11

Two Multi-Site Randomized Control Trials: Two Multi-Site Randomized Control Trials:

Bottom-Up vs. Top-Down Approaches Bottom-Up vs. Top-Down Approaches to Scaling Up PALSto Scaling Up PALS

Is Response to Word-Problem Is Response to Word-Problem Intervention among Students with MD Intervention among Students with MD

Moderated by Concurrent RD?Moderated by Concurrent RD?

Doug Fuchs and Lynn FuchsDoug Fuchs and Lynn FuchsVanderbilt UniversityVanderbilt University

22

Bottom-Up vs. Top-Down Approaches Bottom-Up vs. Top-Down Approaches to Scaling Up PALSto Scaling Up PALS

Doug Fuchs, Kristen McMaster, Laura Saenz,Doug Fuchs, Kristen McMaster, Laura Saenz,Devin Kearns, Lynn Fuchs, Loulee Yen, Devin Kearns, Lynn Fuchs, Loulee Yen,

Don Compton, and Chris Lemons Don Compton, and Chris Lemons Vanderbilt UniversityVanderbilt University

Chris SchatschneiderChris SchatschneiderFlorida State UniversityFlorida State University

R305G04104 R305G04104 Institute of Education SciencesInstitute of Education Sciences

33

Purpose of PALSPurpose of PALS Supplements the general education core programSupplements the general education core program Implemented 3 times per week in reading; 2 times per Implemented 3 times per week in reading; 2 times per

week in mathweek in math Creates a “routine” for teachers to differentiate Creates a “routine” for teachers to differentiate

instruction by creating many simultaneous peer-instruction by creating many simultaneous peer-mediated lessons rather than one teacher-directed mediated lessons rather than one teacher-directed lessonlesson

PALS Reading: kindergarten, first grade, grades 2-6, PALS Reading: kindergarten, first grade, grades 2-6, high schoolhigh school

PALS math: kindergarten, first grade, grades 2-6PALS math: kindergarten, first grade, grades 2-6

44

PALS Research

• Based on Juniper Gardens Classwide Peer Tutoring• Over 15 years of experimental research• Title I and Non-Title I schools• Urban and suburban schools• High, average, and low achievers• Students with learning disabilities • “Validated Practice” status (USDE, WWC, BEE)

55

Grades 2-6 PALS Grades 2-6 PALS

Partner ReadingPartner Reading

Paragraph ShrinkingParagraph Shrinking

Prediction RelayPrediction Relay

66

Partner Reading

• Conducted for 11-12 minutes• Stronger reader reads aloud for 5 minutes• Weaker reader reads same text aloud for 5 minutes• Weaker reader retells story for 1-2 minutes• Readers read quickly, correctly, and with expression• Coaches listen, correct mistakes, and mark points (1

point for each correctly read sentence and 10 points for story retell)

77

Paragraph Shrinking• Conducted for 10 minutes• For 5 minutes:

• Stronger reader reads new text aloud, summarizing paragraph by paragraph

– Name the most important who or what (1 point)– Name the most important thing about the who or

what (1 point)– Shrink it to 10 or fewer words (1 point)

• For next 5 minutes:• Weaker reader reads new text aloud, summarizing

paragraph by paragraph (as above)• Coach listens, corrects mistakes, and marks points

88

Prediction Relay

5 minutes, stronger reader read new text− Makes prediction (1)− Reads half page (1) − Checks prediction (1)− States main idea (3)− Makes new prediction− Continues to read

5 minutes, weaker reader continues on in new text, with the same activities

Coach listens, corrects mistakes, and marks points

99

Two Kinds of PALS ResearchTwo Kinds of PALS Research

Randomized Controlled TrialsRandomized Controlled TrialsStudy schools include Title I and no Title I.Study schools include Title I and no Title I.Classrooms randomly assigned within Classrooms randomly assigned within schools to PALS and control groups.schools to PALS and control groups.HA, AA, LA (including LD) students HA, AA, LA (including LD) students targeted in each classroom.targeted in each classroom.Fidelity of treatment implementation.Fidelity of treatment implementation.Individually administered pre-/posttests by Individually administered pre-/posttests by trained examiners.trained examiners.

1010

0

20

40

60

80

LearningDisabled

LowAchieving

AverageAchieving

HighAchieving

Experimental Control

Imp

rove

men

tO

ver

16 W

eek

sImprovement in Reading

1111

Two Kinds of PALS ResearchTwo Kinds of PALS Research

School-Improvement ProjectsSchool-Improvement Projects

Title I schools implement PALS school-wide.Title I schools implement PALS school-wide.

Our TA funded by Nashville’s Title I officeOur TA funded by Nashville’s Title I office

No fidelity of treatment assessed.No fidelity of treatment assessed.

Group administered high-stakes tests.Group administered high-stakes tests.

1212

Report Card Scores Based on Students’ Performance on the TCAP (CTB/McGraw-Hill)

GOWER

MATH 76.0 (61) 107.8 (12)

‘93 ‘94Subject

Language Arts 61.7 (60) 91.8 (33)

READING 74.0 (60) 112.5 (25)

Science 74.4 (58) 95.6 (24)

Social Studies 60.1 (61) 81.4 (53)

Note 1: A score of 100 means that students of a school are progressing at a rate equivalent to that of the national rate.

Note 2: The numbers in parentheses represent Gower’s standing in relation to the Metro Schools’ other 66 elementary schools.

