact of learning characteristics on, 84, 121instructional practices and, 28, 32–33mixing problem types and, 235models of, 222NCTM and, 27–28patterns and, 109place values and, 110–111reasoning for, 234strategy examples of, 32, 70t, 209f, 210tstructured dialogue sheet for, 181f

Procedural fluency, 31accuracy requirements and error patterns for, 110building of, 228–230conceptual understanding and, 93, 163, 172–174, 218t,

223–230for understanding, 225

Procedure-first instruction, 225Process standards

communication as, 29–30, 115connections between ideas as, 30–31, 285



382 Index

modeling and, 34NCTM and, 5–6, 31–32, 114proof skills as, 28–29f, 36representation and, 30, 114

Processing disabilitiessummary of, 73t, 296ttypes of, 85–88

Productive disposition, 31, 32procedural fluency and, 163

Productive struggle in learning mathematics, 231–235, 236

Proficiency stage of learningcontent strands and, 31, 94fluency in, 118illustration of, 163foverview of, 116teaching strategies and, 15–16understanding and, 147–149, 281see also Monitoring and charting performance

Progress Monitoring Assessments, 100–101Prompts

for math writing, 180fpurpose of, 226tquestions or, 187recognition, 190ffor students, 114, 115, 172

Purposeful content focus, 253–254

RD/MD, see Reading difficulties and mathematics difficulties, as related

Reading, research on, 264Reading difficulties and mathematics difficulties (RD/

MD), as related, 88Reading disabilities, 88

summary of, 73t, 296tReason abstractly and quantitatively, 33

practice of, 286Reasoning and proof skills

development of, 49hypotheses and, 42–43for problem solving, 234process standards as, 28–29f, 36strategy examples for, 50t, 161

Receptive responseassessment and, 120–122examples of, 124fpractice activities and, 190f

Recognitionrecognition prompt, 190fresponse opportunities versus, 187–189

Record, 294ffor place values, 368f, 369fteachers’ notes as, 294f, 360t

Reflections: How to Support the NCTM Teaching Practices with EIAs activity, 237

Reforms, see Instructional reformsRegrouping errors, 110–111Reinforcement, see FeedbackRelational understanding, 19, 26Repeated abstract composite grouping, 54, 56tRepeated addition process, 256fRepresentation

of fractions, 59–60learning characteristics and, 74–75f, 86of mathematics, 172multiplicative reasoning and, 55fprocess standards and, 30, 114of solutions, 86ftypes of, 220, 221fuse and connection of, 218–223, 236

Representational-level understandingassessment centers and, 125drawings and pictures for, 129modeling for, 196–198

Representations, categories for, 219Research, 10

on basic fact automaticity, 165on error patterns, 109to improve instruction practices, 244–245on math intervention, 264on mathematics education, 93on numeracy development, 184about special needs, 242on struggling learners, 30support for EIAs and, 241–245on working memory, 81–82

Response cards, 182, 183fResponse formats

assessment from, 288–290receptive and expressive as, 131recognition-type of, 187

Response opportunitiesanchor, 252f, 259–263feedback and, 180–189, 232f, 235, 259providing for, 181, 182–185, 227see also Practice opportunities

Response to intervention (RTI)assessment and evaluation with, 100–103MTSS and, 239–246see also Multi-tiered systems of supports (MTSS)/

Response to intervention (RTI)Responsive instruction, planning and implementing of,

3–4, 9–10case study and, 362–374guidance for, 298hypotheses as guide for, 283

Retrieval skills, 81Role playing demonstration, 113Rounding, 167RTI, see Response to interventionRubric

examples of, 114f, 115ffor fluency development, 174tas formative assessments, 114

Scaffoldingconnections and, 210across continuum of instructional choices, 149–150with emphasis, 146f, 147–149examples of, 146f, 148f, 150ffeedback during, 152group instruction and, 148finstructional strategies and practices, 145–147, 201tiers as, 271–272visual cues as, 262t

Schema-based instruction, 192Schematic representations, 220, 221fSchool performance, see Mathematics achievementSchool-wide practices, 269–270SEAL, see Stages of Early Arithmetic Learning

instructionSelective attention, see Attention disabilitiesSelf-correcting materials, practice opportunities with,

185fSelf-evaluation/monitoring, metacognition and, 85Self-monitoring, 206

example of strategy for, 209fSelf-observation, 11Self-reflection inventory, 153–154, 216, 237, 246, 278Self-regulation, 243tSemi-concrete to representational understanding, 104

Process standards—continued



Index 383

Seven anchors model, 252fSharing, learning intentions, 156, 160–161Skills, 16

application of, 171assessment of, 106cluster of, 284tconcepts and, 143t, 207, 287discrimination as, 83

SOLO, see the Structure of the Observed Learning Outcome taxonomy

Solving linear equations, 168fSpecial education

instruction and, 92learning barrier accommodation in, 8MTSS and, 273practices in, 244

Special education teachers, see TeachersStages of Early Arithmetic Learning (SEAL) instruction,

43–48Standard algorithm

double digit addition with, 225f, 227, 229fas multiplicative strategy, 55f, 350f

Standard procedures, 21, 28Statistics, processes and, 23Story problems, see Problem solving; Word problemsStrategic competence, 31, 163Strategy instruction

addition strategies and, 48, 50mnemonics and, 82–83problem-solving and, 32see also Cuing