1313

Typical Teacher Support Typical Teacher Support in PALS Researchin PALS Research

Support was ongoing and on-site: RAs gave in-class Support was ongoing and on-site: RAs gave in-class assistance 1x or 2x per wk during training and assistance 1x or 2x per wk during training and implementation. implementation.

RA support is costly, unlikely in wider implementations, RA support is costly, unlikely in wider implementations, an obstacle to scaling up.an obstacle to scaling up.

Absent such support, quality of implementations suffer.Absent such support, quality of implementations suffer. How to separate an intervention from its support How to separate an intervention from its support

system? How to scale-up (“export” the intervention from system? How to scale-up (“export” the intervention from A to B) without researchers’ nurturance?A to B) without researchers’ nurturance?

There’s also a matter of transcending time….There’s also a matter of transcending time….

1414

Purpose of the Present StudyPurpose of the Present Study

With Grades 2-6 PALS as a “prop,” and With Grades 2-6 PALS as a “prop,” and students’ reading achievement as the students’ reading achievement as the criterion, does a “bottom-up” approach criterion, does a “bottom-up” approach beat a “top-down” approach to scaling up?beat a “top-down” approach to scaling up?

Do these approaches affect teachers’ Do these approaches affect teachers’ sustainability of PALS?sustainability of PALS?

1515

METHODMETHOD

1616

Participating SitesParticipating Sites

1717

Participating TeachersParticipating Teachers Two cohorts of 3Two cohorts of 3rdrd-, 4th-, and 5th-grade teachers :-, 4th-, and 5th-grade teachers :

− Cohort 1Cohort 1 Entered study in 2006-07Entered study in 2006-07

− Cohort 2Cohort 2 Entered study in 2007-08Entered study in 2007-08

Two years of study participation:Two years of study participation:− Year 1Year 1

Assigned randomly to PALS or ControlAssigned randomly to PALS or Control− Year 2Year 2

PALS Teachers selected Top Down or Bottom Up PALSPALS Teachers selected Top Down or Bottom Up PALS Control Teachers remained in Control groupControl Teachers remained in Control group

1818

Teachers by Study Group in Teachers by Study Group in Year 2Year 2

Top DownTop Down Bottom UpBottom Up ControlControl

Cohort 1Cohort 1

TennesseeTennessee 55 55 77

MinnesotaMinnesota 55 55 44

TexasTexas 66 77 77

Total Total 1616 1717 1818

Cohort 2Cohort 2

TennesseeTennessee 1212 88 99

MinnesotaMinnesota 66 99 12*12*

TexasTexas 88 44 77

Total Total 2626 2121 28*28*

TOTAL TOTAL

TeachersTeachers4242 3838 46*46*

*This number is based on all MN controls; 10 were eliminated, right?*This number is based on all MN controls; 10 were eliminated, right?

1919

Students by Study Group in Students by Study Group in Year 2Year 2

Top DownTop Down Bottom UpBottom Up ControlControl

Cohort 1Cohort 1

TennesseeTennessee 6060 6060 6868

MinnesotaMinnesota 6161 6565 4949

TexasTexas 6161 9090 8888

Total Total 182182 215215 205205

Cohort 2Cohort 2

TennesseeTennessee 136136 8686 102102

MinnesotaMinnesota 7272 109109 151*151*

TexasTexas 9595 4646 7373

Total Total 303303 241241 326*326*

TOTAL TOTAL

StudentsStudents485485 456456 531*531*

2020

Study Conditions: Year 1Study Conditions: Year 1 Control – Teachers implemented core language Control – Teachers implemented core language

arts curriculumarts curriculum PALS – Teachers implemented with fidelity:PALS – Teachers implemented with fidelity:

− 3 times/week for 35-40 min (about 54 sessions)3 times/week for 35-40 min (about 54 sessions)− Coaches and Readers: higher-performing readers Coaches and Readers: higher-performing readers

paired with lower-performing readerspaired with lower-performing readers− Four PALS Activities:Four PALS Activities:

Partner Reading (10 min)Partner Reading (10 min) Retell (2 min)Retell (2 min) Paragraph Shrinking (10 min)Paragraph Shrinking (10 min) Prediction Relay (10 min)Prediction Relay (10 min)

2121

Study Conditions: Year 2Study Conditions: Year 2 Control – Teachers implemented core language arts Control – Teachers implemented core language arts

curriculumcurriculum ““Top Down” (TD) PALSTop Down” (TD) PALS

− Teachers did PALS “by the book”Teachers did PALS “by the book”− Fidelity of PALS implementation was emphasizedFidelity of PALS implementation was emphasized

““Bottom Up” (BU) PALS Bottom Up” (BU) PALS − Teachers implemented core components of PALSTeachers implemented core components of PALS− Customization was strongly encouraged and Customization was strongly encouraged and

supportedsupported

222222

BU PALS: Core ElementsBU PALS: Core Elements

48 sessions minimum48 sessions minimum 35 minutes per session minimum35 minutes per session minimum 10 minutes of Partner Reading10 minutes of Partner Reading 10 minutes of Paragraph Shrinking10 minutes of Paragraph Shrinking A motivational peer reinforcement systemA motivational peer reinforcement system

232323

BU PALS: RequirementsBU PALS: Requirements

Teachers asked to:Teachers asked to:− Conduct core elements of PALS as designedConduct core elements of PALS as designed− Develop changesDevelop changes

Match to curriculum, students’ needs, teaching Match to curriculum, students’ needs, teaching stylestyle

Create a type of PALS for the long termCreate a type of PALS for the long term

2424

Student MeasuresStudent Measures

Academic MeasuresAcademic Measures Test of Word Reading Efficiency Test of Word Reading Efficiency

(TOWRE)(TOWRE) Woodcock Reading Mastery Test-Woodcock Reading Mastery Test-

Revised (WRMT-R) Revised (WRMT-R) − Word Identification SubtestWord Identification Subtest

Wide Range Achievement Test (WRAT) Wide Range Achievement Test (WRAT) − Letter and Word Identification Letter and Word Identification

SubtestsSubtests Comprehensive Reading Assessment Comprehensive Reading Assessment

Battery (CRAB; 2 passages)Battery (CRAB; 2 passages)− Oral reading (1 min & 3 min)Oral reading (1 min & 3 min)− Comprehension (10 open-ended Comprehension (10 open-ended

questions)questions) CBM Maze Task (2 passages)CBM Maze Task (2 passages)

− Correct maze choices made in 2.5 Correct maze choices made in 2.5 minmin

Iowa Test of Basic Skills (ITBS)Iowa Test of Basic Skills (ITBS)− Reading ComprehensionReading Comprehension− VocabularyVocabulary

Student CharacteristicsStudent Characteristics DemographicsDemographics SWANSWAN

− Teachers rated each Teachers rated each student’s abilities to focus student’s abilities to focus attention, control activity, attention, control activity, and inhibit impulsesand inhibit impulses

Teacher ratingsTeacher ratings− Teachers rated each Teachers rated each

student’s effort in reading student’s effort in reading and behavior in the and behavior in the classroomclassroom

2525

Teacher MeasuresTeacher Measures

Classroom MeasuresClassroom Measures PALS CalendarsPALS Calendars PALS FidelityPALS Fidelity Language Arts ObservationLanguage Arts Observation Classroom Atmosphere Classroom Atmosphere

Rating Scale (Wehby)Rating Scale (Wehby) Survey of Enacted Curriculum Survey of Enacted Curriculum

(SEC): English and Language (SEC): English and Language

ArtsArts

Teacher CharacteristicsTeacher Characteristics DemographicsDemographics Berends teacher survey Berends teacher survey

(assesses school climate, (assesses school climate, teacher professional teacher professional development, teacher efficacy, development, teacher efficacy, etc.)etc.)

2626

ProceduresProcedures Pretesting (September-October)Pretesting (September-October) PALS Workshops (September-October)PALS Workshops (September-October)

− Year 1: All teachers attend same workshopYear 1: All teachers attend same workshop− Year 2: Separate TD and BU workshopsYear 2: Separate TD and BU workshops

PALS Implementation (~18 weeks)PALS Implementation (~18 weeks)− Teachers implemented 3 times per week for 35-40 minTeachers implemented 3 times per week for 35-40 min− Weekly classroom visits from project staffWeekly classroom visits from project staff− Three “booster” sessions for TD and BU PALS teachersThree “booster” sessions for TD and BU PALS teachers− Two fidelity observationsTwo fidelity observations

Language arts observations in PALS and Control classroomsLanguage arts observations in PALS and Control classrooms− 45-60 min45-60 min− Momentary time sampling of a variety of reading instructional Momentary time sampling of a variety of reading instructional

componentscomponents− Supplementary field notesSupplementary field notes

Posttesting (March-May)Posttesting (March-May)

2727

Scaling-Up PALS for Grades Scaling-Up PALS for Grades 2-62-6

ResultsResults

2828

Organization of StudyOrganization of Study

COHORT 1 (1st YEAR)

Top-Down PALS (W + B + Tutor)

Top-Down PALS (W + B + Helpers)

Control

Year 1(2006-07)

Year 2(2007-08)

Top-Down PALS

Control

Year 3(2008-09)

COHORT 2 (2nd YEAR)

Top-Down PALS

Bottom-Up PALS

Control

COHORT 1 (2nd YEAR)

Control

Bottom-Up PALS

Top-Down PALS

COHORT 2 (1st YEAR)

COHORT 1 (1st YEAR)

Top-Down PALS (W + B + Tutor)

Top-Down PALS (W + B + Helpers)

Control

Year 1(2006-07)

Year 2(2007-08)

Top-Down PALS

Control

Year 3(2008-09)

COHORT 2 (2nd YEAR)

Top-Down PALS

Bottom-Up PALS

Control

COHORT 1 (2nd YEAR)

Control

Bottom-Up PALS

Top-Down PALS

COHORT 2 (1st YEAR)

2929

Analysis ProceduresAnalysis Procedures Create latent pretest and posttest variables combining 5 reading Create latent pretest and posttest variables combining 5 reading

measures into 1measures into 1 Create a latent change scoreCreate a latent change score

− Produces an “error-free” change valueProduces an “error-free” change value Run 2-level HLM analysesRun 2-level HLM analyses

− Outcome: Latent change scoreOutcome: Latent change score− Variables: Treatment condition (TD, BU, Control); Site (TN, Variables: Treatment condition (TD, BU, Control); Site (TN,

MN, TX); latent pretest scoreMN, TX); latent pretest score− Random effects: Level 2 teacher effects; ICC = .10Random effects: Level 2 teacher effects; ICC = .10

Test comparability of groups on variables plausibly related to Test comparability of groups on variables plausibly related to selection of TD or BUselection of TD or BU