Strengths, 11Structure

of assessment, 289of evaluations, 277grouping and, 204–208, 265of information, 85intensive instructional sessions, 259language experiences and practice opportunities

with, 1for recording, 294fSOLO taxonomy as, 173tstandards, 284use of, 35, 140f

Structure of the Observed Learning Outcome (SOLO) taxonomy, 173t

Structured dialogue cue sheet, 181fStruggling learners

algebra and, 26, 60–61ARC assessment, 104–108assessment for, 97–133assessment-related constructs for, 115–124attention disabilities of, 33, 83–84barriers for, 36, 69–96changing expectations for, 217–237choices continuum for, 137–154curriculum considerations for, 202tdiagnostic interviews for, 112–113EIAs and, 155–216, 156t, 270tengagement and, 151–153error pattern analysis, 109–111fractions and, 22graphic organizers for, 177instruction for, 1–11, 32–33, 91–94, 239–246intention importance for, 160learning characteristics of, 8–9, 71–88, 145metacognitive disabilities and, 33, 69mnemonics use by, 262multiplication and division for, 58research on, 30response opportunities for, 184, 187scaffolding for, 146f

time for, 263visuals use for, 192, 219, 221word problems for, 260

Struggling learners, specific learning needs of, 2–3, 7–9

assessment tasks and, 288case study and, 357–362instructional decisions and, 283performance traits and, 292

Student directed instructioncharacteristics of, 140fcontinuum as teacher directed to, 139–141examples of, 139f, 146f, 147implicitness and, 151–153

Student responses, performance data from, 185–187Student-centered instruction

instructional decisions and, 162student-directed and, 137–138student-directed versus, 137–138

Students interests, authentic contexts of, 213f, 215Students with disabilities

barriers to success for, 69–96identified as, 293, 360tlearning needs of, 92, 251testing accommodations for, 289

Substandardsprogram recommendations and, 158skills cluster and, 284t

Subtechnical words, 175Subtraction

algorithms and, 167, 168with understanding, 173

Success, 1barriers to, 69–96with core instruction, 274curriculum factors and, 91–93determining criteria for, 159–160growth mindset for, 234math anxiety and, 80MTSS and, 277planning for, 248teaching systemically for, 155

Summative assessments, data from, 271Formative versus, 101–102

Supplementary instructionat elementary level, 252MTSS and, 205, 240f, 249f, 272f, 275response opportunities and, 259time for, 264

Supportcuing as, 121determining appropriate levels of, 156teacher features of, 91, 179, 263, 290

Symbolic words, 175–176Symbols

conceptual knowledge and, 198language and, 86, 89–90representations as, 18–19, 81, 250

Systemic instruction frameworkphases of, 156, 157fsee also Teaching systemically

Teach Math Metacognition anchor, 257–258Teacher directed instruction

characteristics of, 140fcooperative learning groups and, 152, 205examples of, 139f, 146f, 147explicitness, instructional levels of and,

255–257to student directed as continuum, 139–141

Teacher self-examination, 38–39



384 Index

Teachersknowledge for, 240notes record by, 297f, 360tin special education, 294to student ratio, 252f, 264–265

Teachingfor deep learning, 4forest and trees analogy and, 15, 36–38incrementally, 77math metacognition, 257–258measurement, 23vocabulary, 174–176

Teaching Mathematics Meaningfully Processas case study, 345–374examples of, 2f, 10, 67f, 248f, 279foverview of, 281–298

Teaching strategiesgeneralization stage of learning and, 120proficiency stage of learning and, 15–16see also Instructional strategies and practices

Teaching systemically, 155–158Teaching to mastery, see MasteryTechnical words, 175Ten Frame, 20fTesting, 289Think-aloud strategies, 82, 113, 208

explicitness in, 221use for story problems, 343–344

Thinking strategies, see Metacognition; Strategy instruction

Toolsappropriate use of, 35–36, 235for cuing, 207drawings as, 30frepresentations as, 219

Traditional regrouping algorithm, 258Transitional multiplicative strategies, 52–55, 56t, 350fTriangle as term, 177, 178f

UDL, see Universal Design for LearningUnderstanding

accuracy in, 117–119algorithms and, 169barriers and, 60–61, 77–78CRA instruction and, 199demonstration of, 105early numeracy and, 63information about student’s, 122learning intentions, 161MDA and levels of, 130fperformance and, 108procedural fluency for, 225proficiency and, 147–149, 281

subtraction with, 173see also Assessment; Learning

Units of measure, see MeasurementUniversal Design for Learning (UDL), 204

core instruction with, 273planning and instructional framework of, 249–250

Universal Screeners, 100

Value in learning, 212Variables, 168f, 241

structured dialogue cue sheet for, 181fVisual cues

CRA instruction and, 194ffor mnemonics, 209fas scaffold, 262tin strategy instruction, 219

Visual diagramsas explicit, 223fas nonexplicit, 222f

Visual models, 22Visual processing disabilities, 86–87

see also Processing disabilitiesVisual representation, 220Visual spatial processing difficulties, 87, 296tVisual vocabulary word strategy, 178fVisuals

cognitive framework as, 210utilization of, 192–204, 256f, 257f

VocabularyEIAs and, 174–180word problems and, 78–79see also Mathematics vocabulary

Wait time, providing, 82, 88What Works Clearinghouse (WWC) practice guide, 61Whole-class instruction

data collection for, 186fdifferentiated instruction for, 365–368groups for, 127planning for, 128

Word problemsassessment with, 125CGI and, 48–49FAST DRAW strategy and, 260tone variable equations, 260t, 261t, 262treasoning for, 234story problems as, 172t, 221f, 222fthink-aloud strategies, 343–344vocabulary and, 78–79

Word walls, 176–178Writings, mathematics learning with, 180WWC, see What Works Clearinghouse practice guide