3030

Descriptive StatisticsDescriptive Statistics

Readi ng Measur e M SD M SD M SD

WRMT- R WI D

wor ds cor r ect 5. 08 11. 51 5. 41 12. 29 4. 47 11. 55

TOWRE SWE

wor ds cor r ect 5. 45 12. 67 6. 08 12. 26 4. 95 12. 02

I TBS RC

i t ems cor r ect 3. 15 18. 68 3. 39 8. 79 2. 88 8. 75

CRAB Fl uency

wor ds cor r ect 22. 18 44. 91 23. 78 42. 30 20. 19 43. 80

CBM Maze

cor r ect compl et i ons 4. 72 8. 25 4. 78 8. 30 4. 52 8. 31

Lat ent change

scor e ( z- scor ed) - 0. 03 0. 97 0. 18 0. 95 - 0. 17 1. 05

Tabl e 1. Pr et est - post t est r aw scor e gai ns

Note: WRMT-R WID = Woodcock Reading Mastery Test, Revised, Word Identification subtest; TOWRE SWE = Test of Word Reading Efficiency, Sight Word Efficiency subtest; ITBS RC = Iowa Test of Basic Skills, Reading Comprehension subtest; CRAB = Comprehensive Reading Assessment Battery; CBM = Curriculum-Based Measurement

Bot t om- UpTop- Down Cont r ol

3131

Regression AnalysisRegression Analysis

Tabl e 2. Mul t i l evel Regr essi on Eff ect s

Compar i son Est i mat e SE t

PALS vs. Cont r ol 0. 247 0. 087 2. 85 **TD vs. Cont r ol 0. 150 0. 097 1. 54BU vs. Cont r ol 0. 343 0. 100 3. 42 ***TD vs. BU - 0. 193 0. 095 2. 03 *

Var i ance SE z

Teacher 0. 102 0. 024 4. 20Resi dual 0. 872 0. 035 24. 79

Not e: Degr ees of f r eedom f or fi xed eff ect s = 109. * = p<. 05; ** = p<. 01; *** = p<. 001

Random Eff ect s

3232

Effects for Study Groups by LA, Effects for Study Groups by LA, AA, and HA StudentsAA, and HA Students

Tabl e 3. Chi l d- l evel eff ect si zes f or PALS ( Hedges' g )

St udent t ype PALS v. Cont r ol

TD v. Cont r ol BU v. Cont r ol BU v. TD

Over al l . 25 . 15 . 34 . 19

Low Achi evi ng . 26 . 08 . 43 . 35

Aver age Achi evi ng . 23 . 19 . 27 . 08

Hi gh Achi evi ng . 28 . 20 . 37 . 18

Not e: Hedges' g eff ect si zes cal cul at ed usi ng r ecommendat i ons f or What Wor ks Cl ear i nghouse r epor t i ng ( I ES, 2007) BU=Bot t om- Up; TD=Top- Down; PALS = Peer - Assi st ed Lear ni ng St r at egi es.

3333

Effect Sizes for PALS

.00

.05

.10

.15

.20

.25

.30

.35

.40

.45

.50

BU P ALS vs. Control

TD P ALS vs. Control

BU P ALS vs. TDP ALS

P ALS vs. Control

Effect Size (Hedges' g)

Overall

Low

Average

High

Effects for Study Groups by LA, Effects for Study Groups by LA, AA, and HA StudentsAA, and HA Students

**

****

3434

Is Response to Word-Problem Is Response to Word-Problem Intervention among Students with MD Intervention among Students with MD

Moderated by Concurrent RD?Moderated by Concurrent RD?

Lynn Fuchs, Sarah Powell, Pamela Seethaler, Paul Cirino, Lynn Fuchs, Sarah Powell, Pamela Seethaler, Paul Cirino, Jack Fletcher, Doug Fuchs, Jack Fletcher, Doug Fuchs,

Carol Hamlett, and Rebecca ZumetaCarol Hamlett, and Rebecca ZumetaVanderbilt University and University of HoustonVanderbilt University and University of Houston

Journal of Educational PsychologyJournal of Educational Psychology, 2009, 2009

Grant #Grant #P01046261P01046261National Institute of National Institute of

Child Health and Human DevelopmentChild Health and Human Development

3535

3636

Study PurposesStudy Purposes Examine the efficacy of tutoring protocols for Examine the efficacy of tutoring protocols for

remediating remediating − Math fact deficits Math fact deficits − Word problem deficitsWord problem deficits

Assess whether treatment efficacy is different forAssess whether treatment efficacy is different for− Students with MD alone versusStudents with MD alone versus− Students with MDRDStudents with MDRD

Determine whether effects are comparable as a function Determine whether effects are comparable as a function of siteof site

− Nashville, where the tutoring protocols were Nashville, where the tutoring protocols were developeddeveloped

− Houston, a site distal to the developersHouston, a site distal to the developers

3737

ParticipantsParticipants 924 students screened in 63 classrooms in 18 schools in Nashville and 924 students screened in 63 classrooms in 18 schools in Nashville and

Houston (similar sample size at each site)Houston (similar sample size at each site)

Inclusion criteria:Inclusion criteria:− WRAT-A: < 26WRAT-A: < 26thth percentile percentile− 5-item word-problem measure: score < 25-item word-problem measure: score < 2− At least 1 (of 2) WASI subtest T score: > 36At least 1 (of 2) WASI subtest T score: > 36

162 students eligible for the study; 133 students remained at posttesting162 students eligible for the study; 133 students remained at posttesting

Blocking on site (Nashville and Houston) and MD status (MD vs. MDRD), Blocking on site (Nashville and Houston) and MD status (MD vs. MDRD), students randomly assigned to tutoring conditions:students randomly assigned to tutoring conditions:

− Math Facts Tutoring (“Math Flash”)Math Facts Tutoring (“Math Flash”)− Word Problem Tutoring (“Pirate Math”)Word Problem Tutoring (“Pirate Math”)− ControlControl

3838

ParticipantsParticipants

Treatment groups comparable on all variablesTreatment groups comparable on all variables MD vs. MDRD differences (across treatment MD vs. MDRD differences (across treatment

groups) as expectedgroups) as expectedMDMD MDRD MDRD

AgeAge 99 99

FemaleFemale 40%40% 48%48%

Sub. LunchSub. Lunch 68%68% 90%90%

Spec. Ed.Spec. Ed. 8%8% 28%28%

WASI IQWASI IQ 9292 8585

WRAT-AWRAT-A 8888 8181

WRAT-RWRAT-R 105105 7878

3939

Examined Efficacy Examined Efficacy of Two Tutoring Protocolsof Two Tutoring Protocols

Both Tutoring ProtocolsBoth Tutoring Protocols

Delivered individuallyDelivered individually 48 sessions: 3 per week for 16 weeks48 sessions: 3 per week for 16 weeks 20-30 minutes per session20-30 minutes per session Scripted lessons, which tutors studied (not read)Scripted lessons, which tutors studied (not read) Motivational system to ensure on-task behavior and Motivational system to ensure on-task behavior and

hard, accurate workhard, accurate work Each session audiotaped; tapes sampled and coded for Each session audiotaped; tapes sampled and coded for

fidelity, which was high for both tutoring conditionsfidelity, which was high for both tutoring conditions

4040

Examined Efficacy Examined Efficacy of Two Tutoring Protocolsof Two Tutoring Protocols

The exclusive focus of Math Flash was The exclusive focus of Math Flash was math factsmath facts

The primary focus of Pirate Math was The primary focus of Pirate Math was word problemsword problems

− but it also addressed foundational skills but it also addressed foundational skills (math facts, procedural calculations, and (math facts, procedural calculations, and algebra skills)algebra skills)

4141

Pirate Math TutoringPirate Math Tutoring

48 sessions: 3 per week for 16 weeks48 sessions: 3 per week for 16 weeks

20-30 minutes per session20-30 minutes per session

Scripted lessons, which tutors study (not read)Scripted lessons, which tutors study (not read)

Four unitsFour units

Foundational Skills for Word ProblemsFoundational Skills for Word Problems

Total Word ProblemsTotal Word Problems

Difference Word ProblemsDifference Word Problems

Change Word ProblemsChange Word Problems

4242

Pirate Math: Introductory Pirate Math: Introductory UnitUnit

Teach students:Teach students:− Efficient counting strategies to answer Efficient counting strategies to answer

math factsmath facts− 2-digit procedural calculations2-digit procedural calculations− How to solve for X in addition and How to solve for X in addition and

subtraction equations (a+b=c; x-y=z)subtraction equations (a+b=c; x-y=z)− How to check workHow to check work

4343

Introductory Unit: Introductory Unit: Counting UpCounting Up

COUNTING UP Addition

1. Put the bigger number in your head

and say it.

2. Count up the smaller number on your

fingers.

3. Your answer is the last number you

say. To ADD, you CAN reverse the numbers!

Example: 5 + 2 = ? Example: 2 + 5 = ?

1. “5.” 1. “5.” 2. (Hold up 1 finger.) “6.” 2. (Hold up 1 finger.) “6.”

(Hold up 2 fingers.) “7.” (Hold up 2 fingers.) “7.” 3. “The answer is 7.” 3. “The answer is 7.”

COUNTING UP Subtraction

1. Put the minus number in your head

and say it. 2. Count up on your fingers to the

number you started with.

3. Your answer is the number of fingers

you have up.

To SUBTRACT, do NOT reverse the numbers. The minus number always goes first.

Example: 5 - 2 = ?

1. “2.” 2. (Hold up 1 finger.) “3.”

(Hold up 2 fingers.) “4.” (Hold up 3 fingers.) “5.”

3. “The answer is 3.”

4444

Introductory Unit: Introductory Unit: Finding X in All 3 Finding X in All 3

Positions of EquationPositions of Equation

If X is at the end of a number sentence, If X is at the end of a number sentence, do what the problem tells you to do (e.g., do what the problem tells you to do (e.g., 3 + 2 = X; 6 – 2 = X)3 + 2 = X; 6 – 2 = X)

If X is not at the end, and it’s an “X minus” If X is not at the end, and it’s an “X minus” problem, add (e.g., X – 2 = 4).problem, add (e.g., X – 2 = 4).

If X is not at the end, and it’s not a X If X is not at the end, and it’s not a X minus problem, subtract (e.g., X + 2 = 8; 5 minus problem, subtract (e.g., X + 2 = 8; 5 – X = 2; 7 + X = 12).– X = 2; 7 + X = 12).

4545

Introductory Unit: Introductory Unit: Checking WorkChecking Work

CHECKING YOUR WORK

1. Sense Does the work make sense?

2. Lining Up

Did I line up the numbers correctly?

3. Math

Did I add or subtract correctly?

4. Labels

Did I use a label?

5. Signs

Did I use signs correctly?

4646

Remaining Units:Remaining Units:Word-Problem LessonsWord-Problem Lessons

Following Unit 1, four activities per session.Following Unit 1, four activities per session.

1. Flash-card warm up1. Flash-card warm up

2. Conceptual/strategic lesson using schema-broadening 2. Conceptual/strategic lesson using schema-broadening instruction instruction

3. Sorting practice on identifying problem types3. Sorting practice on identifying problem types

4. Paper/pencil review4. Paper/pencil review

4747

1. Math Fact Flash Card Warm Up1. Math Fact Flash Card Warm Up

Math Fact flash cards comprise 200 addition and Math Fact flash cards comprise 200 addition and subtraction facts subtraction facts

− Sums 0-18Sums 0-18− Subtrahends 0-18Subtrahends 0-18

Tutor shows flash card to student: Know it or Count Up!Tutor shows flash card to student: Know it or Count Up!− If student answers correctly, flash card placed in If student answers correctly, flash card placed in

correct pile.correct pile.− If student answers incorrectly, tutor asks student to If student answers incorrectly, tutor asks student to

“Count Up”; once correct, goes in correct pile.“Count Up”; once correct, goes in correct pile.− Student graphs score on graph.Student graphs score on graph.

4

+ 5

11

- 6

4848

2. Lesson2. LessonPirate Math RUNPirate Math RUN

Students use “RUN” Students use “RUN” strategy for every strategy for every word problem.word problem.

Students learn to Students learn to circle relevant circle relevant information directly information directly in the text or in the text or picture/graph/chart.picture/graph/chart.

RUN! 1. Read the problem. 2. Underline the question. 3. Name the problem type.

4949

2. Lesson2. LessonPirate Math Setting Up WorkPirate Math Setting Up Work

Write the equation that goes with the problem type.Write the equation that goes with the problem type. Figure out what’s missing. Write X in your equation in Figure out what’s missing. Write X in your equation in

the appropriate slot.the appropriate slot. Figure out what numbers are known. Write those Figure out what numbers are known. Write those

numbers in the appropriate slots.numbers in the appropriate slots. Write the math signs.Write the math signs. Find X.Find X. Make sure your answer has a number and a label.Make sure your answer has a number and a label.

5050

2. Lesson2. LessonProblem Types with TransferProblem Types with Transfer

Problem types at grade 2: Problem types at grade 2: Total, Difference, and Total, Difference, and Change Change

Transfer features:Transfer features:− Irrelevant informationIrrelevant information− MoneyMoney− Double-digit Double-digit

calculationscalculations− Finding relevant Finding relevant

information in graphs information in graphs and pictures.and pictures.

− Combining problem Combining problem types.types.

5151

2. Lesson2. LessonPirate Math ChangePirate Math Change

Change problems with Change problems with a starting amount that a starting amount that increases or decreases increases or decreases (a (a changechange) to make it ) to make it a new amount.a new amount.

““Sarah had 10 pencils. Sarah had 10 pencils. Then she gave 4 Then she gave 4 pencils to Pamela. How pencils to Pamela. How many pencils does many pencils does Sarah have now?”Sarah have now?”

St +/- C = ESt +/- C = E

CHANGE

1. How many do you start with? (St) 2. How many do you change? (C)

Is there an increase? + Is there a decrease? -

3. How many do you end with? (E)

St + or - C = E

4. Write the number sentence.

5. Find X!

5252

““Sarah had 10 pencils. Then, she gave 4 pencils to Sarah had 10 pencils. Then, she gave 4 pencils to Pamela. How many pencils does Sarah have Pamela. How many pencils does Sarah have

now?”now?”

5353

““Sarah had 10 pencils. Then, she gave 4 pencils to Pamela. Sarah had 10 pencils. Then, she gave 4 pencils to Pamela. How many pencils does Sarah have now?”How many pencils does Sarah have now?”

Recognize problem type: Change problem.Recognize problem type: Change problem.Write equation for Change problems: St +/- C = E.Write equation for Change problems: St +/- C = E.Identify missing information (E). Write that in the appropriate slotIdentify missing information (E). Write that in the appropriate slot

St +/- C = ESt +/- C = E XX

Identify the important given numbers (St and C). Write those in the Identify the important given numbers (St and C). Write those in the appropriate slots.appropriate slots.St +/- C = ESt +/- C = E10 4 X10 4 X

Write math signs. Write math signs. St +/- C = ESt +/- C = E10 - 4 = X10 - 4 = X

Find X: X is at end so do what problem tells me to do: 10 – 4 = 6; X=6.Find X: X is at end so do what problem tells me to do: 10 – 4 = 6; X=6.Label answer: 6 pencils.Label answer: 6 pencils.

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Lexie had some comic books in her desk. Then she Lexie had some comic books in her desk. Then she bought 8 more. Now, she has 12 comic books. bought 8 more. Now, she has 12 comic books. How many comic books did Lexie have in her How many comic books did Lexie have in her

desk to begin with?desk to begin with?

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Lexie had some comic books in her desk. Then she bought 8 Lexie had some comic books in her desk. Then she bought 8 more. Now, she has 12 comic books. How many comic books more. Now, she has 12 comic books. How many comic books

did Lexie have in her desk to begin with?did Lexie have in her desk to begin with?

Recognize problem type: Change problem.Recognize problem type: Change problem.Write equation for Change problems: St +/- C = E.Write equation for Change problems: St +/- C = E.Identify missing information (St). Write that in the appropriate slotIdentify missing information (St). Write that in the appropriate slot

St +/- C = ESt +/- C = EX X

Identify the important given numbers (St and C). Write those in the appropriate Identify the important given numbers (St and C). Write those in the appropriate slots.slots.St +/- C = ESt +/- C = EX 8 12X 8 12

Write math signs. Write math signs. St +/- C = ESt +/- C = EX + 8 = 12X + 8 = 12

Find X: X is not at end and it’s not an X minus problem, so subtract: 12 – 8 = 4; Find X: X is not at end and it’s not an X minus problem, so subtract: 12 – 8 = 4; X=4.X=4.

Label answer: 4 comic books.Label answer: 4 comic books.

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Alicia has 3 friends in her math class. Alicia has 3 friends in her math class. The chart shows how many stars The chart shows how many stars Alicia and her friends earned on Alicia and her friends earned on

Monday. On Tuesday, Alicia lost 3 Monday. On Tuesday, Alicia lost 3 stars for talking. How many stars stars for talking. How many stars

does she have now?does she have now? Monday’s Star Chart

0 2 4 6 8 10

David

Trish

Number of Gold Stars

Milo

Alicia

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Recognize problem type: Change problem.Recognize problem type: Change problem.

Identify transfer features: Irrelevant information (cross it Identify transfer features: Irrelevant information (cross it out) and relevant information in a graph (number the out) and relevant information in a graph (number the graph). graph).

Write equation for Change problems: St +/- C = E.Write equation for Change problems: St +/- C = E.

Identify missing information (St).Identify missing information (St). Write that in the Write that in the appropriate slotappropriate slot

St +/- C = ESt +/- C = E X X

Identify the important given numbers (St and C). Write Identify the important given numbers (St and C). Write those in the appropriate slots.those in the appropriate slots.

St +/- C = ESt +/- C = E8 3 X8 3 X

Write math signs. Write math signs. St +/- C = ESt +/- C = E8 - 3 = X8 - 3 = X

Find X: X is at end, so do what the problem says: 8 – 3 = 5; Find X: X is at end, so do what the problem says: 8 – 3 = 5; X=5.X=5.Label answer: 5 stars.Label answer: 5 stars.

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2. Lesson2. LessonPirate Math TotalPirate Math Total

Total problems have Total problems have two parts that are two parts that are combined for a combined for a totaltotal..

Total amount is the Total amount is the entire or combined entire or combined amount.amount.

““Sarah has 5 pencils. Sarah has 5 pencils. Pamela has 3 pencils. Pamela has 3 pencils. How many pencils do How many pencils do the girls have in all?”the girls have in all?”

P1 + P2 = TP1 + P2 = T

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2. Lesson 2. Lesson Pirate Math DifferencePirate Math Difference

Difference problems Difference problems compare two amounts compare two amounts to find the to find the differencedifference between them.between them.

““Sarah has 7 pencils. Sarah has 7 pencils. Pamela has 12 Pamela has 12 pencils. How many pencils. How many more pencils does more pencils does Pamela have than Pamela have than Sarah?”Sarah?”

B – s = DB – s = D

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3. Sorting3. Sorting

Student sorts word problems by problem Student sorts word problems by problem type for 2 minutes.type for 2 minutes.

Tutor reads cards to student.Tutor reads cards to student. Student places cards on Sorting Mat.Student places cards on Sorting Mat. At end of 2 minutes, tutor counts number At end of 2 minutes, tutor counts number

of correctly sorted cards and uses of correctly sorted cards and uses correction procedure for up to 3 incorrectly correction procedure for up to 3 incorrectly sorted cards.sorted cards.

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3. Sorting3. SortingSorting Mat

?

Total

Difference

Change

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3. Sorting3. Sorting

Maria and Jackie picked 16 flowers. Jackie

picked 7 flowers. How many flowers did Maria

pick?

Maria picked 8 more flowers than Jackie.

Jackie picked 4 flowers. How many flowers did

Maria pick?

Maria picked 11 flowers. Then Jackie took 4 of

them for her Mom. How many flowers does Maria have now?

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4. Paper/pencil review 4. Paper/pencil review

* 10 math facts* 10 math facts* 4 double-digit calculations* 4 double-digit calculations* 1 word problem* 1 word problem

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MotivationMotivation during during Pirate Math Pirate Math

Students earn coins Students earn coins throughout lesson for throughout lesson for listening well, working hard, listening well, working hard, following directions, and following directions, and correct work.correct work.

At end of lesson, students At end of lesson, students color footsteps on treasure color footsteps on treasure map equaling amount of map equaling amount of coins earned.coins earned.

When students color 16 When students color 16 footsteps, they pick a prize footsteps, they pick a prize from treasure box and from treasure box and receive a new map.receive a new map.

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Examined Efficacy Examined Efficacy of Two Tutoring Programsof Two Tutoring Programs

The exclusive focus of math facts tutoring The exclusive focus of math facts tutoring was was math factsmath facts

The primary focus of Pirate Math tutoring The primary focus of Pirate Math tutoring was was word problemsword problems

− but it also addressed foundational skills but it also addressed foundational skills (math facts, procedural calculations, and (math facts, procedural calculations, and algebra skills)algebra skills)

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Efficacy: Efficacy: Fluency with Math Facts and Fluency with Math Facts and

Procedural CalculationsProcedural Calculations

On math facts, Pirate Math effects superior improvement On math facts, Pirate Math effects superior improvement compared to control group. No difference between compared to control group. No difference between Pirate Math and math facts tutoring. Notable, because Pirate Math and math facts tutoring. Notable, because Pirate Math only allocates an initial lesson and then 4-6 Pirate Math only allocates an initial lesson and then 4-6 minutes per session on number combinations.minutes per session on number combinations.

On procedural calculations, Pirate Math effects superior On procedural calculations, Pirate Math effects superior improvement compared to control group and compared improvement compared to control group and compared to math facts tutoring. Again, little time spent on to math facts tutoring. Again, little time spent on procedural calculations.procedural calculations.

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EfficacyEfficacy: : AlgebraAlgebra

On algebra, Pirate Math effects superior outcomes On algebra, Pirate Math effects superior outcomes compared to control group and compared to math facts compared to control group and compared to math facts tutoring.tutoring.

Algebraic cognition improved even though students Algebraic cognition improved even though students were severely deficient in math and young.were severely deficient in math and young.

Given strong focus on algebra in high schools, given Given strong focus on algebra in high schools, given graduation requirements for algebra, and given graduation requirements for algebra, and given emphasis in NMAP, introducing algebra earlier in the emphasis in NMAP, introducing algebra earlier in the curriculum may represent a productive innovation.curriculum may represent a productive innovation.

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Correct RepresentationCorrect Representation

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Correct RepresentationCorrect Representation

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Incorrect RepresentationIncorrect Representation

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Over Efficacy Results: Word ProblemsOver Efficacy Results: Word Problems

Work on these foundation skills (MFs, procedural calculations, Work on these foundation skills (MFs, procedural calculations, algebra), combined with schema-broadening instruction, also algebra), combined with schema-broadening instruction, also produced differential growth on WP outcomes compared to MF produced differential growth on WP outcomes compared to MF tutoring group and compared to control group.tutoring group and compared to control group.

MF tutoring did not result in improvement on WPs. MF tutoring did not result in improvement on WPs.

Lack of transfer suggests that source of difficulty is not diverting Lack of transfer suggests that source of difficulty is not diverting attention from the complex mathematics to the MFs embedded in attention from the complex mathematics to the MFs embedded in those problems, but rather failing to comprehend the relations those problems, but rather failing to comprehend the relations among the numbers embedded in the narratives or to process the among the numbers embedded in the narratives or to process the language in those stories adequately. language in those stories adequately.

Suggests that MFs may not be the bottleneck for WP performance. Suggests that MFs may not be the bottleneck for WP performance. Instead, mathematics disability represents a more complicated Instead, mathematics disability represents a more complicated pattern of difficulty, implicating language (as has been suggested pattern of difficulty, implicating language (as has been suggested elsewhere).elsewhere).

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Is Tutoring Differentially Efficacious Is Tutoring Differentially Efficacious Depending on MD Status (MD vs. MDRD)?Depending on MD Status (MD vs. MDRD)?

Why We Hypothesized MD students would be more Why We Hypothesized MD students would be more responsive to tutoring than MDRD studentsresponsive to tutoring than MDRD students

For MFs For MFs − A key deficit among students with reading difficulty is A key deficit among students with reading difficulty is

phonological processing and phonological processing and − Phonological processing deficits are linked with difficulty in Phonological processing deficits are linked with difficulty in

automatic retrieval of MFs. automatic retrieval of MFs. For WPsFor WPs

− Using text to construct a WP model involves language Using text to construct a WP model involves language − Language profiles of students with MDRD are depressed Language profiles of students with MDRD are depressed

compared to students with MD. compared to students with MD.

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Is Tutoring Differentially Efficacious Is Tutoring Differentially Efficacious Depending on MD Status (MD vs. MDRD)?Depending on MD Status (MD vs. MDRD)?

No evidence of differential responsiveness to intervention No evidence of differential responsiveness to intervention as a function of difficulty status on any outcome. as a function of difficulty status on any outcome.

Raises questions about the tenability of the MD/MDRD Raises questions about the tenability of the MD/MDRD subtyping scheme and suggests the need to pursue other subtyping scheme and suggests the need to pursue other avenues for subtyping mathematics disability.avenues for subtyping mathematics disability.

Even so, across tutoring conditions and sites, students with Even so, across tutoring conditions and sites, students with MD outperformed students with MDRD at pre- and posttest. MD outperformed students with MDRD at pre- and posttest.

Additional work to examine the tenability of the MD/MDRD Additional work to examine the tenability of the MD/MDRD subtyping scheme is warranted, even as research pursuing subtyping scheme is warranted, even as research pursuing alternative frameworks proceeds. alternative frameworks proceeds.

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These Tutoring Protocols These Tutoring Protocols Are TransportableAre Transportable

No MD/MDRD by treatment by site effects.No MD/MDRD by treatment by site effects.

No treatment by site effects.No treatment by site effects.

Tutoring protocols were comparably Tutoring protocols were comparably effective in Nashville and Houston, for MD effective in Nashville and Houston, for MD and for MDRD students.and for MDRD students.

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Overall ConclusionsOverall Conclusions

MF tutoring enhances fluency with MFs MF tutoring enhances fluency with MFs with transfer to procedural calculations but with transfer to procedural calculations but without transfer to algebra or WPs.without transfer to algebra or WPs.

For a comparable amount of tutoring time, For a comparable amount of tutoring time, WP tutoring (with work on foundational WP tutoring (with work on foundational skills) enhances WP skill, fluency with skills) enhances WP skill, fluency with MFs, procedural calculations, and MFs, procedural calculations, and algebra.algebra.

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For Materials, Contact:For Materials, Contact:Flora MurrayFlora Murray

flora.murray@vanderbilt.eduflora.murray@vanderbilt.edu

Vanderbilt UniversityVanderbilt University

228 Peabody College228 Peabody College

Department of Special EducationDepartment of Special Education

Nashville, TN 37203Nashville, TN 37203

(615) 343-4782(615) 343-4782